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MECHANICS AND HEAT

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PUBLISHERS OF BOOKS F O R^

Coal Age v Electric Railway Journal
Electrical World v Engineering News-Record
American Machinist v Ingenieria Internacional
Engineering g Mining Journal ^ Power
Chemical 6 Metallurgical Engineering
Electrical Merchandising

PHYSICS FOR TECHNICAL STUDENTS

MECHANICS

AND

HEAT

BY
WILLIAM BALLANTYNE ANDERSON, PH. D.

ASSOCIATE PROFESSOR OP PHT8IC8, IOWA STATE COLLEGE

FIRST EDITION
SIXTH IMPRESSION

McGRAW-HILL BOOK COMPANY, INC.
NEW YORK: 370 SEVENTH AVENUE

LONDON: 6 & 8 BOUVERIE ST., E. C. 4
1914

PREFACE

The present volume is the outgrowth of mimeograph notes
which the author has used in connection with a course of lectures
given during the past six years. Since the author has also con-
ducted the recitations for several sections during this time, the

1 successive revisions of the notes have been made by one viewing
the work from two angles, that of class instructor, as well as that
of lecturer. It is believed that in this way a keener realization

>^ of the student's difficulties, and a better appreciation of what
parts should be revised, have been obtained than would have

fc. been possible without this two-fold contact.

^{ We now have a large and rapidly increasing number of students
who are interested primarily in the practical side of education.

^ With the needs of these students in mind, the practical side of the
subject has been emphasized throughout the book. This
method, it is believed, will sustain interest in the subject by
showing its application to everyday affairs, and will, it is hoped,
be appreciated by .both students and instructors in Agriculture
and Engineering. In this connection, attention is directed to
sections 18, 19, 20, 29, 30, 39, 44, 54, 56, 60, 62, 63, 76, 80, 83,
108, 109, 111, 134, 138, 170, 185, 189, 190, 195, 200, 204, 205,
206, 218 and Chapters VII, XII, XVII, and XVIII.

More space than usual has been devoted to the treatment of
Force, Torque, Translatory Motion, and Rotary Motion. It is
felt that the great importance of these topics, which underlie
so much of the subsequent work of the student, warrants such
treatment. Probably everyone who has taught the theory of
electrical measuring instruments, for example, has realized that
the student's greatest handicap is the lack of a thorough grasp of
the fundamental principles of mechanics. The student who has
'^thoroughly mastered elementary mechanics has done much
toward preparing himself for effective work in technical lines.

The sketches, which are more numerous than is usual in such a
text, are chosen with special reference to the help they will be in
enabling the student to readily grasp important or difficult
principles. Wherever possible, every principle involved in the

443914

vi PREFACE

text is brought up again in a problem; so that in working all of
the problems a review of practically the entire book is obtained.
For a complete course, the text should be accompanied by lectures
and laboratory work.

In the treatment of many of the subjects, the author is indebted
to various authors of works in Physics, among whom may be
mentioned Professors Spinney, Duff, Watson, and Crew. The
order in which the different subjects are treated is that which
seems most logical and most teachable, and was given much
thought.

Thanks are due Professor G. M. Wilcox, of the Department of
Physics, Armour Institute, and Professor W. Weniger, of the
Department of Physics at Oregon Agricultural College, for their
careful reading of the original mimeograph notes and for the
numerous suggestions which they offered. I wish also to thank
my colleagues, Professor H. J. Plagge and Professor W. Kunerth,
for reading of the manuscript and proofs, and for valuable sug-
gestions. Thanks are also due to Professor W. R. Raymond of
the English Department of this College for reading much of
the manuscript during revision, and to Professor J. C. Bowman
of the same department, for reading practically all of the manu-
script just before it went to press.

IOWA STATE COLLEGE, W B A

March, 1914. W.J5. A.

CONTENTS

PAGE
PREFACE v

PART I

MECHANICS

CHAPTER I

MEASUREMENT 1

Section 1. The three fundamental quantities. 2. Units and
numerics. 3. Fundamental units. 4. Standards of length, mass,
and time. 5. The metric system. 6. Conversion of units. 7.
Measurement of length. 8. The vernier caliper. 9. The mi-
crometer caliper. 10. The micrometer microscope. 11. Meas-
urement of mass, inertia. 12. Measurement of time.

CHAPTER II

VECTORS 11

Section 13. Scalars and vectors denned. 14. Representation of
vectors by straight lines. 15. Addition of vectors, resultant.
16. The vector polygon. 17. Vectors in equilibrium. 18. The
crane. 19. Resolution of vectors into components. 20. Sailing
against the wind. 21. Sailing faster than the wind.

CHAPTER III

TRANSLATORY MOTION 23

Section 22. Kinds of motion. 23. Speed, average speed, velocity
and average velocity. 24. Acceleration. 25. Accelerating force.
26. Uniform motion and uniformly accelerated motion. 27.
Universal gravitation. 28. The law of the inverse square of the
distance. 29. Planetary motion. 30. The tides. 31. Accelera-
tion of gravity and accelerating force in free fall. 32. Units of
weight and units of force, compared. 33. Motion of falling
bodies; velocity — initial, final and average. 34. Distance fallen
in a given time. 35. Atwood's machine. 36. Motion of projec-
tiles; initial velocity vertical. 37. Motion of projectiles; initial
velocity horizontal. 38. Motion of projectiles; initial velocity
inclined. 39. Time of flight and range of a projectile. 40.
Spring gun experiment. 41. The plotting of curves. 42. New-
ton's three laws of motion. 43. Action and reaction, inertia

viii CONTENTS

PAGE

force, principle of d'Alembert. 44. Practical applications of
reaction. 45. Momentum, impulse, impact and conservation of
momentum. 46. The ballistic pendulum.

CHAPTER IV

ROTARY MOTION 59

Section 47. Kinds of rotary motion. 48. Torque. 49. Resultant
torque and an tiresultant torque. 50. Angular measurement. 51.
Angular velocity and angular acceleration. 52. Relation between
linear and angular velocity and acceleration. 53. The two condi-
tions of equilibrium of a rigid body. 54. Moment of inertia and
accelerating torque. 55. Value and unit of moment of inertia. 56.
Use of the flywheel. 57. Formulas for translatory and rotary
motion compared.

CHAPTER V

UNIFORM CIRCULAR MOTION, SIMPLE HARMONIC MOTION 72

Section 58. Central and centrifugal forces and radial acceleration.
59. Bursting of emery wheels and flywheels. 60. The cream
separator. 61. Efficiency of cream separator. 62. Elevation of
the outer rail on curves in a railroad track. 63. The centrifugal
governor. 63a. The gyroscope. 64. Simple harmonic motion.
65. Acceleration and force of restitution in S.H.M. 66. Period
in S.H.M. 67. The simple gravity pendulum. 68. The torsion
pendulum.

CHAPTER VI

WORK, ENERGY, AND POWER 89

Section 69. Work. 70. Units of work. 71. Work done if the line
of motion is not in the direction of the applied force. 72. Work
done by a torque. 73. Energy — potential and kinetic. 74. Trans-
formation and conservation of energy. 75. Value of potential and
kinetic energy in work units. 76. Energy of a rotating body. 77.
Dissipation of energy. 78. Sliding friction. 79. Coefficient of
friction. 80. Rolling friction. 81. Power. 82. Units of power.
83. Prony brake.

CHAPTER VII

MACHINES 110

Section 84. Machine defined. 85. Mechanical advantage and
efficiency. 86. The simple machines. 87. The lever. 88. The
pulley. 89. The wheel and axle. 90. The inclined plane. 91.
The wedge. 92. The screw. 93. The chain hoist or differential
pulley. 94. Center of gravity. 95. Center of mass. 96. Stable,
unstable and neutral equilibrium. 97. Weighing machines.

CONTENTS ix

PART II
PROPERTIES OF MATTER

CHAPTER VIII

PAGE
THE THREE STATES OF MATTER AND THE GENERAL PROPERTIES OP

MATTER 137

Section 98. The three states of matter. 99. Structure of matter.
100. Conservation of matter. 101. General properties of matter.
102. Intermolecular attraction and the phenomena to which it
gives rise. 103. Elasticity, general discussion.

CHAPTER IX

PROPERTIES OF SOLIDS 144

Section 104. Properties enumerated and defined. 105. Elasticity,
elastic limit and elastic fatigue of solids. 106. Tensile stress, and
tensile strain. 107. Hooke's law and Young's modulus. 108.
Yield point, tensile strength, breaking stress. 109. Strength of
horizontal beams. 110. Three kinds of elasticity of stress and of
strain; and the three moduli. 111. The rigidity of a shaft and the
power transmitted.

CHAPTER X

THE PROPERTIES OF LIQUIDS AT REST 155

Section 112. Brief mention of properties. 113. Hydrostatic pres-
sure. 114. Transmission of pressure. 115. The Hydrostatic
paradox. 116. Relative densities of liquids by balanced columns.
117. Buoyant force. 118. The principle of Archimedes. 119.
Immersed floating bodies. 120. Application of Archimedes'
principle to bodies floating upon the surface. 121. Center of
buoyancy. 122. Specific gravity. 123. The hydrometer. 124.
Surf ace tension. 125. Surface a minimum. 126. Numerical value
of surface tension. 127. Effect of impurities on surface tension of
water. 128. Capillarity. 129. Capillary rise in tubes, wicks, and
soil. 130. Determination of surface tension from capillary rise in
tubes.

CHAPTER XI

PROPERTIES OF GASES AT REST 177

Section 131. Brief mention of properties. 132. The earth's atmos-
phere. 133. Height of the atmosphere. 134. Buoyant effect,
Archimedes' principle, lifting capacity of balloons. 135. Pressure
of the atmosphere. 136. The mercury barometer. 137. The
aneroid barometer. 138. Uses of the barometer. 139. Boyle's
law. 140. Boyle's law tube, isothermals of a gas. 141. The
manometers and the Bourdon gage.

x CONTENTS

CHAPTER XII

PAGE

PROPERTIES OF FLUIDS IN MOTION 194

Section 142. General discussion. 143. Gravity flow of liquids.
144. The siphon. 145. The suction pump. 146. The force pump.
147. The mechanical air pump. 148. The Sprengel mercury
pump. 149. The windmill and the electric fan. 150. Rotary
blowers and rotary pumps. 151. The turbine water wheel. 152.
Pascal's law. 153. The hydraulic press. 154. The hydraulic
elevator. 155. The hydraulic ram. 156. Diminution of pressure
in regions of high velocity. 157. The injector. 158. Ball and
jet. 159. The curving of a baseball.

PART III
HEAT

CHAPTER XIII

THERMOMETRY AND EXPANSION 217

Section 160. The nature of heat. 161. Sources of heat. 162.
Effects of heat. 163. Temperature. 164. Thermometers. 165.
The mercury thermometer. 166. Thermometer scales. 167.
Other thermometers. 168. Linear expansion. 169. Coefficient
of linear expansion. 170. Practical applications of equalities and
differences in coefficient of linear expansion. 171. Cubical expan-
sion; Charles's law. 172. The absolute temperature scale. 173.
The general law of gases. 174. The thermocouple and the
thermopile.

CHAPTER XIV

HEAT MEASUREMENT, OR CALORIMETRY 243

Section 175. Heat units. 176. Thermal capacity. 177. Specific
heat. 178. The two specific heats of a gas. 179. The law of
Dulong and Petit. 180. Specific heat, method of mixtures. 181.
Heat of combustion. 182. Heat of fusion and heat of vaporiza-
tion. 183. Bunsen's ice calorimeter. 184. The steam calorimeter.
185. Importance of the peculiar heat properties of water. 186.
Fusion and melting point. 187. Volume change during fusion.
188. Regelation. 189. Glaciers. 190. The ice cream freezer.

CHAPTER XV

VAPORIZATION 260

Section 191. Vaporization defined. 192. Evaporation and ebulli-
tion. 193. Boiling point. 194. Effect of pressure on the boiling
point. 195. Geysers. 196. Properties of saturated vapor.
197. Cooling effect of evaporation. 198. Wet-and-dry-bulb
hygrometer. 199. Cooling effect due to evaporation of liquid

CONTENTS xi

PAGE

carbon dioxide. 200. Refrigeration and ice manufacture by
the ammonia process. 201. Critical temperature and critical
pressure. 202. Isothermals for carbon dioxide. 203. The Joule-
Thomson experiment. 204. Liquefaction of gases. 205. The
cascade method of liquefying gases. 206. The regenerative
method of liquefying gases.

CHAPTER XVI

TRANSFER OF HEAT 283

Section 207. Three methods of transferring heat. 208. Convec-
tion. 209. Conduction. 210. Thermal conductivity. 211. Wave
motion, wave length and wave velocity. 212. Interference of
wave trains. 213. Reflection and refraction of waves. 214.
Radiation. 215. Factors in heat radiation. 216. Radiation and
absorption. 217. Measurement of heat radiation. 218. Trans-
mission of heat radiation through glass, etc. 219. The general
case of heat radiation striking a body.

CHAPTER XVII

METEOROLOGY 302

Section 220. General discussion. 221. Moisture in the atmos-
phere. 222. Hygrometry and -hygrometers. 223. Winds, trade
winds. 224. Land and sea breezes. 225. Cyclones. 226. Tor-
nadoes.

CHAPTER XVIII

STEAM ENGINES AND GAS ENGINES 311

Section 227. Work obtained from heat — thermodynamics. 228.
Efficiency. 229. The steam engine. 230. Condensing engines.
231. Expansive use of steam, cut-off point. 232. Power. 233.
The slide valve mechanism. 234. The indicator. 235. The steam
turbine. 236. Carnot's cycle. 237. The gas engine — fuel, carbu-
retor, ignition and governor. 238. Multiple-cylinder engines.
239. The four-cycle engine. 240. The two-cycle engine.

INDEX . . 335

PART I
MECHANICS

_\J

MECHANICS AND HEAT

CHAPTER I
MEASUREMENT

1. The Three Fundamental Quantities. — The measurement
of physical quantities is absolutely essential to an exact and scien-
tific study of almost any physical phenomenon. For this reason,
Measurement is usually the topic first discussed in a course in
Physics. The popular expressions, "quite a distance," a "large
quantity," etc., are too indefinite to satisfy the scientific mind.
A physical quantity may be defined as anything that can be
measured. The measurement of length, mass, and time are of
special importance and will therefore be considered first.
Indeed, almost all physical quantities may be expressed in
terms of one or more of these three quantities, for which reason
they are called the fundamental quantities. In the case of some
physical quantities this is at once apparent. Thus, to measure
the area of a piece of land, it is, as a rule, only necessary to
measure the distance across it north and south (say LI) and then
east and west (L2). The product of these two dimensions, Z/iL2,
is then an area. If it is required to find how many "yards" of
earth have been removed in digging a cellar, not only the length
and width must be known, but also the depth (L3). The result
evidently involves a length (i.e., distance) only, since volume =
LiL2L3. Coal, grain, etc., are measured in terms of mass. If
the quantity involved is the time between two dates it is, of
course, measured in terms of time. If a train goes from one city
to another in a known time T, its average velocity is the distance
between the two points (i.e., a length) divided by the time
required to traverse that distance, or

Velocity =|

A force may be measured in terms of the rate at which it
changes the velocity of a body of known mass upon which it acts.
Velocity, as we have just seen, is a quantity involving both

2 MECHANICS AND HEAT

length and time; hence, force must be a quantity involving all
three fundamental quantities. In like manner it may be shown
that other physical quantities, e.g., power, work, electric
charge, electric current, etc., are expressible in terms of one or
more of the three fundamental quantities — length, mass, and
time.

2. Units and Numerics. — In order to measure and record
the value of any quantity, it is necessary to have a unit of that
same quantity in which to express the result. Thus if we meas-
ure the length of a board with a foot rule and find that we must
apply it ten times, and that the remainder is then half the length
of the rule, we say that the length of the board is 10^ ft. If
this same board is measured with a yard stick, 3| yds. is the
result; while, if the inch is the unit, 126 inches is the result.
Here the foot, the yard, or the inch is the Unit, and the 10^,
3^, or 126 is the Numeric. Evidently the larger the unit, the
smaller the numeric, and vice versa. Thus, in expressing a
weight of 2 tons as 4000 Ibs., the numeric becomes 2000 times
as large because the unit chosen is 1/2000 as large as before.

3. Fundamental Units. — In the British System of measure-
ment, which is used in practical work in the United States and
Great Britain, the units of length, mass, and time are respec-
tively ihefoot, the pound, and the second. It is often termed the
foot-pound-second system, or briefly the " F.P.S." system. Since,
as has been pointed out, nearly all physical quantities may be
expressed in terms of one or more of the above quantities, the
above units are called Fundamental Units. (The fundamental
units of the metric system are given in Sec. 5.)

4. Standards of Length, Mass, and Time. — If measurements
made now are to be properly interpreted several hundred years
later, it is evident that the units involved must not be subject
to change. To this end the British Government has had made,
and keeps at London, a bronze bar having near each end a fine
transverse scratch on a gold plug. The distance between these
two scratches, when the temperature of the bar is 62° Fahren-
heit, is the standard yard. At the same place is kept a piece of
platinum of 1 Ib. mass. This bar and this piece of platinum
are termed the Standards of length and mass respectively.
The standard for time measurement is the mean solar day, and
the second is then fixed as the 1/60X1/60X1/24, or 1/86400
part of a mean solar day.

* '

MEASUREMENT 3

The Day — Sidereal, Solar, and Mean Solar. — Very few things so com-
monplace as the day, are so little understood. The time that elapses
between two successive passages of a star (a true star, not a planet)
across the meridian (a north and south line), in other words the time
interval from "star noon" to "star noon," is a Sidereal Day. From
"sun noon" to "sun noon" is a Solar Day. The longest solar day is
nearly a minute longer than the shortest. The average of the 365 solar
days is the Mean Solar Day. The mean solar day is the day commonly
used. It is exactly 24 hours. The sidereal day, which is the exact
time required for the earth to make one revolution on its axis, is
nearly four minutes shorter than the mean solar day.

The cause for the four minutes difference between the sidereal day and
the solar day may be indicated by two or three homely illustrations. If
a silver dollar is rolled around another dollar, without slipping, it will be
found that the moving dollar makes two rotations about its axis,
while making one revolution about the stationary dollar. The moon
always keeps the same side toward the earth, and for this very reason
rotates once upon its axis for each revolution about the earth. Compare
constantly facing a chair while you walk once around it. You will find
that you have turned around (on an axis) once for each revolution about
the chair. If, now, you turn around in the same direction as before,
three times per revolution, you will find that you face the chair but
twice per revolution. For exactly the same reason the earth must
rotate 366 times on its axis during one revolution about the sun, in order
to "face" the sun 365 times. Consequently the sidereal day is, using
round numbers, 365/366 as long as the mean solar day, or about four
minutes shorter.

Variation in the Solar Day. — If the orbit of the earth around the sun
were an exact circle, and it, further, the axis of rotation of the earth
were at right angles to the plane of its orbit (plane of the ecliptic), then
all solar days would be of equal length. The orbit, however, is slightly
elliptical, the earth being nearer to the sun in winter and farther from it
in summer than at other seasons; and the axis of the earth lacks 23°. 5
of being at right angles to the plane of the ecliptic.

Let S, (Fig. 1) represent the sun, E, the earth on a certain day, and E',
the earth a sidereal day later (distance EE' is exaggerated). Let the
curved arrow indicate the rotary motion of the earth and the straight
arrow, the motion in its orbit. When the earth is at E, it is noon
at point A; i.e., AS is vertical; while at E', the earth having made
exactly one revolution, the vertical at A is AB, and it will not be noon
until the vertical (hence the earth) rotates through the angle 0. This
requires about four minutes (0 being much smaller than drawn), causing
the solar day to be about four minutes longer than the sidereal day.
The stars are so distant that if AS points toward a star, then AB,
which is parallel to it, points at the same star so far as the eye can detect.

4 MECHANICS AND HEAT

Hence the sidereal day gives, as above stated, the exact time of one
revolution of the earth.

When the earth is nearest to the sun (in December) it travels fastest;
i.e., when AS is shortest, EE' is longest. Obviously both of these
changes increase 0 and hence make the solar day longer. The effect
of the above 23°.5 angle, in other words, the effect due to the
obliquity of the earth's axis, is best explained by use of a model.
We may simply state, however, that due to this cause the solar day in
December is still further lengthened. As a result it is nearly a minute
longer than the shortest solar day, which is in September.

When the solar days are longer than the mean solar day (24 hour day)
the sun crosses the meridian, i.e., "transit" occurs,
later and later each day; while when they are
shorter, the transit occurs earlier each day. In
February, transit occurs at about 12:15 mean so-
lar time (i.e., clock time), at which date the alma-
nac records sun "slow" 15 minutes. In early
November the sun is about 15 minutes "fast."
These are the two extremes.

6. The Metric System. — This system is in common use in
most civilized countries except the United States and Great
Britain, while its scientific use is universal. The fundamental
units of the Metric System of measurement are the centimeter,
the gram, and the second. It is accordingly called the centimeter-
gram-second system, or briefly the "C.G.S." system. This
system far surpasses the British system in simplicity and facility
in computation, because its different units for the measurement
of the same quantity are related by a ratio of 10, or 10 to some
integral power, as 100, 1000, etc. The centimeter (cm.) is the
1/100 part of the length of a certain platinum-iridium bar when at
the temperature of melting ice. This bar, whose length (between
transverse scratches, at 0°C.) is 1 meter (m.), is kept at Paris by
the French Government. The gram is the 1/1000 part of the mass
of a certain piece of platinum (the standard kilogram) kept at the
same place. The milligram is 1/1000 gm., and the millimeter
(mm.) is 1/1000 meter. The second is the same as in the British
system. The above meter bar and kilogram mass are respectively
the Standards of length and mass in the Metric System.

6. Conversion of Units. — In this course both systems of
units will be used, because both are frequently met in general
reading. Some practice will also be given in converting results
expressed in terms of the units of one system into units of the

MEASUREMENT 5

other (see problems at the close of this chapter). To do this it
is only necessary to know that 1 inch = 2.54 cm. and 1 kilogram
( = 1000 gm.) = 2.2046 Ibs., or approximately 2.2 Ibs. These two
ratios should be memorized, and perhaps also the fact that the
meter = 39.37 in. From the first ratio it will be seen that the
numeric is made 30.48 (or 12X2.54) times as large whenever a
certain length is expressed in centimeters instead of in feet. The
relation between all other units in the two systems can readily
be obtained if the above two ratios are known.

7. Measurement of Length. — The method employed in
measuring the length of any object or the distance between any
two points, will depend upon the magnitude of the distance to
be measured, and the accuracy with which the result must be
determined. For many purposes, either the meter stick or the
foot rule answers very well; while for other purposes, such as
the measurement of the thickness of a sheet of paper, both are
obviously useless. For more accurate measurements, several
instruments are in use, prominent among which are the vernier
caliper, the micrometer caliper, and the micrometer microscope.

8. The Vernier Caliper. — In Fig. 2 is shown a simplified
form of the vernier caliper from which the important principle
of the vernier may be readily comprehended. This vernier cali-

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FIG. 2.

per consists of a bar A, having marked near one edge a scale in
millimeter divisions Rigidly attached to A is the jaw B,
whose face F is accurately perpendicular to A, and parallel to
the face of jaw D, attached to bar C. C may be slid along A
until D strikes B, if there is nothing between the jaws. While
in this position, a scale of equal divisions is ruled upon C having
its zero line in coincidence with the zero line of A, and its tenth
line in coincidence with the ninth line on A. The scale on C is
called the vernier scale and that on A, the main scale. Obvi-
ously, the vernier divisions are 1/10 mm. shorter than the main

6 MECHANICS AND HEAT

scale divisions; i.e., they are 9/10 as long, since 10 vernier divi-
sions just equal 9 scale divisions.

To measure the length of the block E, place it between the
jaws D and B, as shown. Since the two zero lines coincide when
the jaws are together, the length of the block must be equal to
the distance between the two zeros, or 3 mm., plus the small dis-
tance a. But if line 2 on the vernier coincides with a line on the
main scale, as shown, then a is simply the difference in length
between 2 vernier divisions and 2 main scale divisions, or 0.2
mm. The length of E is then 3.2 mm.

If C were slid to the right 1/10 mm., line 3 on the vernier
would coincide with a main scale line, and a would then equal 0.3
mm.; so that the distance between the jaws would be 3.3 mm.
Evidently, the above 1/10 mm. is the least motion of C that can
be directly measured by the vernier. This distance (1/10 mm.)
is called the sensitiveness of this vernier. If the divisions on A
had been made 1/20 inch, and 25 vernier divisions had been
made equal to 24 main scale divisions, then the sensitiveness or
difference between the length of a main scale division and a ver-
nier division would be 1/500 inch. For the vernier divisions,
being 1/25 division shorter than the main scale divisions (i.e.,
24/25 as long), are 1/25X1/20 or 1/500 inch shorter.

This arrangement of two scales of slightly different spacing,
free to slide past each other, is an application of the Vernier
Principle. This principle is much employed in making measur-
ing instruments. Instead of having 10 vernier spaces equal to
9 spaces on the main scale, the ratio may be 25 to 24 as men-
tioned, or 50 to 49, 16 to 15, etc., according to the use that is to
be made of the instrument. In the case of circular verniers
and scales on surveying instruments, the above-mentioned ratio
is usually 30 to 29 or else 60 to 59, because they are to be read in
degrees, minutes, and seconds of arc. If the vernier principle is
thoroughly understood, there should be no difficulty in reading
any vernier, whether straight or circular, in which a convenient
ratio is employed.

9. The Micrometer Caliper. — The micrometer caliper (Fig.
3) consists of a metal yoke A, a stop S, a screw B whose threads
fit accurately the threads cut in the hole through A, and a sleeve
C rigidly connected to B. When B and S are in contact, the
edge E of C is at the zero of scale D; consequently the dis-
tance from S to B, in other words the thickness of the block F

MEASUREMENT 7

as sketched, is equal to the distance from this zero to E. If
the figure represents the very common form of micrometer cali-
per in which the "pitch" of B (i.e., the distance B advances for
each revolution) is 1/2 mm., D is a scale of millimeter divi-
sions, and the circumference of C at E is divided into 50 equal
divisions; then the thickness of F is 4.5 mm. plus the slight dis-
tance that B moves when E turns through 6 of its divisions,
or 6/50 of a revolution. But 6/50X1/2 mm. = 0.06 mm.; so
that the thickness of E is 4.5+0.06 or 4.56 mm. It should be
explained that if the instrument is properly adjusted, then, when
B and S are in contact, the zero of E and the zero of D coincide.
Accordingly if the zero of E were exactly in line with scale D,
then 4.5 would be the result. As sketched, however, it is 6/50
of a revolution past the position of alignment with D, which

FIG. 3.

adds 0.06 mm. to the distance between B and S as already
shown.

If C were turned in the direction of arrow a through 1/50 revo-
lution, then line 7 of E, instead of line 6, would come in line
with D, and B would have moved 1/50X1/2 mm., or 0.01 mm.
farther from S. This, the least change in setting that can be
read directly without estimating, is called the Sensitiveness of
an instrument (see Sec. 8). Thus the sensitiveness of this
micrometer caliper is 0.01 mm.

10. The Micrometer Microscope. — The micrometer micro-
scope consists of an ordinary compound microscope, having
movable crosshairs in the barrel of the instrument where the
magnified image of the object to be measured is formed. These
crosshairs may be moved by turning a micrometer screw similar
to B in Fig. 3.

If it is known how many turns are required to cause the cross-
hairs to move over one space of a millimeter scale, placed on

8 MECHANICS AND HEAT

the stage of the microscope, and also what part of a turn will
cause them to move the width of a small object also placed on
the stage, the diameter of the object can be at once calculated.

11. Measurement of Mass, Inertia. — Consider two large pieces
of iron, provided with suitable handles for seizing them, each one
resting upon a light and nearly frictionless truck on a level steel
track, and hence capable of being moved in a horizontal direc-
tion with great freedom. If a person is brought blindfolded and
permitted to touch only the handles, he can very quickly tell
by jerking them to and fro horizontally, which one contains the
greater amount of iron. If one piece of iron is removed and
replaced by a piece of wood of the same size as the remaining
piece of iron, he would immediately detect that the piece of
wood moved more easily and would perhaps think it to be a very
small piece of iron. The difference which he detects is certainly
not difference in volume, as he is not permitted either to see or
to feel them; neither is it difference in weight, since he does not
lift them. It is difference in Mass that he detects. Hence
Mass may be denned as that property of matter by virtue of which
it resists being suddenly set into motion, or, if already in motion,
resists being suddenly brought to rest.

Inertia and Mass are synonymous; inertia being used in a
general way only, while mass is used in a general, qualitative
way and also in a quantitative way. Thus we speak of a large
mass, great inertia, a 5-lb. mass, etc., but not of 5 Ibs. inertia.

If it were possible, by the above method, for the person to
make accurate determinations, and if he found that one piece
had just twice as much mass as the other, then upon weighing
them it would be found that one piece was exactly twice as
heavy as the other. In other words, the Weight of any body is
proportional to its Mass. The weight of a body is simply the
attractive pull of the earth upon it; hence we see that the pull of
the earth upon any body depends upon the mass of the body, and
therefore affords a very convenient, and also very accurate
means of comparing masses.

Thus the druggist, using a simple beam balance, "weighs
out" a pound mass of any chemical by placing a standard pound
mass in one pan and then pouring enough of the chemical into
the other pan to exactly "balance" it. That is, the amount of
chemical in one pan is varied until the pull of the earth on the
chemical at one end of the beam is made exactly equal to the

MEASUREMENT 9

pull of the earth on the standard pound mass at the other end.
He then knows, since the pull of the earth on each is equal,
that their weights, and consequently their masses, are equal.
Weights, and hence masses, may be compared also by means
of the steel-yard, the spring balance, and the platform scale.
These devices will be discussed later in the course.

The mass of a body is absolutely constant wherever it is
determined, while its weight becomes very slightly less as it is
taken up a mountain or taken toward the equator. This is
due partly to the fact that the body is slightly farther from the
earth's center at those points, and partly to the rotary motion of
the earth (see centrifugal force, Sec. 58). The polar diameter of
the earth is about 27 miles less than its equatorial diameter.
A given object weighed at St. Louis and then at St. Paul with
the same spring balance should show an increase in weight at
the latter place; whereas if weighed with the same beam balance
at both places, there should be no difference in the weights
read. The weight of the object actually does increase, but the
weight of the counterbalancing standard masses used with the
beam balance also increases in the same proportion.

12. Measurement of Time. — A modern instrument for
measuring time must have these three essentials: (1) a device1
for measuring equal intervals of time, i.e., for time "spacing,"
(2) a driving mechanism, (3) a recording mechanism. In
the case of the clock, (1) is the pendulum, (2) is the mainspring or
weights, train of wheels and escapement, and (3) is the train of
wheels and the hands. In the watch, the balance wheel and
hairspring take the place of the pendulum.

The necessity for the pendulum or its equivalent, and the
recording mechanism, is obvious. Friction makes the driving
mechanism necessary. The escapement clutch attached to the
pendulum is shaped with such a slant that each time it releases
a cog of the escapement wheel it receives from that wheel a
slight thrust just sufficient to compensate for friction, which
would otherwise soon bring the pendulum to rest. If the
pendulum, as it vibrates, releases a cog each second, and if
the escapement wheel has 20 cogs, the latter will, of course,
make a revolution in 20 seconds. It is then an easy matter to
design a connecting train of geared wheels and pinions between
it and the post to which the minute hand is attached, so that
the latter will make one revolution in an hour. In the same

10 MECHANICS AND HEAT

way the hour hand is caused to make one revolution in twelve
hours.

In the hourglass of olden times, and in the similar device, the
clepsydra or water dropper of the Ancient Greeks, only the time
"spacing" is automatic. The observer became the driving
mechanism by inverting the hourglass at the proper moment;
and by either remembering or recording how many times he had
inverted it, he became also the recording mechanism.

Other time measurers, in which only time spacing is present,
are the earth and the moon. The rotation of the earth about
its axis determines our day, while its revolution about the sun
determines our year. The revolution of the moon about the
earth determines our lunar month, which is about 28 days.

PROBLEMS
jr.

1. What is the height in feet and inches of a man who is 1 m. 80 cm. tall?
Reduce 5 ft. 4.5 in. to centimeters.

2. What does a 160-lb. man weigh in grams? In kilograms? Reduce
44 kilograms 240 grams to pounds.

3. Reduce 100 yds. to meters. What part of a mile is the kilometer?

4. A cubic centimeter of gold weighs 19.3 gm. Find the weight of 1 cu.
ft. of gold in grams. In pounds.

5. One cm.3 of glycerine weighs 1.27 gms. How many pounds will 1
gaUon (231 in.3) weigh?

6. If a man can run 100 yds. in 10 sec., how long will he require for the
100 meter dash? Assume the same average velocity for both.

7. If, in Fig. 2, the main scale divisions were 1/16 inch, and 20 vernier
divisions were equal to 19 divisions on the main scale, other conditions being
as shown, what would be the length of El

8. The pitch of a certain micrometer caliper is 1/20 inch and the screw
head has 25 divisions. After setting upon a block and then removing it,
7 complete turns and 4 divisions are required to cause the screw to advance
to the stop. What is the thickness of the block?

9. Between the jaws of a vernier caliper (Fig. 2) is placed a block of such
length that line 5 of the vernier scale coincides with line 10 of the main scale,
and consequently the zero of the vernier scale is a short distance to the right
of line 5 of the main scale. If the main scale divisions are 1/2 mm., and 25
vernier divisions are equal in length to 24 main scale divisions, what is the
length of the block?

10. What is the sensitiveness (see Sec. 8) of the vernier caliper in problem
7? In problem 9? What is the sensitiveness of the micrometer caliper in
problem 8?

CHAPTER II

<\ V r

VECTORS

13. Scalars and Vectors Defined. — All physical quantities
may be divided into two general c\assesf^Scalars and Vectors.
A scalar quantity is one that is fully specified if its magnitude
only is given; while to specify a vector quantity completely,
not only its magnitude, but also its direction must be given.
Hence vectors might be called directed quantities.

Such quantities as volume, mass, work, energy, and quantity
of heat or of electricity, do not have associated with them any
idea of direction, and are therefore scalars. Force, pressure,
and velocity, must have direction as well as magnitude given or
they are not completely specified; therefore they are vectors.
Thus, if the statement is made that a certain ship left port at a
speed of 20 miles per hour, the motion of the ship is not fully
known. The statement that the ship's velocity was 20 miles an
hour due north, completely specifies the motion of the ship, and
conveys the full meaning of velocity. This distinction between
speed and velocity is not always observed in popular language,
but it must be observed in technical work.

If two forces FI and F2 act upon a body, say a boat in still
water, they will produce no effect, if equal and opposed; i.e.,
if the angle between the two forces is 180°. If this angle is zero,
i.e., if both forces act in the same direction, their Resultant F
(Sec. 15), or the single force that would produce the same effect
upon the boat as both FI and Fz, is simply their sum, or

F = F1+FZ (1)

If FI is greater than FZ, then when the angle between them is
180°, that is when ^i and F2 are oppositely directed, we have

F=F,-F2 (2)

The resultant F has in Eq. 1 its maximum value, and in Eq. 2
its minimum value. It may have any value varying between
these limits, as the angle between F\ and F2 varies from zero to
180°.

o {

12 MECHANICS AND HEAT

In contrast with the above statements, observe that in scalar
addition the result is always simply the arithmetical sum.
Thus, 15 qts. and 10 qts. are 25 qts.; while the resultant of a
15-lb. pull and a 10-lb. pull may have any value between 5 Ibs.
and 25 Ibs. and it may also have any direction, depending upon
the directions of the two pulls.

Note that such physical objects as a stone or a train are neither
scalars nor vectors. Several physical quantities relating to a
stone are scalars; viz., its mass, volume, and density; while
some are vectors; viz., its weight, and, if in motion, its velocity.

14. Representation of Vectors by Straight Lines. — A very
simple and rapid method of calculating vectors, called the
Graphical Method, depends upon the fact that a vector may be
completely represented by a straight line having at one end an
arrow head. Thus to represent the velocity of a southwest

wind blowing at the rate of 12 miles an hour, a line (a) 2
cm. long, or (6) 4 cm. long, or (c) 2 inches long, may be used as
shown at A, Fig. 4. In case (a), 1 cm. represents 6 miles an
hour; while in case (6) it represents 3 mi. an hour. In case
(c) the scale is chosen the same as in case (a), except that lin.,
instead of 1 cm., represents 6 miles an hour velocity. Any con-
venient scale may be chosen. In each case the length of the
line represents the magnitude of the vector quantity; and the
direction of the line represents the direction of the vector
quantity.

15. Addition of Vectors, Resultant. — The vector sum or
Resultant (see Sec. 13) of two or more forces or other vectors
differs in general from either the arithmetical or the algebraic
sum. By the Graphical method, it may be found as follows.
Choose a suitable scale and represent the first force ^i by a line

VECTORS

having an arrow head as shown at B, Fig. 4. Next, from
the arrow point of this line, draw a second line representing the
second force F2, and from the arrow point of Fz draw a line
representing Fs, etc. Finally connect the beginning of the first
line with the arrow point of the last by a straight line. The
length of this line, say in inches, multiplied by the number of
pounds which one inch represents in the scale chosen, gives the

FIG. 5.

magnitude of the resultant force R. The direction of this line
gives the direction of the resultant force. Obviously, the same
scale must be used throughout. An example involving several
velocities will further illustrate this method of adding vectors.
Although in this course we shall apply the graphical method to
only force and velocity, it should be borne in mind that it may
be, and indeed is, applied to any vector quantity.

A steamboat, which travels 12 miles an hour in still water, is

14 MECHANICS AND HEAT

headed due east across a stream which flows south at the rate of
5 miles an hour. Let us find the velocity of the steamboat.
In an hour, the boat would move eastward a distance of 12
miles due to the action of the propeller, even if the river did not
flow; while if the propeller should stop, the flow of the river alone
would cause the boat to drift southward 5 miles in an hour.
Consequently, if subjected to the action of both propeller and
current for an hour, the steamboat would be both 12 miles far-
ther east and 5 miles farther south, or at D (case A, Fig. 5). By
choosing 1 cm. to represent 4 miles per hr., the " steam"
velocity would be represented by a line a, 3 cm. in length;
while the "drift" velocity of 5 miles an hour to this same
scale, would be represented by a line 6, 1.25 cm. in length. The
length (3.25 cm.) of the line OD or R represents the magnitude
of the steamboat's velocity, and the direction of this line gives
the course of the boat, or the direction of its velocity. The
velocity is then 4X3.25 or 13 miles an hour to the south of east
by an angle 6 as shown. This velocity R, of 13 miles per hour, is
the resultant or vector sum of the two velocities a and 6, and is
evidently the actual velocity of the steamboat.

By the analytical method, the magnitude of the resultant
velocity is given by the equation

# = \/(12)2+(5)2
and its direction is known from the equation

tan 5 = 5/12 = 0.417
from which 0 = 22°.38.

If the steamboat is headed southeast, then a\ and &i (case
B, Fig. 5) represent the "steam" and "drift" velocities re-
spectively, and the magnitude of the resultant velocity R\,
in miles per hr., will be found by multiplying the length of
#1 in centimeters by 4. If the analytical method is employed,
we have from trigonometry,

-Ri2 = ai2+&i2+2a1&i cos 0

Suppose, further, that it is required to find the actual velocity
of a man who is walking toward the right side of the steamboat
at the rate of 2 miles an hour, when the boat is headed as shown
in case B. Let a\, b\, and Ci represent the "steam," "drift,"
and "walking" velocities respectively; then R2 represents the

VECTORS 15

actual velocity of the man as shown in case C, Fig. 5. If the man
walks toward the left side of the boat, his "walking" velocity is
c2 and his actual velocity is R3. In these cases his velocity could
also be found by the analytical method, but not so readily.

16. The Vector Polygon. — In cases A and B (Fig. 5), the
vector triangle is used in determining the resultant; while in
case C, the vector polygon, whose sides are a\, bi, c\ and Rz, is

? so used. In general, however, many vectors are involved, the
closing side of the polygon represents the re-
sultant of all the other vectors.

If a man were to run toward the left and
rear end of the steamboat in the direction Rf
at the speed of 13 miles per hour (case A), he
would appear to an observer on shore to be
standing still with respect to the shore.
Hence his actual velocity is zero. Since R'
is equal to R and oppositely directed, we see
that the three vectors a, 6, and R' would form
exactly the same triangle as a, b, and R, but FlG 5a

for the fact that the arrow head on R' points

in the opposite direction to that on R. Thus vectors forming I f
a closed triangle have a resultant equal to zero.

Again, suppose that the man while walking toward the right
side of the ship, case C (Fig. 5), and therefore having an actual
velocity R2, should throw a ball with an equal velocity R'z in a
direction exactly opposite to that of R2 (i.e., jR'2=— #2). It
will be evident at once that the ball under these circumstances
would simply stand still in the air as far as horizontal motion is
concerned. It will be seen that there are four horizontal
velocities imparted to the ball. First, the "steam" velocity
ai (Fig. 5a), second, the "drift" velocity 61, third, the "walking"
velocity Ci, and fourth, the "throwing" velocity R'2. These four
velocities, however, form a closed polygon and the actual velocity
of the ball is zero. Hence we may now make the general state-
ment that when any number of velocities (or forces or any other
vectors) form a Closed Triangle or a Closed Polygon, their resultant
is zero. This fact is of great importance and convenience in the
treatment of forces in equilibrium and will be made use of in
some of the problems at the close of this chapter.

17. Vectors in Equilibrium. — The method of the preceding
sections applies equally well if the vectors involved are any other

MECHANICS AND HEAT

quantities; e.g., forces, instead of velocities; and the construc-
tions are made in the same way. This method has many impor-
tant applications in connection with forces, among which is the
calculation of the proper elevation of the outer rail on a curve
(Sec. 62) in order that the weight, or better, the thrust of a train
shall be equal upon both rails; and the calculation of the proper
strength for the different parts of bridges and other structures.
In Sec. 16 it was shown that to find in what direction and with
what speed the man must throw the ball in order to make its
actual velocity zero, a line R'z must be drawn equal to R2,
but oppositely directed. R2 is the resultant of the three veloci-

;

FIG. 6.

ties ai, bi, and Ci, while R'z is the Antiresultant (anti = opposed
to) or Equilibrant.

Thus it will be seen that in the graphical method the anti-
resultant of any number of velocities is represented by a line
drawn from the arrow point of the last velocity to the beginning
of the first velocity. In other words, it is represented by the
closing side of the vector polygon. Observe that in this case the
arrow heads all point in the same way around the polygon;
while, if the closing side is the resultant, its arrow head is directed
oppositely to all the others.

The case of several forces in equilibrium, or so-called "bal-
anced forces," is of special importance. The construction is
the same as that shown in Fig. 5a. Suppose that a body floating
in still water is acted upon by four horizontal forces, whose

VECTORS 17

values are represented both in magnitude and direction by the
lines a, b, c, and d of A (Fig. 6). Let it be required to find the
magnitude and direction of a fifth force e', which applied to the
body will produce equilibrium, so that the body will have no
tendency to move in any direction; in other words, let us find
the antiresultant of a, b, c, and d. From B (Fig. 6) we find
the resultant e, or that single force which would exactly replace
a, b, c, and d; i.e., which alone would move the body in the same
direction, and with the same speed as would these four forces.
The construction C shows how e' is found. Obviously, e'
and e alone (D, Fig. 6) would produce equilibrium, and since
e is exactly equivalent to a, b, c, and d, it follows that a, b, c, d,
and e' produce equilibrium. From E (Fig. 6) it will be seen that
the resultant is the same if the vectors a, b, c, and d are taken in
a different order.

This is true for the reason that wherever, in the construction
of the polygon, we choose to draw d, say, the pencil point will
thereby be moved a definite distance to the left. Likewise
drawing 6 moves the pencil a definite distance to the right and
downward. Consequently the final position of the pencil after
drawing lines a, b, c, and d, which position determines the
resultant e, can in no wise depend upon the order of drawing
these lines.

18. The Crane. — The crane, in its simplest form, is shown in
Fig. 7. B is a rigid beam, pivoted at its lower end and fastened
at its upper end by a cable C to a post A. D is the "block and
tackle" for raising the object L whose weight is W. After the
object is raised, the beam B may be swung around horizontally;
and then, by means of the block and tackle, the object may be
lowered and deposited where it is wanted. By shortening the
cable C it is possible to raise the weight higher, but the "sweep"
of the crane is of course shortened thereby.

The traveling crane, used in factories, is mounted on a " car-
riage" which may be run back and forth on a track sometimes
extending the entire length of the building, so that a massive
machine weighing several tons may readily be picked up and
carried to any part of the building.

In choosing the size of the cable and the beam for a crane as
sketched, it is necessary to know what pull will be exerted on C,
and what end thrust on B when the maximum load is being
lifted. These two forces, c and 6, we shall now proceed to find.

MECHANICS AND HEAT

In Sec. 17 it was shown that any number of forces or any other
vectors in equilibrium are represented by a closed polygon.
Three forces in equilibrium will accordingly form a closed tri-
angle. The point 0, at the upper end of the beam B, is obviously
in equilibrium and is acted upon by the three forces W, c, and
6; which forces, graphically represented, must therefore form a
closed triangle. The directions of 6 and c are known but not
their magnitudes. W, however, is fully specified both as to
direction and magnitude. Hence the forces acting upon 0

FIG. 7.

may be represented as in E (Fig. 7), or as in F, since a thrust 6
will have the same effect as an equal pull 6. If L weighs 1 ton,
or 2000 Ibs., its weight W, using as a scale 2000 Ibs. to the cm.,
will be represented by a line 1 cm. in length (G, Fig. 7).
From the lower end of W draw a line b parallel to the beam, and
through the other end of W draw a line c parallel to the cable.
The intersection of these two lines at X determines the magni-
tude of both & and c. For the three forces have the required
directions, and they also form a closed triangle, thus represent-
ing equilibrium. The length of 6 in centimeters times 2000 Ibs.

VECTORS

gives the thrust on the beam. The value of c is found in the
same way. The construction may also be made as shown in H.

The problem will be seen to be simply this: Given one side W
of a triangle, both in direction and length, and the directions only
of the other two sides b and c; let it be required to construct the
triangle.

19. Resolution of Vectors into Components. — The resolution
of a vector V into two components, consists in finding the magni-
tude of two vectors, Vi and Vz, whose directions are given, and ,
whose vector sum shall be the given vector V. It is thus seen to be
the converse of vector addition. The method will be best under-
stood from one or two applications. We shall here apply it
to velocities and forces, but it applies equally well to any other
vector quantity.

FIG. 8.

FIG. 9.

A ship is traveling with a uniform velocity of 20 mi. per hr. in a
direction somewhat south of east. An hour later the ship is
18 mi. farther east and 8.7 mi. farther south than when first
observed. Under such circumstances the velocity of the ship
may be resolved into an eastward component of 18 mi. per hr.
and a southward component of 8.7 mi. per hr. Had the ship
been headed nearly south, the southward component would
have been the larger. We shall next resolve a force into two
components.

Consider a car B (Fig. 8) of weight W, held by a cable C
from running down the inclined track A. Let it be required to
find the pull c that the car exerts upon the cable, and also the
force b that it exerts against the track. The latter is of course
at right angles to the track, but it is not equal to the weight of
the car, as might at first be supposed. In fact, the weight of the
car W, or the force with which the earth pulls upon it, gives
rise to the two forces, b and c. The directions of 6 and c are

Lxiv

MECHANICS AND HEAT

known, but not their magnitudes. In order to find their mag-
nitudes, first draw W to a suitable scale. Then, from the arrow
point of W, draw two lines, one parallel to b and intersecting c,
the other parallel to c and intersecting b. These intersections
determine the magnitudes of both 6 and c, as shown. We may
also determine b and c by the method used in the solution of the
crane problem.

If the cable is attached to a higher point, the construction is as
shown in Fig. 9. It will be noticed that under these conditions
the c component has become larger, and the 6 component smaller,
than in Fig. 8. If the cable is fastened directly above the car,
the 6 component is zero; that is, the car is simply suspended by
the cable.

In case a force is resolved into two components at right angles
to each other, their values may be readily
found by the analytical method. Thus in
Fig. 8, c = W sin 8, and 6 = W cos 6. Note
that 0i = 6.

20. Sailing Against the Wind.— Al-
though sailing "into the wind" by "tack-
ing" has been practised by sea-faring peo-
ple from time immemorial, it is still a puz-
zle to many. Let AB (Fig. 10) represent a
sailing vessel, CD its sail, CE the direction
in which it is headed, and W the direction
of the wind. If the sail CD were friction-
less and perfectly flat, the reaction of the

air in striking it would give rise to a force F strictly at right angles
to the sail. A push (force) against a frictionless surface, whether
exerted by the wind or by any other means, must be normal to
the surface; otherwise it would have a component parallel to
the surface, which is impossible if there is no friction. This
force F may be resolved into the two components 'F\ and F2 as
shown. Although as sketched, F2 is greater than the useful
component FI, nevertheless the sidewise drift of the ship is small
compared with its forward motion, because of its greater resist-
ance to motion in that direction. Making slight allowance for
this leeward drift, we have CE' for the course of the ship.
Obviously, in going from C to £", the ship goes the distance
CH "into" the wind.

In case the boat is moving north at a high velocity, the wind, to

FIG. 10.

VECTORS 21

a person on the boat, will appear to come from a point much more
nearly north than it would to a stationary observer. In other
words, the angle between the plane of the sail and the real direc-
tion of the wind, is always greater than the angle between this
plane and the apparent direction of the wind as observed by an
occupant of the boat. It is, however, the apparent direction or,
perhaps better, the relative velocity of the wind, that determines
the reacting thrust upon the sail. Hence strictly, W (Fig. 10)
should represent the apparent direction of the wind. It is a
matter of common observation that, to a man driving rapidly
north, an east wind appears to come from a point
considerably north of east.

Because of the very slight friction of the wind
on the sail, F' is more nearly the direction of the
push on the sail. The useful component of F',
which drives the ship, is obviously slightly less
than FI as found above for the theoretical case of
no friction.

21. Sailing Faster Than the Wind. — It is pos-
sible, strange though it may seem, to make an
iceboat travel faster than the wind that drives pIG
it. Let AB (Fig. 11) represent the sail (only)
of an iceboat which is traveling due north, and v the velocity of
the wind. If the runner friction were zero, so that no power would
be derived from the moving air, the air would move on unchanged
in both direction and speed. Considering the air that strikes at A,
this would evidently require the sail to travel the distance A A '
while the wind traveled from A to B'. Hence the velocity v' of
the boat would be AA'/AB' times that of the wind, or v'/v =
AA'/AB'. The slight friction between the runners and the ice
reduces this ratio somewhat; nevertheless, under favorable cir-
cumstances, an iceboat may travel twice as fast as the wind.
Velocities as high as 85 mi. per hr. have been maintained for
short distances.

PROBLEMS

1. A balloon is traveling at the rate of 20 miles an hour due southeast.
Find its eastward and southward components of velocity by both the graph-
ical and analytical methods.

2. Find the force required to draw a wagon, which with its load weighs
2500 Ibs., up a grade rising 40 ft. in a distance of 200 ft. measured on the grade.
Neglect friction, and use the graphical method.

22 MECHANICS AND HEAT

3. Find Rlt case B (Fig. 5) if 0=60°. (Cos 60° =0.5). Use the ana-
lytical method.

4. A boat which travels at the rate of 10 mi. an hr. in still water, is headed
S.W. across a stream flowing south at the rate of 4 mi. an hr. A man on the
deck runs at the rate of 7 mi. an hr. toward a point on the boat which is due
east of him. Find the actual velocity of the man with respect to the earth,
and also that of the boat. Use graphical method.

6. By the graphical method, find the resultant and antiresultant of the
following four forces: 10 Ibs. N., 12 Ibs. N.E., 15 Ibs. E., and 8 Ibs. S.

6. If the beam B (Fig. 7) is 30 ft. in length and makes an angle of 45° with
the horizontal, and the guy cable C is fastened 15 ft. above the lower end of
B, what will be the thrust on B and the pull on C if the load L weighs 3000
Ibs. ? Use the graphical method.

7. After a man has traveled 4 miles east, and 4 miles N., how far must he
travel N.W. before he will be due north of the starting place, and how far
will he then be from the starting place? Solve by both the graphical method
and the analytical method.

8. A certain gun, with a light charge of powder, gives its projectile an
initial (muzzle) velocity of 300 ft. per sec. when stationary. If this gun is
on a car whose velocity is 100 ft. per sec. north, what will be the muzzle
velocity of the projectile if the gun is fired N.? If fired S.? If fired E.?

9. A south wind is blowing at the rate of 30 mi. per hr. Find, by the
graphical method and also by the analytical method, the apparent velocity
of the wind as observed by a man standing on a car which is traveling east
at the rate of 40 mi. per hr.

10. The instruments on a ship going due north at the rate of 20 miles an
hour record a wind velocity of 25 miles per hour from the N.E. What is the
actual velocity of the wind? Use the graphical method.

11. A tight rope, tied to two posts A and B which are 20 ft. apart, is pulled
sidewise at its middle point a distance of 1 ft. by a force of 100 Ibs. By two
graphical methods (Sec. 18 and 19) find the pull exerted on the posts. Solve
also by the analytical method.

12. Neglecting friction, find the pull on the cable and the thrust on the
track in drawing a 1000-lb. car up a 45° incline. The cable is parallel to the
track.

13. Find the pull and the thrust (Prob. 12) if the cable is (a) horizontal;
(6) inclined 30° above the horizontal.

•

TRANSLATORY MOTION

CHAPTER III

22. Kinds of Motion. — All motion may be classed as either
translatory motion or rotary motion, or as a combination of these
two. A body has motion of translation only, when any line
(which means every line) in the body remains parallel to its
original position throughout the motion. It may also be defined
as a motion in which each particle of the body describes a path
of the same form and length as that of every other particle, and
at the same speed at any given instant; so that the motion of
any one particle represents completely the motion of the entire
body. Thus if A, B, C, and D represent the positions of a

FIG. 12.

FIG. 13.

triangular body (abc) at successive seconds, it will be noted that
in case a moves a greater distance in the second second
than it does in the first that 6 and c and all other particles do
also.

In pure rotary motion there is a series of particles, e.g., those
in the line AB (Fig. 13) which do not move. This line is called
the axis of rotation of the body. All other particles move in
circular paths about this axis as a center, those particles farthest
from the axis having the highest velocity.

Having obtained a clear notion of rotary motion, we may con-
sider a body to have pure translatory motion if it moves from
one point to another by any path, however straight or crooked,
without any motion of rotation. The rifle ball has what is

24 MECHANICS AND HEAT

termed Screw motion. The motion of a steamship might seem to
be pure translatory motion, and indeed it closely approximates
such motion when the sea is calm. In a rough sea its motion is
very complicated, consisting of a combination of translatory
motion, with to-and-fro rotation about three axes: In the "roll-
ing" of a ship, the axis is lengthwise of the ship or longitudinal.
The "pitching" of a ship is a to-and-fro rotation about a trans-
verse axis. As the ship swerves slightly from its course, it
rotates about a vertical axis.

Both translatory and rotary motion may be either uniform,
or accelerated; that is, the velocity may be either constant or
changing. Accelerated motion is of two kinds, uniformly accel-
erated and nonuniformly accelerated. Thus there are three types
each of both translatory and rotary motion. Before discussing
these types of motion, it will be necessary to define and discuss
velocity and acceleration.

23. Speed, Average Speed, Velocity, and Average Velocity. —
As already mentioned (Sec. 13), speed is a scalar quantity and
velocity is a vector quantity. Both designate rate of motion;
but the former does not take into account the direction of the
motion, whereas the latter does.

Average speed, which may be designated by s (read " barred s ")
is given by

D
« "J (3)

in which D is thejtotal distance traversed by a body in a given
time t. Average velocity v is given by the equation

- ^ ft\

v = y (4)

in which d is the distance from start to finish measured in a
straight line, and t is the time required. Observe that d has,
in addition to magnitude, a definite direction, and is therefore a
vector; whereas D is simply the distance as measured along the
path traversed, which may be quite tortuous, and is therefore a
scalar. The Speed of a body at any given instant is the distance
which the body would travel in unit time if it maintained that
particular rate of motion; while the Velocity of the body at that
same instant has the same numerical value as the speed, and
is defined in the same way except that it must also state the

TRANSLATORY MOTION 25

direction of the motion. An example will serve to further illus-
trate the significance of the above four quantities.

Suppose that a fox hunt, starting at a certain point, termi-
nates 10 hrs. later at a point 20 miles farther north. Suppose
further that during this time the dog travels 100 miles. Then
d (Eq. 4) is 20 miles due north (a vector), D (Eq. 3) is 100 miles
(scalar), ~v is 2 miles an hour north (vector), and ¥ is 10 miles an
hour (scalar). If the dog's speed s at a given instant is 15 miles
an hour (often written 15 mi./hr. and called 15 mi. per hr.),
then an hour later, if he continues to run at that same speed, he
will be 15 miles from this point as measured along the trail;
whereas if the dog's velocity at that same instant is 15 miles per
hour east, then, an hour later, if he maintains that same velocity,
he will be at a point 15 miles farther east.

If the hunter travels 40 miles, while a friend, traveling a
straight road, travels only 20 miles in the ten hours, then the
hunter's average speed is twice that of his friend and only two-
fifths that of the dog; whereas the average velocity ID is the same
for all three, viz., 2 miles an hour. We thus see that the aver-
age velocity of a body is that velocity which, unchanged in
either magnitude or direction, would cause the body to move from
one point to the other in the same time that it actually does
require.

24. Acceleration. — If a body moves at a uniform speed in a
straight line it is said to have uniform velocity, and its velocity
is the distance traversed divided by the time required. If its
speed is not uniform its velocity changes (in magnitude), and
the rate at which its velocity changes is called the acceleration, a.
If the velocity of a body is not changing at a uniform rate, then
the change in velocity that occurs in a given time, divided by
that time, gives the average rate of change of the velocity of
the body, or its average acceleration for that time. Since the
second is the unit of time usually employed, we see that the
average acceleration is the change (gain or loss) in velocity per
second. The acceleration of such a body at any particular
instant is numerically the change in velocity that would occur
in 1 sec. if the acceleration were to have that same value for
the second; i.e., if the velocity were to continue to change at
that same rate for the second.

If the velocity is increasing, the acceleration is positive; if
decreasing, it is said to be negative. Thus the motion of a train

26 MECHANICS AND HEAT

when approaching a station with brakes applied, is accelerated
motion. As it starts from the station it also has accelerated
motion, but in this case the acceleration is positive, since it is in
the direction of the velocity; while in the former case, the
acceleration is negative.

If the acceleration of a body is constant, for example if the body
continues to move faster and faster, and the increase in velocity
each succeeding second or other unit of time is the same, its
motion is said to be uniformly accelerated. Thus if the velocity
of a body expressed in feet per second, e.g., the velocity of a
street car, has the values 10, 12, 14, 16, 18, etc., for successive
seconds; then the acceleration a for this interval is constant,
and has the value 2 ft. per sec. per sec., or

a = 2 ft. per sec. per sec. (also written 2 — ^,)

sec.2

If a certain train is observed to have the above velocities for
successive minutes, then the motion of the train is uniformly
accelerated, since its acceleration is constant; but it is less than
above given for the street car, in fact, 1/60 as great, or 2 ft. per
sec. per min. ; that is,

2 ft

a = 2 ft. per sec. per min. (also written , )

sec. mm.

This means that the gain of velocity each minute is 2 ft. per sec.
A freely falling body, or a car running down a grade due to its
weight only, are examples of uniformly accelerated motion. In
order that a body may have accelerated motion, it must be acted
upon by an applied or external force differing from that required
\ to overcome all friction effects upon the body.

25. Accelerating Force. — Force may be defined as that which
produces or tends to produce change in the velocity of a body, to
which it is applied; i.e., force tends to accelerate a body. A force
may be applied to a body either as a push or a pull. It has been
shown experimentally that it requires, for example, exactly twice
as great a force to give twice as great an acceleration to a given
mass which is perfectly free to move; and also that if the mass be
doubled it requires twice as much force to produce the same ac-
celeration. In other words, the force (F) is proportional to the
resulting acceleration (a), and also proportional to the mass

TRANSLATORY MOTION 27

(M) of the body accelerated. These facts are expressed by the
equation

F = Ma (5)

For, to increase a n-fold, F must be increased n-fold; in other
words, the resulting acceleration of a body is directly propor-
tional to the applied force, and is also inversely proportional to
the mass of the body.

Eq. 5 is sometimes written F = kMa. If the units of force,
mass, and acceleration are properly chosen (see below), k becomes
unity and may be omitted.

Units of Force. — Imagine the masses now to be considered, to
be perfectly free to move on a level frictionless surface, and let
the accelerating force be horizontal. Then the unit force in the
metric system, the Dyne, is that force which will give unit mass
(1 gm.) unit acceleration (1 cm. per sec. per sec.); while in the
British system, unit force, the Poundal, is that force which will
give unit mass (1 Ib.) -unit acceleration (1 ft. per sec. per sec.).
Thus, to cause the velocity of a 10-gm. mass to change by 4 cm.
per sec. in 1 sec.; i.e., to give it an acceleration of 4 cm. per sec.
per sec., will require an accelerating force of 40 dynes, as may be
seen by substituting in Eq. 5.

The relation between these units and the common gravita-
tional units, the gram weight and the pound weight, will be ex-
plained under the study of gravitation (Sec. 32) ; but we may here
simply state without explanation that 1 gram weight is equal to
980 dynes (approx.), and that 1 pound weight is equal to 32.2
poundals (approx.).

In general, only a part of the force applied to a body is used
in accelerating it, the remainder being used to overcome friction
or other resistance. The part that is used in producing accelera-
tion is called the Accelerating Force. It should be emphasized
that Eq. 5 holds only if F is the accelerating force. Thus if a
stands for the acceleration in the motion of a train, and M for
the mass of the train, then F is not the total pull exerted by the
drawbar of the engine, but only the excess pull above that needed
to overcome the friction of the car wheels on axle bearings and
on the track, air friction, etc. If an 8000-lb. pull is just sufficient
to maintain the speed of a certain train at 40 miles an hour on a
level track, then a pull of 9000 Ibs. would cause its speed to in-
crease, and 7000 Ibs., to decrease. The accelerating force, i.e.,

28 MECHANICS AND HEAT

the F of Eq. 5, would be 1000 Ibs., i.e., 32,200 poundals, in each
case.

In the case of a freely falling body, the accelerating force is
of course the pull of the earth upon the body, or its weight; while
in the case of a lone car running down a grade, it is the component
of the car's weight parallel to the grade (see Fig. 8), minus the
force required to overcome friction, that gives the accelerating
force. We may now make the statement that when a body is in
motion its velocity will not change if the force applied is just suffi-
cient to overcome friction; while if the force is increased, the
velocity will increase, and the acceleration will be positive and
proportional to this increase or excess of force. If the applied
force is decreased so as to become less than that needed to
overcome friction, then, of course, the velocity decreases, and the
acceleration is negative and proportional to the deficiency of
the applied force.

•" 26. Uniform Motion and Uniformly Accelerated Motion. —
This subject will be best understood if discussed in connection
with a specific example. Suppose that a train, traveling on a
straight track and at a uniform speed from a town A to a town B
20 miles north of A, requires 30 minutes time. In this case its
velocity

_ distance traversed _ d _20 miles
time required = t = 30 min.

or 2/3 of a mile per min. north. Since the velocity is constant,
the train is said to have Uniform Motion. If the track is level,
the pull on the drawbar of the engine must be just sufficient to
overcome friction, since there is no acceleration and hence no
accelerating force. Thus, uniform motion may be defined as
the motion of a body which experiences no acceleration. This
train would have to be a through train; for if it is a train that
stops at A, its velocity just as it leaves A would be increasing;
i.e., there would be an acceleration. Consequently there would
have to be an accelerating force; that is, the pull on the drawbar
would have to be greater than the force required to overcome
friction. In this case the motion would be accelerated motion.

In case the accelerating force is constant, for example, if the
pull on the drawbar exceeds the force required to overcome
friction by, say 4000 Ibs. constantly for the first minute, then the
acceleration (a) is constant or uniform, and the motion for this

TRANSLATOR? MOTION 29

first minute would be Uniformly Accelerated Motion. For, from
F = Ma (Eq. 5), we see that if the accelerating force F (here
4000 Ibs.) is constant, a will also be constant; i.e., the velocity of
the train will increase at a uniform rate. As a rule, this excess
pull is not constant, so that the acceleration varies, and the train
has nonuniformly accelerated motion.

Let us further consider the motion of the above train if the
accelerating force is constant, and its motion, consequently,
uniformly accelerated. Suppose that its velocity as it passes a
certain bridge is 20 ft. per sec. and that we represent it byy0;
while its velocity 10 seconds later (or t sec. later) is 34.6 ft. per
sec., represented by vt. Its total change of velocity in this time
t is vt — vu, hence the acceleration

vt-Vo 34.6-20
a = — -. — = — 10 = 1.46 ft. per sec. per sec. (6)

It is customary to represent the velocity first considered by va,
and the velocity t seconds later by vt, as we have here done. If
we first consider the motion of the train just as it starts from A,
i.e., as it starts from rest, then v0 is zero, and vt is its velocity t
seconds after leaving A. If t is 60 sec., then vt is the velocity of
the train 60 seconds after leaving A.

Let us suppose that one minute after leaving A (from rest)
the velocity of the train is 60 miles per hour. This is the same
as 1 mile per min. or 88 ft. per sec. The total change in velocity
in the first minute is then 60 miles per hour, and hence the accel-
eration is 60 miles per hour per minute, or

a = 60 miles per hr. per min.
This same acceleration is 1 mile per minute per minute or

a = l mi. per min. per min.
It is also 88 ft. per second per minute, or

a = 88 ft. per sec. per min. =77^ ft. per sec. per sec.

= 1.46 ft. per sec. per sec.

This equation states that the change of velocity in one minute
is 88 ft. per sec., while in one second it is of course 1/60 of
this, or 1.46 ft. per sec. Ten seconds after the train leaves A,

30 MECHANICS AND HEAT

its velocity is 10X1.46 or 14.6 ft. per sec. Observe that when
v0 is zero, Eq. 6 may be written

vt = at (7)

27. Universal Gravitation. — Any two masses of matter exert
upon each other a force of attraction. This property of matter is
called Universal Gravitation. Thus a book held in the hand
experiences a very feeble upward pull due to the ceiling and other
material above it; side pulls in every direction due to the walls,
etc.; and finally, a very strong downward pull due to the earth.
This downward pull or force is the only one that is large enough
to be measured by any ordinary device, and is what is known as
the weight of the body.

That there is a gravitational force of attraction exerted by
every body upon every other body, was shown experimentally
by Lord Cavendish. A light rod with a small metal ball at
each end was suspended in a horizontal position by a vertical
wire attached to its center. A large mass, say A, placed near
one of these balls B and upon the same level with it, was found
to exert upon the ball a slight pull which caused the rod to rotate
and twist the suspending wire very slightly. Comparing this
slight pull on B due to A, with the pull of the earth upon 5, i.e.,
with B's weight, Cavendish was able to compute the mass of the
earth. In popular language, he Weighed the Earth.

From the mass of the earth and its volume Lord Cavendish
determined the average density of the earth to be about 5.5
times that of water. The surface soil and surface rocks —
sandstone, limestone, etc. — have an average density of but 2.5
times that of water. Hence the deeper strata of the earth are
the more dense, and consequently as a body is lowered into a
mine and approaches closer and closer to the more dense mate-
rial, its weight might be expected to increase. The upward
attraction upon the body exerted by the overlying mass of earth
and rocks should cause its weight to decrease. The former more
than offsets the latter, so that there is a slight increase in the
weight of a body as it is carried down into a deep mine.

Newton's Law of Gravitation. — Sir Isaac Newton was the first to express
clearly the law of universal gravitation by means of an equation. He
made the very logical assumption that the attractive gravitational
force (F) exerted between two masses MI and M 2, when placed a distance

<r W* ^

- frKtt

TRANSLATORY MOTION 31

d apart, would be proportional to the product of the masses, and
inversely proportional to the square of the distance between them (Sec.
28), i.e.,

F_kMM*
d*

If, in this equation, MI and M2 are expressed in grams, the distance in
centimeters, and F in dynes, then k, the proportionality constant or
proportionality factor (Sec. 28) is shown by experiment to be
0.0000000666. If Mi, Mz, and d are all unity, then F = k. In other
words, the gravitational attraction between two 1-gm. masses when 1 cm.
apart is 0.0000000666 dynes. Since the dyne is a small force, this will
be seen to be a very small force. Lord Cavendish used this equation in
computing his results.

28. The Law of the Inverse Square of the Distance.— This
law is one of the most important laws of physics and has many
applications, a few of which we shall now consider. We are
all familiar with the fact that as we recede from a source of light,
for example a lamp, the intensity of the light decreases. That
the intensity of illumination at a point varies inversely as the
square of the distance from that point to the light source, has
been repeatedly verified by experiment, and it may also be dem-
onstrated by a simple line of reasoning as follows: Imagine a
lamp which radiates light equally in all directions, to be placed
first at the center of a hollow sphere of 1 ft. radius, and later at
the center of a similar hollow sphere whose radius is 3 ft.
In each case the hollow sphere would receive all of the light
emitted by the lamp, but in the second case this light would be
distributed over 9 times as much surface as in the first. Hence,
the illumination would be 1/9 as intense, and we have therefore
proved that the intensity of illumination varies inversely as the
square of the distance from the lamp.

An exactly similar proof would show that the same law applies
in the case of heat radiation, or indeed in the case of any effect
which acts equally in all directions from the source. This
law has been shown to hold rigidly in the case of the gravita-
tional attraction between bodies, for example between the differ-
ent members of the solar system.

Proportionality Factor. — In all cases in which one quantity is propor-
tional to another, the fact may be stated by an equation if we introduce
a proportionality factor (k). Thus the weight of a certain quantity of

32 MECHANICS AND HEAT

water is proportional to its volume; i.e., 3 times as great volume will
have 3 times as great weight, and so on. We may then write

W«V, but not W=V
We may, however, write

W = kV

in which k is called the proportionality factor. In this case k (in the
English system) would be numerically the weight of a cubic foot of
water, or 62.4 (1 cu. ft. weighs 62.4 Ibs.), V being the number of cubic
feet whose weight is sought.

We may add another illustration of the use of the proportionality
factor. We have just seen that the illumination (/) at a point varies
inversely as the square of the distance from the source. We also know
that it should vary as the candle power (C.P.) of the source. Hence we
may write

. C.P. .CP.

/---, or 7 = *-^-

A third illustration has already been given at the close of Sec. 27.

29. Planetary Motion. — The earth revolves about the sun once
a year in a nearly circular orbit of approximately 93,000,000
miles radius. The other seven planets of the solar system have
similar orbits. The planets farthest from the sun have, of course,
correspondingly longer orbits, and they also travel more slowly;
so that their "year" is very much longer than ours. Thus
Neptune, the most distant planet, requires about 165 years to
traverse its orbit, while Mercury, which is the closest planet to
the sun, has an 88-day "year." The moon revolves about the
earth once each lunar month in an orbit of approximately 240,000
miles radius. Several of the planets have moons revolving
about them while they themselves revolve about the sun.

If a stone is whirled rapidly around in a circular path by means
of an attached string, we readily observe that a considerable pull
must be exerted by the string to cause the stone to follow its
constantly curving path (Sec. 58). In the case of the earth and
the other planets, it is the gravitational attraction between planet
and sun that produces the required inward pull. Our moon is
likewise held to its path by means of the gravitational attraction
between the earth and the moon. The amount of pull required
to keep the moon in its course has been computed, and found to
be in close agreement with the computed gravitational pull that

TRANSLATORY MOTION 33

the earth should exert upon a body at that distance. In comput-
ing the latter it was assumed that the inverse square law (Sec.
28) applied.

Since the moon is approximately 60 times as far from the center
of the earth as we are, it follows that the pull of the earth upon a
pound mass at the moon is (I/GO)2 or 1/3600 pound. By means
of the formulas developed in Sec. 58, the student can easily show
that this force would exactly suffice to cause the moon to follow
its constantly curving path if it had only one pound of mass. Since
the mass of the moon is vastly greater than one pound, it requires
a correspondingly greater force or pull to keep it to its orbit, but
its greater mass also causes the gravitational pull between it and
the earth to be correspondingly greater so that this pull just
suffices.

30. The Tides. — A complete discussion of the subject of tides
is beyond the scope of this work, but a brief discussion of this
important phenomenon may be of interest. Briefly stated, the

a ^^_

/" Moon "~^\

FIG. 14. FIG. 15.

$%£ &<

main cause of tides is the fact that the gravitational attraction of (> t
the moon upon unit mass is greater for the ocean upon the side of
the earth toward it, than for the main body of the earth; while ,VLJ
for the ocean lying upon the opposite side of the earth, it is less.
This follows directly from a consideration of the law of inverse
squares (Sec. 28).

This difference in lunar gravitational attraction tends to heap
the water slightly upon the side of the earth toward it and also
upon the opposite side; consequently if the earth always presented
the same side to the moon, these two "heaps" would be perma-
nent and stationary (Fig. 14). As the earth rotates from west
to east, however, these two "heaps" or tidal waves travel from

34 MECHANICS AND HEAT

east to west around the earth once each lunar day (about 24 hrs.
50 min.), tending, of course, to keep directly under the moon.
Due to the inertia of the water, the tidal wave lags behind
the moon; so that high tide does not occur when the moon is
overhead (Fig. 14), but more nearly at the time it is setting, and
also when it is rising (Fig. 15). Since the moon revolves about
the earth from west to east in approximately 28 days, we see
why the lunar day, moonrise to moonrise, or strictly speaking,
"moon noon" to "moon noon," is slightly longer than the solar
day (Sec. 4).

Every body of the solar system, so far as known, except Nep-
tune's moon, revolves in a counterclockwise direction both about
its axis and also in its orbit as viewed from the North Star. Hence
the arrows a, b, c, and d respectively represent the motion of the
moon, rotation of the earth, motion of the tides, and apparent
motion of the moon with respect to the earth. Consequently,
according to this convention, the moon rises at the left and sets
at the right, which is at variance with the usual geographical
convention.

Although the sun has a vastly greater mass than the moon, its
much greater distance from the earth reduces its tidal effect to
less than half that of the moon. During new moon, when the
sun and moon are on the same side of the earth, or at full moon,
when on opposite sides, their tidal effects are evidently additive, and
therefore produce the maximum high tides known as Spring
tides. During first quarter and last quarter their tidal effects are
subtractive, giving the minimum high tides or Neap tides. For
if the sun were in the direction S (Fig. 14) it would tend to pro-
duce high tide at e and/, and low tide at c and h.

On small islands in mid-ocean, the tidal rise is but a few feet;
while in funnel-shaped bays facing eastward, such as the Bay of
Fundy, for example, it is from 40 to 50 feet.

If the earth were completely surrounded by an ocean of uniform depth,
the above simple theory would explain the behavior of the tides. Under
such circumstances tides would always travel westward. The irregular
form and varying depth of the ocean make the problem vastly more
complex. Thus the tide comes to the British Isles from the south-
west. (Ency. Brit.). This tide, which is simply a large long wave pro-
duced by the true tidal effect in a distant portion of the open ocean, first
reaches the west coasts of Ireland and England, and then, passing
through the English Channel, reaches London several hours later.

TRANSLATORY MOTION 35

31. Acceleration of Gravity and Accelerating Force in Free
Fall. — Since the earth exerts the same pull upon a body whether
at rest or in motion, it will be evident that the accelerating force
in the case of a falling body is simply its weight W, and hence we
have from Eq. 5, Sec. 25.

W = Ma, or W=Mg (8)

in which M is the mass of the falling body, and g is its accelera-
tion. It is customary to use g instead of a to designate the
acceleration of gravity, i.e., the acceleration of a freely falling
body. From Eq. 8 we see that g = W/M, and since a mass n
times as large has n times as great weight, g must be constant;
i.e., a 10-lb. mass should fall no faster than a 1-lb. mass, neglect-
ing air friction. If it were not for air friction, a feather would fall
just as fast as a stone. This has been demonstrated by placing
a coin and a feather in a glass tube ("guinea and feather" ex-
periment) and then exhausting the air from the tube by means
of an air pump. Upon inverting the tube, it is found that the
coin and the feather fall equally fast; hence they must both ex-
perience the same constant acceleration. From Eq. 8 it follows
that g varies in value with change of altitude or latitude just
as does the weight W of a body (Sec. 11).

Since the acceleration of gravity, g, represents the rate at
which any falling body gains velocity, it is at once evident that
it is a very important constant. Its value has been repeatedly
determined with great care, and it has been found that

0 = 980.6 cm. per sec. per sec. (9)

for points whose latitude is about 45°. For points farther north
it is slightly greater than this (983.2 at pole); and for points
farther south, slightly less (978 at equator.) The above equa-
tion states that in one second a falling body acquires an addi-
tional velocity of 980.6 cm. per sec. Since 980.6 cm. per sec. =
32.17 ft. per sec., we have

gr = 32.17 ft. per sec. per sec. (9a)

We may define the Acceleration of Gravity as the rate of change
of velocity of a freely falling body; hence it is numerically the
additional velocity acquired by a body in each second of free
fall. If it were not for air friction, a body would add this 32.17
ft. per sec. (980.6 cm. per sec.) to its velocity every second, h<?w-

36 MECHANICS AND HEAT

ever rapidly it might be falling. Though a close study of the
effects of air friction upon the acceleration is beyond the scope
of this course, we readily see that when a falling body has ac-
quired such a velocity that the air friction resisting its fall is
equal to one-third of its weight, then only two-thirds of its weight
remains as the accelerating force. Its acceleration would then,
of course, be only two-thirds g. When a falling body, for exam-
ple a hailstone, has acquired such a velocity that the air friction
encountered is just equal to its weight, then its entire weight is
used in overcoming friction, the accelerating force acting upon
it has become zero, and its acceleration is zero; i.e., it makes no
further gain in velocity.

32. Units of Weight and Units of Force Compared. — From
Eq. 5 (Sec. 25) we see that the logical unit of force is that force
which will give unit mass unit acceleration, or unit change of
velocity in unit time. Hence, in the metric system, unit force,
or the Dyne (See also "Units of Force," Sec. 25), is that force
which will give one gram mass an acceleration of 1 cm. per sec.
per sec., i.e., a change in velocity of 1 cm. per sec. in a
second. In the case of a gram mass falling, the accelerating force
is a gram weight, and the velocity imparted to it in one second
is found by experiment (in latitude 45°) to be 980.6 cm. per sec.
(Sec. 31); whence g equals 980.6 cm. per sec. per sec. It
follows at once, then, that a gram weight equals 980.6 dynes,
since it produces when applied to a gram mass 980.6 times as
great an acceleration as the dyne does. Likewise in the British
system, unit force (the Poundal) is that force which will give unit
mass (the pound) unit velocity (1 ft. per sec.) in unit time (the
second). But in the case of a pound mass freely falling, the
accelerating force is one pound weight, and this force, as experi-
ment shows, imparts to it a velocity of 32.17 ft. per sec. in one
second. It follows at once that one pound force, or one pound
weight, equals 32.17 poundals, since it produces 32.17 times
as great acceleration with the same mass (see Eq. 5).

The poundal and the dyne are the absolute units of force. The
pound, ton, gram, kilogram, etc., are some of the units of force in
common use. Forces are measured by spring balances and
other weighing devices.

In Eq. 8, the weight is expressed in absolute units; in which
case W = Mg. If W is expressed in grams weight or pounds
weight, then we have simply W = M (numerically), i.e., a 100-

TRANSLATORY MOTION 37

gm. mass weighs 100 grams, or 98,060 dynes. Likewise a 10-lb.
mass weighs 10 Ibs., or 321.7 poundals (latitude 45°).

The Engineer's Units of Force and Mass. — In engineering work
the pound is used as the unit of force instead of the poundal.
Transposing Eq. 8, Sec. 31, we have M = W/g. Now in physics,
W is expressed in poundals, M being in pounds, while in engi-
neering work W is expressed in pounds. Since the pound is 32.17
times as large a unit as the poundal, M must be expressed in
the engineering system in a unit 32.17 times as large as the
pound mass (close approximation). This 32.17-lb. mass is
sometimes called the Slug.

As a summary, let us write the equation F = Ma, and the simi-
lar equation restricted to gravitational acceleration; namely,
W = Mg, indicating the units for each symbol in all three sys-
tems— the Metric, the British, and the Engineering systems.

Metric System:

F = Ma and W = Mg, i.e., F'or W (dynes)

= M (gm.) X« or g (cm. per sec. per sec.).

British System:

F = Ma and W = Mg, i.e., F or W (poundals)

= M(lbs.) Xa or g (ft. per sec. per sec.).

Engineering System:

W
F = — a=(Ma) and W = Mg, i.e., F or W (pounds)

= Af(slugs) X« or g (ft. per sec. per sec.).

Thus, practically, the engineering system differs from the
British system in that the units of mass, force, and weight are
32.17 times as large as the corresponding units in the British system.

Some regret that the engineering system was ever introduced.
It is now firmly established, however, and the labor involved in
mastering this third system is very slight, indeed, if the British
system is thoroughly understood. Furthermore, this system
has in some cases certain advantages.

Observe that the word "pound" is used for the unit of mass
and also for one of the units of force. Having defined the pound
force as the weight of a pound mass, we may (and frequently do)
use it (the pound force) as the unit in measuring forces which
have absolutely nothing to do with either mass or weight. Thus
in stretching a clothes line with a force of, say 50 Ibs., it is clear
that this 50-lb. force has nothing to do with the mass or weight
of the clothes line, or post, or anything else. The pound force

443914

38 MECHANICS AND HEAT

is used almost exclusively as the unit of force in engineering
work. Objection to its use as a unit is sometimes made because
of the fact that the weight of a 1-lb. mass varies with g. Since
g varies from 978 at the equator to 983.2 at the poles (Sec. 31),
we see that the weight of a 1-lb. mass (or any other mass) is about
1/2 per cent, greater at the poles than at the equator. This
slight variation in the value of the pound force may well be
ignored in practically all engineering problems. If the standard
pound force is defined as the weight of a 1-lb. mass in latitude 45°
(g = 980.6), it becomes as definite and accurate as any other unit
of force.

33. Motion of Falling Bodies; Velocity — Initial, Final, and
Average. — The initial velocity of a body is usually represented by
v0 (Sec. 26), and the final velocity by vt. An example will serve
the double purpose of illustrating exactly what these terms mean
as applied to falling bodies, and also of showing how their numer-
ical values are found.

Suppose that a body has been falling for a short time before we
observe it and that we wish to discuss its motion for the succeed-
ing eight seconds of fall. Suppose that its initial velocity v0,
observed at the beginning of this eight-second interval, is 20 ft.
per sec. Its final velocity vt at the close of this eight-second
interval would be found as follows. It will at once be granted
that the final velocity vt will be equal to the initial velocity plus the
acquired velocity. But by definition (Sec. 31), g is numerically
the velocity acquired or gained in one second of free fall. Hence
in two seconds the acquired velocity would be 2g, in 3 seconds 3g,
and in t seconds the velocity acquired would be gt. Accordingly

Vt = v0+gt (10)

In the present problem vt = 20+32.17X8 = 277. 36 ft. per sec.

Average velocity is commonly represented by » (read "barred
v"), and in the case of falling bodies it is equal to half the sum of
the initial and final velocities. Hence

_ Vo+vt v0+(vo+gt) 1

v = — ^ — = ^ '2 =v0+wt (11)

In general, the average velocity of a train would not be even
approximately equal to half the sum of the initial and final
velocities. We ought therefore to prove the validity of Eq. 11.

TRANSLATORY MOTION

We readily see that the average value of all numbers from 40 to
100 is 140-^2 or 70. If the velocity of a train is 10 feet per sec.,
and each succeeding minute it gains 2 feet per second, then its
velocities for the succeeding minutes are respectively 10, 12, 14,
16, 18, 20, 22 feet per second, and its average velocity would
be, under these special circumstances, one-half the sum of
the initial and final velocities. Adding all these numbers and
dividing by 7 gives an average of 16, but one-half the sum of the
first and last is also 16.

We may now make the general statement that one-half the
sum of the first and last of a series
of numbers gives a correct value for
the average, provided the successive
values of the numbers in the series
differ by a constant amount. Now
the velocity each successive second is
g feet per second (approximately 32
feet per second) greater than for the
preceding second; consequently, in all
cases of falling bodies, the average veloc-
ity is half the sum of the initial and
final velocities, as given in Eq. 11.

The above facts are shown graph i- \

cally in Fig. 16, in which the succes-
sive lines 1, 2, 3, 4, 5, . . . t repre-
sent the velocities of a body after
falling 1, 2, 3, 4, . . . t seconds re- pIQ 16

spectively. Observe that the velocity

at any time, e.g., after 6 seconds, consists of two parts; that
above the horizontal dotted line being the initial velocity va,
and that below, the acquired velocity (or gt), at that instant.
It will be evident, as the figure shows, that the average velocity
will be attained when half of the time, viz., 4 seconds, has
elapsed, and hence v = v0+%gt; whereas the final velocity vt is
attained after the whole time t has elapsed, and is therefore
v0+gt, as given above (Eq. 10).

In case the body falls from rest, v0 is zero, and the conditions
would be represented by only the portion of Fig. 16 below the
dotted line. In this case the entire velocity vt at any instant
would be merely gt or that acquired previous to that instant, and
the average velocity 1 for a given time t would be %gt.

MECHANICS AND HEAT

34. Distance Fallen in a Given Time. — In general, the distance
d traversed by any body in a given time is its average velocity v
times this time, or d = jjt. Introducing the value of v from Eq.
11 gives,

d = vt=(v0-\ — n-)t = v0t-}-%gt~ (12)

If v0 = Q, i.e., if the body falls from rest, and the distance it falls
in seconds is wanted, then, from Eq. 12,

d = $gt2 (13)

If v0 = Q, Eq. 10 may be written t = —. Substituting this value
of t in Eq. 13, we obtain

vt = V2gd = V2gh

(14)

In this equation, vt is the velocity acquired by a body in falling
from rest through a distance d (or k).

It will be observed that v0t of Eq. 12 is the distance which the
body with initial velocity v0 would
travel in t seconds if there were no
acceleration; while %gt* is the dis-
tance it would travel in this same
time if there had been no initial ve-
locity, i.e., had it fallen from rest.
The distance it actually does travel,
since there are both initial velocity
and acceleration, is simply the sum
(vector sum) of these two. If a
person throws a stone vertically up-
ward with a velocity v0, then the
distance from that person's hand
to the stone after t seconds will be
v0t — % gt2. For evidently the dis-
tances the stone would go, due to
its initial velocity alone, and due to

falling alone, are directly opposite as indicated by the minus
sign. Finally, if a person on a high cliff throws a stone at an
angle of 45° (upward) from the horizontal with a velocity of
20 ft. per second, let us find the distance from his hand to the
stone 3 seconds later. Due to its initial velocity alone, it would
be 60 ft. distant, represented by line a (Fig. 17), while due to

FIG. 17.

TRANSLATORY MOTION 41

falling alone it would be approximately 144 ft. distant, repre-
sented by line b. Hence due to both, we have, by vector con-
struction, HS (about 100 ft.) as the distance from his hand to the
stone after 3 seconds of its flight. The actual path of the stone
is HCS.

Distance Traversed in a Given Time. — Equations 10, 11, 12, 13,
and 14, which are derived from a consideration of a particular
kind of uniformly accelerated motion, namely, that of falling
bodies, become perfectly general by substituting in them the
general symbol a in place of the particular symbol g to represent
the acceleration. Making this substitution, these equations,
taken in order, become

vt = v0+at ClOo)

(lla)
(12a)

d = \atz (13a)

vt=V2ad=V2ah 4*S *«£-''* (I4a)
>

The equations just given apply to the motion of a car when
coasting on a uniform grade, or to the motion of any body when
acted upon by a constant accelerating force. In the case of a
car on a uniform grade, the accelerating force is, barring friction,
the component of the car's weight which is parallel to the grade
(Fig. 8, Sec. 19), and is therefore constant.

Aside from the motion resulting from gravitational attraction,
there are very few examples of uniformly accelerated motion.
Such motion, however, is very roughly approximated by many
bodies when starting from rest; e.g., by a train, a steamship, a
sailboat, or a street car. In all these cases the accelerating force,
that is, the amount by which the applied force exceeds friction,
decreases rapidly as the speed increases; consequently the accel-
eration decreases rapidly, and the motion is then not even
approximately uniformly accelerated. ^

35. Atwood's Machine. — If we attempt to make an experi-
mental study of the motion of freely falling bodies we find that
the time of fall must be taken very small, or the distance fallen
will be inconveniently large. Thus in so short a time as three
seconds, a body falls somewhat more than 144 feet. Hence, in
all devices for studying the laws of falling bodies and verifying

MECHANICS AND HEAT

experimentally the equations expressing these laws, the rapidity
of the motion is reduced. Thus a wheel or a marble rolling
down an inclined plane experiences an acceleration much smaller
than if allowed to fall freely. For in the latter case the acceler-
ating force is the full weight of the marble or the wheel;
while in the former case it is only the component of the weight
parallel to the incline. This reduction of the acceleration makes
it possible to study the motion for a period of several seconds.
In the Atwood Machine, shown in its simplest form in Fig. 18,
the reduction in the acceleration is attained in an en-
tirely different way. A and B are two large equal
masses connected by a light cord passing over a light
wheel as shown. If a small additional mass C is
placed on A, it will cause A to descend and B to ascend.
Suppose that A and B are each 150-gm. masses and
that C is a 10-gm. mass. If we neglect the slight mass
and opposing friction of the wheel, it is clear that the
weight of C is the accelerating force that must accel-
erate A, B, and C — an aggregate mass equal to 31
' timfcs the mass of (7; while if C were permitted to fall
freely, its weight would have to accelerate itself only.
Hence the acceleration under these circumstances is
1 1/31 of that of free fall or g, or 1/31 X980 = 31.6 cm.
fci per sec. per sec., which is about 1 ft. per sec. per
FIG. 18. sec. With this value for the acceleration, we see
from Eq. 13a that A would "fall" only about 4.5
feet in 3 seconds. By experiment also we find that A "falls"
4.5 feet in 3 seconds, thus verifying Eq. 13a.

The above acceleration may also be calculated by means of the
equation F = Ma, in which F is the weight of C in dynes and M is
the combined mass of A, B, and C in grams. A pendulum or
other device beating seconds is an essential auxiliary. If by
means of an attached thread, C is removed after one second of
"fall," A's velocity, since no accelerating force is then being
applied, will be constant, and will have the value 31 cm. per
sec. (see above) ; while if in another test C remains 3 seconds, A's
velocity at the end of the 3 seconds will be 93 cm. per sec., as
may easily be observed. This verifies the equation vt = at
(Eq. 7, Sec. 26).

36. Motion of Projectiles: Initial Velocity Vertical.— If a
rifle ball is fired vertically upward, it experiences a downward

TRANSLATORY MOTION 43

force (its weight) which slows it down, giving rise to a negative
acceleration. This decrease in velocity each second is of course
32.17 ft. per sec.; so that if the muzzle velocity is 1000 ft. per sec.,
the velocities after 1, 2, 3, 4, etc., to t seconds are, respectively,
1000 — g (or 968), 1000 -20, 1000-30, 1000-40 (or 872 ft. per sec.),
etc., to 1000 — gt. Since the velocity of the bullet is zero when
it reaches its highest position, the number of seconds CO that
the bullet will continue to rise is found by placing 1000 — gt
equal to zero and solving for t. CCompare Sec. 39.) This gives
2 = 31 sec., approximately. The bullet requires just as long to
fall back, so that its time of flight is 62 seconds. To get the
height to which it rises, which is obviously the distance it falls
in 31 seconds, let t be 31 in Eq. 13 and solve for d. We may also
use the relation v = \/2gh (Eq. 14) to find h if v is known, or
vice versa. Here v = 1000 ft. per sec., since, neglecting air fric-
tion, the bullet, in falling, strikes the ground with the same
velocity with which it was fired.

Throughout the discussion of projectiles no account will be
taken of the effect of air friction, which effect is quite pronounced
on very small projectiles (Sec. 39). In approximate calcula-
tions, the distance a body falls in the first second will be taken as
16 ft. instead of 16.08, and 0 will be taken as 32 instead of 32.17
ft. per sec. per sec. If a rifle ball is fired vertically downward,
e.g., from a balloon, with a velocity v0, its velocity will increase by
32 ft. per sec. every second (ignoring air friction), so that t
seconds later its velocity will be v0-\-gt. In this case the distance
traversed in the first t seconds is v0t+^gt2 (Eq. 12); while if
the initial velocity is upward, the distance from the rifle to the
rifle ball after t seconds is v0t — %gt2, as explained in Sec. 34.

37. Motion of Projectiles: Initial Velocity Horizontal.— If
a projectile is fired horizontally, it experiences, the instant it
leaves the muzzle A of the gun (Fig. 19), a downward pull (its
weight) which gives it a downward component of velocity of 32
ft. per sec. for every second of flight. This causes it to follow the
curved path AB'C' . . . F'. If it were not for gravitational
attraction, the bullet at the end of the first, second, third, . . .
etc., seconds would be at the points B, C, D, . . . etc., respec-
tively (AB = BC = CD = 1000 f t.) , instead of at B', C', etc. . . .

To find the velocity of the bullet at any time t, say when at
F' 5 sec. after leaving the muzzle of the gun, we simply find the
vector sum v' of its initial velocity and its acquired velocity, as

44 MECHANICS AND HEAT

shown in Fig. 19 (left lower corner). The downward velocity
acquired in 5 sec. would of course be gt, or 160 ft. per sec.
(that is, 32X5), and we will assume 1000 ft. per sec. as the initial
horizontal muzzle velocity.

It will be evident that the horizontal component of velocity
(1000 ft. per sec.) must be constant, for the pull of gravity has no
horizontal component to either increase or decrease the horizontal
component of velocity. This, of course, is true whether the initial
velocity is vertical, horizontal, or aslant. Hence, neglecting
friction, it is always only the vertical component of velocity of a
projectile that changes.

To find the distance that the bullet will "fall" in going the first
1000 ft., i.e., its distance BB' (Fig. 19) from the horizontal line
of firing AF, apply Eq. 13. From this equation we see that a
body falls approximately 16 ft. in one second, 64 ft. in two sec.,
and 144 ft. (i.e., 16 X32) in 3 sec. Hence 55' = 16 ft., CC" = 64ft.,

FIG. 19.

and DD' = 144 ft., etc. To correct for this falling of the bullet,
the rear sight is raised, causing the barrel to point slightly above
the target. The greater the distance to the target, the more the
sight must be raised; the settings for the different distances being
marked on it.

In accordance with the above statements, it follows that if a
bullet is dropped from a tower erected on a level plain, and another
bullet is fired horizontally from the same place at the same in-
stant, then the two bullets will reach the ground at the same
instant, whether the second one is fired at a high or low speed.
This fact can be verified experimentally (Sec. 40).

38. Motion of Projectiles : Initial Velocity Inclined. — If a rifle
ball is fired from a point A (Fig. 20), in a direction AQ making an
angle 8 with the horizontal, it describes a curved path which may
be drawn as follows. Since distance is a vector, to find where the
projectile will be after a time t, we simply obtain the vector sum
of the distance traversed in t seconds due to its initial velocity
and the distance traversed in t seconds of free fall from rest, as

TRANSLATORY MOTION 45

was done in Sec. 34 (Fig. 17). Hence on the line AQ, which has
the direction of the initial velocity, lay off the distances AB, BC,
CD, DE, etc., each representing 1000 ft. (for a muzzle velocity
of 1000 ft. per sec.). From B, C, D, E, etc , draw the lines BB',
CCr, DD', EE', etc., representing respectively the distances fallen
in 1, 2, 3, 4, sec. Then here, just as in Fig. 19, we have BB' = 16
ft., CC' = 64 ft., DD' = U4 ft., etc. The curve AB'C'D'E',
etc., represents the path of the projectile. For consider any
point, e.g., K'. Due to its initial velocity alone, the projectile
would go from A to K (10,000 ft.) in 10 seconds. Due to gravity
alone it would fall a distance KK', or 1600 ft., in 10 seconds.
Hence, due to both, it covers the distance AK', the vector sum of
the distances AK and KK', as shown.

Note that the straight line AK' gives not only the magnitude
but also the direction of the distance from A to the projectile

FIG. 20.

after ten seconds of flight. Note also that AK is the v0t, and
that KK' is the %gtz of Eq. 12 (Sec. 34).

39. Time of Flight and Range of a Projectile.— The Range is
the horizontal distance A Q' (Fig. 20), or the distance from the point .
from which the projectile is fired to the point at which it again
reaches the same level. The Time of Flight is the time required
to traverse this distance.

To find how long the projectile will continue to rise, in other
words, to find the time ti that will elapse before its vertical com-
ponent of velocity (vv] will be zero, place vv — gti = Q (i.e., ti = vv/g
= v0 sin 6/g} and solve for ti (compare Sec. 36). It was shown in
Sec. 37 that only the vertical component of velocity changes.
Since the vertical component of velocity is zero at this time ti,
the projectile must be at the middle of its path (/', Fig. 20).
Therefore the time of flight.

T = 2tt (15)

MECHANICS AND HEAT

The vertical component of velocity vv = v0 sin 6, and the hori-
zontal component of velocity Vh = v0 cos 6 (see left upper corner
Fig. 20). If y0=1000 ft. per sec., then, as the projectile leaves
the gun, vv = about 240 ft. per sec., and VH = about 970 ft. per sec.
If the angle 6 is known, these two components of the velocity may
be accurately found by the use of tables of sines and cosines.
The graphical method may also be used. When £=1 sec., i.e.,
1 sec. after the projectile leaves the gun (see Fig. 20), vv = 208 ft.
per sec. Another second later vv is 32 ft. per sec. less, and when
t = 240/32, or approximately 8 sec. after the gun is fired, the
vertical component of velocity is zero. That is, in 8 sec. the bul-
let reaches the horizontal part of its path at /', at which point
its vertical component of velocity is clearly zero. Since ti is
8 sec., the time of flight T CEq. 15) is 16 sec.

Obviously, the range (R) is given by the equation,

ft = VhXT = v0 cos ex2ti = v0 cos 9X2v0 sin B/g (16)

Here the range is 15,520 (i.e., 16 X 970) ft. The Maximum Height
reached, or /'/", is %gtz, in which t is the ti of Eq. 15. For at
/' the path is horizontal, and it was pointed out in Sec. 37
that a bullet fired horizontally would reach the ground in the
same time as would a bullet dropped from the same point.
Hence I' I" = 16 X 82 = 1024 ft.

Effect of Air Friction on Velocity and Range. — Thus far, in the study of
the motion of projectiles, we have neglected the effects of air friction;
so that the resulting deductions apply strictly to a projectile traveling
through a space devoid of air or any other substance, i.e., through a
vacuum. The theoretical range so found is considerably greater than the
actual range, since the friction of the air constantly decreases the veloc-
ity of the projectile (see table below), and therefore causes it to strike
the earth much sooner than it otherwise would. Below is given the ve-
locity of an Army Rifle projectile in feet per second at various distances
from the muzzle.

Distance
in yds.

Velocity in ft.
per sec.

Distance
in yds.

Velocity in ft.
per sec.

100
200
300
400
600
800

1780
1590
1420
1265
1044
923

1000
1200
1400
1600
1800
2000

830
755
690
630
575
530

TRANSLATORY MOTION 47

The angle (0, Fig. 20) which the barrel of the gun makes with the
horizontal is called the Angle of Elevation. Obviously, if the angle of
elevation is small, increasing it will increase the range. It can be shown
by the use of calculus that the theoretical maximum range is obtained
when this angle is 45°. The trigonometric proof is given below. For
heavy cannon (12-in. guns), the angle of fire for maximum range is
nearly the same as the theoretical, namely, 43°; while for the army rifle
it is about 31°. This difference is due to the greater retarding effect of
air friction upon the lighter projectile.

In firing at targets 1/4 mi. distant or less, such as is usually the case
in the use of small arms, there is not a very marked difference between
the theoretical and the actual path of the projectile. The maximum
range of the new army rifle is about 3 miles. It may be of interest to
note that its range in a vacuum (angle of elevation 45°) would be about
24 miles, and that the bullet at the middle of its flight would be about
6 miles above the earth, and would strike the earth with its original
muzzle velocity.

The artillery officer who directs the firing at moving ships at a distance
of 5 miles or more, especially during a strong wind, must make very
rapid and accurate calculations or he will make very few "hits." Many
other things concerning the flight" of projectiles, which are of the utmost
importance to the artillery man, must be omitted in this brief discussion.

Angle of Elevation for Maximum Range. — Since sin 20 = 2 sin 0 cos 0
(trigonometry), Eq. 16 may be written

2sin0cos0_ 2sin 20

g ~v° g

Now the maximum value of the sine of an angle, namely, unity, occurs
when the angle is 90°. Therefore when 20 = 90°, i.e., when 0 = 45°,
sin 20 is a maximum; hence the range R is also a maximum, which was
to be proved.

40. Spring Gun Experiment. — From the discussion given in
Sec. 38, it is seen that if a target at B, or at C, or at D, or at any
other point on AQ (Fig. 20), is released at the instant the trigger
is pulled, it will by falling reach B' (or C' , or D', etc., as the case
may be) just in time to be struck by the bullet. This may be
shown experimentally by the use of a spring gun, using wooden
balls for both projectile and target. The target ball is held by an
electrical device which automatically releases it just as the
projectile ball leaves the muzzle of the gun. The two balls meet
in the air whether the projectile ball is fired at a high or low veloc-
ity. If the target is placed at the same height as the spring gun,

MECHANICS AND HEAT

and the latter is fired horizontally, the two balls will reach the
floor at the same instant.

41. The Plotting of Curves. — The graphical method of presenting
data is found very useful in all cases in which a series of several observa-
tions of the same phenomenon has been made. Coordinate or cross
section paper is used for this purpose. Usually a vertical line at the
left of the page is called the axis of ordinates, and a horizontal line at
the bottom of the page is called the axis of abscissae. To construct a
curve, plot as abscissae the quantity that is arbitrarily varied, and as
ordinates the corresponding values of the particular quantity that is
being studied. This can be best illustrated by an example.

To plot the results given in the table, Sec. 39, choose a suitable scale
and lay off 200, 400, etc., upon the axis of abscissae (Fig. 21) to represent

800 1000 1200

DISTANCE IN YARDS
FIG. 21.

1400 1GOO 1300 2000

the distance (from muzzle of gun) in yards, and 400, 800, etc., on the axis
of ordinates to represent the velocity of the bullet in feet per second.
From the table we see that the velocity for a range of 100 yds. is 1780
ft. per sec. A point A at the center of a small circle (Fig. 21) gives this
same information graphically, for the abscissa of A is 100 and its ordi-
nate is 1780. Point B, whose abscissa is 200 and whose ordinate is
1590, fully represents the second pair of values (200 and 1590) in the table.
In like manner the points C, D, etc., are plotted. Through these points
a smooth curve is drawn as shown.

Use of the Curve. — It will be observed that the smooth curve passing
through all of the other points does not pass through D'. The fact that
a point does not fall on the curve indicates a probability of error either
in taking the data or in plotting the results. In this case a defective
cartridge may have been used at the 500-yd. distance. A second trial

TRANSLATORY MOTION 49

from that same distance with a good cartridge would probably give a
velocity of 1130 ft. per sec. as we would expect from the curve.

To find the velocity at a distance of 900 yds., note that the vertical
line at 900 strikes the curve at H. But the ordinate of H is 850. Hence
we know without actually firing from that distance, that the velocity of
the projectile when 900 yds. from the muzzle is 850 ft. per sec. This
method of finding values is called Interpolation. Such use of curves for
detecting errors and for interpolating values makes them very valuable.
They also present the data more forcibly than does the tabulated form,
for which reason debaters frequently use them. In the physical labora-
tory and in engineering work curves are almost indispensable.

If there were also negative velocities to be plotted, i.e., velocities
having a direction opposite to that of the bullet, they would be desig-
nated by points at the proper distance below the axis of abscissae. This
axis would then be near the middle of the coordinate sheet instead of
at the bottom as shown.

42. Newton's Three Laws of Motion. — Sir Isaac Newton, the
great English mathematician and physicist, formulated the fol-
lowing fundamental laws of motion which bear his name.

1. A body at rest remains at rest, and a body in motion con-
tinues to move in the same direction and at the same speed,
unless acted upon by some external force.

2. The acceleration experienced by a given mass is propor-
tional to the applied force (accelerating force), and is hi the
direction of the applied force.

3. Action and reaction are equal, and oppositely directed.

The first law refers to the inert character of matter, the prop-
erty of inertia by virtue of which any body resists any change in
velocity, either in magnitude or direction. It is really impossible
to have a body perfectly free from the effects of all external
forces, but the more we eliminate these effects by reducing fric-
tion, etc., the more readily do we observe the tendency of a body
to keep in motion when once started. The second law states
the fact with which we have already become familiar in the dis-
cussion of the equation F = Ma (Sec. 25). The third law is a
statement of the fact that whenever and wherever a force is applied
there arises an equal and oppositely directed force. This law will
be further considered in the next section.

43. Action and Reaction, Inertia Force, Principle of d'Alem-
bert. — If we press with the hand upon the top, bottom, or side
of a table with a force of, say 10 Ibs., we observe that the table
exerts a counter push or force exactly equal to the applied force,

50 MECHANICS AND HEAT

but oppositely directed. If the applied force is increased, the
counter force, or Reaction, is inevitably increased. If, in order to
push a boat eastward from a bank, the oarsman exerts a west-
ward thrust (force) upon a projecting rock by means of his oar,
the eastward reacting thrust of the rock that arises dents the oar
and starts the boat eastward. If an eastward pull is exerted on a
telephone pole, the guy wires to the westward tighten.

If a horse exerts a 300-lb. pull or force FI upon the rope at-
tached to a canal boat a moment after starting, then the backward
pull that the canal boat exerts upon the other end of the rope
cannot possibly be either more or less than 300 Ibs. Many peo-
ple cling tenaciously to the erroneous belief that the forward
pull of the horse must be at least slightly greater than the back-
ward pull of the boat or the latter would not move. Many
people also think that the winning party in a tug-of-war contest
must exert a greater pull on the rope than does the losing party,
which is certainly not the case. For this reason, we shall discuss
very carefully the problem of the horse and canal boat. The
applied force FI in this case overcomes two forces; one, the fric-
tion resistance, say 100 Ibs., encountered by the boat in moving
through the water, the other (200 Ibs.), the backward pull exerted
by the boat because, by virtue of its inertia, it resists having its
speed increased. Note that we are here dealing with four forces.
The 100 Ibs. of the forward pull exerted by the horse just balances
the 100-lb. backward pull of water friction on the boat; while the
other 200 Ibs. of forward pull or force /i exerted by the horse,
just balances the resisting pull or force /2 that the boat offers to
having its speed increased. Obviously the accelerating f orce/i =
— /2 = Ma, in which M is the mass of the canal boat and a is its
acceleration. The minus sign indicates that the forces are oppo-
sitely directed.

From this discussion, we arrive at the conclusion that the for-
ward pull exerted upon any body is exactly equal in magnitude
to the backward pull or resisting force exerted by the body. Thus
here, if the horse had exerted a 400-lb. pull, we cannot escape the
conclusion that the backward pull of the boat would have been
400 Ibs.; 100 Ibs. being the pull of water friction resistance as
before, and 300 Ibs. backward pull arising from the resistance the
boat offered to having its speed increased. Since the accelerating
force would be 300 Ibs. in this case, instead of 200 Ibs. as before,
the acceleration would be 1/2 greater than before.

TRANSLATORY MOTION 51

The above backward pull or force that any body, by virtue
of its inertia, exerts in resisting change of velocity, has been very
appropriately called Inertia Force. The inertia force is always
numerically equal to the accelerating force that gives rise to it,
and is always oppositely directed. If the canal boat were to run
onto a sand bar, the friction would produce a large negative
accelerating force, and the resistance the boat offered to decrease
of speed would develop an equal forward, or Driving Inertia
Force, that would carry the boat some distance onto the bar, even
though the horse had ceased to pull. Had the sand bar been
more abrupt, then both the negative accelerating force and the
driving inertia force would have been greater than before, but
they would still have been exactly equal.

The above fact, that all the forces exerted both upon and by
any body under any possible circumstances are balanced forces,
i.e., that the vector sum of all the forces exerted upon and by a body
is invariably zero, is known in mechanics as the Principle of
d'Alembert. In common language, we frequently speak of
unbalanced forces. In physics, even, it is frequently found
convenient to use the term, but in such cases we are simply
ignoring the inertia force. Strictly speaking, then, there is no
such thing as unbalanced forces, if all forces, including inertia
force, are taken into account. In the above case of the canal
boat, the only external forces acting upon the boat to affect its
motion are the forward pull exerted by the horse, and the back-
ward pull exerted by the water friction. These external forces
are clearly unbalanced forces. In this sense, and in this sense
only, may we correctly speak of unbalanced forces.

44. Practical Applications of Reaction. — A horse cannot draw
a heavy load on a slippery road unless sharply shod. In order to
exert a forward pull on the vehicle, he must exert a backward
push on the ground. A train cannot, by applying the brakes,
stop quickly on a greased track because of the inability of the
wheels to push backward on the axle, and therefore on the car,
without pushing forward on the track. The wheels cannot,
however, exert much forward push on a greasy rail.

A steamship, by means of its propellers, forces a stream of
water backward. The reaction on the propellers pushes the ship
forward. One of the best suggestions to give a person who is
learning to swim is to tell him to push the water backward. The
reaction forces the swimmer forward.

52 MECHANICS AND HEAT

An aeroplane, by means of its propellers, forces a stream of air
backward. The reaction on the propellers forces the aeroplane
forward. The forward edge of each plane or wing is slightly
higher than the rear edge. This causes the planes to give the
air a downward thrust as the machine speeds horizontally through
it. The reaction to this thrust lifts on the planes and supports
the weight of the machine.

Suppose that an aeroplane, traveling 50 miles per hour, sud-
denly enters a region in which the wind is blowing 50 miles per
hour in the same direction. Under these circumstances the air
in contact with the planes, having no horizontal motion with
respect to the planes, fails to give rise to the upward reacting
thrust just mentioned, and the aeroplane suddenly plunges
downward. Such regions as these, described by aeronauts as
"holes in the air," are very dangerous. It is interesting to note
in this connection that birds face the wind, if it is blowing hard,
both in alighting and in starting, thus availing themselves of the
maximum upward thrust of the air through which their wings
glide.

45. Momentum, Impulse, Impact, and Conservation of Momen-
tum.— The Momentum of a moving body is denned as the product
of the mass of the body and its velocity, or

Momentum = Mv (17)

The impulse of a force is the product of the force and the time
during which the force acts, or

Impulse =Ft (18)

An impulse is a measure of the ability of a force to produce
motion or change of motion. We readily see that a force of 100
Ibs. acting upon a boat for 2 sec. will produce the same amount
of motion as a force of 200 Ibs. acting for 1 sec. The term
" impulse" is usually applied only in those cases in which the
force acts for a brief time, e.g., as in the case of collision or
impact of two bodies, the action of dynamite or powder in blast-
ing, the firing of a gun, etc., and the force is then called an
impulsive force.

We shall now show that an impulse is numerically equal to
the momentum change which it produces in a body, i.e., Ft = Mv.
Observe that a "bunted" ball loses momentum (mainly), while
a batted ball loses momentum and then instantly acquires even

TRANSLATORY MOTION 53

greater momentum in the opposite direction, due to the impulse ap-
plied by the bat. Obviously, the total change in the momentum
of the ball, in case it returns toward the pitcher, is the product
of the mass of the ball and the sum of its "pitched" and "batted"
velocities. If a force F acts upon a certain mass M, it imparts
to the mass an acceleration, determined by the equation F = Ma',
while if this force acts for a time t, the impulse Ft = Mat. But
the acceleration of a body multiplied by the time during which
it is being accelerated gives the velocity acquired. Hence

Ft = Mat = Mv (19)

It should be emphasized that v here represents the change in
velocity produced by the impulse Ft.

We shall next show that when two free bodies are acted upon
by an impulse, for example in impact or when powder explodes
between them, then the change of momentum of one body is
exactly equal but opposite in sign to the change in momentum of
the other. In other words the total momentum of both bodies is,
taking account of sign, exactly the same before and after impact.
This law is very appropriately called the law of the Conservation
of Momentum.

Theoretical Proof of the Conservation of Momentum. — Let us now
study the effects of the impact in a rear end collision, caused by
a truck A of mass Ma and velocity va overtaking a truck B of
mass Mb and velocity Vb. Let v'a and v'b be the velocities after
impact. During the brief interval of impact t, truck A pushes
forward upon B with a variable force whose average value may
be designated by Fb. During this same time t, truck B pushes
backward upon A with a force equal at every instant to the
forward push of A upon B (action equals reaction). Conse-
quently the average value Fa of this backward push must equal
Fb, and therefore

Fbt=-Fat (20)

The minus sign in this equation indicates that the forces are
oppositely directed. In fact Fa, being a backward push, is
negative.

Since an impulse is equal to the change in momentum which it
produces, and since the change in velocity of A is v'a — va, and that
of B is v'b — Vb, we have

Fat = Ma(v'a - va) and Fbt = Mb(v'b - vb)

54 MECHANICS AND HEAT

Hence, from Eq. 20, we have

Mb(v'b-vb] = -Ma(v'a-va),
or

Mb(v'b-vb}+Ma(v'a-va}=Q (21)

From the conditions of the problem, we see at once that vb is less
than v'b, and that va is greater than v'a. Accordingly, in Eq. 21,
the first term, which represents the momentum change of
truck B, is positive; while the second term, which represents
the momentum change of truck A (momentum lost), is negative.
Since these two changes are numerically equal but opposite in
sign, the combined momentum of A and B is unchanged by the
impact, thus proving the Conservation of Momentum.

Observe in equation 21 that the changes in velocity vary
inversely as the masses involved. Thus if B had 3 times as
great mass as A, its change (increase) in velocity would be only
1/3 as great as the change (decrease) in the velocity of A.

Briefer Proof. — The above concrete example has been used in
the proof for the sake of the added clearness of illustration. We
are now prepared to consider a briefer, and at the same time more
general proof. In every case of impact of two bodies, whatever
be their relative masses, or their relative velocities before impact,
the impulsive force acting on the one, since action is equal to
reaction, is equal to, but oppositely directed to that acting upon
the other. Since these two forces are not only equal but also act
for the same length of time, the two impulses are equal, and they
are also oppositely directed. But, since an impulse is equal to
the change in momentum (Mv) produced by it, it follows that
the momentum changes of the two bodies are equal but oppositely
directed, and that their sum is therefore zero. In other words,
the momentum before impact is equal to the momentum after impact,
thus proving the Conservation of Momentum.

Experimental Proof. — Consider two ivory balls A and B of
equal mass suspended by separate cords of equal length. Let
A be displaced through an arc of say 6 inches and then be released.
As A strikes B it comes to rest and B swings through an equal
6-inch arc. This shows that the velocity of B immediately after
impact is equal to the velocity of A immediately before impact.
But A and B have equal mass, hence the total momentum is the
same before and after impact, as is required by the law of the
conservation of momentum.

TRANSLATORY MOTION

46. The Ballistic Pendulum. — The ballistic pendulum affords
a simple and accurate means of determining the velocity of a rifle
ball or other projectile. It consists essentially of a heavy block
of wood P (Fig. 22), of known mass M, suspended by a cord of
length L. In practice, four suspending cords so arranged as to
prevent all rotary motion are used.

As the bullet b of mass m and velocity v strikes P, it imparts to
P a velocity V which causes it to rise through the arc AB, thereby
raising it through the vertical height h. After impact, the mass
of the pendulum is M + m. From the conservation of momentum
we know that the momentum of the bullet before impact, or mv,

'/////////,

FIG. 22.

will be equal to the momentum of the pendulum (with bullet
embedded) after impact, or (M+m)V, i.e.,

(M+m)V

(22)

The values of m and M are found by weighing, and V is found
from V=^2gh (Eq. 14). For, as we shall presently prove, the
velocity which enables the pendulum to swing through arc AB,
or the equal velocity which it attains in returning from B to A,
is that velocity which it would acquire in falling through the
vertical height h. All other quantities being known, Eq. 22 may
then be solved for v, the velocity of the bullet.

Velocity Dependent upon Vertical Height Only. — We shall now show
that the velocity acquired by a body in descending through a given
vertical height h by a frictionless path, is independent of the length or
form of that path. Thus, if it were not for friction, the velocity of a
sled upon reaching the foot of a hill of varying slope would be exactly
that velocity which a body would acquire in falling through the vertical
height of the hill.

56 MECHANICS AND HEAT

In Fig. 22 (upper right corner) let DE be an incline whose slant height
is, say, four times its vertical height DE' or h', i.e., DE = 4h'. Let the
body C, starting from rest, slide without friction down the incline, and
let C", also starting from rest, fall without friction. Let us prove that
the velocity (vt) of C as it reaches E is equal to the velocity (v't) that C'
acquires in falling to E'. Note that the vertical descent is the same
for both bodies.

The component FI of C"s weight W is the accelerating force acting
upon C. From similar triangles we have

and therefore C"s acceleration a is 0/4. From Eq. 14a we have for
the velocity of C at E, vt= \/2od = \^X4/i= ^2^ But from Eq.

14 we have, for the velocity v't of C" as it reaches E', v't = X/20/j';
therefore Vt = v't, which was to be proved.

Further, it is obvious that the same reasoning would apply had h' been
chosen larger, say equal to DF'. Accordingly, the velocity of C upon
reaching F, would equal the velocity of C" upon reaching F'. This shows
that the increase in C"s velocity while going from E to F is equal to the
increase in the velocity of C' in going through the equal vertical distance
E'F' (or EH).

Let us now consider the path a b c . . . k, Fig. 22 (lower right corner),
whose slope is not uniform. By subdividing this path into shorter and
shorter portions, in the limit each portion ab, be, cd, etc., would be straight,
and therefore abdi, etc., become triangles similar to triangle EFH in the
figure just discussed. From the discussion of triangle EFH already
given, we see that the velocities acquired by a body in sliding without
friction through the successive distances ab, be, cd, etc., are equal respect-
ively to the velocities that would be acquired by a body falling through
the corresponding successive distances hi, h2, h3, etc. But the sum of
one series is the distance ok, while the sum of the other series is h",
the vertical height of ok.

Consequently the total velocity acquired by a body in sliding
from a to k, or in general down any frictionless path, is equal to
the velocity that would be acquired in free fall through the dis-
tance h", or in general through the vertical height of the path.

We now see that V of Eq. 22 is given by the relation V = ^2gh.
If h is measured, V is known, and therefore v of Eq. 22 is deter-
mined. In practice, h is too small to be accurately measured and
is therefore expressed in terms of d and L (see figure) .

TRANSLATORY MOTION 57

0 PROBLEMS

1. The distance by rail from a town A to a town B, 120 miles east of A,
is 240 miles. The speed of a train going from A to B is 30 miles an hour
for the first 120 miles, and 20 miles an hour for the remainder. Find the
average speed and average velocity of the train for the run.

2. A train starts from rest at a town A and passes through a town B
5.5 miles to the eastward at full speed. The excess pull upon the drawbar
above that required to overcome friction (i.e., the accelerating force) is
kept constant, so that the motion from A to B is uniformly accelerated.
The train requires 22 minutes to make the trip. Find its average velocity
and maximum velocity in mi. per min.; mi. per hr.; and ft. per sec.

3.' Express the acceleration of the train (Prob. 2) in miles per hr. per
min.; miles per min. per min.; and feet per sec. per sec.

4.1 What is the velocity of the train (Prob. 2) 15 sec. after leaving A?
2 min. after leaving A?

^ 5. How long will it take a 2-ton pull to give a train of 40 cars, weighing
50 tons each, a velocity of 1 mi. per min. (i.e., 88 ft. per sec.) on a level
track? Neglect friction.

6. Compare the intensities of illumination due to an arc lamp at the
two distances, 1/2 block, and 2 blocks.

7. A 50-lb. stone falls 16 ft. and sinks into the earth 1 ft. Find its
negative acceleration, assuming it to be constant for this foot. Find
the force required to penetrate the earth. Suggestion: Since the velocity
of the stone during fall changes uniformly from zero to its "striking"
velocity, and during its travel through the earth from striking velocity
to zero, it follows that its average velocity in air and its average velocity
in earth are the same, and that each is equal to 1/2 the striking velocity.
See Sec. 33 and Sec. 45.

y 8. If an elevator cable pulls upward with a force of 1200 Ibs. on a 1000-
Ib. elevator, what is the upward acceleration? How far will it rise in 2
sec.? Suggestion: Find the accelerating force and express it in poundals,
not pounds (see Sec. 32). Neglect friction.

9. How much would a 1 50-lb. man weigh standing in the above eleva-
tor if the pull on the cable were increased so as to make the acceleration
the same as in problem 8?

10. A car that has a velocity of 64 feet per sec. is brought to rest in 10
sec. by applying its brakes. Find its average negative acceleration; and
by comparing this acceleration with g, show graphically at what average
slant a passenger standing in the car must lean back during this 10 sec.

11. If the car (Prob. 10) weighs 30 tons, what is the forward push exerted
by its wheels upon the rails while it is being brought to rest?

12. Prove that the weight of a gram mass is 980.6 dynes, and that a
force of 1 pound is equal to 32.17 poundals of force.

13. Reduce 2.5 tons to poundals; to dynes.

14. How far does a body travel in the first second of free fall from rest?
In the second? In the third? In the fifth?

16. What is the gravitational pull of the earth upon a mass of 1 ton at
the moon?

58 MECHANICS AND HEAT

16. How far will a body fall in 7 sec.; and what will be its average and
final velocities?

17. A car on a track inclined 30° to the horizontal is released. How
far will it travel in the first 7 sec.; and what will be its average and final
velocities (neglecting friction) ? Compare results with those of problem 16.

18. How long will it take a body to fall 400 meters?

19. If a rifle ball is fired downward from a balloon with a muzzle
velocity of 20,000 cm. per sec., how far will it go in 4 sec. ? If fired upward,
how far will it go in 4 sec.?

20. A baseball thrown vertically upward remains in the air 6 sec.
How high does it go? Observe that the times of ascent and descent are
equal, neglecting friction.

21. A stone is thrown upward from the top of an 80-ft. cliff with a
velocity whose vertical component is 64 ft. per sec.' What time will
elapse before it strikes the level plain at the base of the cliff?

22. With what velocity does a body which has fallen 2000 ft. strike the
ground?

23. A man 500 ft. south of a west-bound train which has a velocity of
60 miles per hour, fires a rifle ball with a muzzle velocity of 1000 ft. per
sec. at a target on the train. Assuming the aim to be accurate, how much
will the bullet miss the mark if the rifle sight is set for close range?

24. A stone is dropped from a certain point at the same instant that
another stone is thrown vertically downward from the same point with
a velocity of 20 ft. per sec How far apart are the two stones 3 sec. later?

25. A rifle ball is fired at an angle of 30° above the horizontal with a
muzzle velocity of 1200 ft. per sec. Neglecting air friction, find the range
and time of flight.

26. If the rifle ball (Prob. 25) is fired horizontally from the edge of the
cliff (Prob. 21), when and where will it strike the plain on the level of the
base of the cliff ?

27. If a 20-ton car A, having a velocity of 5 mi. an hr., collides with and
is coupled to a 30-ton car B standing on the track, what will be their com-
mon velocity after impact?

28. If the above car A when it strikes B rebounds from it with a
velocity of 1 mile per hour, find the velocity of B after collision. Observe
that the total change of A's velocity is 6 miles per hour. Will B'B change
be more or less, and why?

29. A 2-gram bullet fired into a 2-kilo ballistic pendulum of length 2
meters produces a horizontal displacement d=10 cm. (Fig. 22). Find the
velocity of the bullet in cm. per sec. and ft. per sec.

CHAPTER IV
ROTARY MOTION

47. Kinds of Rotary Motion. — As has previously been stated
(Sec. 22), a body has pure rotary motion if a line of particles,
called the axis of rotation, remains stationary, and all other
particles of the body move in circular paths about the axis as a
center. Familiar examples are the rotation of shafts, pulleys,
and flywheels. Rotary motion is of the greatest importance in
connection with machinery of all kinds, since it is much more
common in machines than reciprocating motion. The study
of rotary motion is much simplified by observing the striking
similarity in terms to those that occur in the discussion of trans-
latory motion.

Translatory motion, as we have seen (Sec. 22), may be either
uniform or accelerated; and the latter may be either uniformly ac-
celerated or nonuniformly accelerated motion. The accelera-
tion may also be either positive or negative. Likewise there
are three kinds of rotary motion: (a) uniform rotary motion,
e.g., the motion of a flywheel or line shaft after it has acquired
steady speed; (6) nonuniformly accelerated motion, e.g., the
usual motion of a flywheel when the power is first turned on (or
off); and (c) uniformly accelerated rotary motion, e.g., the
motion which a flywheel would have if the torque (Sec. 48)
furnished in starting had the proper value to cause its increase of
rotary speed to be uniform.

48. Torque. — Torque may be defined as that which produces,
or tends to produce, rotary motion in a body, just as force is
that which produces, or tends to produce, motion of translation
in a body. The magnitude of a torque is force times "lever
arm" (Eq.25), and its direction depends upon both the direction
and the point of application of the force. A torque is not simply
a force, for it is readily seen that any force directed either
toward or away from the axis, e.g., force a (Fig. 23a), has no
tendency to produce rotation. A torque tending to produce
rotation in a counterclockwise direction is called a positive

MECHANICS AND HEAT

torque, while a torque which is oppositely directed is called
negative.

Fig. 23a represents the grindstone shown in Fig. 23 as viewed
from a point in line with the axle. The torque due to the force
a alone is zero. The torque due to the force b alone is bXOP
(i.e., b.OP), and is negative. The torque due to force c alone is
also negative, and its magnitude is c.OE. For the thrust c
equals the pull cf, which may be thought of as exerted upon a
cord c'P. Evidently the pull of such a cord would be just as
effective in producing rotation, at the instant shown, if attached
to E on a crank OE, as if attached to P on the crank OP. Thus
when we define torque as force times "lever arm," or

T = Fr (25)

we must interpret the "lever arm" r to mean the perpendicular
distance from the axis of rotation to the line of action of the force.

FIG. 23.

FIG. 23a.

The force may be expressed in dynes, poundals, pounds, etc.,
and the lever arm in centimeters, inches, feet, etc.; so that
torque may be expressed in dyne-centimeter units, or in poundal-
feet, or pound-feet units, etc. If several torques, some positive
and some negative, act simultaneously upon a flywheel, the fly-
wheel will start (or, if in motion, increase its speed) clockwise,
provided the negative torques exceed the positive torques;
whereas it will start, or, if in motion, increase its speed counter-
clockwise, provided the positive torques are the greater. If the
positive and negative torques just "balance," then the fly-
wheel will remain at rest; or if already in motion, its speed will
not change.

ROTARY MOTION 61

77Ae Couple. — Two equal and oppositely directed forces
which do not have the same line of action (F and/'1', upper sketch,
Fig. 24) constitute a Couple. The torque developed by this
couple is equal to the product of one of the forces, and the dis-
tance AC between them, and is entirely independent of the posi-
tion (in the plane of the figure) of the pivot point about which
the body rotates. The torque due to this particular couple is
also counterclockwise (positive) whether the pivot point is at
A, B, C, D, or at any other point. If A is the pivot point, then
the force F produces no torque, while Ff produces the positive
torque F'XAC (i.e., F'.AC). If B is the pivot point, then both
forces produce positive torques; but,
since the lever arm for each is then
only \ AC, the total torque is the
same as before. If D is the pivot
point, then F' produces a negative
torque, and F, a positive torque; but,
since F acts upon a lever arm which
is longer than that of F' by the dis-
tance AC, it follows that the sum of
these two torques about D is F.AC as
before, and is also positive.

If three men A, B, and C by pushing Fio. 24.

with one hand and pulling with the

other apply respectively upon the wheel E (Fig. 24) the cou-
ples represented by FI and Ft, F3 and F4, and F6 and F6, then
each man will contribute an equal positive torque helping to
rotate the wheel. For, as sketched, the forces are all equal,
and the distances a, b, and c are equal; consequently the three
torques are equal. But from the above discussion we see that
the torques due to these three similar couples will be equal about
any point in the plane of the wheel, and hence about its axis.

49. Resultant Torque, and Antiresultant Torque. — Let the
forces a, b, c, and d, Fig. 23a, be respectively 20, 12, 14, and 40
pounds, and let OP = 1 ft., OE = 8 in., and OF = 4 in. The torque
due to a is zero; that due to b is 12X1 or 12 lb.-ft., or 144 lb.-in.,
negative; the torque due to c is 14X8 or 112 lb.-in., negative, and
that due to d is 40 X4 = 160 lb.-in. positive. The sum of all these
torques, that is the one torque that would be just as effective in
producing rotation as all of these torques acting simultaneously,
is 96 lb.-in. or 8 lb.-ft., a negative torque. Consequently, one

62 MECHANICS AND HEAT

force, say h, acting in the direction 6, but of magnitude 8 Ibs.,
would produce just as great a torque as would all four forces,
a, b, c, and d acting together. This torque may be called the
Resultant of the other four torques. If the force h is reversed
in direction, it produces a positive torque of 8 lb.-ft., called the
Antiresultant torque. This antiresultant torque, acting with the
torques due to a, b, c, and d, would produce equilibrium. Ob-
viously, this antiresultant torque, instead of being an 8-lb. force
on a 1-ft. arm, might, for example, be a 4-lb. force on a 2-ft. arm,
or a 16-lb. force on a 6-in. arm.

50. Angular Measurement. — Angles may be measured in
degrees, minutes, and seconds, in revolutions, or in radians. In
circular measure, an angle is found by dividing the subtended
arc by the radius, that is,

If the arc equals the radius, then the angle is of course unity, and
is called one Radian. Thus angle AOC (Fig. 25) is one radian
because arc ABC equals the radius r. The
angle AOB, or 8, is 1/2 radian because the
arc AB is 1/2 the radius r. Since the cir-
cumference of a circle is 2nr, it follows that
there are 2ir radians in 360°, or the radian
equals 57°.3. In the study of Mechanics,
angles, angular velocity, and angular ac-
FIG. 25. celeration are almost always expressed in

terms of radians instead of degrees.

61. Angular Velocity and Angular Acceleration. — Angular
velocity is the angle traversed divided by the time required; or,
since the unit of time is usually the second, it is numerically the
angle turned through in one second. If a certain flywheel makes
600 revolutions per min. (written 600 R.P.M.), its angular
velocity

w = 10 rev. per sec., or 62.8 (i.e., 10X27r) radians per sec.

If the rotary speed of the flywheel is constant during the one
minute, the above 62.8 radians per sec. is its actual angular
velocity at any time during that minute; whereas if its speed
fluctuates, then 62.8 radians per sec. is simply the average angular
velocity co (read "barred omega") for the minute considered.

ROTARY MOTION 63

Again, suppose that the above flywheel, starting from rest and
uniformly increasing its speed, makes 600 revolutions in the first
minute. Its average angular velocity w is 62.8 rad. per sec. as
before; but, since its initial velocity is zero, its angular velocity
ojt at the close of the first minute must be twice the average, or
125.6 rad. per sec. (Compare v of Sec. 33.) Since this angular
velocity is acquired in one minute, the angular acceleration (a) is
given by the equation

CO*

a = — = 125.6 radians per sec. per min.

In one second the wheel will acquire 1/60 as much angular veloc-
ity as it does in 1 min.; hence we may also write

a = 2.09 radians per sec. per sec.

which means that in one second the increase in angular velocity
is 2.09 radians per sec. Evidently, at a time t seconds after
starting, the angular velocity <at=od. Thus 5 seconds after
starting co = 10.45 radians per sec.

To summarize (see also Sees. 52 and 57), we have, in transla-
tory motion,

distance traversed d

Average velocity - time required > or V = T

In rotary motion

angle traversed _ 0
Average angular velocity = ^time required >or" = J (27>

. gain in velocity •»• — »

Acceleration (trans, motion) = —r. — — ^rm or a

time required ' t

gain in angl. velocity co, — co0 ,,

Angular aceelerat.on = time r°quired p*. or «- -y- (28)

52. Relation between Linear and Angular Velocity and
Acceleration. — If, due to a constant accelerating torque, a body
starts from rest with a constant angular acceleration a, and, in a
time t, rotates through an angle 6 and acquires an angular veloc-
ity w, then it will be true that any mass particle in this body at a
distance r from the axis travels, in this time t , a distance d = rd
(note that arc = r0, Eq. 26), acquires in this time a linear velocity
v = rw, and experiences during this same time a linear accelera-
tion a = ra.

Proof: Dividing both sides of the equation d = r6 by t, gives

64 MECHANICS AND HEAT

v = r-. = ru. If a body starts from rest with uniform acceleration,

its average velocity v is of course only half as great as its final
velocity v; hence v = 2v. Likewise w = 2co. Hence, since v = ro>,
it follows that v = rw. Now a = v/t; therefore, dividing both
sides of the equation v = rco by t, gives a = ru/t = ra. Accordingly

d = rd, v = rw, and a = ra (29)

If 6 is given in radians, co in radians per second, and a in
radians per second per second, then if r is given in feet, d will be
expressed in feet, v in feet per second, and a in feet per second per
second. From Eq. 29, we see (1) that the distance which a belt
travels is equal to the product of the radius (?•) of the belt wheel
over which it passes, and the angle 6 (in radians) through which
this wheel turns; (2) that the linear velocity of the belt is equal
to r times the angular velocity of the wheel in radians per second,
and (3) that the linear acceleration which the belt experiences in
starting, is equal to r times the angular acceleration of the belt
wheel expressed in radians per second per second.

Let it be required to find the angular velocity co of the drivers
of a locomotive when traveling with a known velocity v. From
Eq. 29 we have co = y/r; hence, dividing the linear velocity of the
locomotive expressed in feet per second by the radius of the driver
in feet, we obtain co in radians per second.

53. The Two Conditions of Equilibrium of a Rigid Body.—
If the resultant of all of the forces acting upon a body is zero,
the First Condition of Equilibrium is satisfied (Sec. 17), and the
body will remain at rest, if at rest, or continue in uniform motion
in a straight line if already in motion. If, in addition, the result-
ant of the torques acting upon the body is zero, the Second Con-
dition of Equilibrium is satisfied, and the body will remain at rest,
if at rest, or if already rotating its angular velocity will neither
increase nor decrease. Forces which satisfy the first condition
of equilibrium may not satisfy the second. The general case of
several forces acting upon various points of the body, and in
directions which do not all lie in the same plane, is too complex
to discuss here. The simpler but important case of three forces
all lying in the same plane will now be considered.

A body acted upon by three forces which lie in the same plane
is in equilibrium if (a) the three forces when represented graphic-
ally form a closed triangle (first condition of equilibrium); and

ROTARY MOTION 65

(&) if the lines of action of these three forces meet in a point
(second condition of equilibrium). Thus the body A (Fig. 26)
is in equilibrium, since the three forces a, b, and c, form a closed
triangle as shown, and they also (extended if necessary) meet
at the point E.

The three forces a', &', and c' which act upon the body B (Fig.
27), when graphically represented form a closed triangle and
therefore have zero resultant. Consequently they have no tend-
ency to produce motion of translation in the body, but they do
tend to produce rotation. For the forces b' and c' meet at D,
about which point the remaining force a' clearly exerts a clock-
wise torque; hence the second condition of equilibrium is not
fulfilled.

That forces a, b, and c (Fig. 26) produce no torque about E is

FIG. 26. FIG. 27.

evident, since all three act directly away from E. It may not
be equally evident that they produce no torque about any other
point in A, such as F. That such is the case, however, may be
easily shown. The two forces a and b have a resultant, say c",
which is equal to c but oppositely directed (since the three force
a, 6, and c are in equilibrium); hence a and b may be replaced
by c" acting downward at E. But obviously c and c" would
produce equal and opposite torques about F, or about any other
point that may be chosen. Hence three forces which form a closed
triangle and also meet in a point have no tendency to produce either
translation or rotation of a body.

Applications to Problems. — A ladder resting upon the ground at
the point A (Fig. 28) and leaning against a frictionless wall at B,
supports at its middle point a 200-lb. man whose weight is

MECHANICS AND HEAT

represented by W. Neglecting the weight of the ladder, let us
find the thrusts a and b. Since the ladder is in equilibrium, the
three forces a, b, and W which act upon it must meet at a point
and must also form a closed triangle. The thrust b must be
horizontal, since the wall is frictionless, and it therefore meets
W produced at C. The upward thrust of the ground on the
ladder must also pass (when extended) through C; i.e., it must
have the direction AC. To find the magnitude of a and of 6,
draw W to & suitable scale, and through one end of W draw a line
parallel to 6, and through the other end draw a line parallel to a.
The intersection of these two lines determines the magnitude of
a and of 6, as explained under Fig. 7,
Sec. 18.

Since the crane beam in the problem
at the close of this chapter is acted
upon by three forces, and since it is
also in equilibrium, the problem may
be solved by the method of this sec-
W tion.

54. Moment of Inertia and Accel-
erating Torque. — The mass of a body
may be defined as that property by vir-
tue of which the body resists a force
tending to make it change its velocity.
The Moment of Inertia of a body, e.g., of a flywheel, is that property
by virtue of which the flywheel resists a torque tending to make
it change its angular velocity. Consider a steam engine which is
belted to a flywheel connected with a buzz saw, as in the case of a
small saw mill. The difference between the tension on the tight
belt and the slack belt, times the radius of the pulley over which
the belt passes, gives the applied torque. If the applied torque
is just sufficient to overcome the opposing torque due to friction of
bearings, and the friction encountered by the saw, then the speed
remains uniform; while if the applied torque exceeds this value,
the angular velocity increases, and its rate of increase, that is, the
angular acceleration, is proportional to this excess torque. If the
applied torque is less than the resisting torque, the angular accel-
eration is negative, that is, the flywheel slows down, and the
rate at which it slows down is proportional to the deficiency in
torque. Compare with accelerating force, Sec. 25.

The relation between the moment of inertia 7 of a flywheel, th<3

28.

ROTARY MOTION 67

accelerating torque, and the resulting angular acceleration a, is
given by the following equation,

Accelerating torque = la, i.e., T = Ia (30)

Compare with F = Ma (Eq. 5, Sec. 25). If we apply a known
torque and determine a experimentally, we may find the numer-
ical value of I from Eq. 30. If the torque is expressed in dyne-
centimeters (i.e., the force in dynes and the lever arm in centi-
meters) and a in radians per sec. per sec., then I will be.expressed
in C.G.S. units (see also Sec. 55). From Eq. 30 we see that unit
torque will give a body of unit moment of inertia unit angular
acceleration; while from Eq. 5, we see that unit force will give
unit mass unit linear acceleration.

The moment of inertia of two similar wheels is found to be
proportional to the products of the mass and the radius squared
for each (Eq. 31, Sec. 55). Hence we find fly
wheels made with large mass and large radius,
and with the greater part of the mass in the
rim, for which part the radius is largest.

65. Value and Unit of Moment of Inertia. —
We shall now determine the relation between
the C.G.S. unit of moment of inertia (Sec.
54) and the mass and radius of the rotating
body, say a wheel. We shall first determine FIG. 29.

the expression for the moment of inertia of a
particle of mass mi at a distance r\ from the axis of rotation
(Fig. 29). Let us consider only this mass mi, ignoring, for the
time being, the mass of the rest of the wheel. To further sim-
plify the discussion, let the force F\ that produces the accelerating
torque T\, act upon m\ itself, so that

miari

AirM, -jj\crC
since a = ria (Eq. 29, Sec. 52).

But this same accelerating torque = /ia (Eq. 30), in which I\
is the moment of inertia of nil about the axis through 0, and a
is its angular acceleration about the same axis.
Consequently

or /i = Wxri2 (31)

68 MECHANICS AND HEAT

Likewise, the moment of inertia 72 of ra2 (see Fig. 29) can be
shown to be ra2r22, and that of m3 to be W3r32, etc. Now if we
add together the moments of inertia of all the mass particles of
the wheel we have for the moment of inertia of the entire wheel

This may be briefly written

I=2mr2 (32)

in which Zrar2 (read sigma mr2) signifies a summation of rar2 for
all of the mass particles in the wheel.

If, in Eq. 31, all quantities are expressed in C.G.S. units,
then m will be expressed in grams, r in centimeters, and hence
7 will be expressed in gm.-cm.2 units. If units of the F.P.S.
system are used, 7 will be expressed in lb.-ft.2 units. Thus a
2000-lb. flywheel having practically all of its mass in the rim of
mean radius 5 feet, would have a moment of inertia I = Mr2
(approx.) = 50,000 (i.e., 2000 X52) lb.-ft.2 For the r of Eq. 32
is practically the same (i.e., 5 ft.) for every mass particle in the
wheel, and the combined mass of all these particles is M or
2000 Ibs.

The moment of inertia of an emery wheel or grindstone of
radius r and mass M is obviously less than Mr2; for in this case
the mass is not mainly concentrated in the "rim," since many of
the mass particles move in circles of very small radius r. It can
be shown by the use of higher mathematics that the moment of
inertia of such disc-like bodies is

(33)
For a sphere of radius r and mass M

I = %Mr2 (34)

66. Use of the Flywheel. — The purpose of a flywheel, in
general, is to "steady" the motion. Thus, in the above-
mentioned case of the saw mill (Sec. 54), if the applied torque
furnished by the steam engine is greater than all the resisting
friction torques, this excess torque, or accelerating torque,
causes the speed of the flywheel to increase; while if the saw
strikes a tough knot, so that the friction torques exceed the
applied torque, then the flywheel helps the engine to run the saw,

ROTARY MOTION 69

and in so doing is slowed down. Indeed the flywheel, when its
speed is increasing, is storing up energy, which is again handed
on to the saw when its speed decreases.

It is a matter of common observation that a heavy wheel,
when being set in motion with the hand, offers an opposing
backward inertia torque; while if we attempt to slow down its
motion, it offers an opposing forward inertia torque, or Driving
Inertia Torque. It is just this driving inertia torque, developed
by the flywheel when slowing down, that helps the engine to run
the saw through the tough knot. Compare this with the driving
inertia force that pushes the canal boat onto the sand bar (Sec.
43). If one were to shell some corn with the ordinary hand corn
sheller, both with and without the flywheel attached, he would be
very forcibly impressed with the fact that, at times, the flywheel
assists with a driving torque.

In the case of "four cycle" gas engines (Chap. XVIII) which
have one working stroke to three idle strokes (i. e., the three
strokes during which the gas is not pushing upon the piston),
it must be clear that the flywheel runs not only the machinery,
but also the engine itself, during these three strokes. Doing
this work, i.e., supplying the driving torque during the three
idle strokes, necessarily slows down the flywheel, but this lost
speed is regained during the next stroke, or working stroke,
when the explosion occurs. If the flywheel is too light, this
fluctuation in speed is objectionably great. Since, in the case
of steam engines, every stroke is a working stroke, lighter fly-
wheels suffice than for gas engines of the same horse power and
speed.

The flywheel of a high speed gas engine need not have so
great moment of inertia as is required for a lower speed engine
furnishing the same horse power. In each case, to be sure, the
flywheel "carries" the load during the three idle strokes, but
the time for these three idle strokes is shorter for the high speed
engine. (Flywheel design will be considered in Sec. 76).

57. Formulas for Translatory and Rotary Motion Compared.
— Below will be found a collection of formulas applied to
translatory motion, and opposite them the corresponding for-
mulas for rotary motion. The similarities and differences in
these two sets of formulas should be observed. All of these
formulas should be thoroughly understood, and most of them
may be memorized with profit.

70 MECHANICS AND HEAT

Translatory Motion Rotary Motion
d 6

-•-T

vt = at or, v0-\-at ut=

-v =v0+%at w =

N Vt —V0 Vt Wj— co° wt

a(oTg) = — j — OTJ a= — ^ — ory

d = v~t B = ut

d =t* 9 = %at2

F =Ma T =Ia

F is accelerating force. T is accelerating torque.

Kinetic energy = \Mvi Kinetic energy = |/co2

(Energy is discussed in Chap. VL.)

PROBLEMS

1. Reduce 2.5 revolutions (a) to radians; (6) to degrees. Express the
angle between north and northeast in (c) radians; (d) degrees J and (e)
revolutions.

2. A shaft makes 1800 R.P.M. Find « in radians per sec.; in degrees
per sec.

3. Through how many degrees will a shaft rotate in 3 min., if w=20
radians per sec.?

4. A flywheel, starting from rest with uniformly accelerated angular
motion, makes 15 revolutions in the first 10 sec. What is its average
angular velocity (a) in revolutions per sec.? (b) In radians per sec.?
(c) In degrees per sec.? (d) What is its velocity at the close of the first
10 sec.?

6. What is the angular acceleration of the flywheel of (problem 4) (a) in
radians per sec. per sec.? (b) In radians per sec. per min.?

6. A belt which travels at the rate of 30 ft. per sec. drives a pulley whose
radius is 3 in. What is the angular velocity for the pulley ?

7. A small emery wheel acquires full speed (1800 R.P.M.) 5 sec. after
starting. Assuming the angular acceleration to be constant, find its
value for this 5 sec.

8. Through what angle does the emery wheel rotate (Prob. 7) in the
first 5 sec.?

9. Find the total torque produced by the forces a, b, c, and d (Sec. 49) if
o and b are both reversed in direction.

10. A locomotive has a velocity of 30 miles per hr. one minute after
starting, (a) What is its average acceleration for this minute? (b)
What is the average angular acceleration of its drivers, which are 6 ft. in
diameter?

11. A crane (Fig. 7, Sec. 18) is lifting a load of 2400 Ibs. Find the
thrust of the beam B against the post A and the pull on cable C due to this

ROTARY MOTION

load, if B is 30 ft. in length and inclines 30° to the vertical, and if C is
attached to B at a point 10 feet from O, and to A at a point 20 feet
above the foot of B. Use the graphical method and compare with the
ladder problem, Sec. 53.

12. The arms AO and OB of the bell crank (Fig. 30) are equal. Find
the pull F, and also the thrust of 0 on its bearings.

13. Find the required pull and thrust (Prob. 12) if F has the direction BC.
Compare ladder problem, Sec. 53.

14. The belt which drives a 1600-lb. flywheel, whose rim has an average
radius of 2 feet (assume mass to be all in the rim), passes over a pulley of
1-ft. radius on the same shaft as the flywheel. The average pull of the
tight belt exceeds that of the slack belt by 100 Ibs. Neglecting friction,
how long will it take the flywheel to acquire a velocity of 600 R.P.M.
First find /, then a, etc.

CHAPTER V

UNIFORM CIRCULAR MOTION, SIMPLE HARMONIC
MOTION

58. Central and Centrifugal Forces, and Radial Acceleration. —
If a body moves in a circular path with uniform speed, it is said
to have Uniform Circular Motion. If a stone, held by a string,
is whirled round and round in a horizontal circular path, it has
approximately uniform circular motion. In order to compel
the stone to follow the curved path, a certain inward pull must be
exerted upon the string by the hand. This pull is termed the
Centripetal or Central force. The opposing pull or force exerted
by the stone by virtue of its inertia (which inertia in accordance
with Newton's first law tends to make it move in a straight line
tangent to the circle), is exactly equal to this central force in mag-
nitude, and is termed the Centrifugal force.

If the string breaks, both the central and centrifugal forces
disappear, and the stone flies off in a straight line tangent to its
path at that instant. The pull upon the string causes the stone
to change its velocity (not in magnitude but in direction) and is
therefore an accelerating force and equal to Ma, in which M is
the mass of the stone and a, its acceleration. Hence to find the
pull upon the string it will be necessary to weigh the stone to get
M, and also to compute the acceleration a. Observe that the
applied accelerating force is the pull of the string; while the cen-
trifugal force is really the inertia force that arises due to the resist-
ance the stone offers to having its velocity changed (in direction).

Here, as in all possible cases that may arise, the accelerating
force and the inertia force are equal in magnitude but oppositely
directed, and they disappear simultaneously (Sec. 43) . The simul-
taneous disappearance of the central and centrifugal forces at the
instant the string breaks, is in complete accord with the behavior
of all reactions. Thus, so long as we push down upon a table,
we experience the upward reacting thrust; but the instant we
cease to push, the reacting thrust disappears.

Centrifugal force has many important applications, for
example, in the cream separator (Sec. 60), the centrifugal gov-

UNIFORM CIRCULAR MOTION

ernor (Sec. 63), the centrifugal pump, and centrifugal blower
(Sec. 150). It is this force which causes too rapidly revolving
flywheels and emery wheels to "burst" (Sec. 59), and it is also
this force which necessitates the raising of the outer rail on curves
in a railroad track (Sec. 62). The centrifugal clothes dryer
used in laundries, and the machine for separating molasses from
.sugar, used in sugar refineries, both operate by virtue of this
principle. The centrifugal force due to the velocity of the earth
in its orbit prevents the earth from "falling" to the sun (Sec.
29), while the centrifugal force due to its rotation about its axis
causes the earth to flatten slightly at the poles and bulge at the

"^ .

FIG. 31.

equator. The polar diameter is about 27 miles less than the
equatorial diameter.

To find the Radial Acceleration a, construct a circle (Fig. 31)
whose radius r represents the length of the string. Let S repre-
sent the stone at a certain instant (£ = 0), at which instant it is
moving west with a velocity v0. After a time t (here t is chosen
about 1/2 sec.), the stone is at Si, and its velocity vt is the same
in magnitude as before, but is directed slightly south of west.
Its velocity has evidently changed, and if this change is divided
by the time t in which the change occurred, the result is by
definition (Sec. 24) the acceleration a.

74 MECHANICS AND HEAT

This change in velocity, or the velocity acquired, is readily
found by drawing from S (Fig. 31, upper sketch) two vectors,
SA and SB, to represent v0 and vt respectively, and then con-
necting A and B. Obviously the acquired velocity is that velocity
which added (vectorially) to v0 gives vt; consequently it is repre-
sented by the line AB. Acquired velocity, however, is always
given by the product of acceleration and time, or at; hence,
the velocity AB = at.

The triangles OSSi and SAB are similar, since their sides are
perpendicular each to each; and if 6 is very small, arc SS\ may be
considered equal to chord SSi. But SSi is the distance the stone
travels in the time t, or v0t. Hence, from similar triangles,

at v0t v0z

Since F = Ma, the central force, usually designated as Fc, is
given by the equation

F,=^ (36)

As already stated, the centrifugal force and the central force are
equal in magnitude but oppositely directed, hence, Fc (Eq. 36)
may stand for either. If M is the mass of the stone S in pounds,
v its velocity in feet per second, and r is the length of the string
in feet; then Fc is the pull on the string in poundals, not pounds
(see Sees. 25 and 32). If Mis given in grams, v in centimeters
per second, and r in centimeters, then Fc is the pull in dynes,
not grams of force. By means of this equation we may compute
the forces brought into play in the operation of the centrifugal
clothes dryer, cream separator, steam engine governor, or in the
case of a fast train rounding a curve.

In many cases it is found more convenient to use a formula
involving angular velocity in revolutions per second instead of
linear velocity. If a body, e.g. a wheel, makes n revolutions per
second, its ''rim" velocity, or the distance traversed in one second
by a point on the rim of the wheel, is n circumferences or 2irrn.
Substituting this value for v in Eq. 36 we have

(37)

For, since one revolution is 2r radians, w = 2,irn, and o>2 =

UNIFORM CIRCULAR MOTION 75

Central Force Radial. — That Fc is radial is apparent in the
above case, since the force must act in the direction of the string.
That this is equally true in the case of a flywheel or cream separa-
tor, or in all cases of uniform circular motion, may be seen from
a discussion of Fig. 32. For if the central force Fc acting upon a
particle P which is moving to the left in the circle, had the direc-
tion a, there would be a component of this force, a', acting in the
direction of the motion, and hence tending to increase the
velocity; if, on the other hand, Fc acted in the direction b, there
would be a component of the force, b', acting in such a direction
as to decrease the velocity. But if
P has uniform circular motion, its
velocity must neither increase nor
decrease; hence neither of these com-
ponents, a' and &', can be present, and
Fc must therefore be radial.

That the acceleration is radial can
be shown in another way. As point
Si (Fig. 31) is taken closer and closer
to S (i.e., as t is chosen smaller and
smaller) vt becomes more nearly par-
allel to va, and AB (see upper sketch, JPIQ- 32.
Fig. 31) becomes more nearly per-
pendicular to v0. In the limit, as Si approaches S, AB
becomes perpendicular to v0, and therefore parallel to r. But
the acceleration has the direction AB, hence it is radial. It
should also be emphasized that the acceleration is linear (not
angular), and is therefore usually expressed either in feet per
second per second or in centimeters per second per second
(Sec. 24).

69. Bursting of Emery Wheels and Flywheels. — The central
force Fc required to cause the material near the rim of a revolving
emery wheel to follow its circular path, is usually enormous.
If the speed is increased until Fc becomes greater than the strength
of the material can withstand, then the material pulls apart,
and we say that the emery wheel "bursts." It is evident that
it does not burst in the same sense that it would if a charge of
powder were exploded at its center. In the latter case the
particles would fly off radially; while in the former they fly off
tangentially. Indeed, the instant the material cracks so that
the central force disappears, the centrifugal force also disappears

MECHANICS AND HEAT

(Sec. 58), and each piece moves off in a straight line in the
direction in which it happens to be moving at that instant.

60. The Cream Separator.— The essentials of a cream separator
are, a bowl A (Fig. 33), attached to a shaft B, and surrounded by
two stationary jackets C and E. When B is rapidly revolved
by means of the "worm" gear, as shown, the fresh milk, enter-
ing at G, soon acquires the rotary motion of the bowl, and, due
to its inertia which tends to make it move in a straight line, it
crowds toward the outside of the bowl with a force Fc.

Both the cream and the milk particles tend to crowd outward
from the center of the bowl, but the milk particles being heavier
than cream particles of the same size,
experience the greater force, and a sepa-
ration takes place. In the figure the
cross-hatched portion c represents the
cream, and the space between this and
the bowl, marked m, represents the
milk. Small holes marked a permit the
cream to fly outward into the stationary
jacket E, from which it flows through
the tube F into the cream receptacle, the
holes marked 6, farther from the center
of the bowl than holes a, permit the
skim-milk to fly outward into the sta-
tionary jacket C, from which it flows
through the tube D into the milk re-
ceptacle.

The bowls of many of the commercial

separators contain numerous separating chambers designed to
make them more effective. This simple form, however, illus-
trates the features common to all. With a good cream sepa-
rator about 98 or 99 per cent, of the butter fat is obtained; i.e.,
1 to 2 per cent, remains in the skim-milk. In the case of
"cold setting" or gravity separation and skimming as usually
practised, 5 per cent, or more remains in the skim-milk.

61. Efficiency of Cream Separator. — In fresh milk, the cream is
distributed throughout the liquid in the form of finely divided par-
ticles. If allowed to stand for several hours the cream particles,
being slightly lighter than the milk particles, slowly rise to the sur-
face. Thus a separation of the cream from the milk takes place,
and, since it is due to gravitational force, it is termed "gravita-

FIG

UNIFORM CIRCULAR MOTION 77

tional" separation. Calling the mass of one of these cream par-
ticles mi and the mass of an equal volume of milk m», the pull of
the earth (in dynes, Sec. 32) on the cream particles is m\g and
the pull on the milk particles is m^g. The difference between these
two pulls, m2g — mig, or g(m?.—m\) constitutes the separating force.
This slight separating force is sufficient to cause the cream par-
ticle to travel from the bottom of a vessel to the top, a distance of
one foot or so in the course of a few hours.

In the case of the centrifugal separator, the force with which mz
crowds toward the outside of the bowl is 47r2w2rm2 (Eq. 37) while
for the cream particle it is 4ir2n2rmi. The difference between < \
these two forces, or 4?r2n2r (mz — mi), is, of course, the separating
force which causes the cream particle to travel toward the center.
The ratio of this separating force to the separating force in the
case of gravity separation is sometimes called the separator effi-
ciency. Hence

,
Efficiency = — — --- ~ (38)

In the above equation, if the gram and the centimeter are used
throughout as units of mass and length respectively, the separat-
ing force will be expressed in dynes; while if pounds and feet are
used, the force is expressed in poundals, not pounds (see Sec. 58).
The word efficiency is used in several distinctly different ways —
the more usual meaning brought out in Sec. 85, being quite
different from that here given.

62. Elevation of the Outer Rail on Curves in a Railroad Track.
— Let B (Fig. 34) represent a curve in the railroad track ABC.
Suppose that for a short distance this curve is practically a circle
of radius ri with center of curvature at E. Let it be required to
find the "proper elevation" d of the outer rail in order that a car,
when passing that particular part of the curve with a velocity Vi,
shall press squarely against the track, so that its "weight," so-
called, shall rest equally on both rails.

On a level, straight track, the thrust of the car against the
track is simply the weight of the car, and is vertical; whereas on
a curve, the thrust T7! (lower sketch, Fig. 34), is the resultant of
the weight of the car W and the centrifugal force Fc which the
car develops in rounding the curve (Eq. 36, Sec. 58). These
forces should all be considered as acting on the center of mass
0 of the car (Sec. 95). If the velocity of the car is such that

MECHANICS AND HEAT

Mvi2/r (i.e., Fc), has the value shown, then the total thrust T\
will be perpendicular to the track, and consequently the thrust
will be the same on both rails.

If the car were to pass the curve at a velocity twice as great as
that just mentioned (or 2z>0, the centrifugal force would be quad-
rupled, and would therefore be represented by the line OH.
This force, combined with W, would give a resultant thrust 7"
directed toward the outer rail. The inner rail would then bear
no weight, while the thrust Tr on the outer rail would be about
one-half greater than the entire weight W of the car, as the figure

FIG. 34.

shows. The least further increase in velocity would cause the
car to overturn. This theoretical limiting velocity could never be
reached in practice, because either the wheel flanges or the rail
would give way under the enormous sidewise thrust. Indeed
whenever the above-mentioned velocity v\, which may be called
the "proper" velocity, is exceeded, the wheel flanges push out on
the outer rail. If this sidewise push is excessive, a defective

flange may give way and cause a wreck,
velocity should not be much exceeded.

Hence the "proper'

UNIFORM CIRCULAR MOTION 79

From the figure it may be seen that

From the figure we also have

-fi = sin 0i, or di = D sin 0i (40)

Observe that the two angles marked 0i are equal (sides perpen-
dicular each to each). Knowing the values of v\t g, and r\ we
may determine tan 0i from Eq. 39. Having found the value of
tan 0i, we may obtain 0i by the use of a table of tangents. If
the width of the track D is also known, the proper elevation di
of the outer rail may be found from Eq. 40. All quantities in-
volved in Eqs. 39 and 40 must be expressed either in F.P.S.
units throughout, or else in C.G.S. units throughout. Observe
that for radius r\ we use v\, T\, Q\, and di respectively for the
"proper" velocity, thrust, angle, and elevation.

In practice the curvature is not made uniform, but decreases
gradually on both sides of the place of greatest curvature until
the track becomes straight; while the elevation of the outer rail
likewise gradually decreases until it becomes zero, where the
straight track is reached. This construction eliminates the
violent lurching of the car, which would occur if the transition
from the straight track to the circular curve were sudden.

63. The Centrifugal Governor. — The essential features of the
centrifugal governor, or Watt's governor, used on steam engines,
are shown in the simplest form in
Fig. 35. The vertical shaft S,
which is driven by the steam engine,
has attached to its upper end two
arms c and d supporting the two
metal balls A and B as shown.

It will readily be seen that the
weight of the balls tends to bring
them nearer to the shaft, while the
centrifugal force tends to make FIQ> 35.

them move farther from the shaft.

If, then, the speed of the engine becomes slightly greater than
normal, the balls move farther out, c and d rise (see dotted posi-
tion), and by means of rods e and / cause collar C to rise. By

80 MECHANICS AND HEAT

means of suitable connecting levers, this upward motion of C
partially closes the throttle valve. The supply of steam being
reduced, the speed of the engine drops to normal. If, on the
other hand, due to a sudden increase in load, the speed of the
engine drops below normal, then the balls, arms c and d, and
collar (7, all lower. This lowering of C opens the throttle wider
than normal, thereby supplying more steam to the engine, and
restoring the normal speed.

In some engines, when the speed becomes too low, the governor
automatically adjusts the inlet valve so that steam is admitted
during a greater fraction of the stroke. This raises the average
steam pressure on the piston, and the normal speed is regained.
This subject will be further considered under "cut off point"
in the chapter on the steam engine.

63a. The Gyroscope. — Just as a body in linear motion resists change
in direction (Sec. 58), so a rotating flywheel resists any change in direction
of rotation, i.e., it resists any shifting in direction of its axis. By virtue
of this principle, a rapidly rotating flywheel, properly mounted, will
greatly reduce the rolling of a ship, as has been shown by tests. This
principle may also yet be successfully applied in securing greater stability
for aeroplanes.

The Gyroscope in its simplest form is shown in Fig. 35a. This device,
until recent years, was merely an interesting, perplexing scientific toy.
The wheel W rotates as indicated by arrow c at a high speed and with
very little friction on the axle AB. If, now, the end of the axle A is
rested upon the supporting point P, the end B, which is without support,
does not drop in the direction d as it would if the wheel were not rotat-
ing, but moves horizontally round and round the point of support as
indicated by arrow e.

The mathematical treatment of the gyroscope is very difficult; so that
we shall here simply state a few facts with regard to its motion. The
angular velocity w of the wheel W is a vector, and may be represented
at a given instant by the arrow co0, called a rotor. Observe that if W
were a right-handed screw, a rotation in the direction indicated by arrow
c would advance the screw in the direction of arrow w0.

Following this same convention we see that the torque produced by the
weight of the wheel would tend to produce rotation, i.e., would produce
an angular acceleration about the horizontal axis indicated by the rotor
a, and further that the direction of this angular acceleration would be
properly represented by placing the arrow head on the end of a away
from the reader. Note that rotor a lies in the axis of torque. The
rotation of B in the direction of arrow e (horizontal) with a constant
angular velocity «' about a vertical axis through P, is, by this same con-

UNIFORM CIRCULAR MOTION 81

vention (right-handed screw) properly represented by the rotor «'.
From the figure, we see that w» lies in the axis of spin, and a in the
axis of torque. The vertical axis in which lies a' is called the Axis of
Precession. The change in direction of the axis AB is called Precession.

As an aid to the memory, using the right hand, place the middle finger
at right angles to the forefinger and the thumb at right angles to both.
Next point the forefinger in the direction of rotor ua and the middle finger
in the direction of a. It will then be found that the thumb points in
the direction of the rotor «' (i.e., down, not up).

Cause of Precession. — Since rotors are vectors, they may be added
graphically. In the figure, «<, represents the angular velocity of W at
a given instant. Its angular velocity «« a short time t later would be
given by the equation ut=u0+(»a, in which w0 is the angular velocity
acquired in the short time t. But angular velocity acquired (gained)
is at (Eq. 28, Sec. 51). Therefore ut = w0+ at as shown graphically in
Fig. 35a, in which the rotor at is drawn from the arrow point of rotor w0.
The resultant is the closing side, or the new angular velocity w«, which

FIG. 35a.

differs from «„ only in direction. In other words, during this short time
t, the axis has changed in direction through the angle 0, and B has
moved in the direction e. Clearly 6 is the angle of precession in this
time t, and e/t is the precessional angular velocity w'.

Compare this change in direction (not in magnitude) of rotary motion
with the change in direction of linear motion (centrifugal force, Sec. 58).
Observe in the vector diagram given in Fig. 35a that u0, at, and ut
correspond respectively to v0, at, and vt of the vector diagram shown in
Fig. 31.

The Reeling (precession of axis of rotation) of a top when its axis is
inclined, is due to this gyroscopic action. In fact if B is considerably
higher than A (Fig. 35a) when A is placed uponP, the gyroscope becomes
essentially a reeling top.

Due to the rotation of the earth (centrifugal force), the equatorial
diameter is 27 miles greater than the polar diameter. Since the axis of
the earth inclines to the normal to the plane of its orbit around the sun
("Plane of the Ecliptic") by an angle of 23°.5, the gravitational pull of

82 MECHANICS AND HEAT

the sun (and also the moon) on this equatorial protuberance produces a
torque about an axis perpendicular to the earth's axis of spin, just as
the weight of the top (when inclined) produces a torque about an axis
lying on the floor and at right angles to the spindle of the top (axis of
spin).

Thus thf earth reels like a great top once in about 26,000 years. The
earth's axis if extended would sweep, each 26,000 years, around a circle
of 23°. 5 radius with a point in the sky in the direction normal to the plane
of the ecliptic as center of this circle. Consequently, 13,000 years
from now the earth's axis will point in a direction 47° from our present
pole star, Polaris. This reeling of the earth causes the Precession of the
Equinoxes around the ecliptic once in 26,000 years.

Monorail Car. — One of the most wonderful recent mechanical achieve-
ments is the successful operation of a car which runs on a track consisting
of only one rail. By a clever adaptation of the gyroscopic principle of
precession of two wheels having opposite rotation (the "Gyrostat"), the
car is balanced, whether in motion or at rest. In rounding a curve,
the "Gyrostat" causes the car to "lean in" just the right amount
(Sec. 62).

If the passengers move to one side of the car, that side of the car rises,
paradoxical though it may seem, and the equilibrium is maintained. As
the passengers move to the side, a "table " presses on the axis of the wheel ,
which axis is transverse to the car, and through the friction developed
by the rotation of the axis, against the table, the end of the axis is caused
to creep forward (or backward) thus developing a torque about a vertical
axis, and precession about an axis at right angles to both of these, namely,
an axis lengthwise of the car. This precession gives rise to the torque
that raises higher the heavier loaded side of the car. For an extended
discussion of the gyroscope and numerous illustrations and practical
applications, consult Spinney's Textbook of Physics, or Franklin and
MacNutt's Mechanics and Heat.

64. Simple Harmonic Motion. — Simple harmonic motion
(S.H.M.) is a very important kind of motion because it is quite
closely approximated by many vibrating bodies. Thus if a
mass, suspended by a spiral spring, is displaced from its equilib-
rium position and then released, it will vibrate up and down for
some time, and its motion will be simple harmonic motion.
Other examples are the vibratory motions of strings, and reeds
in musical instruments, the vibratory motion of the air (called
sound) which is produced by strings or other vibrating bodies,
and the motion of the simple pendulum.

The vibrations of the string of a musical instrument consist,
as a rule, of a combination of vibrations of the string as a whole,

UNIFORM CIRCULAR MOTION 83

and vibrations of certain portions or segments. Consequently
the motion of a vibrating string is usually a combination of several
simple harmonic motions. We shall here restrict ourselves to
the study of the simpler case of uncombined S.H.M.

The piston of a steam engine executes approximately S.H.M.;
while in the motion of the shadow of the crank pin cast upon a
level floor by the sun when over head, we have a perfect example
of S.H.M. Observe that the motion of the crank pin itself is
not S.H.M., but uniform circular motion. An exact notion of
what S.H.M. is, and a simple de-
duction of its important laws, are
most readily obtained from the fol-
lowing definition, which, it will be
seen, accords with the statement
just made with regard to the crank
pin. S.H.M. is the projection of
uniform circular motion upon a di-
ameter of the circle described by the
moving body.

To illustrate the meaning of the
above definition, let A (Fig. 36) be
a body traveling with uniform speed
in the circular path as shown. Let

DC be any chosen diameter, say a horizontal diameter. From
A drop a perpendicular on DC. The foot B of this perpendicular
is the "projection" of A. Now as A moves farther toward D, B
moves to the left at such a rate as always to keep directly below
A. As A moves from D back through F to C, B constantly
keeps directly above A. Under these conditions the motion of
B is S.H.M.

In the position shown it will be evident that B, in order to
keep under A, need not move so fast as A. When A reaches
E, however. B will be at 0 and will then have its maximum speed,
which will be equal to A's speed. As A and hence B approach
D, the speed of B decreases to zero. In case of the vibrating
mass supported by a spring (mentioned at the beginning of the
section) it is evident that its velocity would be zero at both ends
of its vibration and a maximum at the middle, just as we have
here shown to be the case with B.

66. Acceleration and Force of Restitution in S.H.M.— If the
two bodies A and B move as described in Sec. 64, it is clear that

84 MECHANICS AND HEAT

they both have at any and every instant the same horizontal
velocities. Thus, in the position shown in Fig. 36, we see that

B's velocity (horizontal) must be equal to the horizontal com-
ponent of A's velocity. An instant later, A's horizontal compo-
nent of velocity will have increased, and since B always keeps
directly below (or above) A, we see that B's velocity must have
increased by the same amount. In other words, the rate of
change of horizontal velocity, or the horizontal acceleration, is the
same for both bodies.

Similar reasoning shows that as A passes from E to D, and
consequently B passes from 0 to D, the leftward velocity de-
creases at the same rate for both bodies. As A passes from D
to F and then from F to C, we see that B passes from D to 0 with
ever increasing velocity, and then from 0 to C with decreasing
velocity. To summarize, we may state that at every instant
the horizontal components of A's velocity and acceleration are equal,
respectively, to the actual (also horizontal} velocity and acceleration
of B at that same instant.

We have seen that whenever B moves toward 0, its velocity
increases, while as it moves away from 0, its velocity decreases;
i.e., its acceleration is always toward 0. To impart to B such
motion, obviously requires an accelerating force always pulling
B toward O. We shall presently show that this force, called the
Force of Restitution Fr, is directly proportional to the distance
that B is from 0. This distance is called the Displacement x.

The central force required to cause A to follow its circular
path is

and the horizontal component of this, or — Fh, has the value
shown in Fig. 36. Note that if the vector x, directed to the
right, is positive, then Fh, when directed to the left, is negative.
Now Fh is the accelerating force that gives A its horizontal
acceleration, while Fr is the accelerating force that gives B its
horizontal acceleration; but these two horizontal accelerations
have been shown to be always equal. Hence if A and B are of
equal mass M, it follows from F = Ma (Eq. 5) that Fr=Fh.
From similar triangles (Fig. 36) we have

-Fh/Fc = x/r, i.e., -Fh or -FT = XFC = 4^n2Mx (41)

UNIFORM CIRCULAR MOTION 85

Eq. 41 shows that the force of restitution, acting upon B at any
instant, is proportional to the displacement of B at that same
instant. Accordingly B's accelerating force, and hence its
acceleration or rate of change of velocity, is a maximum when at
C or D, at which points its velocity is zero, and a minimum (in
fact zero) at 0, at which point B has its maximum velocity.
The minus sign indicates that the force of restitution is always
oppositely directed to the displacement. Thus when B is toward
the left from 0, x is negative, but Fh is then positive.

If, then, a body is supported by a spring or otherwise, in such
a manner that the force required to displace it varies directly
as the displacement, we know at once that the body will execute
S.H.M. if displaced and then released. Thus it can easily be
shown, either mathematically (Sec. 67) or experimentally, that
the force required to displace a pendulum bob is proportional
to the displacement, provided the latter is small. Hence we
know that when the bob is released it will vibrate to and fro in
S.H.M.

66. Period in S.H.M.— Solving Eq. 41 for n gives

If a body makes n vibrations per second, its period of vibration,
or the time P required for one complete vibration (a swing to
and fro), is 1/n; hence

P = 2Tr-^ (42)

Eq. 42 gives the period of vibration for any body executing
S.H.M., i.e., for any body for which the force of restitution is
proportional to the displacement x, and in such a direction as to
oppose the displacement. In this equation, M is the mass of
the vibrating body in grams, P the period of vibration in seconds,
and Fr the force of restitution in dynes, when the displacement
is x centimeters. See remark on units below Eq. 36, Sec 58 and
also Sec. 32. Since x and Fr always differ in sign, the expression
under the radical sign is intrinsically positive.

If a heavy mass suspended by a spiral spring requires a force
of 1 kilogram to pull it downward, say, 1 cm. from its equilib-
rium position, it will require a force of 2 kilograms to displace
it (either downward or upward) 2 cm. from its equilibrium posi-

MECHANICS AND HEAT

tion. This shows that the force of restitution is proportional
to the displacement; hence we know that if the mass is pulled
down and suddenly released, it will vibrate up and down and exe-
cute S.H.M. Here Fr or 2 X 1000 X 980 = 1,960,000 dynes when x =
2 cm. Suppose that the mass is 3 kilograms. We may then
find its period of vibration, without timing it, by substituting
these values in Eq. 42. Thus, neglecting the mass of the spring,

= 0.346 sec.

67. The Simple Gravity Pendulum. — The following discussion
applies, to a very close degree of approximation, to the physical
simple pendulum having a small bob
B (Fig. 37) suspended by a light cord
or wire. The length L of the pendu-
lum is the distance from the center
of the bob to the point of suspension.
Consider the force upon the bob at
some particular point in its path. Its
weight, W, or Mg (Fig. 37), may be
resolved into two components, 7^1 in
the direction of the suspending wire,
and FT in the direction of motion, i.e,.
toward A. For small values of 6, A
approaches 0, so that CA may be
called equal to CO, i.e., equal to L,
and Fr may be called the force of
restitution. From similar triangles,

FIG. 37.

FT -X FT -X

W = CA or approx' 'Mg = ~L or

-Mgx
L

(43)

Eq. 43 shows that the force of restitution Fr is proportional
to the displacement, and oppositely directed. Hence the pen-
dulum executes S.H.M. ; and we may therefore substitute the
value of Fr from Eq. 43 in Eq. 42, and obtain the period P of
the pendulum,

2ir\
\ -

-MX

jrr-
Mgx

(44)

UNIFORM CIRCULAR MOTION 87

The maximum value of x, i.e., the distance from 0 to the bob
when at the end of the swing, is called the Amplitude of vibration.
Since x and M cancel out in Eq. 44, the period of a pendulum is
seen to be independent of either its mass or its amplitude. The
latter is true only for small amplitudes. If x is large, CA and
CO are not approximately equal, as is assumed in the above
derivation. A pendulum vibrates somewhat more slowly if
the amplitude is large than if it is small, since CA appreciably
exceeds L when 6 is large, thus making Fr smaller than given
by Eq. 43.

68. The Torsion Pendulum. — The torsion pendulum usually
consists of a heavy disc suspended from its center by a steel
wire, and hence free to rotate in a horizontal plane. When the
disc is rotated from its equilibrium position through an angle
6, it is found that the resisting torque is proportional to the angle
0. In this case, the torque of restitution, or the returning torque,
is (a) proportional to the displacement angle 0, and (6) opposes
the displacement. These are the two conditions for S.H.M. of
rotation. In the case of the balance wheel of a watch, the torque
of restitution due to the hair spring, is proportional to the angle
through which the balance wheel is rotated from its equilibrium
position. Hence the balance wheel of a watch executes S.H.M. ,
and therefore its period is independent of the amplitude of its
rotary vibration.

PROBLEMS

1. If a 2-lb. mass is whirled around 240 times per minute by means of a
cord 4 ft. in length (a) what is the pull on the cord? (6) What is the radial
acceleration experienced by the mass?

2. A mass of 1 kilogram is whirled around 180 times per minute by
means of a cord 1 meter in length. What is the pull on the cord? (a)
in dynes? (6) in grams force? (c) What is tbe radial acceleration?

3. How many times as large does the central force become when the
velocity (Prob. 2) is doubled? When r, the length of the cord, is doubled,
the number of revolutions per second, and also the mass, remaining
the same?

4. An emery wheel 12 in. in diameter makes 2400 R.P.M. Find the
force (in poundals, and also in pounds), acting upon each pound mass of tbe
rim of the wheel tending to "burst" it.

6. At a point where the radius of curvature r (Fig. 34) is 2000 ft., what
is the "proper" elevation of the outer rail for a train rounding the curve
at a velocity of 30 miles per hr., i.e., 44 ft. per sec.? Distance between
rails is 4 ft. 8 in.

6. An occupant of a ferris wheel 20 ft. from its axis observes that he

88 MECHANICS AND HEAT

apparently has no weight when at the highest point. Find his linear
velocity and radial acceleration, and also the angular velocity of the
wheel.

7. Find the maximum velocity of the occupant of a 20-ft. swing if the
pull he exerts upon the swing at the instant the ropes are vertical is one-
half more than his weight.

8. The diameter of a cream separator bowl is 20 cm. Find its
"efficiency" when making 4800 R.P.M.

9. A 4000-gm. mass, when suspended by a spring, causes the spring to
elongate 2 cm. What will be the period of vibration of the mass if
set vibrating vertically? Neglect the mass of the spring.

10. A sprinter passing a turn in the path, where the radius of curvature
is 60 ft., at a speed of 10 yds. per sec., leans in from the vertical by an angle
e. Find tan 6.

11. What is the period of a pendulum (in Lat. 45°) which has a length
of 20 cm.? 100 cm.?

12. What is the length of a pendulum (Lat. 45°) that beats seconds,
i.e., whose full period of vibration is 2 sec.?

13. A pendulum 30 ft. in length has a period of 6.0655 sec. at
London. What is the value of g there?

14. A pendulum whose length is 10 meters makes 567.47 complete
vibrations per hour at Paris. Find the value of g at Paris.

CHAPTER VI
WORK, ENERGY AND POWER

69. Work. — Work is defined as the production of motion
against a resisting force. The work done by a force in moving a
body is measured by the .product of the force, and the distance
the body moves, provided the motion is in the direct-ion of the
force (see Sec. 71). Hence work W may always be expressed by
the equation

W=Fd (45)

Thus the work done by a team in harrowing an acre of ground is
equal to the product of the average force required to pull the
harrow, and the distance the harrow moves. To harrow two
acres would require twice as much work, because the distance
involved would obviously be twice as great. If the applied force
is not sufficient to move the body, it does no work upon the body.
Thus if a man pushes upon a truck, it does not matter how hard
he pushes, nor how long, nor how tired he becomes; he does no
work upon the truck unless it moves in response to the push.

In case F and d are oppositely directed, i.e., in case the body,
due to previous motion or any other cause, moves a distance d
against the force, then work is said to be done by the body against
the force. Thus if a stone is thrown upward, it rises a certain
height because of its initial velocity, and in rising it does work
(Fd) against the force of gravity. As it falls back the force of
gravity does work (Fd) upon the stone in accelerating it.

From the above discussion, we see that work may be applied
in three general ways; viz., (a) to move a body against friction,
(6) to move it against some force other than friction, e.g., as in
lifting a body, and (c) to accelerate a body, i.e., to impart velocity
to it. Observe that in all three cases the applied force does work
against some equal opposing force. In case (a) it is the friction
force Ff, in case (6) the weight W (or a component of the weight),
and in case (c) the inertia force Fi, against which the applied
force does work.

90 MECHANICS AND HEAT

As a train starts upgrade from a station and traverses a distance
d, the pull FI upon the drawbar of the locomotive does work in
each of these three ways. Calling the average total friction
force on the train Ff, the component of the weight of the train
which tends to make it run down grade Fw (see Fig. 8, Sec. 19),
and the average inertia force or resistance which the train offers
to being accelerated Fi} we have

Total work Fld=Ffd+Fwd+Fid (46)

If, at the above distance d from the station, the drawbar of the
locomotive becomes uncoupled from the train while going full
speed up grade, and if the train comes to rest after going a distance
d', it is clear that the driving inertia force F'i of the train (Sec. 43)
does work F'jd' in pushing the train up the grade against F/ and
Fw, so that the work

Fi'd'=Ffd'-\-Fwdf (47)

Observe that d and d' and also Ft and F'i would, in general, be
quite different in value, while the values of F/ and Fw would be
practically the same before and after uncoupling; hence these
same symbols are retained in Eq. 47.

70. Units of Work. — Since force may be expressed in dynes,
grams, poundals, pounds, or tons, and distance in centimeters,
inches, or feet, it follows that work, which is force times distance,
may be expressed in dyne-centimeters or ergs, gram-centimeters,
foot-poundals, foot-pounds, foot-tons, etc. Thus, if a locomotive
maintains a 1-ton pull on the drawbar for a distance of one mile,
the work done is 5280 ft.-tons, or 10,560,000 ft.-lbs. If a 20-lb,
mass is raised a vertical distance of 5 ft., the work done against
gravitational attraction is 100 ft.-lbs. If a force of 60 dynes
moves a body 4 cm., it does 240 ergs (dyne-centimeters) of work.
In scientific investigations, the erg is the unit usually employed;
in engineering calculations, on the other hand, the unit is the foot-
pound. The work done by an electric current is usually com-
puted in joules. One joule is 107 ergs.

In changing from one work unit to another, it must be observed
that work contains two factors. For example, let it be required
to express the above 100 ft.-lbs. of work in terms of ergs. This
may be done in two ways: (1) by reducing the 20-lb. force to
dynes and the 5 ft. to centimeters, and then multiplying the two
results together; or (2) by finding the number of ergs in a foot-

WORK, ENERGY AND POWER 91

pound and then multiplying this number by 100. The foot-
pound is larger than the erg for two reasons : first, 1 foot = 30.48
centimeters, and second, the pound being approximately 453
grams, and the gram force being 980.6 dynes, it follows that the
pound force = 445,000 dynes. The foot-pound is therefore
30.48 X445,000 or 13,563,000 ergs. Therefore 100 ft.-lbs. of work
is 1,356,300,000 or 1.356X109 ergs.

71. Work Done if the Line of Motion is not in the Direction of
the Applied Force. — In Sec. 69 it was shown that work = Fd
provided F and d have either the same direction or opposite direc-
tions, i.e., provided the angle between the applied force and the
direction of motion is either zero or 180°. If this angle is zero,
then work is done by the force; while if it is 180°, work is done
against the force. If this angle is 90°, no work is done either by
or against the force. Thus if a team is pulling a wagon westward,
it is perfectly obvious that a man, walking along side the wagon
and pushing north upon it, neither helps nor hinders the team.

FIG. 38.

If he pushes directly forward, the above angle is zero, and in
traveling a distance d while pushing with a force F he helps the
team by an amount of work Fd; while if he pulls back the angle
is 180°, and he adds Fd to the work the team must do.

If he pulls slightly to the south of west with a force F (Fig. 38,
top view of wagon) he does an amount of work which is less than
Fd. Resolving F into components FI and Fz, respectively
parallel and perpendicular to the line of motion, we see that F2
simply tends to overturn the wagon, while FI is fully effective
in helping the team. The work done by F is then Fid, but
Fi=F cos e, hence

W=Fld=Fdcose (48)

As 6 approaches 90°, cos 6, and hence the work done, approaches
zero. As 0 decreases, i.e., as the man pulls more nearly west,
cos B approaches its maximum value, unity (when 0 = zero), and
the maximum work (Fd) is obtained. Since cos 180°= —1, we

92 MECHANICS AND HEAT

see that when F is a backward pull on the wagon, then W= — Fd.
The negative sign indicates that the work instead of being done
by the man, is added work done by the team.

72. Work Done by a Torque. — If the force F (Fig. 39) pushes
the crank through an arc AB, the work done is force times dis-
tance, or W=FXAB. But by definition

, arc AB , ...

6 = — = , from which AB = rd;

hence

W=FXAB=Fr6

But since torque (T) equals force times radius,

W=Td (49)

In rotary motion, it is usually more convenient to compute
work by means of Eq. 49 than by means of Eq. 45. If F is ex-
pressed in pounds and r in feet, i.e., if
the torque is expressed in pound-feet,
and B in radians, then Td gives the work
done in foot-pounds. Thus, for example,
if F is 10 Ibs., r is 2 ft., and 6 is 0.6 radi-
ans, the work done is 12 ft.-lbs. If T6
is expressed in C.G.S. units (dyne, cen-
p on timeter, and radian), the resulting work

is given in ergs.

73. Energy — Potential and Kinetic. — The energy of a body
may be denned as the ability of the body to do work. The
potential energy of a body is its ability to do work by virtue of its
position or condition. The Kinetic Energy of a body is its ability
to do work by virtue of its motion.

The weights of a clock have potential energy equal to the work
they can do in running the clock while they descend. Likewise
the main spring of a clock or watch, when wound, has potential
energy equal to the work it can do as it unwinds. The water in a
mill pond has potential energy. Powder and coal have potential
energy before ignition. A bended bow has potential energy.
When the string of the bow is released and the arrow is in flight,
the energy then possessed by the arrow is kinetic. Any mass in
motion has kinetic energy.

The immense amount of kinetic energy possessed by a rapidly
moving train is appreciated only in case of a derailment or a

WORK, ENERGY AND POWER 93

collision. The kinetic energy of a cannon projectile enables it to
do work in piercing heavy steel armor plate even after a flight
of several miles, during all of which flight it does work against
the air friction upon it. The work done upon the armor plate
of the target ship is Fd; in which F is the enormous force (average
value) required to push the projectile into the plate, and d is
the distance to which it penetrates.

74. Transformation and Conservation of Energy. — Energy
may be transformed from potential to kinetic energy and vice
versa, or from kinetic energy into heat, or by a suitable heat
engine, e.g., the steam engine, from heat to kinetic energy; but
whatever transformation it experiences, in a technical sense, none
is lost. In practice, energy is lost, as far as useful work is con-
cerned, in the operation of all machines, through friction of bear-
ings, etc. This energy spent in overcoming friction is not actually
lost, but is transformed into heat energy which cannot be profit-
ably reconverted into mechanical energy. In all cases of energy
transformation, the energy in the new form is exactly equal in
magnitude to the energy in the old form. This fact, that energy
can neither be created nor destroyed, is referred to as the law of
the Conservation of Energy. This law is of great importance, as
will appear from time to time. It condemns as visionary all
perpetual motion machines purporting to furnish power without
having a source of energy. Further, since it is impossible to
entirely eliminate friction, a perpetual motion machine neither
using nor furnishing power is seen to be an impossibility. The
kinetic energy of the moving parts of such a machine would soon
be transformed by friction into heat, and no longer exist as visible
motion.

The conservation of energy is one of the well-established laws
of Physics, and is frequently used as a basis in the derivation of
equations, and in various lines of reasoning such as just given
with regard to perpetual motion machines. From the conserva-
tion of energy, we see that to give a body a certain amount of
energy, whether potential or kinetic, an exactly equivalent
amount of work must be done on the body.

We may now state in slightly different form than that used in
Sec. 69, the fact that the work done upon a body may be used in
three ways: (a) to move the body against friction; (6) to give the
body potential energy; and (c) to give the body kinetic energy.
These three amounts of work done by the locomotive upon the

94 MECHANICS AND HEAT

train (Sec. 69) are represented respectively by the three terms of
the right-hand member of Eq. 46. Since F^d is the work done by
the locomotive in accelerating the train, i.e., in giving it its veloc-
ity and hence its kinetic energy, it follows, from the conservation
of energy, that Ffd is the kinetic energy of the train just as it
reaches the point at the distance d from the station. Hence,
when uncoupled at this point, this kinetic energy does an equal
amount of work F'4' in forcing the train on up the grade. Eq. 47
shows that this work is used partly (F/d') in driving the train on
against friction, and partly (Fwd') in giving the train more
potential energy.

It should be emphasized, that in the transformation of kinetic
energy into potential energy, and vice versa, work is always done.
To illustrate, suppose that a gun of length d feet fires a projectile
of weight W pounds vertically to a height h feet. Designating by
F the average force (in pounds) with which the powder, upon
exploding, pushes upon the projectile, and ignoring all friction
effects (see Dissipation of Energy, Sec. 77) we have Fd foot-pounds
for the work done in giving the projectile its kinetic energy, and
Wh foot-pounds (force times distance) for the work done by the
kinetic energy of the projectile in raising itself to the height h, in
which position its potential energy (Ep) is a maximum and has
the value Wh foot-pounds. This maximum potential energy (Ep
max.) is the ability the projectile has to do work by virtue of
its elevated position, and it does this work Wh (force times
distance) while descending, in causing the velocity of the pro-
jectile to increase, thereby increasing its kinetic energy. This
kinetic energy (Ek) at the instant of striking is of course a
maximum (Ek max.), and, by the conservation of energy, it
must be equal to the work Wh done by gravitational attraction
in giving it this energy.

To summarize, we have, in accordance with the conservation
of energv, the following successive energy transformations:
Fd (work done by powder) = E max. (at muzzle) = work Wh
(done while rising) = Ep max. or Wh (at highest point) = work
Wh (done while descending) = Ek max. (at striking).

As the projectile rises, its kinetic energy decreases, while its
potential energy increases; but, from the conservation of energy,
we see that at any instant, Ep-\-Ek = Wh = Ep max. Thus, when
the projectile is at a height \h, it is evident that Ep = \Wh;
hence, at that same instant, it must be that Ek = %Wh. If h

WORK, ENERGY AND POWER 95

were 10,000 ft., and d, 10 ft., then F would be 1000 times the weight
of the projectile (since Fd= WK). Likewise, if a 1-ton pile driver
falls 20 ft. (19 ft. before striking) and drives the pile 1 ft., the
average force on the pile is, barring friction effects, 20 tons. The
above discussion applies to the similar energy transformations
that occur in the operation of a pile driver, and in the vibration of
a pendulum.

76. Value of Potential and Kinetic Energy in Work Units.—
From the preceding sections, we see that the potential energy, or \
the kinetic energy possessed by a body, is equal to the work (Fd) \
required to give it that energy. Accordingly, the equation express-
ing the potential energy, or the kinetic energy of a body is very
simply obtained by properly expressing this work (Fd). In
deriving the equation for potential energy, it is customary to take
for this work, the work (WK) done in raising a mass M a certain
distance against gravitational force; while for the kinetic energy
equation, use is made of the work done by gravitational force on
a mass M in falling a certain distance. This is done for two
reasons: first, because gravitational potential energy is the kind
of potential energy with which we have to deal very largely in
calculations, while the kinetic energy of falling bodies is of prime
importance; and second, because of the fact that the gravitational
force upon a body, i.e., its weight, is sensibly constant regardless
of change of height or velocity of the body, which fact very much
simplifies the derivations.

The Potential energy of a mass M, when raised a height h
(Fig. 40), is equal to the work done in raising it, or force times
distance. Here the force is W or Mg, and the distance is h, so
that

Ep = Mgh (50)

Since Mg expresses the force either in dynes or poundals (Sec.
32) and h is the distance either in centimeters or feet, depending
upon which system is used, the work, and hence the potential
energy, is expressed in either ergs or foot-poundals. If the work
is wanted in foot-pounds, the weight must be expressed in
pounds and the distance in feet. The potential energy is then
given by

Ep = Mh (50a)

Note that a mass of M pounds weighs M pounds, not Mg pounds
(Sec. 32).

96 MECHANICS AND HEAT

The Kinetic energy of a moving body would naturally be
expected to depend upon the mass of the body and upon the rapid-
ity of its motion, i.e., upon its velocity. Suppose that the body of
mass M (Fig. 40) falls the distance h. Its kinetic energy after
having fallen that height must, according to the law of con-
servation of energy, be equal to the work done upon it by gravity
while falling, or force Mg times the distance h. Its kinetic energy
is then Mgh, which, by Eq. 50, is just the potential energy that
it has lost during its fall. Substituting for h its value for falling
bodies given in Eq. 13, Sec. 34, namely, h = %gt2, gives

Wvz (51)

If the English system is used, since the weight or force is expressed
in poundals, the result obtained by substituting the mass and
velocity of the moving body, in Eq. 51,
is expressed in foot-poundals, not foot-
pounds. If M is the mass of the body in
grams, then Mg is the force in dynes, and if
h is expressed in centimeters, Mgh, and
hence the kinetic energy %Mv2, is expressed
in dyne-centimeters or ergs.

76. Energy of a Rotating Body. — Any
mass particle of a rotating body, e.g., a fly-
wheel, has the kinetic energy %mv2, in
which m is the mass of the particle and v its
velocity. Hence the kinetic energy of the

niE

, . \~Ek^2-Mv* entire wheel is the sum of all the quantities
77M% \mvi for each and all of its mass particles.
FIG. 40. Now the particles near the rim of the fly-

wheel have much higher velocities and hence

much greater amounts of kinetic energy than those near the axis,
so that the actual summation of the kinetic energy for all particles
cannot be effected without the use of higher mathematics. We
readily see, however, that two wheels of equal mass M, having
equal angular velocity co, will possess different amounts of kinetic
energy if the mass is mainly in the rim of one and in the hub of
the other. Here, as in so many other cases, a very simple method
of deriving the expression for the kinetic energy comes from the
use of the law of the conservation of energy.

From this law we know that the kinetic energy Ek of the fly-
wheel, when it has acquired the angular velocity co, must be

WORK, ENERGY AND POWER 97

equal to the work T6 (Eq. 49, Sec. 72) done by the applied torque
in giving it this kinetic energy, i.e., in imparting to it this angular
velocity w. Hence Ek=T8, which, by a few simple substitu-
tions, may be brought into a form involving only the moment of
inertia / of the wheel, and its angular velocity w. From Eq.
30, Sec. 54, T = la,

also a= y (Sec. 51), 8= at (Eq. 27, Sec. 51), and w = |

Since the wheel starts from rest with uniform acceleration, its
average angular velocity w must be one-half its maximum
angular velocity u>, as explained in Sec. 52. Making successively
these substitutions, we have

Ek=T9 = Iad = %Iuz (52)

If we use C.G.S. units exclusively, then Td (Eq. 52) gives the
work in ergs (Sec. 72) required to produce the kinetic energy
l/co2, which energy must therefore also be expressed in ergs. I
is then, of course, expressed in C.G.S. units or gm.-cm.2 units
(Sec. 55), and co in radians per second. If we use the F.P.S.
system throughout, then Td is expressed in foot-poundals (Sec.
72), £/a;2 in foot-poundals, o> in radians per second, and / in
lb.-ft.2 units (Sec. 55).

Let us now apply Eq. 52 to find the kinetic energy of the 1-ton
flywheel mentioned in Sec. 55, when co = 20 radians per sec., i.e.,
when the flywheel is making slightly more than 3 revolutions per
second. The moment of inertia of the wheel was found in
Sec. 55 to be 50,000 lb.-ft.2, whence, from Eq. 52, we have Ek =
| 50,000 X202= 10,000,000 foot-poundals or 310,000 ft.-lbs.
Dividing this energy (310,000 ft.-lbs.) by 550 (550 ft.-lbs. per
sec. is one horse power, Sec. 82) gives 562, which shows that the
above flywheel, when rotating at the rate of 20 radians per second,
has enough kinetic energy to furnish 1 horse power (H.P.) in
driving the machinery for 562 seconds, or nearly 10 minutes,
before coming to rest.

In case the angular velocity of a flywheel, connected with a
gas engine, decreases from wi just after an explosion stroke, to
co2 just before the next explosion stroke, then the energy Ek
which it gives up in carrying the load during the three idle strokes
(Sec. 56) is

#* = $/wi2-iW, or -|/(Wl2-co22) (53)

98 MECHANICS AND HEAT

If the wheel makes 2 revolutions per sec., i.e., if the piston makes
4 strokes per sec., then the 3 idle strokes will last 3/4 second; so
that if the engine were a 10-H.P. engine, the work W which the
flywheel would have to do in this 3/4 second would be 550 X
10X3/4 or 4125 ft.-lbs. Evidently this work W equals E of Eq.
53, or

TF = I/Can2 -co22) (54)

Eq. 54 is usually employed in computing the proper moment of
inertia / for a flywheel working under certain known conditions.
Thus, if we know the horse power of a certain gas engine, the aver-
age angular velocity co of its flywheel shaft, and the permis-
sible speed variation a>i — wa, we can compute both W (in foot-
poundals) and coi2— co22; then, substituting these two quantities
in Eq. 54, we may solve for /. Having found the value of / in
lb.-ft.2 units, we may, by using the equation I = Mr2 (Sec. 55),
choose a certain value for the radius r of the flywheel and then
solve for its mass M; or we may choose a value for M and then
find the proper value for r in order to make the wheel meet the
above requirements.

If a small car and a hoop of equal mass are permitted to run down the
same incline, it will be found that upon reaching the bottom of the
incline the velocity of the hoop will be about 7/10 that of the car.
Suppose that these velocities are 7 ft. per sec. for the hoop and 10 ft.
per sec. for the car. The potential energy at the top of the incline was
the same for both bodies, hence the kinetic energy upon reaching the
bottom must be the same for both (conservation of energy). The hoop
has kinetic energy of both translation and rotation, while the car,
neglecting the slight rotational energy of its wheels, has only energy of
translation. Consequently we have

in which the left member is the energy of the hoop, and the right member
that of the car. Solving, we find that half of the energy of the hoop is
rotational energy, that is, experiment shows that %Iw* = %Mv2 for the
hoop.

Mathematical Proof. — Since I = Mr2 (the mass M of the hoop considered
to be all in its "rim" of radius r (see below, Eq. 32, Sec. 55), and
since v = ru (Eq. 29, Sec. 52), we have

which was to be proved.

WORK, ENERGY AND POWER 99

A sphere, or a wheel with a massive hub, would travel more nearly as
fast as the car, because in such case the mass would not be all concen-
trated in the "rim," and consequently the moment of inertia, and
therefore the rotational energy, would be less than for the hoop.

77. Dissipation of Energy. — The fact that the energy of a body,
whether potential or kinetic, always tends to disappear as such,
is a matter of common observation, and is referred to as the prin-
ciple of the Dissipation of Energy. Thus a body, for example a
stone, in an elevated position has potential energy. If released,
the stone falls, and at the instant of striking the ground its
energy is kinetic. An instant later the stone lies motionless upon
the ground, both its potential energy and kinetic energy having
disappeared.

The results of many carefully performed experiments lead to
the conviction that in the above case no energy has been lost (see
conservation of energy) ; but that, due to air friction while falling,
and friction against the ground as it strikes, the stone has slightly
warmed itself, the air, and the ground; and that the amount of
heat energy so developed is exactly equal to the original potential
energy of the stone. This example illustrates the general trend
of energy change throughout nature; viz., the potential energy of
a body tends to change to kinetic energy., and its kinetic energy tends
to change into heat energy. The relation between heat and other
forms of energy will be further considered in the study of heat, but
it might here be mentioned that 778 ft.-lbs. of work used in
stirring 1 pound of water will warm it 1° F. Attention is also
called to the fact that the hands may be warmed by rubbing them
together, and that primitive man lighted his fires by vigorously
rubbing one piece of wood against another.

A vibrating pendulum, a rotating flywheel, or a moving train
soon loses its motion if no power is applied. These are good
examples of the dissipation of energy. In all such cases, the
potential energy or the kinetic energy of the body is transformed
into heat through the work done by the body against friction.

78. Sliding Friction. — If one body is forced to slide upon
another, the rubbing together of the two surfaces gives rise to a
resisting force which always opposes the motion and is called
friction. It may also be called the force of friction. Either sur-
face may be that of a solid, a liquid, or a gas. Thus in drawing a
sled on a cement walk, the friction is between two solids, steel and
cement, In the passage of a. boat through water, the friction is be-

100 MECHANICS AND HEAT

tween a solid and a liquid, i.e., between the sides and bottom of
the boat and the water. In the case of the aeroplane, there is fric-
tion between the canvas planes or wings and the air through which
they glide. If the wind in the higher regions of the atmosphere
has either a different velocity or a different direction than the
surface wind, there will be friction between them. In all cases,
the work (Fd) done against friction is the product of the frictional
force and the distance of sliding, and is transformed into heat
energy (Sec. 77). Bending a piece of wire back and forth rapidly,
heats it because of the Internal Friction between its molecules,
which are thereby forced to slide past each other. Internal
friction in liquids causes them to become heated when stirred,
and also gives rise to viscosity. The greater viscosity or molecu-
lar friction of syrups makes them flow much more slowly than
water.

A smooth board or iron plate appears rough under the micro-
scope due to innumerable slight irregularities. The cause of
friction is the fitting together or interlocking of these irregularities
of one surface with those of the other over which it slides. It is
easily observed that it takes a greater force to start the sliding of
a body than to maintain it. The former force must overcome the
backward drag of Static Friction; the latter, that of Kinetic
Friction. The greater value of static friction is probably due to
the better interlocking of the irregularities of the two surfaces
when at rest than when in motion relatively to each other. This
view is supported by the fact that when the velocity of sliding is
very small the kinetic friction differs very little from the static.

The so-called "Laws of Friction" are: (a) the friction is directly
proportional to the force pressing the surfaces together; (6) it is
independent of the area of the surfaces in contact; and (c) it is
independent of the velocity of sliding. These laws are approxi-
mately true between wide limits. Thus the force required to
draw a sled will be approximately doubled by doubling the load,
will be very little affected by change in the length of runner
(within reasonable limits), and will remain about the same though
the velocity is varied from 1 mile per hour or less, to several times
that value.

To reduce the waste of power and also the wearing of ma-
chinery due to friction, lubricating oils are used. The film of
oil between the two rubbing surfaces prevents their coming into
such intimate contact, and thus prevents, in a large measure, the

WORK, ENERGY AND POWER 101

interlocking of the above-mentioned irregularities. During the
motion, the particles of oil in this film glide over each other with
very little friction, and the total friction is thus reduced by sub-
stituting, in part, liquid friction for sliding friction. The resist-
ance which a shaft bearing offers to the rotation of the shaft, is
evidently sliding friction, and is therefore reduced by proper oiling.

In general, friction is greater between two surfaces of the same
material than it is between those of different materials. Thus
bearings for steel shafts are sometimes made of brass, and fre-
quently of babbitt, to reduce friction. Babbitt metal is an alloy
of tin with copper and antimony, as a rule. Sometimes lead is
added. On the other hand, iron brake shoes are used on iron
wheels to obtain a large amount of friction, and pulleys are
faced with leather to prevent belt slippage.

The wasteful effects of friction are usually apparent, but the
beneficial effects are probably not so generally appreciated. If
it were not for friction, it would be impossible to transmit power
by means of belts, or to walk upon a smooth surface. Further-
more, all machinery and all structures which are held together
by nails, screws, or by bolts (unless riveted), would fall to pieces
instantly if all friction were eliminated.

79. Coefficient of Friction.— The Coefficient of Kinetic Friction
is defined as the ratio of the force required to move a body slowly
and with uniform velocity along a plane, to the force that presses
it against the plane. Thus, if a force of 30 Ibs. applied in a hori-
zontal direction is just sufficient to move a body of mass 100 Ibs.
slowly and with uniform velocity over a level surface, then the
coefficient of kinetic friction of that particular body upon that
particular surface is 30/100 or 0.3.

A very simple piece of apparatus for finding the coefficient of
friction is shown in Fig. 41. B is a board, say of oak, which
may be inclined at such an angle that the block C, say of walnut,
will slide slowly down the plane due to its weight. Let this angle
be 6. Resolving W, the weight of the block C, into two compo^.
nerits, one component Fi urging it along the plane, and the other
F2 pressing it against the plane, we have by definition F\/Fz as
the coefficient of friction. Fi/Fz, however, is also tan 6, hence
for this type of apparatus the

Coeff. of f riction =Fi/F» = tan 6.
From the figure it is seen that h/d is also tan 6; so that if in this

102

MECHANICS AND HEAT

particular case h/d = 1/3, the coefficient of friction for walnut on
oak is 0.33 for the particular specimens tested.

The coefficient of friction of metal on metal is, as a rule, some-
what greater than 0.2 for smooth, dry surfaces. Oiling may
reduce this to as low as 0.04.

If the coefficient of friction between the locomotive drivers and
the rail is 0.2, then the maximum pull, or "tractive effort," which
the locomotive can exert upon the drawbar, is about 0.2 of the
weight carried by the drivers. Any attempt to exceed this,
results in the familiar spinning of the drivers. For the same
reason, the maximum resistance to the motion of a car that can
be obtained by setting the brakes, is about 0.2 of the weight of
the car. Any attempt to exceed this force results in sliding, with
the production of the so-called "flat" wheel.

The Coefficient of Static Friction is defined as the ratio between
the force required to start a body to slide, and the force pressing it

FIG. 41.

against the plane. Since it requires a greater force to start
sliding than to maintain it, the coefficient of static friction is
larger than the coefficient of kinetic friction for the same materials.
The probable reason for this difference is the better interlocking
of the surfaces in the case of static friction (Sec. 78).

80. Rolling Friction. — It is a matter of common knowledge
that to draw a 1000-lb. sled, having steel runners, along a steel
track would require a much greater force than to draw a 1000-lb.
truck, having steel wheels, along the same track. In the former
case sliding friction must be overcome; in the latter case, rolling
friction. The fact that rolling friction is so much smaller than
sliding friction has led to the quite common use of ball bearings
in machinery. Thus the wheel of a bicycle or of an automobile
supports the axle by means of a train of very hard steel balls of
uniform size, which are free to roll round and round in a groove

WORK, ENERGY AND POWER 103

on the inside of the hub as the wheel turns. The axle rests with
a similar groove upon these balls and is thereby prevented from
direct rubbing (sliding friction) against the hub. Recent Ameri-
can practice favors rollers instead of balls for automobile "anti-
friction" bearings. By means of ball bearings, the coefficient
of friction, so-called, may be reduced to about 1/2 per cent.

In drawing the above truck on the steel track, the resistance
encountered is due to the fact that the steel wheel makes a. slight
depression in the rail, and is itself slightly flattened by the weight.
Since the material in the rail is not perfectly elastic, the minute
" hill " in front of the wheel is larger than the one behind it. The
wheel is constantly crushing down a small "hill" A in front of it
(shown greatly exaggerated in Fig. 42), and the energy required to
do this is always greater than the
energy applied by the small "hill"
B that is springing up behind it.

Since the thrust a, due to "hill"
A, is greater than the thrust 6, due
to B, the general upward thrust of
the rail against the wheel inclines
very slightly backward from the
vertical as shown. If the weight
W, and the pull F necessary to J?IQ 42

make the wheel roll, are both

known, the thrust T can easily be determined For, since
the wheel is in equilibrium, the three forces W} F, and T,
acting upon it must form a closed vector triangle. If, then, W
and F are drawn to scale as shown, the closing side T of the tri-
angle represents the required thrust. In the case of car wheels
on a steel track, F is about 1 per cent, of W, so that the angle 0
is really much smaller than shown. In the case of a rubber wheel
rolling on a steel rail, the depression of the rail would be prac-
tically zero; but in this case there would be a "bump" on the
wheel itself just in front of the flat portion, which would have to
be crushed down as the wheel advanced. To be sure, the spring-
ing out of the rubber "bump" just behind the flat portion would
help the wheel forward just as the rising of the minute hill on
the rail just behind the wheel would help it forward (in case the
rail is depressed). Since rubber is not perfectly elastic, the
energy required to crush the one "bump" is greater than that
obtained from the other formed by the rubber in springing out

104 MECHANICS AND HEAT

again behind the wheel. The difference between these two
amounts of energy is the energy used in overcoming rolling
friction.

If the wheel and the rail are made of very hard steel, friction is
reduced, because the depression made is less; but the danger of
accidents from the breaking of brittle rails is increased. In the
case of a wagon being drawn on the level along a soft spongy
road, the conditions are the same as those just discussed, except
that the "hill" is more marked in front of the wheel, and the ris-
ing of the hill behind the wheel is extremely sluggish indeed. For
this reason, rolling friction is a vastly greater factor in wagon
traffic than in railway traffic, and for the same reason, slight
grades, which would be prohibitive in railway traffic, are in wagon
traffic of small importance as compared with the character of the
road bed.

The friction upon the axle of the car is simply sliding friction,
but the amount of energy required to overcome it is very much
less than if the sliding were directly upon the rail itself, by
means of a shoe, for example. If the diameter of the axle is 1/10
that of the wheel, the distance of sliding between the axle and the
hub is clearly 1/10 the distance traversed by the car. Hence
we see that the work required to overcome this friction is only
1/10 as much as it would be if the sliding were directly upon the
rail, and if oil were sufficiently cheap to maintain as good lubrica-
tion between rail and shoe as is maintained on axles.

81. Power. — Power is denned as the rate of doing work; con-
sequently average power is the work done divided by the time
required to do the work, or, proper units being chosen,

P = W/t (55)

If the work done in t seconds is divided by t, the result is the work
done in one second. Hence power is numerically the work done
per unit time (usually the second). Thus if a man lifts a 50-lb.
weight to a height of 6 ft. in 2 sec., he does 300 ft.-lbs. of work.
Dividing this amount of work by the time required to do it
gives the power or 150 ft.-lbs. per sec. Also multiplying the
force, 50 Ibs., by the velocity, 3 ft. per sec., gives likewise 150
ft.-lbs. per sec. For, since distance d = vt, we have

WORK, ENERGY AND POWER 105

or power is equal to the force applied multiplied by the velocity of
motion of the body to which it is applied, provided the motion
is in the direction of the force. Thus, multiplying the pull on the
drawbar of a locomotive in pounds, by the velocity of the loco-
motive in feet per second, gives at any instant the power de-
veloped by the locomotive in foot-pounds per second.

82. Units of Power. — Since power is the rate of doing work, it
must be expressed in terms of work units and time units, e.g.,
ergs per second, foot-pounds per second, foot-pounds per minute,
etc. The horse power (H.P.) is one of the large power units in
common use.

1 H.P. = 550 ft.-lbs. per sec. =33,000 ft.-lbs. per min.

Since the pound force or pound weight increases with g, it
follows that the horse power becomes a larger unit with increase of
g. Strictly, the standard H.P. is 550 ft.-lbs. per sec. at latitude
45° (g = 980.6) . At latitude 60°, e.g., in central Sweden and Nor-
way, g is about 1/10 per cent, greater than at latitude 45°, so
that the H.P. there used is about 1/10 per cent, larger unit than
the standard H.P., unless corrected. Such correction is not made
in practice, because it is small in comparison with the fluctuations
in power that occur during a test of an engine or motor.

If a 140-lb. man ascends a stairway at the rate of 4 ft. (verti-
cally) per sec., the work done per second, i.e., the power he de-
velops, is 560 ft.-lbs. per sec., or slightly more than 1 H.P.

If a span of horses, pulling a loaded wagon weighing 2 tons up
a hill rising 1 ft. in 10, travels at the rate of 5 ft. per sec., then,
since the load rises 1/2 ft. per sec., the power developed by the
two horses in working against gravity alone is,

4000X0.5 ft.-lbs. per sec., or 3.63 H.P.

Considering also the work done against friction, it will be seen
that each horse would probably have to develop more than 2 H.P.
The above unit (550 ft.-lbs. per sec.) expresses the power which
a horse can develop for long periods of time, e.g., for a day. It
is^a rather high value for the average horse. On the other hand,
for very short periods (1/2 min. or so), a horse may develop 6 or
8 H.P. This accounts in part for the fact that a 30-H.P. auto-
mobile, stalled in the sand, may readily be drawn by a 4-horse
team. It may be mentioned in passing that the French H.P.
of 75 kilogram-meters per sec. is 541 ft.-lbs. per sec.

106 MECHANICS AND HEAT

Other units of power are the watt (one joule per sec.), and the
kilowatt (1000 watts). These units are used extensively in
expressing electrical power. The H.P. equals approximately 746
watts, or in round numbers, 3/4 kilowatt.

From Eq. 55 we see that work equals power times time. A
span of horses working at normal rate for ten hours does 20 H.P.-
hours of work. A good steam engine will do 1 H.P.-hour of
work for every 1.5 Ibs. of coal burned. If the lighting of a certain
building requires 2 kilowatts (K.W.) then the energy used in five
hours is 10 K.W.-hours. This energy is recorded by the watt-
hour meter, commonly called a recording wattmeter, and costs
usually about ten cents per K.W.-hour. A 32-candle-power
"carbon" lamp (i.e., a lamp whose filament is made of carbon)
requires about 100 watts, while a "tungsten" lamp having the
same candle power requires only about 40 watts. Observe
in this connection that it is not power that is bought or sold, but
energy, which is the product of the power and the time.

83. Prony Brake. — Various devices have been used to test the
power of steam engines and motors. With some of them the test
may be made while the engine is doing its regular work, while
others require that the regular work cease during the test. The
Prony Brake, in fact all brakes, are of the latter class, and are
known as absorption dynamometers. The former devices are
termed transmission dynamometers.

Since W =Td (Eq. 49), and $=ut,

P = W/t=T ut/t = Tu (57)

Hence to find the power of a motor, for example, it is merely nec-
essary to find what torque it exerts, and then multiply this by its
angular velocity co, or 2-irn, in which n is the number of revolu-
tions per second as determined by a speed indicator held against
the end of the motor shaft. A strap pressed against the pulley of
the motor shaft would be pulled in the direction of rotation with a
certain force F. If r is the radius of the pulley, then Fr gives
the torque of Eq. 57. Multiplying this torque by w, as above
found, would give the power of the motor in foot-pounds per
second, provided n is given in revolutions per second, F in pounds,
and r in feet. Dividing this result by 550 would then give the
power of the motor in H.P. If n were given in revolutions per
minute (R.P.M.), it would be necessary to divide by 33,000 in-
stead of by 550.

WORK, ENERGY AND POWER

107

A simple form of the Prony Brake, suitable for testing small
motors or engines, is shown in Fig. 43. The pulley A of the motor
shaft is clamped between two pieces of wood, B and C, as shown.
The end D of C is attached to a spring balance E. As the pulley
turns, it tends to rotate the brake with it, but is prevented by
the upward pull F exerted by E on D. The force, say FI, re-
quired to make the surface of the pulley slide past the wood,
times the radius r\ of the pulley, gives the driving torque F\TI
tending to rotate the brake in a clockwise direction. Since the
brake does not rotate, we see that the opposing torque, that is,
the above pull F times its lever arm r, or Fr, must equal the
torque Ftfi. Accordingly the former torque (Fr), which is easily
found, may be used in Eq. 57.

Fia. 43.

If B and C are lightly clamped together, this torque will be
very small, making the power small (Eq. 57); while if clamped
too tightly, the motor may be so greatly slowed down that the
power is again too small. The proper way to make the test is
to gradually tighten the clamp until the electrical instruments
show that the motor is using its rated amount of electrical power,
and then take simultaneous readings of E and the speed indicator.
From these readings the H.P. of the motor is found as above out-
lined. Likewise in testing a steam engine, the clamp should
be tightened until both the speed and the steam consumption are
normal.

In testing large engines or motors with the Prony Brake, D
rests on a platform scale, and pulley A, in some cases, has a rim
projecting inward which enables it to hold water when revolving,

108 MECHANICS AND HEAT

due to the centrifugal force thereby developed. Water applied
in this or some other way prevents undue heating. The clamp
also differs slightly from that shown.

A convenient form of brake for testing small motors is the
Strap Brake. A leather strap attached to one spring balance is
passed down around the motor pulley and then up and attached
to another spring balance. Evidently when the motor is running,
the two spring balances will register different forces. The
difference between these two forces multiplied by the radius of the
pulley, is the opposing torque. But this torque is equal to the
driving torque. This driving torque, multiplied by the angular
velocity w, gives the power (Eq. 57).

PROBLEMS

1. How much work is required to pump a tank full of water from a
40-ft. well, the tank being 10 ft. long, 5 ft. wide, and 8 ft. deep, and resting
upon a platform 20 ft. above the ground? The pipe enters at the bottom of
the tank. Assume that half of the work is done against friction, the other
half against the force of gravity. 1 cu. ft. of water weighs 62.4 Ibs. Sketch
first.

2. A horse drawing a sled exerts a pull of 120 Ibs. upon the sled at an
angle of 20° with the road bed. How much work is required to draw the
Bled 1/4 mile? Cos. 20° =0.94.

3. A 10-lb. force applied to an 18-in. crank turns it through 4000°.
How much work is done?

4. A plow that makes 12 furrow widths to the rod, i.e., which makes
16.5-in. furrows, requires an average pull of 300 Ibs. How much work,
expressed in ft .-Ibs., is done in plowing one acre?

6. What is the potential energy of a 20-kilogram mass when raised 3 ft.?
Express the result in ft.-lbs. and also in ergs.

6. What is the kinetic energy of a 200-lb. projectile when its velocity is
1600 ft. per sec.?

7. If a force of 1961.2 dynes causes an 8-gm. mass to slide slowly and
with uniform velocity over a level surface, what is the coefficient of fric-
tion?

8. A sled and rider, weighing 100 Ibs., reaches the foot of a hill 64 ft.
high with a velocity of 50 ft. per sec. How much work must have been
done against friction on the hill?

9. At the foot of the hill (Prob. 8) is a level expanse of ice. Neglecting
air friction, how far will the sled (vel. 50 ft. per sec.) travel on this ice
before coming to rest, assuming the coefficient of friction to be 0.03?

10. How much coal would be required per acre in plowing the land
(Prob. 4) with a steam plow? Assume that 6 Ibs. of the coal burned can
do 1 H.P.-hour of work, and that half of this work is done in pulling the
engine, and the other half in pulling the plow.

11. A 200-lb. car A and a 50-lb. car B when at rest on the same level

WORK, ENERGY AND POWER 109

track are connected by a stretched spring whose average tension for 3
seconds is 2 Ibs. greater than that necessary to overcome the friction of
running the cars. Find the momentum and the kinetic energy of each
car at the close of the 3-sec. interval.

12. What is the average H.P. developed by the powder, if the projectile
(Problem 6) takes 0.02 sec. to reach the muzzle, i.e., if the pressure pro-
duced by the powder acts upon the projectile for 0.02 sec.?

13. What is the average force pushing the projectile (Prob. 6) if the
cannon is 20 ft. in length?

14. A runaway team, pulling 200 Ibs., develops 10 H.P. How fast must
they travel?

15. How fast must a 400-lb. bear climb a tree in order to develop 2 H.P.?

16. What is the kinetic energy of a 3-ton flywheel when making 180
R.P.M., if the average diameter of its rim is 12 ft.? Assume the mass to
be all in the rim.

17. What is the cost of fuel for a locomotive for each ton of freight
that it hauls 1000 miles? Assume that the average pull per ton of the
loaded train is 30 Ibs., that the train itself weighs as much as its load, and
that the locomotive develops 1 H.P.-hr. from each 4 Ibs. of coal. The
coal costs $4.00 per ton.

18. A horse, drawing a sulky and occupant at the rate of 1 mile in 2
min., exerts a 10-lb. pull upon the sulky. How much more power must the
horse furnish than if it were to travel at the same rate without sulky or
rider?

19. A steam engine being tested with a Prony Brake makes 300 R.P.M.
and exerts at the end of the brake arm, 4 ft. from the axis, a force of 500
Ibs. Find its H.P.

20. Assuming that 20 per cent, of the energy can be utilized, how many
H.P. can be obtained from a 20-ft. waterfall in a river whose average width,
depth, and velocity at a certain point, are respectively 50 ft., 4 ft., and 5 ft.
per sec.?

21. It is desired to reduce the speed fluctuation between successive
explosions of the 10-H.P. gas engine (Sec. 76) to 1 per cent, of the average
speed. If the average radius of the rim of the flywheel is 3 ft., how heavy
must the flywheel be? Assume the mass to be all in the rim. Also assume
in Eq. 54 that coi is 1/2 per cent, greater than u, and that co2 is 1/2 per
cent, less than «.

CHAPTER VII
MACHINES

84. Machine Defined. — A machine is usually a device for
transmitting power, though it is sometimes (e.g., the dynamo) a
device for transforming one kind of energy into another. Many
machines are simply devices by means of which a force, applied at
one point, gives rise at some other point to a second force which,
in general, differs from the first force both in magnitude and direc-
tion. The force applied to the machine is called the Working
Force, and the force against which the machine works is called
the Resisting Force.

It is at once apparent that whatever power is required to over-
come friction in the machine itself, is power lost in transmission.
Nevertheless, transmission of power through the machine may be
profitable. Thus, in shelling corn with a corn sheller, the power
required to separate the kernels, to mutilate the cobs more or less,
and to overcome friction of the bearings, must be furnished by the
applied power; while if the corn were shelled directly by hand,
only the power required to separate the kernels would have to be
applied. Since power is force times velocity (Eq. 56), it is readily
seen that a person's hand can apply a great deal more power to a
crank than it can if pressed directly on the kernels. For both the
force and the velocity may easily be much greater in case the
crank is used. Again, though a block and tackle may transmit
only 60 per cent, of the applied power, it is profitable to use it in
lifting heavy masses that could not be lifted directly by hand. In
the case of the threshing machine, the power applied by the belt
from the steam engine is transmitted by the threshing machine to
the cylinder, to the blower, and to numerous other parts of the
machine.

We shall here study only what are known as the Simple
Machines. The most complicated machines consist almost
entirely in a grouping together of the various simple machines
described in the following sections. The study of the simple
machines consists mainly in learning the meaning of the efficiency

MACHINES

111

and the two mechanical advantages of each machine, and in find-
ing their numerical values from data given. Hence the necessity
for first having a clear definition of each of these three terms.

85. Mechanical Advantage and Efficiency. — The Actual Me-
chanical Advantage of a machine is the ratio of the resisting or
opposing force F0, to the force Fa applied to the machine, or

Act. Mech. Adv. = F0/Fa

The Theoretical Mechanical Advantage is the ratio of the dis-
tance d through which Fa acts, to the distance D through which
Fo acts, or

Theor. Mech. Adv. = d/D

The Efficiency (E) of a machine is the ratio of the useful work
W (i.e., F0D) done by the machine, to
the total work Wa (i.e., Fad) done upon
the machine, or

E =

Fad

(57a)

FIG. 44.

To illustrate the meaning of the
above terms, consider the common
windlass for drawing water from a
well (Fig. 44). Let the crank, whose
length (K) is 2 ft., rotate the drum of
6-in. radius (r) upon which winds the
rope that pulls up the bucket of water.
The hand, applying the force Fa
through the distance d, does the work
Fad upon the machine; while the

bucket, resisting with a force F0 (its weight) through a distance
D, has an amount of work FoD done upon it by the machine (the
windlass).

From inspection we see that, since R=4r, d must equal 4Z),
and the theoretical mechanical advantage is therefore 4. While
the theoretical mechanical advantage may be found from the
dimensions as here done, the actual mechanical advantage must
always be found from actual experiment. If the hand must apply
a 10-lb. force to lift a 30-lb. bucket, the actual mechanical advan-
tage is 3. If the hand applying this 10-lb. force moves 2 ft.,
the bucket would rise 6 inches or 1/2 ft., and the work done upon

112 MECHANICS AND HEAT

the machine would be 20 f t.-lbs. ; while that done by the machine
would be 15 f t.-lbs. (30X1/2). The efficiency (Eq. 57 a) would
then be 15/20, or 75 per cent.

Observe that the efficiency is also equal to the ratio of the two
mechanical advantages, the actual to the theoretical. This is
always true. For, since there is friction, the work done by the
machine is less than that done upon it; i.e., the efficiency
F0D/Fad, or E, is less than one. F0D/Fad = E may be put in the
form

F0/Fa = EXd/D '|5£ (58)

The left member of this equation is the actual mechanical advan-
tage, while the right member is E times the theoretical mechanical
advantage (note that E is never more than unity) ; whence the
efficiency E is the ratio of the two mechanical advantages, which
was to be proved. If it were possible to entirely eliminate
friction, then the work done "upon" and "by" the machine would
be equal (from the conservation of energy), and therefore E would
be unity. Consequently the efficiency would be 100 per cent., and
the theoretical mechanical advantage d/D would be equal to
the actual mechanical advantage F0/Fa. In other words, the
theoretical mechanical advantage is the ratio that we would find
for F0/Fa from the dimensions of the machine, neglecting friction.
This condition of zero friction is closely approximated in some
machines.

86. The Simple Machines. — The Simple Machines are devices
used, as a rule, to secure a large force by the application of a
smaller force. These machines are the lever, the pulley, the
wheel and axle, the inclined plane, the wedge, and the screw.
Throughout the discussion of the simple machines the symbols
Fa, F0, d, and D will be employed in the same sense as in Sec. 85.
It may be well to now reread the last three sentences of Sec. 84.
Observe that the theoretical mechanical advantage of any simple
machine, or any combination of simple machines for that matter,
is d/D. Thus, if in the use of any combination of levers and pul-
leys, it is observed that the hand must move 20 ft. to raise the load
1 ft., we know at once that the theoretical mechanical advantage
is 20.

87. The Lever. — The lever is a very important and much used
simple machine. Indeed, as will be shown later, all simple ma-
chines may be divided into two types: the lever type and the

MACHINES 113

inclined-plane type. Though the lever is usually a straight bar
free to rotate about a support P, called the fulcrum or pivot point,
it may take any form. Thus a bar bent at right angles and
having the pivot at the angle as shown at N(Fig. 45), is a form of
lever that is very widely used for changing a vertical motion or
force to a horizontal one and vice versa.

There are three general classes of levers, sometimes called 1st
class, 2nd class, and 3rd class, depending upon the relative posi-
tions of the fulcrum or pivot P, and the points A and B, at which
are applied Fa and F0 respectively (see Fig. 45). In the class
shown at K, P is between the other two points; in the class
shown at L, F0 is between; and in the class shown at M, Fa is
between. In all three cases, the applied torque about P is
FaXAP, and, since the lever is in equilibrium (neglecting its

P^ ,~-.~-yP Fg\}d ~~B~7D~ P

FIG. 45.

weight and also neglecting friction), this torque must equal the
opposing torque due to F0, or F0XBP. Hence FaXAP =
F0XBP, from which, noting that for zero friction the two me-
chanical advantages are equal (see close of Sec. 85), we have

F AP
Theor. Mech. Adv. = F° = gp (59)

The theoretical mechanical advantage may be found in another
way. Let the force Fa move point A a distance d (all three classes) .
The point B will then move a distance D, and from similar trian-
gles the theoretical mechanical advantage d/D is seen to be equal
to AP/BP, just as in Eq. 59. By measuring AP and BP, the
theoretical mechanical advantage is known. Thus if in any case
AP equals 3XBP, it is known at once and without testing, that,

114 MECHANICS AND HEAT

neglecting friction, 10 Ibs. applied at A will lift 30 Ibs. resting at
B. Friction in levers is small, so that the actual mechanical
advantage is almost equal to the theoretical, and the efficiency
is therefore nearly 100 per cent.

Obviously, in using a crowbar to tear down a building, the
resisting force F0 is not in general a weight or load. Nevertheless,
since the simple machines are very commonly used in raising
weights, it has become customary to speak of F0 as the "load,"
or the weight lifted, and Fa as the "force," although both are
of course forces. "Resistance" seems preferable to "load"
and we shall call BP (for all three classes) the "resistance arm,"
and AP the "force arm." The latter is sometimes called the
"power arm," but this seems objectionable inasmuch as we are
dealing with force, not power.

From the figure, it will be seen that the force arm may be either
equal to, greater than, or less than the resistance arm in levers of
the type shown at K; while in the type shown at L, it is either
equal to, or greater than the resistance arm; and in the type shown
at M , it is either equal to, or less than, the resistance arm. Conse-
quently the theoretical mechanical advantage (AP/BP) may
have for the first-mentioned type (K) any value; for the next type
(L), one or more than one; and for the last type (AT), its value
is one or less than one. Observe that the theoretical mechanical
advantage is always given by the ratio of the force arm to the
resistance arm (AP/BP}, whatever the type of lever may be.
The lever arm of a force is always measured from the pivot point.

The crowbar, in prying up a stone, may be used as a lever
either as shown at K or at L. A fish pole is used as a lever of the
type shown at M, if the hand holding the large end of the pole
remains at rest, while the other hand moves up or down. A
pump handle is usually a lever of the type shown at K. The
forearm is used as a lever of type M when bending the arm, and
type K when straightening it. A pair of scissors, a pair of nut-
crackers, and a pair of tweezers represent, respectively, classes
K, L, and M .

88. The Pulley. — The theoretical mechanical advantage of
the pulley when used as shown in Fig. 46 is unity. For evi-
dently Fa must equal F0 (neglecting friction) in order to make the
two torques equal. But the theoretical mechanical advantage,
if we neglect friction, is F0/Fa (see last three sentences of Sec.
85). From an actual test in raising a load, it will be found that

MACHINES

115

Fa exceeds F0, hence the actual mechanical advantage is less
than one. Again, if Fa moves its rope downward a distance d,
the weight W will rise an equal distance D, and d/D, or the theo-
retical mechanical advantage, from this viewpoint is also seen
to be one.

Such a pulley does not move up or down, and is called a fixed
pulley. Observe that this pulley may be looked upon as a lever
of the class shown at K (Fig. 4.5) with equal arms r and r' . Al-
though with such a pulley F0 is less than the applied force Fa, the
greater ease of pulling downward instead of upward more than
compensates for the loss of force.

The movable pulley is shown in Fig. 47. With this arrangement

FIG. 46.

FIG. 47.

the pulley rises with the lifted weight. Since both ropes A and B
must be equally tight (ignoring friction), F0 = 2Fa, or F0/Fa, the
theoretical mechanical advantage, is 2. This may be seen in
another way by considering point C as the fulcrum for an instant,
and 2r as the lever arm for Fa, and only r as the lever arm for
F0. It is also evident that if rope B is pulled up 1 ft. the weight
W will rise only 1/2 ft., i.e., d/D, the theoretical mechanical advan-
tage, is 2.

A group of several fixed and movable pulleys arranged as
shown in Fig. 48 with a rope passing over each pulley is called a
Block and Tackle. In practice, the pulleys A and B are placed
side by side on the same axle above; in like manner C and D are

116

MECHANICS AND HEAT

placed on one axle below. The slightly different arrangement
shown in the sketch is for the purpose of showing more clearly
the separate parts of the rope. The rope abcde is continuous,
one end being attached to the ring E and the other end being
held by the hand.

If the applied force Fa on rope a is say 10 Ibs., and the pulleys
are absolutely frictionless, then the parts of the rope b, c, d, and
e would all be equally tight, and hence each would exert an up-
ward lift on W of 10 Ibs., giving a total of 40 Ibs.
The theoretical mechanical advantage is then
(neglecting friction), F0/Fa = 40/W = 4, or the
number of supporting ropes. Again, if W is
raised 1 ft. (D), each rope b, c, d, and e will have
1 ft. of slack, so that a will have to be pulled
down a distance 4 ft. (d) to take up all of the
slack. In other words, the hand must move 4
ft. to raise W 1 ft. Hence the theoretical mechani-
cal advantage from this viewpoint is 4 (i.e.
d/D=4). Observe that here, with a theoretical
mechanical advantage of 4, the weight moves
1/4 as far, and hence 1/4 as fast as the hand.
This general fact concerning simple machines is
epitomized in the following statement: "What is
gained in force is lost in speed, and vice versa."

If friction causes each pulley A, B, C, and D to
require 1 Ib. pull to make it revolve, then if the
pull applied to a were 10 Ibs., the tension on 6
would be only 9 Ibs. ; on c, 8 Ibs. ; on d, 7 Ibs. ; and
on e, 6 Ibs. The total lift exerted on W, i.e., F0,
would therefore be 9+8+7+6, or 30 Ibs.; hence
the actual mechanical advantage F0/Fa would be 3.
Since the efficiency is the ratio of the actual to the
theoretical mechanical advantage, it is here 3/4, or
75 per cent. The efficiency may readily be found
in another way. If the hand moves downward a distance of 4 ft.
while exerting a force of 10 Ibs., then the work done upon the ma-
chine is 40 ft.-lbs., but it has been shown that, due to friction, this
force can raise only 30 Ibs. one ft., i.e., the work done by the ma-
chine is only 30 ft.-lbs. The efficiency is then 4Q . ' , — =75 per
cent, as above. A considerably higher efficiency than this may

FIG. 48.

MACHINES

117

be obtained if the rope is very flexible, and if the pulley bearings
are smooth and well oiled.

89. The Wheel and Axle.— The Wheel and Axle (Fig. 49) con-
sists of a large wheel A of radius R rigidly attached to an axle B
of radius r. A rope a is attached to the rim of the wheel and
wound around it a few turns. Another rope, attached to the axle,
is secured to the weight W that is to be lifted.

Viewed as a lever with the axis as pivot, the theoretical mechan-
ical advantage is clearly the ratio of the two lever arms, or R/r.
If this ratio is, say 5, the rope a will have
to be pulled down a distance (d) of 5 ft.
to lift the weight a distance (D) of 1 ft.,
giving a theoretical mechanical advantage
(d/D) of 5. If from a test, the load lifted
is only 4 times as great as the applied force,
then the actual mechanical advantage is 4,
and the efficiency (by Eq. 58) is 4/5 or 80
per cent.

Observe that the wheel and axle and the
windlass (Fig. 44) are exactly alike in prin-
ciple. It may also be added that practi-
cally the only difference between the cap-
stan and the windlass is that the drum is
vertical in the capstan and horizontal in the windlass.

90. The Inclined Plane. — Let a rope, pulling with a force Fa,
draw the block E of weight W up the Inclined Plane AC (Fig. 50).
Resolving W into two components (Sec. 19), the one (Fi) normal,
the other (Fz) parallel to the plane, and noting that Fa equals
Fz (if we ignore friction), we have for the theoretical mechanical
advantage

= I/sin 8

Fia. 49.

Again, if Fa draws the block from A to C, it lifts the block only
the vertical height BC, and the theoretical mechanical advantage,
d/D, is AC/BC, or I/sin 6, as before. Observe that

AC slant height
-- -

mi_ TVT
Theo. M.

The less steep the grade, the greater the theoretical mechanical
advantage, but the block must be drawn so much the farther in
order to raise it a given vertical distance.

118

MECHANICS AND HEAT

If the pull Fa urging the block up the incline, is horizontal,
then, as the block travels from A to C (Fig. 51), Fa acts in its
own direction through distance AB (i.e., d) and the weight W is
raised the distance BC (i.e., D). Hence in this case

Theo. M. Adv. =

hor. distance

r^-r = cot0=1/tan0

BC vert, height

The equation just given may be derived in another way.
From Fig. 51 we see that the pull on the rope, or F'a, must be of
such magnitude that its component Fa parallel to AC shall equal
the force Fa of Fig. 50. Drawing F3 equal to F'a but in the oppo-
site direction, we have

Theo. M. Adv. ----

AB
coie=BC

The inclined plane is frequently used for raising wagon loads
and car loads of material, for example, at locomotive coaling

FIG. 50.

FIG. 51.

stations, and for many other purposes. A train in ascending a
mountain utilizes the inclined plane, by winding this way and that
to avoid too steep an incline. On a grade rising 1 ft. in 50, the
locomotive must exert upon the drawbar a pull equal to 1/50 part
of the weight of the train in addition to the force required to over-
come friction.

91. The Wedge. — In Fig. 52 the wedge is shown as used in
raising the corner of a building. Fa represents the force exerted
upon the head of the wedge by the hammer, and F0 the weight of
the corner of the building. If Fa acts through the distance d
(the length of the wedge), i.e., if Fa drives the wedge "home,"
then the building will be lifted a distance D (the thickness of
the wedge), and F0 will resist through a distance D. Hence

Theo. M. Adv. =

length of

D thickness of wedge

MACHINES

119

If the hand exerts a force of 20 Ibs. upon a sledge hammer
through a distance of 40 inches, and the hammer drives the wedge
1 inch, i.e., Fa acts through 1 inch, then Fa (average value) equals
20X40 or 800 Ibs. For, in accordance with the conservation of
energy, the work done (force times distance) in giving the ham-
mer its motion must be equal to the work it does upon the wedge,
and, since the distance the wedge moves in stopping the hammer is
1/40 as great as the distance the hand moves in starting it, the
force involved must be 40 times as great, or 800 Ibs. as already
found. If the wedge is 1 in. thick and 8 in. long it could, neglect-
ing friction, lift 8X800 or 6400 Ibs. In practice, friction is very
great in the case of the wedge, so that the weight lifted would be
very much less than 6400 Ibs., say 1600 Ibs. Accordingly, if the
weight resting upon this particular wedge were 1600 Ibs., then

FIG. 52.

each blow of the hammer would drive the wedge 1 inch and raise
the building 1/8 in.

The actual mechanical advantage of the wedge would then be
1600 Ibs. -f- 800 Ibs. or 2, the theoretical mechanical advantage
8 in.-i-l in. or 8, and consequently the efficiency would be 2-r-8
or 25 per cent. For wedge and hammer combined, the actual
mechanical advantage would be 1600 Ibs. -7-20 Ibs. or 80, and
the theoretical mechanical advantage, 40 in. -^ 1/8 in., or 320.
Observe that the latter ratio (320) is the distance that the hand
(not the wedge) moves, divided by the distance that the building
is raised. Thus we see that the great value of the mechanical
advantage is due to the great force developed in suddenly stopping
the hammer when it strikes the wedge, rather than to the wedge

120

MECHANICS AND HEAT

itself. A wedge would be of little or no value, if used directly,
that is, if pushed "home" by the hand.

If the weight on the wedge were 5 times as great (5 X 1600 Ibs.)
it would require 5 times as much force to drive it, and the hammer
would be stopped more suddenly in furnishing this force. In
fact, the same blow would drive the wedge 1/5 as far as before, or
1/5 inch.

92. The Screw. — The screw consists of a rod, usually of metal,
having upon its surface a uniform spiral groove and ridge, the
thread. It is a simple device by which a torque may develop a
very great force in the direction of the length of the screw. For

example, by using a wrench to
turn the nut on a bolt which
passes through two beams, the
bolt draws the two beams for-
cibly together. The principle of
the screw will be readily under-
stood from a discussion of the
jackscrew, a device much used
for exerting very great forces,
such as in raising buildings.

The Jackscrew (Fig. 53) con-
sists of a screw S, free to turn in
a threaded hole in the base A,
and having at its upper end a
hole through which the rod BC

may be thrust as shown. Consider a force Fa applied at C at
right angles to the paper and directed inward (i.e., away from the
reader). Let it be required to find the weight F0 that the head
of the jackscrew will lift. The distance which the screws rise for
each revolution is called the pitch p of the screw. Evidently
for each revolution of the point C, the weight lifted, i.e., the
corner of the building, rises a distance p. In doing this, however,
the force Fa applied to C acts through a distance 2nr. Hence

o_r
Theo. M. Adv. = (d/D) = —

In the jackscrew, friction is large, consequently the actual
mechanical advantage is much less than the theoretical. The
actual mechanical advantage would be found by dividing the

FIG. 53.

MACHINES

121

weight of the corner of the building (i.e., F0) by the force Fa
necessary to make C revolve.

Both the wedge and the jackscrew involve the principle of the
inclined plane. This is obvious in the case of the wedge. In the
case of the jackscrew, the thread in the base is really a spiral
inclined plane up which the load virtually slides. The long rod
BC makes the mechanical advantage much greater than it is for
the inclined plane. Observe that all other simple machines
involve the principle of the lever. Thus there are two types of
simple machines, the inclined-plane type and the lever type.

93. The Chain Hoist or Differential Pulley. — The Chain
Hoist or Differential Pulley (Fig. 54) is a very convenient and

FIG. 54.

FIG. 55.

simple device for lifting heavy machinery or other heavy objects.
It consists of three pulleys A, B, and C, connected by an endless
chain of which the portions c and e bear the weight and a and 6
hang loose. The two upper pulleys A and B, which differ slightly
in radius, are rigidly fastened together, and each has cogs which
mesh with the links of the chain. Designating the radius of A
by r and that of B by r', let us find the expression for the theo-
retical mechanical advantage.

122 MECHANICS AND HEAT

Evidently if rope a is pulled down by Fa a distance 2irr (i.e., d),
A will make one revolution, e will be wound upon A a distance
2irr, and c will be unwound from B a distance 27rr'. Now the
latter distance is slightly smaller than the former, so that the
total length of e and c is shortened, causing pulley C, and conse-
quently the load W, to rise the distance D. The above shortening
is2irr — 2-nrr, or27r(r — r'), and Crises only 1/2 this distance. Hence

Theo. M. Adv. = d/D = ^^ = ^75 (60)

Eq. 60 shows that if r and rf are made nearly equal, then D
becomes very small and the mechanical advantage, very large.
In practice, a ratio of 9 to 10 works very well, i.e., having, for
example, 18 cogs on B and 20 on A. In such case, the above-
mentioned shortening would be two links per revolution (i.e., per
20 links of pull), and the rise D would be one link, giving a theo-
retical mechanical advantage of 20/1 or 20.

In the chain hoist there is sufficient friction to hold the load
even though the hand releases chain a. This is a great conven-
ience and safeguard in handling valuable machinery. Likewise
in the case of either the wedge or the jackscrew, friction is great
enough to enable the machine to support the load though the
applied force Fa is withdrawn. This convenience compensates
for the low efficiency which, we have seen, is the direct result
of a large amount of friction.

The Differential Wheel and Axle is very similar in principle to the chain
hoist. It differs from the wheel and axle shown in Fig. 49, in that the
axle has a larger radius at one end than at the other.

If the force Fa (Fig. 55) pulls rope a downward a distance (d) of
2irR (R being the radius of the large wheel), then, exactly as in the
chain hoist, rope e is wound onto the large part of the axle a distance
2irr and rope c is unwound from the smaller part of the axle a distance
27i-r'. The shortening of ropes c and e is 2*r— 2*r' or 2w(r— r'), and the
weight rises a distance (D) equal to 1/2 of this distance, or ir(r—r').

We thus have

rrtl. -n/r A 1 d 2irR 2R /ni\

Theo. M. Adv. = D=ir(r-r^=r^ (61)

94. Center of Gravity.— The Center of Gravity (C.G.) of a
body may be defined as that point at which the entire weight of
the body may be considered to be concentrated, so far as the torque
developed by its weight is concerned. This is equivalent to the

MACHINES

123

statement that the C.G. of a body is the point at which the body
may be supported in any position without tending to rotate due
to its weight. For its entire weight acts at its C.G., and hence,
under these circumstances, at its point of support, and therefore
develops no torque. The conditions that obtain when a body is
supported at its C.G. will now be discussed.

Let Fig. 56 represent a board whose C.G. is at X. Bore a small
hole at X and insert a rod as an axis. Through X pass a vertical
plane at right angles to the plane of the paper as indicated by the
line AX. Now the positive torque due to a mass particle mi
is its weight m\g times its lever arm r\. Proceeding in the same
way with m* and all other particles
to the left of the line AX, and adding
all of these minute torques, we ob-
tain the total positive torque about X.
In the same way we find the total
negative torque about the same point
due to m3, w4, etc. Since the body
balances if supported at X, the total
positive torque must equal the total
negative torque, and for this reason,
the entire weight behaves as a single
downward pull W acting at its C.G.
This concept greatly simplifies all dis-
cussions and problems relating to the

C.G. of bodies, and will be frequently used. For example, if
the rod is withdrawn from X and inserted at A, we see at
once that the downward pull W, and the reacting upward pull
of the supporting rod, will produce no torque, since they lie
in the same straight line. If, however, the rod is inserted at B,
the negative torque would be TFr, in which r is the horizontal
distance between X and B. If free to do so, the board would
rotate until B and X were in the same vertical line. In other
words, a body always tends to rotate so that its C.G. is directly
below the point of support.

This tendency suggests a very simple means of finding the C.G.
of an irregular body, such as C (Fig. 56) . Supporting the body at
some point as D, determine the plumb line (shown dotted).
Next, supporting it at E, determine another plumb line. The
intersection X of these two lines is the C.G. of the body. Why?

Effect of C.G. on Levers. — If the center of gravity of the lever

FIG. 56.

124 MECHANICS AND HEAT

AB (sketch K, Fig. 45) of weight W, is to the left of P a distance r,
then the weight of the lever produces a positive torque Wr,
which torque added to that due to Fa, which is also positive, must
equal that due to F0, which is negative. Thus we see that in
ignoring the weight of the lever we introduce into Eq. 59 a slight
error. This error is negligibly small in the case of lifting a heavy
load with a light lever. In any given case it can be seen at a
glance whether this torque due to the weight of the lever helps or
opposes Fa, remembering that all torques should be computed
from the fulcrum P.

95. Center of Mass. — The center of mass (C.M.) of any body is
ordinarily almost absolutely coincident with its C.G. Indeed the two
terms are frequently used interchangeably. That the two points may
differ widely under some circumstances, may be seen by considering two
bodies of equal mass, one on the surface of the earth, the other 1000
miles above the surface. Since the two masses are equal, their common
center of mass would be half way between the bodies, or 500 miles above
the earth. Although the two masses are equal, the weight of the lower
body would be roughly 3/2 times that of the upper one (inverse square
law), and the center of gravity of the two, which is really the "center
of weight," would be nearer the lower body. In fact, since the weight
of the lower body is 3/2 times that of the upper one, its '-lever arm,"
measured from it to the C.G., would be 2/3 as great as for the upper
body. The C.G. would therefore be 400 miles above the earth, or 100
miles lower than the center of mass. As a rule, however, the C.M. of a
body is practically coincident with its C.G.

Center of Population. — The center of population of a country is very
closely analogous to the center of mass of a body, and is also a matter of
sufficient interest to warrant a brief discussion. To simplify the discus-
sion, let us use an illustration. Suppose that we have found that the
center of population of the cities (only) of the United States is at Cin-
cinnati. Through Cincinnati draw a north and south line A, and an
east and west line B. Now multiply the population of each city east of
line A by its distance from A and find the sum of these products. Call
this sum Si. Next find the similar sum, say $2, for all cities west of A.
Then 81 = 82. Proceed in exactly the same way for all cities north of
line B, obtaining S3; and finally for all cities south of B, obtaining 84-

It may be of interest to know that the center of population of the
United States, counting all inhabitants of both city and country, was
very close to Washington, D. C., in 1800. It has moved steadily west-
ward, keeping close to the 39th parallel of latitude, until in 1900 it was
in Indiana at a point almost directly south of Indianapolis and west of
Cincinnati.

MACHINES 125

The mass particles of a body bear the same relation to its center of
mass as does the population of the various cities to the center of popu-
lation of them all. The subject is further complicated, however, by the
fact that we are dealing with three dimensions in the case of a solid
body, so that the distances must be measured from three intersecting
planes (compare the corner of a box) instead of from two intersecting
lines.

If a rod of negligible weight connecting a 4-lb. ball M and a 1-lb.
ball m (Fig. 57) receives a blow Fa at a point 1/5 of its length from the
larger ball, which point is the center of mass, it will be given motion of
translation, but no rotation. For, since the two
"lever arms" (distance from ball to C.M.) are inver- °~ ~ ,
sely proportional to their respective masses, the balls,
due to their inertia, produce equal (but opposing)
torques about their common C.M. when experiencing
equal accelerations. But if the balls experience equal
acceleration, the rod does not rotate. If such a
body were thrown, the two balls would revolve about
their common center of mass, which point would ,
trace a smooth curve. We may extend this idea to $0= —i
any body of any form. That is to say, any free body
is not caused to rotate by a force directed toward (or FIG. 57.
away from) its center of mass.

Let us again look at the problem in a slightly different way: Evi-
dently the two torques about the point (C.M.) which receives the blow
(Fa) must be equal and opposite. These torques are produced by the
inertia forces F0 and F'0 which M and m, respectively, develop in oppos-
ing acceleration. Since F'0 acts upon 4 times as long a lever arm (meas-
ured from C.M.) as does F0, it must be 1/4 as large as that force to
produce an equal torque, and it will therefore impart to the 1-lb. mass
m an acceleration exactly equal to that imparted by F0 to the 4-lb. mass
M. If, however, the balls experience equal accelerations the rod will
not rotate.

The mass of the earth is about 80 times that of the moon, so that the
moon's "lever arm" (about their common C.M.) is 80 times as long
as that of the earth, and the C.M. of the two bodies is therefore at a
point 1/81 of the distance between them (about 3000 mi.), measured
from the earth's center toward the moon. Since the radius of the earth
is about 4000 miles, we see that the C.M. of the earth and the moon is
about 1000 miles below the surface of the earth on the side toward the
moon. This point travels once a year around the sun in a smooth
elliptical path; while the earth and the moon, revolving about it (the
C.M.), have very complicated irregular paths.

96. Stable, Unstable, and Neutral Equilibrium.— The Equi-
librium of a body is Stahk if a slight rotation in any direction

126 MECHANICS AND HEAT

raises its center of gravity; Unstable if such rotation lowers its
C.G.; and Neutral if it neither raises nor lowers it. The cone,
placed on a level plane, beautifully illustrates these three kinds
of equilibrium.

When standing upon its base, the cone represents stable
equilibrium, for tipping it in any direction must raise its C.G.
To overturn it with A as pivot point, its C.G. must rise a distance
h as shown (Fig. 58), and the work required (in foot-pounds)
A would be the entire weight of the

cone in pounds times h in feet, since
its weight may be considered to be
concentrated at its C.G. If the cone
'is inverted and balanced upon its
apex, its equilibrium is unstable; for
the least displacement in any di-
rection would lower its center of
piG g8 gravity and it would fall. Finally,

the cone (also the cylinder) lying

on its side is in neutral equilibrium, for rolling it about on a
level plane neither raises nor lowers its C.G.

The equilibrium of a rocking chair is stable if the C.G. of the
chair and occupant is below the center of curvature of the rockers.
For in such case rocking either forward or backward raises the
C.G. Accordingly a chair with sharply curved rockers is very apt
to upset, since the center of curvature is then low. To guard
against this, a short portion of the back end of the rockers is
usually made straight, or better still, given a slight reverse
curvature.

Equilibrium on an Inclined Plane. — To avoid circumlocution in
the present discussion let us coin the phrase "Line of Centers"
to indicate the plumb line through the C.G. of a body. If the
plane (Fig. 58) is inclined, the cone will be in stable equilibrium
so long as the line of centers falls within its base. The instant the
plane is tipped sufficiently to cause the line of centers to fall
without its base, the cone overturns.

A loaded wagon on a hillside is in stable equilibrium so long
as the line of centers (Fig. 59) falls within the wheel base.
Because of lurching caused by the uneven road bed, it is unsafe
to approach very closely to this theoretical limit. A load of hay
is more apt to upset on a hillside than is a load of coal, for two
reasons. The C.G. is higher than in the case of the coal, and

MACHINES

127

also the yielding of the hay causes the C.G. to shift toward the
lower side, as from C to D, so that the line of centers becomes
DE (Fig. 59).

If the line of centers falls well within the base, a body is not
easily upset, whether on an incline or on a level surface. Manu-
facturers recognize this fact in making broad bases for vases,
lamps, portable machines, etc. Ballast is placed deep in the
hold of a ship in order to lower its C.G. and thereby make it more
stable in a rough sea.

FIG. 59.

97. Weighing Machines. — The weighing of a body is the
process of comparing the pull of the earth upon that body with
the pull of the earth upon a standard mass, e.g., the kilogram or
the pound, or some fraction of these, as the gram or the ounce,
etc. This comparison is not made directly with the pull of the
earth upon the standard kilogram mass kept at Paris, or with the
standard pound mass kept at London, but with more or less ac-
curate copies of these, which may be called secondary standards.
We shall here discuss briefly the beam balance, steelyard, spring
balance, and platform scale. Each of these weighing devices,
except the spring balance, consists essentially of one or more
levers, and in the discussion of each a thorough understanding
of the lever will be presupposed.

The Beam Balance consists essentially of a horizontal lever or
beam, resting at its middle point on a "knife-edge" pivot of
agate or steel, and supporting a scalepan at each end, also on

128 MECHANICS AND HEAT

knife-edges. Usually a vertical pointer is rigidly attached to the
beam. The lower end of the pointer, moving over a scale, serves
to indicate whether the load in one scalepan is slightly greater
than that in the other. The body to be weighed is placed,
say, in the left pan, and enough standard masses from a set of
"weights" are placed in the right pan to "balance" it. If too
much weight is placed in the right pan, the right end of the beam
will dip. Obviously if the balance is sensitive, a very slight excess
weight will produce sufficient dip, and consequently sufficient
motion of the pointer to be detected. The Sensitiveness of the
balance depends upon two factors, the position of the C.G. of

the beam and pointer, and
the relative positions of the
three knife-edges.

These factors will now be
discussed in connection with
Fig. 60, which is an exagger-
ated diagrammatic sketch of
the beam and pointer only.

If the C.G. of the beam and
GO. pointer is far below the cen-

tral knife-edge as shown, then

a slight dip of the right end of the beam will cause the C.G. to
move to the left a comparatively large distance r, and there-
fore give rise to a rather large opposing restoring torque equal
to the weight W of the beam and pointer times its lever arm r,
or a torque Wr. Observe, as stated in Sec. 94, that so far as the
torque due to the weight of the beam and pointer is concerned,
their entire weight may be considered to be at their C.G. From
the figure, we see that if the C.G. were only 1/2 as far below the
knife-edge, then r would be 1/2 as great, and 1/2 as great ex-
cess weight in the right pan would, as far as this factor is con-
cerned, produce the same dip, and hence the same deflection
of the pointer as before. Accordingly, a sensitive balance is de-
signed so that the C.G. is a very short distance below the cen-
tral knife-edge, and the smaller this distance, the more sensitive
the balance.

Let us now consider the second factor in determining the sensi-
tiveness of a beam balance. If the end knife-edges are much
lower than the middle one, as in the figure, then the slight dip
shortens the lever arm r\ upon which the right pan acts by an

MACHINES 129

amount a while at the same time the length of r2 is very slightly
increased. Consequently, under these circumstances, a compara-
tively large restoring torque arises, and therefore a comparatively
large excess weight in the right pan will be required to produce a
perceptible dip of the beam or deflection of the pointer. Hence
sensitive balances have the three knife-edges in a straight line, or
very nearly so.

We shall now slightly digress in observing that if the three
knife-edges represent in position the three holes in a two-horse
" evener," and if the horse at each end of the evener be represented
in the figure as pulling downward, then the "ambitious" horse
would have the greater load, for, as just pointed out, the lower
end has the shorter arm. If the horses are represented as
pulling upward in the figure, then the horse that is ahead pulls
on the longer lever arm and hence has the lighter load. This is

i 1 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 — >.

b / 15 A 25 3JO )

FIG. 01.

the usual condition; since the middle hole in the evener is usually
slightly farther forward than the end ones.

The Steelyard consists of a metal bar A (Fig. 61), supported in
a horizontal position on the knife-edge B near the heavy end, and
provided with a sliding weight S, and a hook hanging on knife-
edge C for supporting the load W to be weighed. The supporting
hook // is frequently simply held in the hand. In weighing, the
slider is moved farther out, thus increasing its lever arm, until it
"balances" the load. The weight of the load is then read from
the position of the slider on the scale.

The scale may be determined as follows: Remove W and slide
£ back and forth until a "balance" is secured. Mark this posi-
tion of the slider as the zero of the scale. Next put in the place of
W a mass of known weight, say 10 Ibs., and when a balance is
again secured mark the new position of the slider "10 Ibs." Lay
off the distance between these two positions into ten equal spaces
and subdivide as desired the pound divisions thus formed. The

130 MECHANICS AND HEAT

pound divisions should all be of the same length. For, if moving
S one division to the right enables it to balance 1 Ib. more at W,
then moving it twice as far would double the additional torque due
to S, and hence enable it to balance 2 Ibs. more at W. The same
scale may be extended to the right end of the bar.

The steelyard is made more sensitive by having its C.G. a
very small distance below the supporting knife-edge B, for reasons
already explained in the discussion of the beam balance. This is
accomplished by having the heavy end of the bar bent slightly
upward, thereby raising its C.G.

The Spring Balance consists essentially of a spiral steel spring,
having at its lower end a hook for holding the load to be weighed.
Near the lower end of the spring a small index moves past a scale,
and indicates by its position the weight of the load. Since the
spring obeys Hooke's Law (Sec. 107), that is, since its elongation
is directly proportional to the load, a scale of equal divisions is
used just as with the steelyard.

The Platform Scale. — In the platform scale two results must
be accomplished; first, a small "weight" must "balance" the
load of several tons; and second, the condition of balance must
not depend upon what part of the platform the load is placed.
The first result is accomplished by the use of the Compound
Lever. A Compound Lever consists of a combination of two or
more levers so connected that one lever is actuated by a second,
the second by a third, and so on. It is easily seen that the
mechanical advantage of a compound lever is equal to the product
of the mechanical advantages of its component levers taken
separately. Thus, if there are three component levers whose
mechanical advantages are respectively x, y, and z, then the
mechanical advantage of the compound lever formed by combin-
ing them is xyz. The second result, namely, the independence
of the position of the load on the platform, is attained by so ar-
ranging the levers that the mechanical advantage is the same for
all four corners, and therefore for all points of the platform.

The system of levers (only) of a common type of platform scale
is shown in Fig. 62 as viewed cornerwise from an elevated posi-
tion. The four levers EA, ED, FB and GC are beneath the plat-
form (indicated by dotted lines). These levers are supported
by the foundation on the knife-edges A, B, C, D, and they, in
turn, support the platform on the knife-edges A', B' , C', and D'.
The point E is connected by means of the vertical rod EH with

MACHINES 131

the horizontal lever U, which lever is supported at 7 as indicated.
Finally, J is connected by means of the vertical rod JK with the
short arm KL of the horizontal lever (scalebeam) KM. A
"weight," which if placed on the hanger N would "balance"
1000 Ibs. on the platform, is stamped 1000 Ibs. To facilitate
"balancing," the "slider" S (compare the steelyard) may be
slid along the suitably graduated arm LM. If the "dead load"
on N balances the platform when empty, then an additional
pound mass on N will balance 1000 Ibs. mass resting on the plat-
form, provided the mechanical advantage of the entire system of
levers is 1000.

and if also FA=FB=GC=GD, then

FIG. 62.

the downward force at E, and hence the reading of the scale-
beam above, will not depend upon where the load is placed on the
platform. This independence of the position of the load will be
easily seen by assigning numerical values to the above distances.
Let the first four distances each be 6 in. and the second four be
each 6 ft., and let EA and ED be each 18 ft., then if E rises 1 in.
(equals d), F and G will each rise 1/3- in., and A', B', C', and D'
each 1/12 times 1/3, or 1/36 in. (i.e., D). Consequently, the
mechanical advantage d/D is 1 -r- 1/36 or 36, and has the same
value for all four knife-edges A', B', C"'and D', showing that the
recorded weight is independent of the position of the load on the
platform, which was to be proved.

132 MECHANICS AND HEAT

If JI = SHI, then, since the mechanical advantage obtained by
lifting at E is 36, the mechanical advantage at / will be 3 times
36 or 108. Finally, if LM=10LK, then a downward pull at N
has a mechanical advantage of 10 times 108, or 1080. In other
words, 1 Ib. at N will balance 1080 Ibs. placed anywhere on the
platform.

In small platform scales, E connects directly to the scalebeam
above. In practice, the knife-edges are supported (or support
the load, as the case may be) by means of links, which permit
them to yield in response to sudden side wise jarring, and thus
preserves their sharpness and hence the accuracy of the scale.

PROBLEMS

1. It is found that with a certain machine the applied force moves 20 ft.
to raise the weight 6 in. What weight will 100 Ibs. applied force lift, assum-
ing friction to be zero? If the efficiency is 60 per cent, what will the 100
Ibs. lift? What is the theoretical mechanical advantage of the machine?
What is its actual mechanical advantage?

2. If the distance AB (sketch L, Fig. 45) is 36 in., BP is 6 in., and Fa is
100 Ibs., what is F0? That is, what weight can be lifted at Bl

3. A 6-ft. lever is used: (a) as shown in sketch K (Fig. 45), and again (6) as
shown in sketch L, PB being 1 ft. in each case. Find the applied force nec-
essary to lift 1000 Ibs. at B for each case. Explain why the answers differ.

4. What is the theor. m. adv. of the block and tackle (Fig. 48) ? What
would it be if inverted, in which position pulley A would be below and rope
a would be pulled upward?

5. Sketch a block and tackle giving a theor. m. adv. of 3, of 6, and of 7.

6. What applied force would raise 1000 Ibs. by using a wheel and axle,
if the diameter of the wheel were 4 ft., and that of the axle 6 in., (a) neglect-
ing friction, (6) assuming 90 per cent, efficiency?

7. A hammer drives a wedge, which is 2 in. thick and 1 ft. in length, a
distance of 1/2 in. each stroke. The wedge supports a weight of 1 ton and
the hand exerts upon the hammer an average force of 20 Ibs. through a dis-
tance of 3 ft. each stroke. What is the theor. m. adv. of the wedge? Of
both wedge and hammer?

8. Find the theoretical and also the actual mechanical advantage of a
jackscrew of 30 per cent, efficiency, whose screw has 10 threads to the inch
and is turned by a rod giving a 2-ft. lever arm.

9. Neglecting friction, what pull will take a 200-ton train up a 1 per cent,
grade (i.e., 1 ft. rise in 100 ft.)?

10. What is the value of the actual m. adv., and also what is the efficiency
of the combination mentioned in problem 7?

11. If the jackscrew (Prob. 8) is placed under the lever at A (sketch L,
Fig. 45), what lift can be exerted at B (of the lever) by applying a 50-lb.
pull at the end of the jackscrew lever? Let lever arm BP be 2 ft. and BA, 3 ft.

MACHINES 133

12. What H.P. does the locomotive (Prob. 9) develop in pulling the train,
if its velocity is 40 ft. per sec., and if the work done against friction equals
that done against gravity?

13. In a certain chain hoist the two upper pulleys, which are rigidly
fastened together, have respectively 22 and 24 cogs, (a) What is its
theoretical mechanical advantage? (6) What load could a 150-lb. man lift
with it, assuming an efficiency of 30 per cent.?

14. Let two levers, which we shall designate as G and H, be represented
respectively by the sketches K andL (Fig. 45), except that G is Shifted to the
right so that B of G comes under A of H, thus forming a compound lever.
Neglecting friction (a) what downward force at A of lever G will lift 1 ton
at B of lever H, provided AB and PB are respectively 6 ft. and 1 ft. for
both levers? (6) What is the theoretical mechanical advantage of G, and
of H, and also of both combined? Sketch first.

15. A barrel is rolled up an incline 20 ft. in length and 6 ft. in vertical
height by means of a rope which is fastened at the top of the incline, then
passes over the barrel, and returns from the upper &ide of the barrel in a
direction parallel to the incline. What theoretical mechanical advantage is
obtained by a man who pulls on the return rope?

16. A man, standing in a bucket, pulls himself out of a well by means of
a rope attached to the bucket and then passing over a pulley above and re-
turning to his hand. What theoretical mechanical advantage does he have?

17. The drum of an ordinary capstan for house moving is 16 in. in diame-
ter, and the sweep, to which is hitched a horse pulling 200 Ibs., is 12 ft.
long. Find the pull on the cable, assuming no friction in the drum bearings.

18. If in the lever BP, sketch M, Fig. 45, AB=AP, what weight can be
lifted at B if the block and tackle shown in Fig. 48 lifts on A of the lever,
and if the pull on rope a of the block and tackle is 100 Ibs.? Neglect friction.

19. The weight of a 24-ft. timber is to be borne equally by three men who
are carrying it. One man is at one end of the timber while the other two
lift by means of a crossbar thrust under the timber. How far from the end
should the crossbar be placed?

20. If the lever AB (sketch K, Fig. 45) be a plank 20 ft. long and weigh-
ing 100 Ibs., and if PB be 2 ft., what downward force at A will lift 1000 Ibs.
at B, (a) if we consider the weight of the plank? (&) If we neglect it?

21. A 20-ft. plank which weighs 120 Ibs. lies across a box 4 ft. in width,
with one end A projecting 7 ft. beyond the box. How near to the end A of
the plank can a 60-lb. boy approach without upsetting the plank? How
near to the other end may be approach?

22. How far from the end of the timber should the crossbar be placed
(Prob. 19) if there are two men lifting on each end of it; one man lifting on
the end of the timber as before?

23. In Fig. 62, let BB' (etc.) equal 4 in., BF (etc.) equal 5 ft., AE (and
DE) equal 15 ft., HI = 5 in., JI =20 in., LK = 1 . 5 in., and LM = 30 in. What
weight at N will balance 2 tons on the platform of the scale?

PART II
PROPERTIES OF MATTER

CHAPTER VIII

THE THREE STATES OF MATTER AND THE GENERAL
PROPERTIES OF MATTER

98. The Three States of Matter. — Matter exists in three dif-
ferent states or forms: either as a solid, as a liquid, or as a gas.
Liquids and gases have many properties in common and are some-
times classed together as fluids.

We are familiar with the general characteristics which distin-
guish one form of matter from another. Solids resist change of
size or shape; that is, they resist compression or extension, and
distortion (change of shape). Solids therefore have rigidity,
a property which is not possessed by fluids. Liquids resist com-
pression, but do not appreciably resist distortion or extension.
For these reasons a quantity of liquid assumes the form of the
containing vessel. Gases are easily compressed, offer no resist-
ance to distortion, and tend to expand indefinitely. Thus a
trace of gas introduced into a vacuous space, for example, the
exhausted receiver of an air pump, will immediately expand and
fill the entire space. Most substances change from the solid to
the liquid state when sufficiently heated; thus ice changes to
water, and iron and other metals melt when heated. If still
further heated, most substances change from the liquid state to
the gaseous state; thus, when sufficiently heated, water changes
to steam, and molten metals vaporize. Indeed, practically all
substances may exist either in the solid, the liquid, or the gaseous
state, depending upon the temperature and in some cases upon the
pressure to which the substance is subjected.

We commonly speak of a substance as being a solid, a liquid, or
a gas, depending upon its state at ordinary temperatures. Thus
metals (except mercury), minerals, wood, etc., are solids; mercury,
water and kerosene are liquids; and air and hydrogen are
gases. Mercury may be readily either vaporized or frozen, and
air can be changed to a liquid, and this liquid air has been frozen
to a solid. Some substances, e.g., those which are paste-like or
jelly-like, are on the borderline and may be called semifluids, or

137

138 MECHANICS AND HEAT

semisolids. It is interesting to note that mercury and bromine
are the only elements which are liquid at ordinary temperature.

99. Structure of Matter. — All matter, whatever its form, is
supposed to be composed of minute particles called molecules.
Thus iron (Fe) is composed of iron molecules, chlorine (Cl) of
chlorine molecules, and iron chloride (FeCl2) of iron chloride
molecules. These molecules are composed of atoms — like
atoms in the case of an element, for example, iron, and unlike
atoms in the case of compounds. Thus, the iron chloride mole-
cule (FeCl2) consists of one atom of iron (Fe) and two atoms of
chlorine (Cl).

Molecular Freedom. — In the case of a solid, the molecules that
compose it do not easily move with respect to each other. This
gives the solid rigidity which causes it to resist any force tending
to make it change its shape. In liquids, the molecules glide
readily over each other, so that a liquid immediately assumes the
shape of the containing vessel. In gases, the molecules have even
greater freedom than in liquids, and they also tend to separate
so as to permeate the entire available space as mentioned in the
preceding section.

Divisibility of Matter. — Any portion of any substance may be
divided and subdivided almost without limit by mechanical
means,(but so long as the molecule remains intact, the substance is
unchanged chemically. Thus common salt (NaCl), which is a
compound of the metal sodium (Na) and chlorine, may be ground
finer and finer until it is in the form of a very fine dust, and still
preserve the salty taste. This powdered salt may be used for
curing meats, and chemically it behaves in every way like the
unpowdered salt. If, however, through some chemical change the
molecule is broken up into its separate atoms, namely, sodium and
chlorine, it no longer exists as salt, nor has it the characteristics
of salt. Hence we may say that the molecule is the smallest
portion of a substance which can exist and retain its original
chemical characteristics. Certain phenomena indicate that the
molecule is very small — probably a small fraction of one-mil-
lionth of an inch in diameter.

The Kinetic Theory of Matter. — According to this theory, which
is generally accepted, the molecules of any substance, whether in
the solid, the liquid, or the gaseous state, are in continual to-and-
fro vibration. In solids, the molecule must remain in one place
and vibrate; in liquids and gases it may wander about while

THE THREE STATES OF MATTER 139

maintaining its vibration. This vibratory motion of translation
is supposed to give rise to the diffusion of liquids and gases (Sees.
112 and 131).

Form certain experimental facts, a discussion of which is
beyond the scope of this work, the average distance through which
a hydrogen molecule vibrates, or its "mean free path,'! is esti-
mated to be about 7/1,000,000 inch, if the hydrogen is under ordi-
nary atmospheric pressure, and at the temperature of melting
ice. This distance is smaller for the molecules of other gases,
and presumably very much smaller in the case of liquids and solids.
As a body is heated, these vibrations become more violent. This
subject will be further discussed under "The Nature of Heat"
(Sec. 160), and the "Kinetic Theory of Gases" (Sec. 171).

Brownian Motion. — About 80 years ago, Robert Brown dis-
covered that small (microscopic) particles of either organic or
inorganic matter, held in suspension in a liquid, exhibited slight
but rapid to-and-fro movements. In accordance with the ki-
netic theory of matter, these movements may be attributed to
molecular bombardment of the particles.

100. Conservation of Matter. — In spite of prolonged research
to prove the contrary, it still seems to be an established fact that
matter can be neither created nor destroyed. If several chemicals
are recombined to form a new compound, it will be found that,
the weight, and therefore the mass, of the compound so formed, is
the same as before combination. When a substance is burned,
the combined mass of the substance and the oxygen used in com-
bustion is exactly equal to the combined mass of ash and the
gaseous products of combustion. When water freezes, its den-
sity changes, but its mass does not change. Matter then,
like energy, may be transformed but neither destroyed nor
created.

101. General Properties of Matter. — There are certain proper-
ties, common to all three forms of matter, which are termed
General Properties. Important among these are mass, volume,
density, gravitational attraction, intermolecular attraction, and
elasticity.

As a rule, any portion of matter has a definite mass and a defi-
nite volume. Dividing the mass of a body by its volume gives its
Density, i.e.,

140

MECHANICS AND HEAT

In the case of a solid of regular form, its volume may be deter-
mined from measurement of its dimensions. Its mass, whatever
its shape, would be obtained by weighing. (For the method of
obtaining the density of an irregular solid see Sec. 122.) Below
are given the densities of several substances in the C.G.S.
system, i.e., in grams per cubic centimeter. The density of
water is practically 1 gm. per cm.3, or, in the British system, 62.4
Ibs. per cu. ft. Densities are usually expressed in the C.G.S.
system.

AVERAGE DENSITIES OF A FEW SUBSTANCES

Solids (gm

per cm.3) Liquids (gm.

per cm.3)

Gases (gm. per cm.3)

Gold
Lead

.. 19.30
11 36

Mercury
Bromine
Glycerine. . . .
Milk (whole) .

Sea-water
Water, 4° C..
Cream, about
Alcohol . . .

... 13.60
... 3.15
. .. 1.26
1.028 to
1.035
... 1 . 025
. .. 1.00
... 1.00
. 0.80

Chlorine
Carbon dioxide.
Oxygen
Air
Nitrogen
Marsh gas
Steam, 100° C..
Hydrogen

0.0032
0.002
0.0014
0.0013
0.00125
.0.0007
0 . 0006
0.00009

Silver
Iron
Marble....
Aluminum.
Ice
Cork

.. 10.53

.. 7.80
.. 2.75
. . 2 . 60
.. 0.917
.. 0.25

In general, metals are very dense, as the table shows. Liquids
are less dense, and gases have very small densities. Ice floats in
water, from which it appears that the density of water decreases
when it changes to the solid state. Paraffine, on the contrary,
becomes more dense when it solidifies. The densities of different
specimens of the same substance usually differ slightly. The
approximate values of those in italics should be memorized. With
the exception of steam, the densities given for the gases refer in
each case to the density of the gas when at 0° C. and under stand-
ard atmospheric pressure (Sec. 136).

Solids, liquids, and gases all have weight, which shows that
gravitational attraction acts between them and the earth. The
other two general properties, inter molecular attraction and elastic-
ity, will be discussed in the following sections.

102. Intermolecular Attraction and the Phenomena to Which
1 gives Rise. — It requires a very great force to pull a metal bar
in two, because of the Intermolecular Attraction of its molecules.
If, however, the ends of the bars are now carefully squared and
then firmly pressed together, it will be found upon removing the
pressure that a very slight force will separate them. This ex-
periment shows that this molecular force, which is called Cohesion,

THE THREE STATES OF MATTER 141

and which gives a metal or any other substance its tensile
strength, acts through very small distances. Two freshly cleaned
surfaces of lead cohere rather strongly after being pressed firmly
together. The fact that lead is a soft metal, permits the two
surfaces to be forced into more intimate contact, so that the
molecular forces come into play.

By gently hammering gold foil into a tooth cavity, the dentist
produces a solid gold filling. Gold is not only a fairly soft metal
but it also does not readily tarnish. Because of these two proper-
ties, the molecules of the successive layers of foil are very readily
brought into intimate contact, and therefore unite.

Welding. — In welding together two pieces of iron, both pieces
are heated to make them soft, and they are then hammered to-
gether to make them unite. The "flux" used prevents oxidation
in part, and also floats away from between the two surfaces what-
ever scale or oxide does form, thus insuring intimate contact
between them.

At is cohesion that enables the molecules of a liquid to cling
jfogether and form drops. This will be further considered under
\/" Surface Tension" (Sec. 124). If a clean glass rod is dipped into
water and then withdrawn, a drop of water adheres to it. Obvi-
ously the weight of the drop of water is sustained by the molecular
attraction between the glass molecules and the water molecules.
This force is called Adhesion, whereas the force which holds the
drop together is Cohesion as already stated. That is, the force
of cohesion is exerted between like molecules, adhesion between
unlike molecules.

Two pieces of wood may be held firmly together by means of
glue. One surface of the thin layer of glue adheres to one piece
of wood, and the opposite surface adheres to the other. When
the two pieces of wood are torn apart, the line of fracture will .'>
occur at the weakest place. If the fracture occurs between glue
and wood in such a way that no glue adheres to the wood, then
the adhesion between glue and wood is weaker than the cohesion
of either substance. If the layer of glue is torn apart so that
a portion of it adheres to each piece of wood, then cohesion
for glue is weaker than adhesion between glue and wood. Fi-
nally, if portions of the wood are torn out because of adhering to
the glue, which often happens, it shows that the adhesion be-
tween glue and wood is stronger than the cohesion of wood (at
that point).

142 MECHANICS AND HEAT

As a rule, cohesion is stronger than adhesion. The adhesion
between the layer of gelatine and the glass of a photographic
plate furnishes a striking exception to this rule. Sometimes, in
becoming very dry, this gelatine film shrinks with sufficient force
to tear itself loose from the glass at some points, while at other
points bits of the glass are torn out, leaving the glass noticeably
rough to the touch. A thin layer of fish glue spread upon a
carefully cleaned glass plate produces, as it dries, a similar and
even more marked effect.

103. Elasticity, General Discussion. — When a force is applied
to a solid body it always produces some change either in its length,
„.; its volume, or its shape. The tendency to resume the original
condition upon removal of the applied force is called Elasticity.
When a metal bar is slightly stretched by a force, it resumes its
original length upon removal of the force, by virtue of its Tensile
Elasticity. If the bar is twisted, its recovery upon removal of
the applied torque is due to its Elasticity of Torsion, Rigidity, or
Shearing, as it is variously termed. If the bar is subjected to
enormous hydrostatic pressure on all sides, its volume decreases
slightly. Upon removal of the pressure, the tendency to imme-
diately resume its original volume is due to the Volume Elasticity
of the metal of which the bar is made.

If, upon removal of the distorting force, the body regains
immediately and completely its original shape or size, it is said
to be perfectly elastic. Liquids and gases are perfectly elastic,
but no solids are. Ivory, glass, and steel are more nearly per-
fectly elastic than any other common solid substances. Such
substances as putty have practically no tendency to recover from
a distortion and are therefore called inelastic. They are also
called plastic, which distinguishes them from brittle inelastic
substances such as chalk.

Through wide ranges, most elastic substances are distorted in
proportion to the applied or distorting force, e.g., .doubling the
force produces twice as great stretch, twist, or shrinkage in
volutne, as the case may be. Such substances are said to obey
Hooke's law (Sec. 107).

Any change in the shape of a body must entail a change in the
relative positions of its molecules, hence elasticity of shape or
rigidity may be considered to be due primarily to the tendency
of the molecules to resume their former relative positions. The
resistance which the molecules offer to being crowded more closely

THE THREE STATES OF MATTER 143

together, or rather their tendency to again spring apart, gives
rise to volume elasticity.

Elasticity is one of the most important properties of substances,
and for this reason it has been very much studied. The subject
will be taken up more in detail in subsequent chapters, especially
under "Properties of Solids." For a more complete study the
reader is referred to advanced works on Physics or Mechanics,
some of which are mentioned in the preface.

PROBLEMS

1. By the use of the table, find the densities of air, sea-water, mercury,
and gold in the British system

2. A rectangular block of wood 4 in. X 2 in. X 1/2 in. weighs 44 gm. Find
its density.

3. Find the weight of 1/2 mi. of 1/8-in. iron wire.

4. A cylindrical metal bar 1 cm. in diameter and 20 cm. in length weighs
165.3 gm. Of what metal is it composed? What is its density?

6. Find the mass of a cubic yard of each of the following substances:
hydrogen, air, water, ice. A cubic foot of water weighs 62.4 Ibs. .

6. How many cubic feet of ice will 50 gal. of water form upon freezing?
Water weighs very closely 62.4 Ibs. per cu. ft., or 8 . 33 Ibs. per gal.

7. A hollow iron sphere 10 cm. in diameter weighs 3 kilos. What is the
volume of the cavity within it?

8. Apiece of brass has a density of 8.4 gm. per cm.3 Assuming that the
volume of the brass is exactly equal to the sum of the volumes of copper and
zinc that compose it, what percentage of the brass, by volume, is zinc?
Density of copper is 8.92, zinc 7.2 gm. per cm.3 Suggestion: Represent
by x the fractional part that is zinc.

9. From the answer to problem 8, find what percentage of the brass by
weight is zinc.

CHAPTER IX
PROPERTIES OF SOLIDS

104. Properties Enumerated and Defined. — The following
properties are obviously peculiar to solids: hardness, brittleness,
malleability, ductility, tenacity or tensile strength, and shearing
elasticity.

Hardness and brittleness often go hand in hand. Thus steel
when tempered "glass hard" is brittle. Glass is both hard and
brittle. Chalk, however, is brittle but not hard. Brittleness
may be defined as the property of yielding very little before
breaking. Thus glass or chalk cannot be bent, twisted, or
elongated appreciably before breaking, and are therefore brittle.

If a substance may be made to scratch another, but cannot be
scratched by it, then the former substance is Harder than the latter.
Ten substances, with diamond at the head of the list, sapphire
next as 9, and talc at the bottom of the list as 1, have been used
as a " scale of hardness." If a certain substance may bescratched
by diamond as readily as it can be made to scratch sapphire,
then the substance is 9.5 in the scale of hardness.

Malleability is that property of a solid by virtue of which it may
be hammered into thin sheets. Gold is very malleable, indeed
it is the most malleable known substance. By placing a thin
sheet of gold between two sheets of "gold beater's skin" it
may be hammered into foil about 1/200000 inch thick. Lead
is malleable. Iron becomes quite malleable when heated to
a white heat. Wrought iron is slightly malleable at ordinary
temperatures.

Ductility is that property of a metal which enables it to be
drawn out into the form of a fine wire. Brass, copper, iron and
platinum are very ductile. Although lead is malleable, it is not
strong enough to be very ductile.

The Tenacity or tensile strength of a metal or other substance,
depends, as stated in Sec. 102, upon the cohesive force between its
molecules. Iron has a large tensile strength — from 40,000 to
60,000 Ibs. per sq. in. Copper and lead have relatively low ten-
sile strengths.

144

PROPERTIES OF SOLIDS 145

106. Elasticity, Elastic Limit, and Elastic Fatigue of Solids. —
If several balls, made of different metals, are successively dropped
upon an anvil from a height of a few inches, it will be found that
the first rebound carries the steel ball nearly to the height from
which it was dropped. The brass ball rebounds less, and the iron
one still less than the brass one. The lead ball does not rebound,
but merely flattens slightly where it strikes the anvil. Ivory
rebounds better than steel. The sudden stopping of the ball by
the anvil requires a large force (F = Ma), which flattens the
ball in each case. If the material is elastic, however, the flat-
tened portion springs out again into the spherical form as soon as
the motion of the ball is stopped, and in so doing throws the ball
into the air. If the ball and anvil were both perfectly elastic the
first rebound would bring the ball back to the point from which
it was dropped. This is a very simple, rough test of elasticity.
If the ball were perfectly elastic, the average force required to
flatten it would be exactly equal to the average force with which
it would tend to restore its spherical shape. Obviously, these
two forces would each act through the same distance, hence, the
work of flattening and the work of restoring would be equal.
But the former work is equal to — is in fact due to — the potential
energy of the ball in its original position, and the latter work is
used in throwing the ball back to the height of the first rebound.
Accordingly, this height should be equal to the distance of fall.
Because of molecular friction, the above restoring force is smaller
than the flattening force even in the case of the ivory ball, which
accounts for its failure to rebound to the original height.

If a straight spring is moderately bent for a short time and is
then slowly released (to prevent vibration), it returns to its
original straight condition. If, however, it is moderately bent
and left for years in this bent condition and is then slowly released,
it will immediately become nearly straight, and then very slowly
recover until it becomes practically straight. It might be said
that the steel becomes "fatigued" from being bent for so long a
time. Accordingly, it is said to be Elastic Fatigue of the steel
(see also Sec. 108) which in this case prevents the immediate
return of the spring to its straight condition. Again, if the spring
is very much bent and then released it will remain slightly bent,
i.e., it will have a slight permanent " set." In such case, the steel
is said to be "strained" beyond the Elastic Limit. All solids
are more or less elastic. Even a lead bar if very slightly bent will

146 MECHANICS AND HEAT

recover; but the elastic limit for lead is very quickly reached, so
that if the bar is appreciably bent, it remains bent upon removal
of the applied force.

106. Tensile Stress and Tensile Strain.— In Sec. 103 a
brief discussion of elasticity was given, in which it was shown that
solids possess three kinds of elasticity. We shall now discuss
more in detail the simplest of these, namely, Tensile Elasticity,
and consider the other two in subsequent sections. Before a
systematic study of the elastic properties of a substance can be
made, it is necessary to understand clearly the meaning
of each of the terms, Stress, Strain, and Modulus.

Whenever an elastic body is acted upon by a force
tending to stretch it, there arises an equal internal
force tending to shorten it. See Principle of d'Alem-

bert (Sec . 43). Thus, in Fig. 63, let B be a steel bar
of length L, say 10 ft., and of cross section A, say 2
sq. in. When an external force F of 20,000 Ibs. is ap-
plied, the bar stretches a distance e (elongation), say
0.04 in. It is at once evident that in this stretched
te condition, which is also an equilibrium condition, the
internal forces due to which the bar tends to resume
its normal length must just equal the 20,000 Ibs.
FIG. 63. which tends to make the bar lengthen; otherwise the
weight W would move downward causing the stretch
of the bar to be still further increased. This internal force di-
vided by the cross section of the bar, in other words, the force
per unit cross section, is called the Tensile Stress. But, since
the internal force that arises is always equal to the applied
force, we have

(62)

cross section A

20000
which here is — „ or 10,000 Ibs. per sq. in.

The increase in length, or the elongation e of the bar, divided
by its original length, in other words, the stretch per unit length,
is called the Tensile Strain. Accordingly, we here have

Tensne

A column, in supporting a load, is subjected to a stress and
suffers a strain, both of which are denned essentially as above.

PROPERTIES OF SOLIDS 147

The stress is the load divided by the cross section of the column,
and the strain is the decrease in length divided by the original
length of the column. It is an observed fact that a column, in
supporting a load in the usual way, is decreased in length by an
amount exactly equal to the stretch that it would experience if
its upper end were fastened to a support and the same load were
suspended from its lower end. In other words, within certain
limits, the elasticities of extension and compression are alike.
It appears, then, that within certain limits, the molecules of an
elastic solid resist having their normal spacing decreased with the
same force that they resist having it increased a like amount.

107. Hooke's Law and Young's Modulus.— If the bar B
(Fig. 63) supports twice as large a load it will stretch twice as
much, and so on for still larger loads, so long as it is not strained
beyond the elastic limit. A glance at the above equations shows
that both the stress and the strain must, then, increase directly
as does the load. This being true, it follows that
Stress

which is known as Hooke's law. If a substance is strained beyond
the elastic limit it does not obey Hooke's law; conversely, if an
elastic body does not obey Hooke's law, it must be strained
beyond the elastic limit.

A spiral spring of steel obeys Hooke's law, i.e., the elongation
is proportional to the load it supports. This property is utilized
in the ordinary spring balance used in weighing. If a certain
torque twists a rod or shaft through an angle of 20°, and if
doubling the torque twists it 40°, then the rod or shaft follows
Hooke's Law for that torque. If 5 times as great a torque twists
the rod say 130° (instead of 100°), it shows that it is strained be-
yond the elastic limit, since for this larger torque it does not
follow Hooke's Law.

The constant of Hooke's Law is called the Stretch Modulus
or Young's Modulus for the substance, when applied to tensile
stress and tensile strain, or

, tensile stress F/A FL .

Young s Modulus ^ = tensile^traln = "e/L = Ae

Substituting the values used in Eqs. 62 and 63, we have

lbs- <* • itt-

148 MECHANICS AND HEAT

The above assumed stretch is about what would be found by
experiment if the bar were very good steel. Hence Young's
modulus for good steel is 30,000,000 Ibs. per sq. in. In the metric
system, the force would usually be expressed in dynes (sometimes
in kilograms), the distance in centimeters, and the cross section
in square centimeters. Young's modulus for steel as expressed
in this system is 1.9X1012 dynes per cm.2 For most substances
Young's modulus is very much smaller than for steel; in other
words, most substances offer less resistance to stretching than
steel does.

If, in Eq. 64, A were unity, and if e were equal to L, i.e., if B had
unit cross section and were stretched to double its original length (assum-
ing that to be possible), then the equation would reduce to E = F.
Hence Young's modulus E is numerically equal to the force that would be
required to stretch a bar of unit cross section to twice its original length,
provided it continued to follow Hooke's law. Although a bar of steel,
or almost any other substance except rubber, would break long before
reaching twice its original length, still this concept is useful. For, by
its use in connection with the above data, we see at once, since a force
of 30,000,000 Ibs. would double the length of a bar of 1 sq. in. cross
section (assuming Hooke's law to hold), that a force of 30,000 Ibs., for
which force Hooke's law would hold, would increase its length 1/1000 as
much, or 1 part in 1000.

108. Yield Point, Tensile Strength, Breaking Stress.— If the
bar B (Fig. 63) is made of steel, it will be found that as the load
is increased the bar will stretch more and more, in accordance
with Hooke's law, until the stress is about 60,000 Ibs. per sq. in.
Upon further increasing the load, it will be found that the bar
begins to stretch very much more — perhaps 50 times more —
than for previous increases of like magnitude. This change in
the behavior of the steel, this very great increase in the strain
produced by a slight increase in the stress, is due to a yielding
of the molecular forces, which yielding permits the molecules
to slide slightly with reference to each other. We may say for
this specimen of steel, that a stress of 60,000 Ibs. per sq. in.
strains it to the elastic limit, and that a slightly greater stress
brings it to the Yield Point.

As soon as the yield point is reached, further increase of load
causes the bar to stretch until the elongation is 25 or 30 per cent,
of the original length, in the case of soft steel. The maximum
elongation for hard steel may be as small as 1 per cent. If the

PROPERTIES OF SOLIDS 149

load is removed after the yield point has been passed, the bar
remains permanently elongated, i.e., it has a Permanent Set.
This elongation is accompanied by a decrease in cross section.
The maximum load required to cause breaking, divided by the
original cross section, gives the Breaking Stress or Tensile Strength
of the Steel

A slight difference in the amount of carbon in steel, changes its elastic
behavior very much. Thus, a certain specimen of steel containing 0.17
per cent, carbon had an elastic limit of 51,000 Ibs. per sq. in. and a
breaking stress of 68,000 Ibs. per sq. in. For another specimen, contain-
ing 0.82 per cent, carbon, the elastic limit was 68,000 Ibs. per sq. in., and
the breaking stress was 142,000 Ibs. per sq. in. The annealing or tem-
pering of steel is also an important factor in determining its elastic
properties.

In addition to iron and carbon, steel may contain various other sub-
stances, important among which are nickel, silicon, and manganese,
which greatly influence its elastic properties and its hardness, e\en
though present in very small quantities (1 to 5 per cent, more or less).
A piano wire, having the enormous tensile strength of 340,000 Ibs. per
sq. in., or 170 tons per sq. in., was found upon analysis to contain 0.01 per
cent, sulphur, 0.018 per cent, phosphorous, 0.09 per cent, silicon, 0.4 per
cent, manganese, and 0.57 per cent, carbon. Because of the great com-
mercial importance of steel, this brief statement concerning its composi-
tion and elastic properties is made here. For further discussion consult
some special engineering work on the subject, or an encyclopedia, such
as "Americana" or "Britannica."

Factor of Safety. — If steel is subjected to a great many repeti-
tions of stresses which are well below its tensile strength, or even
below its elastic limit, it is greatly weakened thereby, and it may
finally break with a load which it would have easily carried
at first. This weakening of material by a great number (several
millions) of repetitions of a stress is said to be due to Elastic
Fatigue. (See also Sec. 105.) Of course in any structure the
stress should always be well below the elastic limit for the material
used. Thus steel whose elastic limit is 50,000 Ibs. per sq. in.
would rarely be subjected to stresses greater than 25,000 Ibs.
per sq. in. In such case the Factor of Safety is 2. Structures or
machine parts which are exposed to vibrations and sudden
stresses or shocks, especially if constructed of very hard steel
or other relatively brittle material, require a much higher factor
of safety. The factor of safety also guards against breakage
(rom flaws in the material.

150 MECHANICS AND HEAT

109. Strength of Horizontal Beams. — If a straight beam of wood or
metal (Fig. 64) of length L, having a rectangular cross section of depth
h and width a, is supported at each end and loaded in the middle as
shown, it will bend slightly. Obviously, in the process of bending, the
material near the upper portion of the beam is compressed, while that
below is stretched. The horizontal layer of particles through the
middle of the beam, that is, through the line, BCD, is called the Neutral
Plane, because this portion is neither compressed nor stretched. The
material at G is stretched only 1/2 as much as that at H, because it is
only 1/2 as far from the neutral plane. Hence if the load is made too

FIG. 64.

great the material at H, called the "outer fiber," is the first to be strained
to the yield point, and when fracture occurs, it starts at this point.
It can be shown by means of advanced mathematics that

<«>

in which d is the deflection of the middle of the beam produced by the
load W, and E is Young's modulus for the material of the beam. Eq.
65 shows that the beam will deflect less, and hence be stronger if placed
on edge than if flatwise.

As an illustration, consider a 2-in. by 6-in. joist such as is sometimes
used to support floors. In changing the joist from the flat to the edge-
wise position, we treble h and make a 1/3 as large. Trebling h makes
h3 27 times as large, consequently ah3 is 1/3 times 27, or 9 times as large
as before. This makes d 1/9 as large. In other words, the beam would
require 9 times as large a load to give the same amount of bend, which
means that the Stiffness of the beam is made 9 times as great by turning
it on edge.

In the edgewise position, however, the distance of the "outer fiber"
(Fig. 64) from the neutral plane is three times as large as before, and
consequently a given bend or deflection produces 3 times as great a
strain on this fiber as before, so that the Strength of the beam is not 9
times as great, but only 3 times as great on edge as flatwise.

PROPERTIES OF SOLIDS 151

Next consider the effect on d of variation in length, all other quantities
remaining the same. If the beam is made 3 times as long, L3 and hence
also d become 27 times as great as before. If the beam is three times as
long, it must bend 9 times as much (i.e., d must be 9 times as great) to
produce the same strain in the material. For to produce the same strain
in the longer beam, it must bend to an arc of the same radius of curvature
as the shorter beam. But, for small arcs, the distance d from the middle
point of the chord to the middle point of its arc varies approximately
as the square of the length of the chord. Consequently, the strain is 3
(not 27) times as great as before, and the beam will therefore support
only 1/3 as great a load as before. This relation will be clearly seen
from an application. Suppose that a pine beam 4 ft. long and 2 in. by
4 in. in cross section, will support 1000 Ibs. at its center. Then if twice
as long it will support 1/2 as much, or 500 Ibs. If 3 times as long it will
support 1/3 as much, and so on.

To summarize, we may state that for rectangular beams supported at
the end and loaded in the middle (or supported in the middle and loaded
at the ends, which amounts to the same thing), the strength varies directly
as the first power of the width and as the second power of the depth; while it
varies inversely as the first power of the length. For such beams, the stiff-
ness varies directly as the first power of the width, and as the cube of the depth
(other things not being varied); while it varies inversely as the cube of
the length.

110. Three Kinds of Elasticity, of Stress, and of Strain; and
the Three Moduli. — In Sec. 103 it was stated that a solid, for ex-
ample a metal bar, may be acted upon by forces in three distinct
ways bringing into play its three elasticities. Thus the metal bar
B (Fig. 65) of length L and cross section A, is acted upon by a force
F which produces an elongation e. Upon removal of this force
it returns to its original length due to tensile elasticity. Bi
illustrates the same bar acted upon by forces from all sides, i.e.,
over its entire surface of area A\. Let us suppose these forces to
be due to hydrostatic pressure, which pressure causes a decrease
V in the original volume (F) of the bar. As soon as the pressure
is removed, the bar returns to its original volume by virtue of its
volume elasticity. B2 illustrates the same bar again, this time with
its lower surface fixed. Consequently the force F applied to its
upper surface of area A2 makes it slide or shear a distance s with
respect to the lower surface. The distance between the two sur-
faces we shall call d. Upon removal of the force F the shear
disappears due to shearing elasticity. In all three cases, recovery
upon removal of the force is practically immediate and complete,
provided the bar has not been strained beyond the elastic limit.

152

MECHANICS AND HEAT

The Three Moduli. — The stress to which a certain material is
subjected, divided by the resulting strain, is constant (Hooke's
Law), and this constant is called the Modulus of Elasticity.
Since there are three kinds of stress and three kinds of strain, it
follows that there must be three moduli.

Stress is always the total applied force F divided by the area
to which it is applied. Thus in the first case (B), tensile stress is
F/A, in the second case (Bi), the hydrostatic stress or volume

stress is F/Aij while in the third
z), the shearing stress is
In the first case, the ten-
sile strain is the change in
length divided by the original
length or e/L; in the second
case, the volume strain is the
change in volume divided by the
original volume, or V'/V; while
in the third case, the shearing
strain is the distance sheared
divided by the distance be-
tween the two shearing surfaces,
or s/d.

FIG. 65.

tensile stress F/A

(64 bis)

Summarizing, then, we have:

, . _ ,,, , , x

The mod. of Tension (Young s modulus) =

mi TT i . . hydrostatic pressure

The Volume modulus (bulk mod.) = -—j-——-— = ^

(66)

mi «. • / ., \ shearing stress

The Shearing modulus (mod. of rigidity) =snearing strain =

(67)

Observe that if s is very small with respect to d, then s/d = 6.
The angle d is called the angle of shear. For this reason the
shearing strain is usually called the angle of shear. To illustrate
shearing, the bar B2 may be considered to be made up of a great
number of horizontal layers of molecules, a few of which layers
are indicated in the sketch. Evidently, when the force F is applied,
and the bar is changed from the rectangular form to the sheared
position, each layer is shifted to the right a slight distance, — s for

PROPERTIES OF SOLIDS 153

the top layer, \ s for the middle layer, and so on. Further-
more, each layer is shifted or displaced very slightly with respect
to the next layer below it, thereby causing a slight change in
the relative positions of the molecules of successive layers. If
F is decreased, the tendency of the molecules to resume their
original relative positions reduces the relative shift between
successive layers, and hence reduces the angle of shear. If F
is removed the angle of shear becomes zero, i.e., the molecules
completely return to their normal relative positions, and the bar
again becomes rectangular, provided it has not been strained,
beyond the elastic limit.

111. The Rigidity of a Shaft and the Power Transmitted.— If
one end A of a shaft is clamped and the other end B is turned
through one revolution by some applied torque, the shaft is
said to be twisted through an angle of 360°. Evidently the layer
of molecules on the end B has been displaced or sheared through
1 revolution with respect to the layer at end A, through 1/2
revolution with respect to the transverse layer through the middle
of the shaft, through 1/4 revolution with respect to the layer 1/4
way from B to A, and so on. Indeed every transverse (circular)
layer in the shaft is sheared slightly with respect to its neighbor.
Obviously this shear is greatest for the particles farthest from the
axis of the shaft. Accordingly it is the "outer fibers" (on the
surface of the shaft) which first give way when it is twisted in
two. Observe that when a bolt is twisted in two, the central
fibers are the last to break. Observe also that fracture in this
case consists in a shearing apart of adjacent layers.

By knowing the values of the shearing modulus and the
shearing strength for the steel used, and with the aid of certain
formulas, the derivation of which requires a knowledge of ad-
vanced mathematics, the engineer can readily compute the
proper size of shaft for a specified purpose. The shaft must be
of such size that the maximum torque to which it is to be sub-
jected shall not strain the outer fibers beyond the "safe" limit.
Although the mathematical treatment of this topic is too com-
plicated for an elementary work, it may be stated that the
strength of a shaft, that is the maximum torque which it can
safely transmit, varies as the cube of its radius, while the "stiff-
ness" varies as the 4th power of the radius. Thus a 2-inch shaft
can transmit 8 times as great a torque as a 1-in. shaft; while, if
the length of the shaft and the applied torque are the same for

154 MECHANICS AND HEAT

both, the smaller shaft will be twisted through 16 times as great
an angle as the larger.

Since power is torque multiplied by the angular velocity
(P=Ta>, Sec. 83), it follows that a given amount of power
can be transmitted by 1/4 as great a torque, and hence by
1/4 as strong a shaft by making the angular velocity 4 times
as great. We may also add that the power which a belt of given
strength can transmit varies directly as the speed of the belt.
For, in this case, P=Fv, in which v is the belt speed, and F is
the difference in tension between the tight and the slack belt.

PROBLEMS

1. A certain steel bar 10 ft. in length and 2 sq. in. in cross section is
elongated 0.22 in. by a 50-ton pull. What is Young's modulus E for this
specimen?

2. A steel wire 3 meters in length and 2 mm. in diameter supports a load
of 10 kilos. How much will the wire elongate under this load, if Young's
modulus for the wire is 1.9X1012 dynes per cm.2?

3. How much will a copper wire 10 meters in length and 2 sq. mm. in cross
section stretch under a load of 3 kilos? Young's modulus for copper is 1.2
X1012 dynes per cm.2

4. A certain shaft A can safely transmit 50 H.P. What power can be
transmitted by a shaft of the same material having twice as great a di-
ameter and 3 times as great an angular velocity as A?

6. An oak timber 3 in. by 12 in. rests edgewise upon two supports which
are 8 ft. apart. How much will the beam bend (deflect at the middle) under
a load of 1000 Ibs. applied midway between the supports? Young's Modulus
for oak is 1,500,000 Ibs. per sq. in.

6. How much would the 1000-lb. load bend the timber (Prob. 5) if the
timber rested flatwise upon the supports?

CHAPTER X
PROPERTIES OF LIQUIDS AT REST

112. Brief Mention of Properties. — Some of the properties
of liquids in addition to the general properties of matter (Sec.
101), are Viscosity, Solvent Action, Diffusion, Osmosis, Pressure
Production, Pressure Transmission, and Surface Tension.

Elasticity. — The only kind of elasticity that liquids or gases
can have is of course volume elasticity (Sec. 110). Liquids (also
gases) are perfectly elastic, that is, however much a liquid is
compressed, upon removing the pressure the liquid expands to
exactly its former volume. There is no such thing as elastic
fatigue or elastic limit for liquids. It requires very high pressure
to produce appreciable compression of a liquid. Thus a pres-
sure of 100 Ibs. per sq. in. applied to a volume of water causes a
shrinkage of only 1 part in 3000.

Viscosity. — If a vessel filled with syrup has a small hole made
near the bottom, the syrup will flow slowly through the hole.
If the vessel were filled with water instead, it would be found that
the water, having less viscosity, would flow much more quickly
through the hole. Syrup is said to be viscous, and water mobile.
Water, however, has some viscosity. Glycerine has greater
viscosity than water but less than molasses. Viscosity arises
from internal friction, that is, friction between the molecules
of the liquid. The greater viscosity of glycerine as compared
with that of water is then due to the fact that glycerine molecules
do not glide over each other so readily as do water molecules.

It may easily be observed that the water on the surface of a
river moves more rapidly than that near the bottom, and also
that the water near the center of the stream moves more rapidly
than that near the shore. This difference in velocity is due to
friction upon the bed of the river (and upon its shores), which
causes the layers very near the bottom to move very slowly.
These slowly moving layers of water, due to friction of water
on water, i.e., due to the viscosity of water, tend to retard the
motion of the layers above. The greatest retarding effect is

155

156 MECHANICS AND HEAT

exerted upon the nearest layers, and the least upon the surface
layer. Hence the velocity of flow gradually increases from the
bottom up.

Solvent Action. — Some solids when placed in certain liquids
slowly disappear. Thus salt readily "dissolves" in water, form-
ing a solution. Paraffine dissolves in kerosene, but not in water;
while salt dissolves in water but not in kerosene. When water
has dissolved all of the salt it is possible for it to hold in solution,
the brine thus formed is said to be a saturated solution of salt.
Solution is usually attended by either evolution or absorption
of heat; i.e., by either heating or chilling action.

Gold, zinc, and some other metals dissolve to a certain extent
in mercury, forming gold amalgam, zinc amalgam, etc. These
amalgams are really solutions of the metals in mercury.

Some liquids dissolve in other liquids. Thus, if some ether
and water are thoroughly stirred together in a vessel and then
allowed to stand a moment, the water, being the heavier, settles
to the bottom and the layer of ether rests upon it. Upon ex-
amination it will be found that there is about 10 per cent, ether
in the water, and about 3 per cent, water in the ether, which
shows that a saturated solution of ether in water is about 10 per
cent, ether, while a saturated solution of water in ether is about
3 per cent, water.

Some liquids dissolve certain gases. Thus water dissolves
air to a slight extent, and at room temperature and atmospheric
pressure, water dissolves 450 times its volume of hydrochloric
acid gas (HC1), or 600 times its volume of ammonia gas (NH3).
What is known commercially as ammonia or as hydrochloric acid
is simply an aqueous solution of the one or the other of these
gases. Pure liquid ammonia is used in ice manufacture (Sec.
200). Hydrochloric acid gas can be condensed to a liquid,
thus forming pure liquid hydrochloric acid, by subjecting it to
very high pressure and low temperature. A given volume of
water will dissolve about an equal volume of carbon dioxide
(CO2) at ordinary pressure and temperature. Under greater
pressure it dissolves considerably more, and is then called soda
water. When drawn from the fountain, the pressure upon it
is reduced, and the escaping CO2 produces effervescence.

Diffusion. — Many liquids if placed in the same vessel, mix
even though of quite different densities. Thus, if some ether is
very carefully introduced onto the surface of some water in such

PROPERTIES OF LIQUIDS AT REST 157

a way as to prevent mixing when introducing it, it will be found
after a time that the heavier liquid (water) has diffused upward
into the ether until the latter contains about 3 per cent, water,
while the ether, although lighter, has diffused downward into the
water.

Osmosis. — Osmosis is the mixing or diffusing of two different
liquids or gases through a membrane that separates them.
Membranes of animal or plant tissue readily permit such diffu-
sion of certain substances through them. Thus a bladder filled
with water does not leak, but if lowered into a vessel of alcohol
it slowly collapses. This shows that the water passes readily
through the bladder; the alcohol less readily, or not at all.
On the other hand, if a rubber bag is filled with water and is
then lowered into a vessel of alcohol, it becomes more and more
distended, and may finally burst. In this case it is the alcohol
which passes most readily through the separating membrane.

If a piece of parchment or other such membrane is tied tightly
across the mouth of an inverted funnel filled with sugar solution,
and the funnel is placed in water, it will be observed that the
solution slowly rises in the stem. By prolonging the stem a
rise of several feet may be obtained. Obviously the pure water
passes more readily through the membrane than does the
sweetened water, or sugar solution. If the solution is 1 . 5 per
cent, sugar (by weight), it will finally rise in the stem about 34
ft. above the level of the water outside.

Since a column of water 34 ft. in height exerts a pressure of
about one atmosphere (Sec. 136), which pressure in this case would
tend to force the solution through the membrane into the water,
it follows, when equilibrium is reached, i.e., when no further rise
of the column occurs, that the Osmotic Pressure developed by the
tendency of the water to pass through the membrane into the solu-
tion, must be one atmosphere for a 1 . 5 per cent, sugar solution.

With weak solutions, the osmotic pressure varies approxi-
mately as the strength of the solution. Thus a 3 per cent, sugar
solution would develop an osmotic pressure of about 2 atmos-
pheres. The osmotic pressure also differs greatly for different
solutions. Thus, for example, if a solution of common salt is
used the osmotic pressure developed will be much more than for
the same strength (in per cent.) of sugar solution.

In accordance with the kinetic theory of matter (Sec. 99)
we may explain osmotic pressure by assuming, in the case cited

158 MECHANICS AND HEAT

above, that the water molecules in their vibratory motion, pass
more readily through the animal membrane (the bladder) than
do the more complicated and presumably larger alcohol mole-
cules. This is the commonly accepted explanation. The fact,
however, that substituting a rubber membrane reverses the ac-
tion, makes it seem probable that something akin to chemical
affinity between the membrane and the liquids plays an impor-
tant role. From this standpoint, we would explain this reversal
in osmotic action by stating that the rubber membrane has
greater affinity for alcohol than for water; while in the case of
animal tissue the reverse is true. Osmosis plays an important
part in the physiological processes of nutrition, secretion by
glands, etc., and in the analogous processes in plant life. Gases
also pass in the same way through membranes. In this way
the blood is purified in the capillary blood-vessels of the lungs
by the oxygen in the adjacent air cells of the lungs.

In chemistry, Dialysis, the process by which crystalloids, such
as sugar and salt are separated from the colloids — starch, gum,
albumin, etc., depends upon osmosis. Crystalloids pass readily
through certain membranes; colloids, very slowly, or not at all.
In case of suspected poisoning by arsenic or any other crystal-
loid, the contents of the stomach may be placed on parchment
paper floating on water. In a short time the crystalloids (only)
will have entered the water, which may then be analyzed.

Pressure and its Transmission. — Liquids exert and also trans-
mit pressure. In deep-sea diving the pressure sustained by the
divers is enormous. By means of our city water mains, pressure
is transmitted from the pumping station or supply tank to all
parts of the system. (This property will be fully discussed in
Sees. 113 and 114. Surface Tension will be considered in Sec.
124.)

113. Hydrostatic Pressure. — The study of fluids at rest is
known as Hydrostatics, and that of fluids in motion, as Hydraulics.
From their connection with these subjects we have the terms
hydrostatic pressure and hydraulic machinery such as hydraulic
presses, hydraulic elevators, etc.

A liquid, because of its weight, exerts a force upon any body
immersed in it. This force, divided by the area upon which
it acts, is called the Hydrostatic Pressure, or

TT , total force

Hydrostatu pressure (average) = — -

PROPERTIES OF LIQUIDS AT REST 159

Note that pressure, like all stresses (Sec. 110), is the total force
applied divided by the area to which it is applied. The unit
in which to express pressure will therefore depend upon the
units in which the force and the area are expressed. Some units of
pressure are the poundal per square inch, the pound per square inch,
the pound per square foot, and the dyne per square centimeter.
Let it be required to find the pressure at a depth h below the
surface of the liquid of density d in the cylindrical vessel of
radius r, Fig. 66. The formula for the pres-
sure on the bottom of the vessel is, by defi-
nition,

total force on the bottom
Pressure = -

The force on the bottom is obviously the

weight W of the liquid, and the area A is

Trr2; so that the pressure is W/irr2. We may

express W in dynes, poundals, or pounds

force, and Trr2 in square centimeters, square

inches, etc. The weight in dynes is Mg, pIG gg

but the mass M in grams is the product of

•jrr2h, the volume of the liquid in cubic centimeters, and d its

density in grams per cubic centimeter. Hence

force W Mg irrVidg ,

Pressure p = -- = -r- = — sr = - ^ = hag dynes per cm. (68)
area A Trr2 Trr2

In the British system, -irr2h would be the volume of the liquid
column in cubic feet, and d the density in pounds per cubic foot;
so that irr^hd would be the weight in pounds, and irr^hdg would
be the weight in poundals. Note that 1 Ib. = g poundals, i.e.,
32.17 poundals (Sec. 32). Accordingly, the pressure produced by
a column of liquid whose height is h feet is hdg poundals per
square foot, or hd pounds per square foot.

114. Transmission of Pressure. — If a tube A (Fig. 67) with
side branches B, C, D and E, is filled with water, it will be
found that the water stands at the same level in each branch
as shown. Further, if A contains four small holes, a, b, c, and
d, all of the same size and at the same level, and covered by valves
a', b', c', and d'} respectively, it will be found that it requires the
same amount of force to hold the valve a' closed against the
water pressure as to hold &', c', or d' closed.

If the branch tube B were removed, everything else being left

160

MECHANICS AND HEAT

just as before, it is evident from symmetry that a small valve
at e in order to prevent water from coming out would have to
resist an upward pressure at e (say p%) equal to the upward
pressure at c, d, etc. With B in place, however, the water does
not come out of e, but is at rest; hence the downward pressure
at e (say pi) due to the column of water in B must just balance
the above-mentioned pressure p2. The pressure, pi, however,
is equal to hdg (Eq. 68). If the pressure at a, 6, c, and d, is
represented by pa, pb, pc, and Pd respectively, we have

The experiment shows, then, that in liquids the pressure (a)
is exactly equal in all directions at a given point (see also experi-
ment below); (6) is transmitted undiminished to all points at the

FIG. 67.

same level; and (c) is numerically hdg, in which h is the vertical
distance from the point in question to the upper free surface of
the liquid causing the pressure.

The above three facts or principles (a), (&), and (c) are funda-
mental to the subjects of hydrostatics and hydraulics. They
are utilized in our city water systems, in hydraulic mining, and
in all hydraulic machinery. They must be reckoned with in
deep-sea diving and in the construction of mill dams and coffer-
dams. In these and hundreds of other ways these principles
find application.

The greater pressure in the water mains in the low-lying por-
tions of the city as compared with the hill sections, is at once ex-
plained by (c), noting that the vertical distance from these points
to the level of the water in the supply tank is greater for these
places than it is on the hills.

An exceedingly simple experimental proof of the principle
(a) may be arranged as follows: A glass jar containing water

PROPERTIES OF LIQUIDS AT REST

161

has placed in it several glass tubes which are open at both ends.
Some of these tubes are bent more or less at the lower end, so
that the lower opening in some cases faces upward, in others
downward, and still others horizontally or at various angles of
inclination. If these openings are all at the same depth, the
fact that the water stands at the same height in all of the tubes,
that is, at the general level of the water in the vessel, shows that
the outward pressure at each lower opening must be the same.
Consequently, since no flow takes place, the inward pressure at
each opening, which is due to the general pressure of the main
body of water, and which is exerted in various directions for the
different tubes, must be the same for all.

Pressure Perpendicular to Walls. — The pressure exerted by a
liquid, against the wall of the containing vessel at any point
is always perpendicular to the wall at that point. For if the
pressure were aslant with reference to the wall at any point, it
would have a component parallel to the wall which would tend
to move the liquid along the wall. We know, however, that the
liquid is at rest; hence the pressure can have no component
parallel to the wall, and is therefore perpen-
dicular to the wall at all points.

115. The Hydrostatic Paradox. — A small
body of liquid, for example the column in
tube B (Fig. 68), may balance a large body
of liquid, such as the column in tube A.
This is known as the Hydrostatic Paradox.
From the preceding sections, we see that the
pressure tending to force the liquid through C
in the direction of arrow 6, is hdg, due to the

column of liquid B, while the pressure tending to force it in the
direction of arrow a is likewise hdg due to the column of liquid A.
Evidently the liquid in C will be in equilibrium and will not tend
to move either to the right or left when these two pressures are
equal, i.e., when h is the same for both columns. Thus, viewed
from the pressure standpoint, we see that there is nothing para-
doxical in the behavior of the liquid. If A contained water and
B contained brine, then the liquid level in A would be higher
than in B (Sec. 116).

116. Relative Densities of Liquids by Balanced Columns. — A
very convenient method of comparing the densities of two liquids,
is that of balanced columns, illustrated in Fig. 69. A U-shaped

FIG- 68.

162

MECHANICS AND HEAT

glass tube, with arms A and B, contains a small quantity of,
mercury C, as shown. If water is poured into the arm A and
at the same time enough of some other liquid, e.g., kerosene, is
poured into the arm B to just balance the pressure of the water
column A, as shown by the fact that the mercury stands at the
same level in both arms; then it is evident that the pressure p2
due to the kerosene, which tends to force C to the left, must
be equal to the pressure p\ due to the water, which tends to force
C to the right. But the former pressure is h2d2g
while the latter pressure is hidig, in which hi and
h2 are the heights of the water and the kerosene
columns respectively, and dt and d2 the respective
densities of the two liquids. Hence

hidig = h2d2g

d2 hi hi

or-r = T-, or d2 = ^-dl

FIG.

The density d\ of water is almost exactly 1 gm.
per cm.3; therefore if hi is found to be 40 cm., and
h2 is found to be 50 cm., then the density of kero-
sene is 4/5 that of water or practically 0.8 gm. per cm.3

117. Buoyant Force. — Any body immersed in a liquid experi-
ences a certain buoyant force. This force, if the body is of
small density compared with the liquid, causes the body to rise
rapidly to the surface. Thus cork floats on water, and iron on
mercury. This buoyant force is due to the fact that the upward
pressure on the body is greater than the downward pressure on it.

Let B, Fig. 70, be a cylindrical body immersed in a vessel of
water. Let AI and A 2 be the areas of the lower and upper ends
respectively, and let pi and p2 be the corresponding pressures.
If AI is 3 times as far below the surface as A2, then pi will equal
3p2. The forces on the sides of B will of course neutralize each
other and produce neither buoyant nor sinking effect. The
entire Buoyant Force of the water uponZ? is, then, Fi—F2, in which
FI is the upward push or force on AI, and F2 the much smaller
downward push on A2. Force, however, is the pressure multi-
plied by the area; i.e.,

I, and F2 = p2 A2, or, since

Buoyant

PROPERTIES OF LIQUIDS AT REST 163

If this buoyant force, which tends to make the body rise, is
(a) greater than the weight W of B, which of course tends to
make it sink, the body will move upward — rapidly if much
greater, and slowly if but little greater. (6) If the buoyant
force is equal to W , then B will remain in equilibrium and float
about in the liquid. Finally (c), if W is greater than the buoy-
ant force, then B will sink to the bottom, and the rapidity with
which it sinks depends upon how much its weight exceeds the
buoyant force.

If the body were of irregular shape such as C, it would be
very difficult to find its area, and also difficult to find the average
vertical components of pressure on the upper
and lower surfaces. It is, nevertheless, ob-
vious that the average downward pressure on
the body would be less than the average up-
ward pressure, and it is just this difference
in pressure that gives rise to the buoyant
force whatever shape the body may have (see
Sec. 118). The horizontal components of
pressure would, of course, have no tendency J^Q 79.

to make the body either float or sink.

118. The Principle of Archimedes. — If any body, whatever
be its shape, e.g., A (Fig. 71), is immersed in a vessel of water, it
will be found to be lighter in weight than if it were weighed in air.
This difference in weight is referred to as the "Loss of Weight"
in water, and is found to be equal to the weight of the water that
would occupy the space now occupied by A. In other words,
the loss of weight in water is equal to the weight of the water dis-
placed. This principle, of course, holds for any other liquid,
and also for any gas (Sec. 134), and is known as the Principle
of Archimedes, so called in honor of the Grecian mathematician
and physicist Archimedes (B. C. 287-212) who discovered it.

Theoretical Proof of Archimedes' Principle. — Imagine the body
A (Fig. 71) to be replaced by a body of water A' of exactly the
same size and shape as A and enclosed in a membranous sack
of negligible weight. It is evident that A' would have no tend-
ency either to rise or to sink. It then appears that this particular
portion of water loses its entire weight, hence it must be true that
the buoyant force exerted upon A' is exactly equal to its weight.
Since this buoyant force is the direct result of the greater average
pressure upon the lower side than upon the upper side of the

164

MECHANICS AND HEAT

body, it can in no wise depend upon the material of which the
body is composed. Consequently, the body A must experience
this same amount of buoyant force, and therefore must lose this
same amount of weight, namely, the weight of the water displaced.
Experimental Proof of Archimedes' Principle. — A small cylin-
drical bucket B is hung from the beam of an ordinary beam
balance, and a solid metal cylinder C (Fig. 72) which accurately
fits and completely fills the bucket is suspended from it. Suf-
ficient mass is now placed in the pan at the other end of the beam
to secure a " balance. " Next a large beaker of water is so placed
that the solid cylinder is immersed. This, of course, buoys it
up somewhat and destroys the "balance." Finally the bucket
is filled with water, whereupon it will be found that exact "bal-

FIG. 71.

FIG. 72.

ance" is restored, i.e., Fi=Fz. This fact shows that the weight
of the water in the bucket just compensates for the buoyant
force that arises from the immersion of the cylinder. In other
words the loss of weight experienced by the cylinder is equal to
the weight of the water which fills the bucket, and is therefore
equal to the weight of the water displaced by the cylinder.

119. Immersed Floating Bodies. — In case the body A (Fig.
71) is denser than water, it will weigh more than the water which
it displaces and will therefore tend to sink. If, however, it has
the same density as water, the buoyant force will be just equal
to its weight, and it will therefore lose its entire weight and float
about in the liquid.

If a tall glass jar is about one-third filled with strong brine
and is then carefully filled with water, the two liquids will mix

PROPERTIES OF LIQUIDS AT REST 165

slightly, so that the jar will contain a brine varying in strength,
and hence in density, from that which is almost pure water at
the top, to a strong dense brine at the bottom. If pieces of resin,
wax, or other substances which sink in water but float in brine
are introduced, they will sink to various depths, depending upon
their densities. Each piece, however, sinks until the buoyant
force exerted upon it is equal to its weight, that is, until the
weight of the liquid displaced is equal to its own weight.

Occasionally the query arises as to whether heavy bodies such
as metals will sink to the bottom of the ocean. They certainly
do, regardless of the depth. To be sure, the enormous pressure
at a great depth compresses the water slightly, making it more
dense, and hence more buoyant. The increase in density due
to this cause, however, even at a depth of one mile amounts
to less than 1 per cent, (closely 3/4 per cent). Since the compres-
sibility of metals is about 1/100 as great as that of water, its
effect in this connection may be ignored. Substances, however,
which are more readily compressed than water, e.g., porous sub-
stances containing air, actually become less buoyant at great
depths.

120. Application of Archimedes' Principle to Bodies Floating
Upon the Surface. — If a piece of wood that is lighter than water
is placed in water, it sinks until the weight of the
water displaced is equal to its own weight. If
placed in brine it will likewise sink until the weight
of the liquid displaced is equal to its own weight;
but it will not then sink so deep. A boat, which with
its cargo weighs 1000 tons, is said to have 1000 tons
"displacement," because it sinks until it displaces
1000 tons of water. As boats pass from the fresh FIG. 73.
water into the open sea they float slightly higher.

If a wooden block B (Fig. 73) is placed in water and comes to
equilibrium with the portion mnop immersed, then the volume
mnop is the volume of water displaced, and the weight of this
volume of water is equal to the entire weight of the block.
Further, if d is 9/10 c, we know that the block of wood displaces
9/10 of its volume of water, hence its density is 0.9 gm. per cm.3
(since a cm.3 of water weighs almost exactly 1 gm.).

Ice is about 9/10 as dense as sea water; consequently icebergs
float with approximately 9/10 of their volume immersed and 1/10
above the surface. If some projecting points are 100 ft. above

166 MECHANICS AND HEAT

the sea, it does not follow, of course, that the iceberg extends 900
ft. below the surface.

121. Center of Buoyancy. — If a rectangular piece of wood is placed in
water in the position shown at the left in Fig. 74, the center of gravity of
the displaced water mnop is at C. This point C is called the Center of
Buoyancy. It is the point at which the entire upward lift or buoyant
force F, due to the water, may be considered as concentrated. The center
of gravity, marked G, is the point at which the entire weight W of the
block of wood may be considered as concentrated. The block in this
position is unstable, since the least tipping brings into play a torque (as

shown at the right in Fig. 74) tending to tip it still
farther. Consequently the block tips over and
floats lengthwise on the water. For the same rea-
son logs do not float on end, but lie lengthwise
on the water.

If a sufficiently large piece of lead were fastened
to the bottom of the block of wood so as to bring
FIG. 74. its center of gravity below its center of buoyancy,

the block would then be stable when floating on end.
Ballast is placed deep in the hold of a vessel in order to lower the center
of gravity. It does not necessarily follow, however, that the center of
gravity of ship and cargo must be below the center of buoyancy of
the ship. For, as the ship rolls to the right, say, the form of the hull is
such that the center of buoyancy shifts to the right, and therefore gives
rise to a righting or restoring torque.

122. Specific Gravity.— The Specific Gravity (S) of a substance
is the ratio of the density of the substance to the density of water
at tne same temperature. Representing the density of water
by d", and the density of the substance referred to by d, we have

S = d/df (69)

Since the value of d' is very nearly one (i.e., one gm. per cm.3)
at ordinary temperatures, it follows that the Specific Gravity
of a substance and its density have almost the same value, but
they must not be considered as identical.

Density, however, is mass divided by volume, so that if we
consider equal volumes of the substance and of water, and repre-
sent the mass of the former by M and that of the latter by M' ,
Eq. 69 may be written

S = d/d' = ~ = M/M' = Mg/M'g = W/W (70)

PROPERTIES OF LIQUIDS AT REST 167

in which W is the weight of a certain volume of the substance
and W the weight of the same volume of water. Hence the
specfic gravity of a substance might be denned as the ratio of
the weight of a certain volume of that substance to the weight of
an equal volume of water.

Specific Gravity of a Liquid. — If a bottle full of liquid, say
kerosene, weighs Wi, and the same bottle full of water weighs
W2, while the empty bottle weighs W3, then W\ — W3 is the weight
W of the kerosene in the bottle, and Wz — Ws is the weight W
of an equal volume of water; hence from Eq. 70 we have for the
specific gravity of kerosene

W _Wi_-TF3
-W'~W2-WS

If a piece of metal which has first been weighed in air, is then
immersed in water and again weighed, it will be found to be
lighter. This "loss of weight" in water, i.e., its weight in air
minus its weight in water, is of course due to the buoyant force
and is equal to the weight of the water displaced. If the piece
of metal is again weighed while immersed in brine, the loss of
weight will be equal to the weight of the brine displaced. This
loss of weight will be greater than the former loss. Dividing
it by the former loss we obtain the specific gravity of the brine.

Specific Gravity of a Solid. — Evidently the volume of any body
immersed in water is exactly equal to the volume of water which
it displaces. Consequently its specific gravity is the ratio of the
weights of these two volumes, or the weight of the body in air
divided by its loss of weight in water. This is a convenient
method for determining the specific gravity of irregular solids,
such as pieces of ore.

If a stone weighs 30 gm. in air and 20 gm. in water, then the
weight of the water it displaces must be 10 gm. ; so that the stone
weighs 3 times as much as the same volume of water and its
specific gravity is, therefore, 3. Since the density of water d'
(Eq. 69) is very slightly less than 1.0 at room temperature, the
density d of the stone would be very slightly less than its specific
gravity.

123. The Hydrometer. — The hydrometer, of which there are
several kinds, affords a very rapid means of finding the specific
gravity of a liquid. It is also sufficiently accurate for most
purposes. The most common kind of hydrometer consists of a

168

MECHANICS AND HEAT

glass tube A (Fig. 75), having at its lower end a bulb B contain-
ing just enough mercury or fine shot to properly ballast it when
floating. From Sec. 120 we see that such an instrument will
sink until it displaces an amount of water equal to its own weight.
To do this it will need to sink deeper in a light liquid than in a
heavy liquid; hence the depth to which it sinks indicates the
specific gravity of the liquid in which it is placed. From a
scale properly engraved upon the stem of the hydrometer, the
specific gravity of the liquid in which it is floating may be read
by observing the mark that is just at the surface.
Thus, if the hydrometer sinks to the point a in a given
liquid, we know that the specific gravity of the liquid
is 1.12, i.e., it is 1.12 times as dense as water. The
scale shown is called a Specific Gravity Scale, because
the specific gravity of the liquid is given directly. It
will be observed that it is not a scale of equal divisions.
The Beaume Scale. — The Beaume Scale, which is
very much used, has on the one hand the advantage
of having equal scale divisions; but on the other hand
it has the disadvantage that it is entirely arbitrary,
and that its readings do not give directly the specific
gravity of the liquid. There are two Beaume scales,
one for liquids heavier than water, the other for liq-
uids lighter than water.

To calibrate a hydrometer for heavy liquids it is
placed in water, and the point to which it sinks is
marked 0°. It is next placed in a 15 per cent, brine
(15 parts salt and 85 parts water, by weight) and the
point to which it sinks is marked 15°. The space be-
tween these two marks is then divided into 15 equal spaces and
the graduation is continued down the stem. If, when placed
in a certain liquid, the hydrometer sinks to mark 20, the spe-
cific gravity of the liquid is 20° Beaume heavy.

For use in light liquids, the point to which the instrument sinks in a
10 per cent, brine is marked 0°, and the point to which it sinks in water
is marked 10°. The space between these two marks is divided into
10 equal spaces, and the graduation is extended up the stem. If, when
placed in a certain liquid, the hydrometer sinks to mark 14, the specific
gravity of the liquid is 14° Beaume light.

124. Surface Tension. — Small drops of water on a dusty or
oily surface assume a nearly spherical shape. Small drops of

FIG. 75.

PROPERTIES OF LIQUIDS AT REST 169

mercury upon most surfaces behave in the same manner. Dew-
drops and falling raindrops are likewise spherical. When the
broken end of a glass rod having a jagged fracture is heated until
soft, it becomes smoothly rounded. These and many other
similar phenomena are due to what is called Surface Tension
(denned in Sec. 126).

Surface tension arises from the intermolecular attraction ^or
cohesion) between adjacent molecules. Some of the effects of
this attraction have already been discussed in Sec. 102. Certain
experiments indicate that these molecular forces do not act ap-
preciably through distances greater than about two-millionths
of an inch. A sphere, then, of two-millionths inch in radius
described about a molecule may be called its sphere of influence,
or sphere of molecular attraction.

Let A, B, and C (Fig. 76) represent respectively a molecule of
water well below the surface, one very near the surface, and one

FIG. 76. FIG. 77.

on the surface; and let the circles represent their respective
spheres of molecular action. Evidently A, which is completely
surrounded by water molecules, will be urged equally in all
directions and hence will have no tendency to move. It will
therefore, barring friction, not require any force to move it about
in the liquid; but, as we shall presently see, it will require a
force to move it to the surface. Accordingly, work is done in
increasing the amount of surface of a liquid (Sec. 126), e.g., as
in inflating a soap bubble. Part of B's sphere of molecular
attraction projects above the surface into a region where there
are no water molecules, and hence the aggregate downward pull
on B exerted by the surrounding molecules is greater than the
upward pull upon it. In the case represented by C, there is no
upward pull, except the negligible pull due to the adjacent mole-
cules of air. Consequently Bt and C, and all other molecules
on or very near the surface, are acted upon by downward (inward)

170 MECHANICS AND HEAT

forces. The nearer a molecule approaches to the surface, the
greater this force becomes.

In Fig. 77, A represents a small water drop and a, b, c, d, etc.,
surface molecules. Since every surface molecule tends to move
inward, the result is quite similar to uniform hydrostatic pressure
on the entire surface of the drop. But such pressure would
arise if the surface layer of molecules were a stretched mem-
branous sack (e.g., of exceedingly thin rubber) enveloping the
drop. This fact, that the surface layer of molecules of any
liquid behaves like a stretched membrane, i.e., like a membrane
under tension, makes the name Surface Tension very appro-
priate. Although there is no stretched film over the drop, the
surface molecules differing in no sense from the inner molecules
except that they are on the surface, it is, nevertheless, very conven-
ient to regard the phenomenon of surface tension as arising from
the action of stretched films, and in the further discussion it will
be so regarded. It must be kept in mind, however, that this
is merely a matter of convenience, and that the true cause of sur-
face tension is the unbalanced molecular attraction just discussed.

When certain insects walk upon the water, it is easily observed
that this "membrane" or ''film" sags beneath their weight.
A needle, especially if slightly oily, will float if carefully placed
upon water. We may note in passing that the weight of the
water displaced by the sagging of the surface film is equal to
the weight of the needle (Archimedes' Principle).

125. Surface a Minimum. — Evidently a stretched film enclosing
a drop of liquid would cause the drop to assume a form having
the least surface, i.e., requiring the least
area of film to envelop it. The sphere
has less surface for a given volume than
JTIG yg any other form of surface. Hence drops

of water are spherical. For the same

reason soap bubbles, which are merely films of soapy water en-
closing air, tend to be spherical. A large drop of water, or
mercury, or any other liquid is not spherical if resting upon a
surface, but is flattened, due to its weight (see Fig. 78). Quite
analogous to this is the fact that a small rubber ball filled with
water and resting upon a plane surface will remain almost spher-
ical; while a large ball made of equally thin rubber would flatten
quite appreciably, due to its greater weight.

If the effect of the weight of the drop is removed, this flatten-

PROPERTIES OF LIQUIDS AT REST 171

ing does not take place even for very large drops. Thus, if a
mixture of alcohol and water having the same density as olive
oil is prepared, it will be found that a considerable quantity of
this oil retains the spherical form when carefully introduced well
below the surface of the mixture.

That a film tends to contract so as to have a minimum area,
and that in so doing it exerts a force, is beautifully illustrated
by the following experiment. If the wire loop B (Fig. 79), to
which is attached a small loop of thread a, is dipped into a soap
solution and withdrawn, it will have stretched across it a film
in which the loop a "floats" loosely as indicated. Evidently
the film, pulling equally in all
directions on a, has no tendency
to stretch it. If, however, the
film within a is broken, the in-
ward pull disappears, whereupon
the outward pull causes the loop Fi^ 79

to assume the circular form

shown at the right (Fig. 79). A loop has its maximum area when
circular; consequently, the annular film between the thread and
the wire must have a minimum area when the thread loop is
circular.

If a piece of sealing wax with sharp corners is heated until
slightly plastic, the corners are rounded, due to surface tension
of the wax; and in this rounding process the amount of surface is
reduced. Glass and all metals behave in the same way when
sufficiently heated. All metals when melted, indeed all sub-
stances when in the liquid state, exhibit surface tension. This
property is utilized in making fine shot by dropping molten lead
through the air from the shot tower. During the fall, the drops
of molten lead cool in the spherical form produced by surface
tension.

126. Numerical Value of Surface Tension. — The Surface
Tension T of a liquid is numerically the force in dynes with which
a surface layer of this liquid one centimeter in width resists being
stretched. There are several methods of finding the surface
tension, in all of which the force required to stretch a certain
width of surface layer is first determined. This force, divided
by the width of the surface layer stretched, gives the value of
the surface tension.

The simplest method of finding the surface tension is the follow-

172 MECHANICS AND HEAT

ing: An inverted U of fine wire 1/2 cm. in width is immersed
in a soap solution (Fig. 80) and then suspended from a sensitive
Jolly balance. (The Jolly balance is practically a very sensitive
spring balance.) Since the film across the U has two surfaces,
one toward and one away from the reader, it is evident that in
raising the U, a surface layer 1 cm. in width must be stretched.
Hence the reading of the Jolly balance (in dynes) immediately
before the film breaks minus the reading after, gives the surface
tension for the soap film in dynes per centimeter. For pure
water, T is approximately 80 dynes per cm. Its value decreases
due to rise in temperature, and also due to the presence of im-
purities (Sec. 127.) Observe that T is numerically the force re-
quired to keep stretched a surface layer having
a width (counting both sides) of 1 cm.

In raising the wire (Fig. 80) a distance of 1
cm., a force of 80 dynes (for pure water) must
be exerted through a distance of 1 cm., that is,
80 ergs of work must be done. But 1 cm.2 of
surface has been formed; showing that 80 ergs of
work are required to form 1 cm.2 of surface. In

FIG 80 other words, 80 ergs of work are -required to

cause enough molecules to move from position
A to that of C (Fig. 76) to form 1 cm.2 of additional surface.

Observe that a soap bubble has an outer and an inner surface.
Between these two surfaces is an exceedingly thin layer of soapy
water. This soapy water, as it flows down between the two
surfaces, forms the drop which hangs below the bubble and at
the same time causes other portions of the bubble to become
thinner and thinner until it finally bursts. The greater vis-
cosity of soapy water, as compared with pure water, causes the
downward flowing to be much slower than with pure water, and
therefore causes a soap bubble to last much longer than a water
bubble.

In blowing a soap bubble, work is done upon the film in in-
creasing its area; on the other hand, if the film is permitted to
contract by forcing air out through the pipestem, work is evi-
dently done by the film. Barring friction, these two amounts
of work must be equal.

In another method of determining surface tension, quite similar
in principle to the one just given, a wire ring suspended in a hori-
zontal position from a Jolly balance is lowered until it rests flat

PROPERTIES OF LIQUIDS AT REST 173

upon the water, and is then raised, say 1/16 inch. In this posi-
tion it would be found that a film tube of water, having the di-
ameter of the ring and a length of 1/16 inch, connects the ring
with the water and exerts upon the ring a downward pull. The
reading of the Jolly balance just before this film breaks, minus
the reading after (or F\, say), gives this downward pull. The
width of surface layer that is stretched is twice the circumference
of the ring, or 4irr. Note that a tube has an outer and an inner
surface. Hence

which may be solved for T.

127. Effect of Impurities on Surface Tension of Water. —
Most substances when dissolved in water produce a marked
decrease in its surface tension. For this reason, parings of cam-
phor move rapidly over the surface of water if

dropped upon it. Let A, B, and C, Fig. 81, be

three pieces of camphor upon the surface of

water. The piece A dissolves more rapidly from

the point a than elsewhere, so that the surface

tension on the end a is reduced more than on the

opposite end, and the piece moves in the direc FIG. 81.

tion of the stronger pull, as indicated by the ar-

row. In the case of C, this same effect at c gives rise to a rotary

motion, as shown; while B describes a curved path due to the

same cause.

128. Capillarity. — If a glass A, Fig. 82, contains water, and
another glass B contains mercury, it may easily be observed that
most of the surface of each is perfectly flat, as shown, but that
near the edge of the glass, the water surface curves upward, while
the mercury surface curves downward. If the glass A were made
slightly oily, the water would curve downward; while if B were
replaced by an amalgamated zinc cup, the downward curvature
of the mercury would disappear. Thus the form of the surface
depends upon both the liquid and the containing vessel.

If a clean glass rod is dipped into water and then withdrawn, it
is wet. This shows that the adhesion between glass and water
exceeds the cohesion between the water particles. For the water
that wets the glass rod must have been more strongly attracted
by the glass than by the rest of the water, or it would not have
come away with the rod. If the glass rod is slightly oily it will

174

MECHANICS AND HEAT

FIG. 82.

not be wet after dipping it into the water. If a clean glass rod
is dipped into mercury and then withdrawn, the fact that no
mercury comes with it shows that the cohesion between mercury
molecules exceeds the adhesion between mercury and glass mole-
cules. It is, indeed, the relative values of cohesion and adhesion
that determine surface curvature at edges. If the cohesion of
the liquid molecules for each other just equals their adhesion for
the substance of which the containing vessel is made, the sur-
face will be flat from edge to edge. If greater, the curvature is
downward (B, Fig. 82), while if smaller, it is
upward (A, Fig. 82). Thus, in the latter
case, the water at the edge rises above the
general level, wetting the surface of the
glass, simply because glass molecules at-
tract water molecules more strongly than

other water molecules do. This phenomenon is most marked
in the case of small tubes (capillary tubes) and is therefore
called capillarity.

129. Capillary Rise in Tubes, Wicks, and Soil. — If clean glass
tubes a and b (Fig. 83) are placed in the vessel of water A, and
c and d in the vessel of mercury B, it will be found that the capil-
lary rise in a and 6, and the capillary depression in c and d is
greater for the tube of smaller bore. Indeed, it will be shown in
the next section, and it is eas-
ily observed experimentally
with tubes of different bore,
that a given liquid rises n times
as high in a tube of 1/n times
as large bore.

Any porous material produces
a marked capillary rise with any
liquid that wets it. There are FIG. 83.

numerous phenomena due to

capillary action, many of which are of the greatest importance.
If one corner of a lump of sugar, or clod of earth, touches the water
surface, the entire lump or clod becomes moist. Due to capil-
larity, the wick of a lamp carries the oil to the flame where it is
burned. If the substratum soil is moist, this moisture, during a
dry time, is continually being carried upward to the roots of plants
by the capillary action of the soil. Capillarity is probably an
important factor, in. connection with osmosis (Sec. 112), in the

PROPERTIES OF LIQUIDS AT REST 175

transference of liquid plant food from the rootlets to the topmost
parts of plants and trees.

Cultivating the soil to the depth of a, few inches greatly reduces
the amount of evaporation, and hence helps retain the moisture
for the use of the plants. For, stirring the ground destroys, in a
large measure, the continuity, and hence the capillary action,
between the surface soil and the moist earth a few inches below.
Consequently the surface soil dries more quickly, and the lower
soil more slowly, than if the ground had not been stirred.

130. Determination of Surface Tension from Capillary Rise
in Tubes. — In Fig. 84, B represents a capillary tube having a
bore of radius r cm., and giving, when placed
in water, a capillary rise of h cm. It may be
considered to be the upward putt of the sur-
face layer / that holds the column of water
in the capillary tube above the level of the
water in the vessel. The weight of this col-
umn is Trr'hdg (see Eq. 68, Sec. 113). The
hemispherical surface layer that sustains this
weight, however, is attached to the bore of FIG. 84.

the tube by its margin abc (as shown at A), so
that the "width of surface" (see Sec. 126) that must support this
weight is 2irr, consequently

T = \rhdg dynes per cm. (71)

The above method is the one most frequently used for deter-
mining surface tension. It is usually necessary first to clean the
tube with nitric acid or caustic soda, or both, and then carefully
rinse before making the test.

PROBLEMS

1. What is the pressure at a depth of 2 mi. in the ocean?

2. A water tank has on one side a hole 10 cm. in diameter. What force
will be required to hold a stopper in the hole if the upper edge of the hole is
4 meters below the water level?

3. What horizontal force will a lock gate 40 ft. in width exert on its sup-
ports if the depth of the water is 18 ft. above the gate, and 6 ft. below it?

4. Express a pressure of 15 Ibs. per in.2 in dynes per cm.2

5. The right arm of a U-tube, such as shown in Fig. 69, contains mercury
only and the left arm some mercury upon which rests a column of brine 60

176 MECHANICS AND HEAT

cm. in height. The mercury stands 5.2 cm. higher in the right arm than
in the left. What is the density of the brine? Sketch first.

6. The weight of a stone in air is 60 gm., in water 38 gm., and in a certain
oil 42 gm. What is the sp. gr. (a) of the stone? (6) of the oil?

7. Two tons increase in cargo makes a boat sink 1.2 in. deeper (in fresh
water). What is the area of a horizontal section of the boat at the water
line?

8. A marble slab (density 2 . 7 gm. per cm.3) weighs 340 Ibs. when immersed
in fresh water. What is its volume?

9. How much lead must be attached to 20 gm. of cork to sink it in fresh
water? Consult table of densities, Sec. 101.

10. What capillary rise should water give in a tube of (a) 1 mm. bore,
(b) 2 mm. bore?

11. A wire ring of 5 cm. radius is rested flat on a water surface and is then
raised. The pull required to raise it is 5 gm. more before the "film" breaks
than it is after. What value does this give for the surface tension?

CHAPTER XI
PROPERTIES OF GASES AT REST

131. Brief Mention of Properties. — Gases have all the prop-
erties of liquids that are mentioned in Sec. 112 (to which section
the reader is referred) except solvent action and surface tension.
Gases have also properties not possessed by liquids, one of which
is Expansibility.

Viscosity. — The viscosity of gases is much smaller than that of
liquids, but it is not zero, nor is it even negligible. In order to
force water to flow rapidly through a long level pipe, the pressure
upon the water as it enters the pipe must be considerably greater
than the pressure upon it as it flows from the pipe. This differ-
ence in pressure is known as Friction Head. It requires a pressure
difference or pressure drop to force water through a level pipe
because of the viscosity of water. To produce the same rate of
flow through a given pipe would require a much greater pressure
drop if the fluid used were molasses instead of water, and very
much smaller drop if the fluid used were a gas. This difference is
due to the fact that the viscosity of water is less than that of
molasses and greater than that of the gas. The slight viscosity
of illuminating gas necessitates a certain pressure drop to force
the required flow through the city gas mains.

Usually in ascending a high tower there is a noticeable, steady
increase in the velocity of the wind; which shows that the higher
layers of air are moving more rapidly than those below (compare
with the flowing of a river, Sec. 112). Indeed, just as in the case
of the layers of water in the river, each layer experiences a forward
drag due to the layer above it and a backward drag due to the
layer below it, and therefore moves with an intermediate velocity.
The lower layers are retarded by trees and other obstructions.

It is probable that the viscosity of gases should not be attrib-
uted to molecular friction but rather to molecular vibration (see
Kinetic Theory of Matter, Sec. 99). Consider a rapidly moving
stratum of air gliding past a slower moving stratum below. As
molecules from the upper stratum, due to their vibratory motion,

177

178 MECHANICS AND HEAT

wander into the lower stratum, they will, in general, accelerate
it; whereas molecules passing from the lower stratum to the upper
will, in general, retard the latter. Thus, any interchange of
molecules between the two strata results in an equalization of the
velocities of the portions of the strata near their surface of separa-
tion. Of course sliding (molecular) friction would produce this
same result, but the fact that a rise in temperature causes the
viscosity to decrease in liquids and increase in gases, points to a
difference in its origin in the two cases. As a gas is heated, the
vibrations of its molecules, according to the Kinetic Theory of
Gases (Sec. 171), become more violent, thus augmenting the above
molecular interchange between the two layers and thereby
increasing the apparent friction between them.

Diffusion. — Diffusion is very much more rapid in the case of
gases than with liquids, probably( because of greater freedom of
molecular vibration. Thus if some carbon dioxide (CO2) is
placed in the lower part of a vessel and some hydrogen (H) in the
upper part, it will be found after leaving them for a moment that
they are mixed due to diffusion; i.e., there will be a large percent-
age of carbon dioxide in the upper portion of the vessel, notwith-
standing the fact that it is more than twenty times as dense as
hydrogen. Escaping coal gas rapidly diffuses so that it may soon
be detected in any part of the room. An example of gas Osmosis
has already been given (see Sec. 112).

Since gases have weight, they produce pressure for the same
reason that liquids do (Sec. 113). Thus the air produces what is
known as atmospheric pressure, which is about 15 Ibs. per sq. in.
In the case of illuminating gas, we have an example of Transmis-
sion of Pressure by gas from the gas plant to the gas jet. Another
example is the transmission of pressure from the bicycle pump to
the bicycle tire.

Elasticity. — Gases, like liquids, are perfectly elastic, i.e.,
after being compressed they expand to exactly their original
volume upon removal of the added pressure. Gases are very
easily compressed as compared with liquids. Indeed, if the pres-
sure upon a given quantity of gas is doubled or trebled, its volume
is thereby reduced very closely to 1/2 or 1/3 its original volume,
as the case may be. The fact that doubling the pressure on a
certain quantity of gas halves the volume, or, in general, increas-
ing the pressure n-fold reduces the volume to \/n the original
volume provided the temperature is constant, is known as Boyle' 's

PROPERTIES OF GASES AT REST 179

Law. This very important gas law will be further considered in
Sec. 139. It may be stated that Boyle's law does not apply
rigidly to any gas, but it does apply closely to many gases, and
through wide ranges of pressure.

Expansibility. — Gases possess a peculiar property not possessed
by solids or liquids, namely, that of indefinite expansibility
(Sec. 98). A given mass of any gas may have any volume,
depending upon the pressure (and also the temperature) to which
it is subjected. If the pressure is reduced to 1/10 its original
value the volume expands 10-fold, and so on. A mass of gas,
however small, always (and instantly) expands until it entirely
fills the enclosing vessel.

The expansibility and also the compressibility of a gas may be
readily shown by the use of the apparatus sketched in Fig. 85.
A is a circular brass plate which is perfectly
flat and smooth on its upper surface. B is
a glass bell jar turned open end down
against A . The lower edge D of B is care-
fully ground to fit accurately against the
upper surface of A, over which some vase-
line is spread. A and B so arranged con- ^
stitute what is called a receiver. The re-
ceiver forms an air-tight enclosure in which
is placed a bottle C, across the mouth of
which is secured a thin sheet of rubber a, thus enclosing some air
at ordinary atmospheric pressure.

By means of the pipe E leading to an air pump, it is possible
to withdraw the air from the space H within the receiver, or to
force air into the space H. In the former case the air pressure
in H is reduced so as to be less than one atmosphere, and the thin
membrane of rubber stretches out into a balloon-like form a\ ;
while in the latter case, that is, when the air in H is com-
pressed, this increased pressure, being greater than the pressure
of the air confined in C, causes the membrane to assume the form
a2. The process by which the air pump is able to withdraw from
H a portion of the air, also depends upon the property of expansi-
bility. A reduction of pressure is produced in the pump, where-
upon the air in H expands and rushes out at E. (This process
will be further considered in Sees. 145 and 147.)

Gas Pressure and the Kinetic Theory. — According to the
Kinetic Theory of Gases (Sec. 171), the pressure which a «gas

180 MECHANICS AND HEAT

exerts against the walls of the enclosing vessel is due to the bom-
bardment of these walls by the gas molecules in their to-and-fro
motion. The fact that the ratio of the densities of any two gases,
e.g., carbon dioxide and hydrogen, when subjected to the same
pressure and temperature, is the same as the ratio of their mo-
lecular weights, shows that a certain volume of hydrogen contains
the same number of molecules as does the same volume of carbon
dioxide or any other gas under like conditions as to pressure and tem-
perature. This is known as Avogadro's Law. It will be recalled
that momentum change is equal to the impulse required to pro-
duce it (Eq. 19, Sec. 45). Consequently, since the hydrogen
molecule is 1/16 as heavy as the oxygen molecule, it will need to
have 4 times as great velocity as the oxygen molecule to produce
an equal contribution toward the pressure. For each impulse
of the hydrogen molecule would then be 1/4 as great as those of
the oxygen molecule, but, because of the greater velocity of the
former, these impulses would occur 4 times as often.

Knowing the density of the gas, it is comparatively easy to
compute the molecular velocity required to produce the observed
pressure. The average velocity of the hydrogen molecule at 0° C.
is, on the basis of this theory, slightly more than 1 mi. per sec.,
while that of the oxygen molecule is 1/4 as great, as already
explained.

The very rapid diffusion of hydrogen as compared with other
gases would be a natural consequence of its greater velocity, and
therefore substantiates the kinetic theory. The observed
increase in pressure resulting from heating confined gases is
attributed to an increase in the average velocity of its molecules
with temperature rise. The kinetic theory of gas pressure
affords a very simple explanation of Boyles' law (close of Sec.
139).

132. The Earth's Atmosphere. — Because of the importance and
abundance of the mixture of gases known as air, the remainder of
the chapter will be devoted largely to the study of it. It may
be remarked that most of the gases are very much like air with
respect to the properties here discussed.

The term "atmosphere" is applied to the body of air that sur-
rounds the earth. Dry air consists mainly of the gases nitrogen and
oxygen — about 76 per cent, of the former and 23 per cent, of the
latter, by weight. The remaining 1 per cent, is principally argon.
In addition to these gases there are traces of other gases, impor-

PROPERTIES OF GASES AT REST 181

tant among which are carbon dioxide (C02) and water vapor.
The amount of carbon dioxide in the air may vary from 1 part in
3000 outdoors (not in large cities), to 10 or 15 times this amount in
crowded rooms. The oxygen of the air in the lungs (see Osmosis,
Sec. 112) is partially exchanged for carbon dioxide and other
impurities of the blood; as a result the exhaled air contains 4 or
5 per cent, carbon dioxide. If the breath is held for an instant
and then carefully and slowly exhaled below the burner of a lamp
(the hands being held in such a position as to exclude other air
from the burner), the flame is quickly extinguished. The air in
this case does not have enough oxygen to support combustion.
Through repeated inhalation, the air in crowded, poorly venti-
lated rooms becomes vitiated by carbon dioxide. Carbon dioxide
escapes from fissures in the earth and forms the deadly "choke
damp" of mines. It also results from the explosion of "fire
damp," or marsh gas (CH4), as it is known to the chemist. If a
candle when carefully lowered into a shaft is extinguished upon
reaching the bottom, the presence of choke damp is indicated.

In nature, even in deserts, air never occurs dry. The amount
of water vapor in the air varies greatly, sometimes running as
high as 1/2 oz. per cubic yard (about 1.5 per cent.) in hot, sultry
weather. As moist air is chilled, its ability to retain water vapor
decreases rapidly and precipitation (Sec. 221) occurs. Conse-
quently during extremely cold weather the air is very dry.

133. Height of the Atmosphere. — As meteors falling toward
the earth strike the earth's atmosphere, the heat developed by
them through air friction as they rush through the upper strata
of rarefied air causes them to become quite hot, so that they
shine for an instant. Suppose that one is seen at the same in-
stant by two observers 40 or 50 miles apart. The meteor will
appear to be in a different direction from one observer than from
the other. This makes possible the calculation of the height of
the point at which the meteor began to glow. But it could not
glow before striking the earth's atmosphere; hence the earth's
atmosphere extends to at least that height.

The duration of twilight after sunset also enables the calcula-
tion of the height of the atmosphere. Fine dust particles float-
ing in the upper regions of the air are, of course, flooded with
sunlight for a considerable time after sunset. The general glow
from these particles constitutes twilight. If an observer at A
(Fig. 85a) looking in the direction AX observes the last trace of

182 MECHANICS AND HEAT

twilight when it is sunset at B, then the intersection X of the
tangents at A and B is the highest point at which there are enough
dust particles to give appreciable twilight effect.

Knowing the angle 0 and the radius of the earth, the height of
X above the earth is readily found. Since twilight lasts until
the sun is 15 or 20 degrees below the horizon, we see that 9 is 15
or 20 degrees. If 6 is 18°, X is about 50 miles above the earth.
Extremely rare air, almost free from dust particles, doubtless
extends far above this height. Estimates of the height of the
atmosphere range from 50 to 200 miles.

The upper strata of air are very rare and the lower strata
comparatively dense due to compression caused by the weight
of the air above; so that upon a mountain 3.5 miles high about
half of the weight of the atmosphere is above and half below.
The entire region above 7 miles contains only 1/4 of the earth's
atmosphere.

FIG. 85a.

134. Buoyant Effect, Archimedes' Principle, Lifting Capacity of
Balloons. — Since air has weight, it produces a certain buoyant effect
just as liquids do, but since it is about 1/800 as dense as water,
the buoyant effect is only 1/800 as great. That air has weight
may easily be shown by weighing a vessel, e.g., a brass globe,
first with air in it, and then weighing it again after the air has
been partially pumped out of it by means of an air pump. The
difference in weight is the weight of the air withdrawn. Galileo
(1564-1642) weighed a glass globe when filled with air at atmos-
pheric pressure, and again after forcing air into it. The observed
increase in weight he rightly attributed to the additional air
forced in.

Archimedes' Principle (Sec. 118) applies to gases as well as to
liquids; therefore any body weighed in air loses weight equal to
the weight of the air displaced by the body. Thus a cubic yard

PROPERTIES OF GASES AT REST 183

of stone, or any other material, weighs about two pounds less in
air than it would in a Vacuum, i.e., in a space from which all air
has been removed. The buoyant force exerted by the air upon a
150-lb. man is about 3/16 lb., i.e., 1/800X150 lb.; since his body
has about the same density as water. Observe that he would
lose practically his entire weight if immersed in water; hence, since
air is about 1/800 as dense as water, he loses 1/800 of his weight
by being immersed in air.

The lifting capacity of a balloon, if it were not for the weight
of the balloon itself and the contained gas, would be the weight of
the air displaced, or approximately 2 Ibs. for each cubic yard of
the balloon's volume. If a balloon is filled with a light gas, e.g.,
with hydrogen, its lifting capacity is much more than if filled
with a heavier gas. The car or basket attached to a balloon
contains ballast, which may be thrown overboard when the
aeronaut wishes to rise higher. When he wishes to descend he
permits some of the gas to escape from the balloon, thereby
decreasing the volume and hence the weight of the air displaced.

135. Pressure of the Atmosphere. — Since the air has weight,
the atmosphere must inevitably exert pressure upon all bodies
with which it comes in contact. This pressure at sea level is
closely 14.7 Ibs. per sq. in., and at an altitude of 3.5 miles,
about half of this value. Ordinarily the atmospheric pressure
is not observable. It seems hard to believe that the human body
withstands a pressure of about 15 Ibs. on every square inch of
surface, which amounts to several tons of force upon the entire
body, without its even being perceptible. It is certain, however,
that such is the case. We may note in this connection that the
cell walls in the tissues of the body do not have to sustain this
pressure, since the cells are filled with material at this same pres-
sure. Thus, the atmospheric pressure of about 15 Ibs. per sq. in.
has no tendency to crush the lung cells when they are filled with
air at this same pressure. Sudden changes in pressure, however,
such as accompany rapid ascent or descent in a balloon, or in a
diving bell, produce great discomfort.

The pressure exerted by water at a depth of about 34 ft. is one
atmosphere (Sec. 136), so that a diver 34 ft. below the surface of a
lake experiences a pressure of 2 "atmospheres," one atmosphere
due to the air, and one due to the water. Divers can work more
than 100 ft. beneath the surface of water, and must then experi-
ence a pressure of 4 or 5 atmospheres, i.e., 60 or 75 Ibs. per sq.

184

MECHANICS AND HEAT

in. The air which the diver breathes must, under these circum-
stances, be also under this same high pressure.

The pressure of the atmosphere acts in a direction which is at
all points perpendicular to the surface of a body immersed in it.
Compare the similar behavior of liquids (Sec. 114). That the
atmospheric pressure may be exerted vertically upward, and that
it may be made to lift a heavy weight, is forcibly shown by the
following experiment.

A cylinder A, having a tight-fitting piston P to which is
attached the weight W, is supported as shown (Fig. 86). If,
by means of an air pump connected to the tube C, the air is
partly withdrawn from the space B, it will be found
that P will rise even if W is very heavy. If it were
possible to remove all of the air from B, producing
in the cylinder a perfect vacuum, the pressure within
the cylinder, and hence the downward pressure on P
would be zero. The upward pressure upon P, be-
ing atmospheric pressure or about 14.7 Ibs. per sq.
^ c \. in., would enable it to lift 147 Ibs., provided it had

an area of 10 sq. in.

If only part of the air is withdrawn from B, so
that the pressure within the cylinder is say 5 Ibs.
per sq. in., P would then exert a lifting force of 14.7
minus 5, or only 9.7 Ibs. for each square inch of its
surface. The pressure of the atmosphere cannot
be computed by use of the formula p = hdg; because
the height is uncertain, and also because the density d varies, be-
ing much less at high altitudes. The pressure is very easily
obtained, however, by means of the barometer described in the
next section.

136. The Mercury Barometer. — There are several different
kinds of barometers. The simplest, and also the most accurate
form is shown in Fig. 87. Various devices found in the practical
instrument for making adjustments, and for determining very
accurately the height of the mercury column (vernier attach-
ment), are omitted in the sketch for the sake of simplicity in
showing the essentials and in explaining the principle involved.
A glass tube A, about 1/3 in. in diameter and 3 ft. in length,
and closed at the end a, is filled with mercury, and then, a
stopper being held against the open end to prevent any mercury
from escaping, it is inverted and placed open end down in a vessel

FIG. 86.

PROPERTIES OF GASES AT REST

185

of mercury B, as shown. Upon removing the stopper, it might
be expected that the mercury would run out until it stood at the
same height inside and outside the tube. Indeed it would do this
if there were at a the slightest aperture to admit the air to the
upper portion of the tube, for then the pressure inside and out-
side the tube would be exactly the same, namely, atmospheric
pressure. If a is perfectly air-tight, it will be found that some
mercury runs out of the tube until the upper surface sinks to a
point c. The height h of the mercury column c to 6, is called
the Barometric Height, and is usually about 30 in. near sea level.
Evidently the space a to c contains no air nor
anything else. Such a space is called a Vacuum.
The downward pressure on the surface of the mer-
cury at c is then zero.

This experiment was first performed in 1643
by Torricelli (1608-1647) and is known as Tarri-
celli's experiment. A few years later the French
physicist Pascal (1623-1662) had the experiment
performed on a mountain, and found, as he had
anticipated, that the column be was shorter there
than at lower altitudes.

Consider the horizontal layer of mercury particles
6 within the tube and on the same level as the sur-
face s outside the tube. The downward pressure
on this layer is hdg in which h is the height of the
column be, and d is the density of mercury (13.596
gm. per cm.3). But the upward pressure on this
layer 6 must have this same value, since the layer is
in equilibrium. The only cause for this upward
pressure, however, is the pressure of the atmosphere upon the
surfaces of the mercury, which pressure is transmitted by the
mercury to the inside of the tube. Hence the pressure of the at-
mosphere is equal to the pressure exerted by the mercury column,
or hdg. The barometric height varies greatly with change of
altitude and also considerably with change of weather. Stand-
ard atmospheric pressure supports a column of mercury 76 cm.
in height, at latitude 45° and at sea level (0 = 980.6); hence
Standard Atmos. Pr. =hdg = 76X13.596X980.6= 1,000,000 dynes
per sq. cm. (approx.).
This is approximately 14.7 Ibs. per sq. in.

Quite commonly the pressure of the atmosphere is expressed

FIG. 87.

186 MECHANICS AND HEAT

simply in terms of the height of the barometric column which it
will support, as "29.8 in. of mercury," "74 cm. of mercury." At
sea level the pressure of the atmosphere is usually about 30 in.
of mercury; at an altitude of 3.5 mi., about 15 in. of mercury;
while aeronauts at still higher altitudes have observed as low a
barometric height as 9 in.

Unless great care is taken in filling the tube (Fig. 87), it will
be found that some air will be mixed with the mercury, and that
therefore the space from a to c, instead of containing a vacuum,
will contain some air at a slight pressure. This counter pressure
will cause the mercury column to be somewhat shorter than it
otherwise would be, and the barometer will accordingly indicate
too low a pressure. If the mercury is boiled in the tube before
inverting, the air will be largely driven out and the error from this
source will be greatly reduced. It will be evident that this slight
counter pressure of the entrapped air, in case a trace of air is left
in the space ac, plus hdg for the column of mercury be, gives the
total downward pressure at b. But this total pressure must equal
the upward pressure at 6, due to the atmosphere as shown.
Hence hdg will give a value for the atmospheric pressure, which is
too small by exactly the amount of pressure on c, due to the
entrapped air.

Since water is only 1/13.6 times as dense as mercury, it follows
that atmospheric pressure will support 13.6 times as long a col-
umn of water as of mercury, or about 13.6X30 in., which is
approximately 34 ft. Accordingly, the pressure required to
force water through pipes a vertical height of 340 ft. is approxi-
mately 10 atmospheres, or 150 Ibs. per sq. in., in addition to the
pressure required to overcome friction in the pipes.

137. The Aneroid Barometer. — The Aneroid Barometer con-
sists of an air-tight metal box of circular form having a corrugated
top and containing rarefied air. As the pressure of the atmos-
phere increases, the center of this top is forced inward, and when
the pressure decreases the center moves outward, due to the
elasticity of the metal. This motion of the center is very slight
but is magnified by a system of levers connecting it with a pointer
that moves over the dial of the instrument. The position of this
pointer upon the dial at a time when the mercury column of a
simple barometer is 75 cm. high is marked 75, and so on for
other points. This type of barometer is light, portable, and
easily read.

PROPERTIES OF GASES AT REST 187

138. Uses of the Barometer. — Near a storm center the atmos-
pheric pressure is low (Sec. 225), consequently a falling barom-
eter indicates an approaching storm. Knowing the barometric
readings at a great number of stations, the Weather Bureau can
locate the storm centers and predict their probable positions a
few days in advance. Thus this Bureau is able to furnish infor-
mation which is especially valuable to those engaged in shipping.

Due to the capricious character of the weather, these predic-
tions are not always fulfilled. Although the forcasting of the
weather a year in advance is absolute nonsense, there are many
who have more or less faith in such forcasts. Of course one is
fairly safe in predicting "cold rains" for March, "hot and dry"
for August, etc., but to fix a month or a year in advance the date
of a storm from the study of the stars (which certainly have
nothing to do with the weather), is surely out of place in this
century.

As stated in Sec. 136, the barometric height decreases as the
altitude increases. Near sea level the rate of this decrease is
about 0.1 in. for each 90 ft. of ascent. At higher altitudes this
decrease is not so rapid because of the lesser density of air in those
regions. A formula has been developed, by the use of which the
mountain climber can determine his altitude fairly well from the
readings of his barometer. An "altitude scale" is engraved on
many aneroid barometers, by means of which the altitude may be
roughly approximated.

139. Boyle's Law. — The volume of a given mass of gas, mul-
tiplied by the pressure to which it is subjected, is found to be
nearly constant if the temperature remains unchanged. This
is known as Boyle's Law and may be written

pV (temp, constant) =K (72)

This important law was discovered by Robert Boyle (1627-
1691) and published in England in 1662. Fourteen years later
it was rediscovered by the French physicist Marriotte. This
illustrates the slow spread of scientific knowledge in those days.
In France it is called Marriotte's Law.

From the equation it may be seen that to cause a certain vol-
ume of gas to shrink to 1/n its original volume will require the
pressure to be increased n-fold, provided that the temperature
remains constant. The equation also shows that if we permit
a certain mass of confined gas to expand to, say, 10 times its

188 MECHANICS AND HEAT

original volume, then the new pressure will be 1/10 as great as
the original pressure. By original pressure and volume we mean
the pressure and volume before expansion occurred. As already
stated, Boyle's Law applies closely to many gases, rigidly to
none.

To illustrate Boyle's Law by a problem, let P (Fig. 88) be
an air-tight, frictionless piston of, say, 10 sq. in. surface and of
negligible weight, enclosing in vessel A a quantity of air at atmos-
pheric pressure, say 15 Ibs. per sq. in.' Let it be required to find
how heavy a weight must be placed upon P to force it down to
position PI, thereby compressing the entrapped air to 1/3 its
original volume.

From Eq. 72, we see that the pressure of the entrapped air in
the latter case will be increased 3-fold and hence will exert upon P
when at PI, an upward pressure of 45 Ibs. per sq. in.
The outside atmosphere exerts a pressure of 15 Ibs.
per sq. in. on P; consequently the remaining 30
Ibs. pressure required to hold P down must be
furnished by the added weight. A pressure of 30
Ibs. per sq. in. over a piston of 10 sq. in. surface
amounts to 300 Ibs. force; hence the added weight
required is 300 Ibs.

I t

FIG. 88. We may explain Boyle's Law in full accord with

the Kinetic Theory of gas pressure (Sec. 131). For
when the volume of the air in the vessel represented in Fig. 88 is
reduced to 1/3 its original volume, the molecules, if they con-
tinue to travel at the same velocity, would strike the piston
three times as frequently, and experience each time the same
amount of momentum change, as in the original condition.
They would therefore produce three times as great pressure
against the piston as they did in the original condition, which,
it will be noted, accords with experimental results.

140. Boyle's Law Tube, Isothermals of a Gas. — A bent glass
tube A (Fig. 89), having the short arm closed at e, and the long
arm open and terminating in a small funnel at 6, is very conven-
ient to use in the verification of Boyle's Law. The method of
performing the experiment is given below.

A few drops of mercury are introduced into the tube and ad-
justed until the mercury level c in the long arm is at the same
height as the mercury level d in the short arm. As more mer-
cury is poured into the tube at 6, the pressure on the air enclosed

PROPERTIES OF GASES AT REST

189

in de is increased, which causes a proportional decrease in its
volume.

If now we plot these values of the pressure as ordinates (Sec.
41) and the corresponding values of the volume as abscissae,
we obtain, provided the room temperature is 20°, the curve
marked 20° in Fig. 90. This is called the Isothermal for air at
20° C.

Method in Detail. — If the barometer reads 75 cm., that is, if
the atmospheric pressure is 75 cm. of mercury, then, since c and
d are at the same level, it follows that the pressure on the en-
trapped air is 75 cm. of mercury. If the tube has 1 sq. cm. cross

\dt

FIG. 89.

section and de is 20 cm., then the corresponding volume of the
air is 20 cm.3 Accordingly the point marked A on the curve
(ordinate 75, abscissa 20) represents the initial state of the
entrapped air. Next, mercury is poured into 6 until it stands at
Ci and di in the tubes. If the vertical distance from c\ to d\ is
25 cm., the pressure upon the air in d\e\ will be 25 cm. more than
atmospheric pressure, or a total of 100 cm. Since this is 4/3
of the initial pressure, the corresponding volume should be 3/4
of the initial volume, or 15 cm.3 Measurement will show that
dtfi is 15 cm.3 Hence point B (ordinate 100 and abscissa 15)

190

MECHANICS AND HEAT

represents the new state of the entrapped air as regards its pres-
sure and volume. When still more mercury is poured in, the
mercury stands at, say, c2 and d%, the vertical distance c2c?2 being
75 cm. The pressure upon the entrapped air (dzez) is now this
75 cm. plus atmospheric pressure, or a total of 150 cm. Since
this pressure is twice the initial pressure, the corresponding vol-
ume is, as we should expect, one-half the original volume, or
10 cm.3 Hence the point on the curve marked C (ordinate 150,
abscissa 10) represents this, the third state of the entrapped air.
To obtain smaller pressures than one atmosphere, a different
form of apparatus shown at the right in Fig. 89 is more conven-

60 80 100

VOLUME IN CM.'
FIG. 90.

ient. A small tube B of, say 1 sq. cm. cross section, is filled with
mercury to within 20 cm. of the top and then stoppered and
inverted in a large tube C which is nearly filled. with mercury.
Upon removing the stopper and pressing the tube down until the
mercury in both tubes stands at the same height, it will be seen
that the volume of the entrapped air (which is now at atmospheric
pressure) is 20 cm.3 If, now, tube B is raised until the mercury
within it stands at d3, and if d3f is 25 cm., then the pressure upon
the entrapped air is 50 cm. ; for this pressure plus the pressure of
the column of mercury d3f must balance the atmospheric pressure
of 75 cm. Since this pressure (50 cm.) is 2/3 of the initial pres-

PROPERTIES OF GASES AT REST 191

sure, the corresponding volume in accordance with Boyle's Law
must be 3/2 of the original volume, or 30 cm.3 Measurement
will show that d&z is 30 cm. Hence the point on the curve
marked D (ordinate 50, abscissa 30) represents this state of the
air. If tube B is raised still farther until the mercury within it
stands 50 cm. higher than in C, then the pressure of the entrapped
air is 25 cm., or 1/3 of the initial pressure, and its volume will be
found to be three times the initial, or 60 cm.3 Hence point E
(ordinate 25, abscissa 60) represents this, the fifth state of the
entrapped air. In the same way points F, G, etc., are determined.
Drawing a smooth curve through these points A, B, (7, etc.,
gives the isothermal for air at 20° C. When we take up the study
of heat we will readily see that the 100° isothermal would be
drawn about as shown (see dotted curve).

Observe that the three rectangles, A-75-0-20, D-50-0-30,
and #-25-0-60 all have the same area and that this area repre-
sents the product of the pressure — 75, 50, or 25 as the case may
be, and the corresponding volumes of the entrapped air for the
three different states which are represented respectively by the
points A, D, and E on the curve. Thus the curve verifies Boyle's
Law as expressed in Eq. 72, and shows that the constant K in
this equation is, for this particular amount of gas, 1500; for 75 X 20,
50X30, and 25X60, each gives 1500.

141. The Manometers and the Bourdon Gage. — Manome-
ters are of two kinds, the Open Tube Manometer, usually used

FIG. 91.

for measuring small differences in pressure, and the Closed
Tube Manometer which may be used to measure the total pres-
sure to which a gas or a liquid is subjected.

The Open Tube Manometer (Fig. 91) consists of a U-shaped
glass tube T, open at both ends and containing some liquid,
frequently mercury. If, when the manometer is connected with
the vessel A containing some gas, it is found that the mercury
stands at the same height in both arms, namely, at a and b, then

192

MECHANICS AND HEAT

FIG. 92.

the pressure of this gas which acts upon a, must be equal to the
pressure of the atmosphere which acts upon 6. If the mercury
meniscus 61 is higher than a\ by a distance hi cm., then the pres-
sure in B is 1 atmosphere + hidg dynes per cm.2, in which d is
the density of the mercury. The pressure of the gas in C is
evidently less than one atmosphere by the amount hzdg. If very
small differences in pressure are to be meas-
ured it is best to employ a light liquid for
the manometer. «

The Closed Tube Manometer (Fig. 92) may
be used for measuring high pressures, such
as the pressure of steam in steam boilers,
city water pressure, etc. Let D represent
a steam boiler containing some water, and

T, an attached closed tube manometer. If the mercury stands
at the same height in both arms a and 6 when valves leading
from D to the outside air are open, it shows that the entrapped
air in the manometer is at one atmosphere pressure. If, upon
closing these valves and heating the water in D, the pressure of
the steam developed forces the mercury down to a' in the left
arm and up to 6' in the right arm,
thereby reducing the volume of the
entrapped air to 1/3 its original
volume, it follows from Boyle's Law
that the pressure on it is increased
3-fold and is therefore 3 atmospheres.
The steam in D is then at 3 atmos-
pheres pressure. It is really slightly
more than this, for the mercury
stands a distance h higher in the
right arm than in the left. The
correction is clearly hdg. That is,
the pressure upon the enclosed air
above b' would be, under these cir-
cumstances, exactly 3 atmospheres

while the steam pressure in the boiler would be 3 atmospheres
plus the pressure hdg due to the mercury column of height h.

The Bourdon Gage. — The essentials of the Bourdon gage, which
is widely used for the measurement of steam pressure and water
pressure, are shown in Fig. 93. The metal tube T, which rs
closed at B, is of oval cross section, CD being the smaller diameter.

FIG. 93.

PROPERTIES OF GASES AT REST 193

If A is connected to a steam boiler, the pressure of the steam
causes the cross section of the tube to become more nearly cir-
cular, i.e., it causes the smaller diameter CD to increase. Ob-
viously, pushing the sides C and D of the tube farther apart will
cause the tube to straighten slightly, thereby moving B to the
right and causing the index 7 to move over the scale as indicated.
By properly calibrating the gage, it will read directly the steam
pressure in pounds per square inch. Most steam gages are of
this type. The same device may be used to measure the pressure
of water, or the pressure of any gas.

The Vacuum Manometer or Vacuum Gage. — If the space above b in
tube T (Fig. 92) were a perfect vacuum (e.g., if that arm of the tube were
first entirely filled with mercury), and if nearly all of the air were
pumped out of D, then T would be a "vacuum" gage. If, under these
circumstances, meniscus 6 stood 0.05 mm. higher than a, it would show
that the pressure of the remnant of the air in D was only equal to that
produced by a column of mercury 0.05 mm. in height. If the " vacuum "
in D were perfect, then a and 6 would stand at the same height.

PROBLEMS

1. What is the pressure of the atmosphere (in dynes per cm.2) when the
mercury barometer reads 74.2 cm.?

2. What is the pressure of the atmosphere (in Ibs. per in.2) when the ba-
rometer reads 28.2 in.?

3. If, in Fig. 89, d3/ = 30 cm. and the barometer reads 74 cm., what is the
pressure on the entrapped air in centimeters of mercury? In atmospheres?

4. An aneroid barometer, at a certain time, reads 29.9 in. at sea level and
29.35 in. on a nearby hill. What is the approximate altitude of the hill?
(Sec. 138.)

6. The liquid (oil of density 0.9 gm. per cm.3) in an open tube manometer
stands 4 cm. higher in the arm which is exposed to the confined gas than it
does in the other arm. What is the pressure exerted by the gas? The
barometric reading is 29 in.

6. A closed tube manometer contains an entrapped air column 8 cm. in
length when exposed to atmospheric pressure, and 3.2 cm. in length when
connected to an air pressure system. What is the pressure of the system?
The mercury stood at the same level in both arms in the first test.

7. If a 1000-lb. weight is rested upon P (Fig. 88), what will be the new
volume of the enclosed air in terms of the old?

8. A certain balloon has a volume equal to that of a sphere of 15-ft.
radius. What weight, including its own, will it lift when the density of the
air is (a) 2 Ibs. per cubic yard? (b) 0.0011 gm. per cm.3? Express the
weight in pounds in both cases.

9. Plot a curve similar to that shown in Fig. 90 and explain how it is
obtained.

CHAPTER XII
PROPERTIES OF FLUIDS IN MOTION

142. General Discussion. — The steady flow of a fluid, either a
liquid or a gas, at a uniform velocity through a level pipe from one
point to another, is always due to a difference in pressure main-
tained between the two points (friction head, see footnote). This
difference in pressure multiplied by the cross section of the pipe
gives the total force which pushes the column of fluid through the
pipe. Since the velocity of this column is neither increasing nor
decreasing, there is no accelerating force, and the above pushing
force must be just equal to the friction force exerted upon the
column by the pipe. If at any point the fluid is increasing in
velocity, an accelerating force F must be present, and part of the
pressure difference (velocity head)1 is used in producing this
accelerating force. F is equal to the mass M of the liquid being
accelerated, multiplied by its acceleration a (Sec. 25, F = Ma),

Just as the canal boat (Sec. 43), by virtue of its inertia, develops
a forward driving inertia force (F = Ma) which pushes it onto the

1 Head of Water. — In hydraulics, the pressure at a point, or the difference
in pressure between two points, is called pressure head, and is measured in
terms of the height (in feet) of the column of water required to produce
such pressure, or pressure difference. To illustrate, suppose that in certain
hydraulic mining operations, the supply reservoir is 600 ft. above the hose
nozzle, and that the velocity of the water as it leaves the nozzle is 100 ft.
per sec. Since a body must fall about 150 ft. to acquire a velocity of 100 ft.
per sec., the head required to impart this velocity to the water would be
150 ft. (see Sec. 143). Consequently the Velocity Head required is 150 ft.
The remainder of the 600-ft. head, namely 450 ft., is used in overcoming
friction in the conveying pipes and hose, and is called Friction Head. As
the water from the reservoir enters the conveying pipes it must acquire
velocity. As the water passes from the pipe into the much smaller hose,
and again as it passes from the hose into the tapering nozzle, it must ac-
quire additional velocity. Thus the total head of 600 ft. is equal to the sum
of the velocity heads of the pipe, the hose, and the nozzle, in addition to
the friction head for all three. If the size of the conveying pipe or hose
changes abruptly (either increases or decreases) eddies will be formed which
cause considerable friction and consequent loss of head. To reduce this
loss, the pipe should flare as it enters tie reservoir.

194

PROPERTIES OF FLUIDS IN MOTION

195

sand bar; so also a moving fluid (e.g., water, steam, or air) exerts
a driving inertia force (F = Ma) against any body that changes
its velocity. It is this inertia force which drives the wind mill,
the steam turbine, and the turbine water wheel, or any other water
wheel which utilizes the velocity of the water.

A thorough understanding of the above principles and their
applications gives one a fair elementary knowledge of the subject
of Hydraulics. A discussion of Fig. 94 will aid in securing such an
understanding.

Let B be a level water pipe communicating with the vertical
pipes C, E, and F, and with the tank A. If B is closed at G, so
that no water flows through it, the water will stand at the same
level, say at a-d-e-f, in the tank and in the vertical pipes. If
G is removed the water will at first flow out slowly, for it will

C j

E j

* c'

Pipii

:i .^S-

—-=--=—

t **"•- ^.

L ^-.^

— _

FIG. 94.

take the force at i, due to the tank pressure, a short interval of
time to impart to all of the water in pipe B a high velocity. After
a few seconds the water will be flowing rapidly and steadily at G,
and the water in the vertical pipes will stand at the different
levels c', d, e', and /'. Observe that c', e', f, and G all lie
in the same straight line. This uniform pressure drop or friction
head loss is due to the fact that the friction is the same in all
parts of B. If the pipe B between c" and e" were rusty ' and
rough, or smaller than elsewhere, the friction head between these
two points would be greater than elsewhere, causing e' to be
lower than shown. In such case c', e' ', and f would not lie
in a straight line.

Observe that removing G has not lowered the level in pipe D,
but has produced a decided drop in C. This difference in level
(hz), corresponding to a difference in pressure of h^dg, cannot be

196 MECHANICS AND HEAT

due to friction head in the short distance ic". This difference in
pressure is mainly due to the pressure head (velocity head)
required to accelerate the water as it passes from the tank, where
it is almost without motion, to the pipe B, where it moves rapidly.
If the pipe B were nearly closed at G, the flow would be slow,
and the friction head small, so that the water would stand nearly
as high in C, E, and F as in the tank A. Just as the heavy flow
from G lowers the water pressure at /" and hence in pipe F, so,
during a fire, when many streams of water are thrown from the
same main, the heavy flow lowers the available pressure at the
hose.

143. Gravity Flow of Liquids. — In the last section, where the
flow of water in level pipes was discussed, it was shown that a
pressure difference sufficient to overcome the friction of the water
on the pipes is always necessary to maintain
such flow. In the case of pipes which are not
level, but have a slight slope, such as tile
drains and sewer pipes, friction between the
water and the pipe is overcome, not by differ-
ence in pressure, but by a component of the
weight of the water itself. The weight of a
car on a grade may be resolved into two
components, one of which is parallel to the
FIG. 95. grade and therefore urges the car down the

grade (Sec. 19). Likewise, the weight of the
water in the tile drain may be resolved into two components, one
of which is parallel to the drain and therefore urges the water
along the drain. If the slope of the drain is increased, the com-
ponent parallel to the drain becomes larger, and the flow becomes
more rapid. The other component which is perpendicular to
the drain does not interest us in the present discussion.

The flowing of the water in a river is maintained in the same
way as in a tile drain. The bed of the river has a certain average
slope down stream. The component of the weight of the water
in the river which is parallel to the bed, constitutes the driving
force that overcomes the friction on the shores and on the bot-
tom. At points where the slope is great this force is great and
"rapids" exist.

Velocity of Efflux, Torricclli's Theorem. — As the water in A (Fig.
95), of depth hi, flows from the orifice B it acquires a velocity v, givea
by the equation v = \//clghi.

PROPERTIES OF FLUIDS IN MOTION 197

Proof: As M pounds of water pass through orifice B, the water level
in A is lowered slightly, and the potential energy of A is reduced by
Mhi foot-pounds or Mghi foot-poundals. The kinetic energy of the
M Ibs. of flowing water is %Mv* (Eqs. 50 and 51, Sec. 75). From the
conservation of energy it follows that this kinetic energy must be equal
to the potential energy lost by the tank; i,e., %Mvz = Mghi, from which
we have vz = 2gh} OTV = \/2gh^. From Eq. 14, Sec. 34, we see that \/2gh
is the velocity acquired by a body in falling from rest through a height
h. By this proof, known as Torricelli's theorem, we have shown that the
velocity oifree efflux produced by a given head h is equal to the velocity
of free fall through this same height h. If a pipe were connected to B of
a length such as to require a friction head of %hi to maintain the flow in
it, then the velocity head would be f hi, and the velocity of flow in the
pipe would be \/2g X |Ai or that acquired by a body in falling a distance

144. The Siphon.— The siphon, which is a U-shaped tube T
(Fig. 96), may be used to withdraw water or other liquids from
tanks, etc. If a siphon is filled with water and stoppered and
then inverted and placed in a vessel of water A, as shown, it
will be found that the water flows from A through T to B.
There must be an unbalanced pressure that forces this water
through T. This pressure may be readily found.

Imagine, for a moment, a thin film to be stretched across the
bore of the tube at C. The pressure tending to force this film
to the right, minus the pressure tending to force it to the left
is evidently the unbalanced pressure which causes the flow in
the actual case. The former pressure is the atmospheric pres-
sure, frequently called B (from barometer) minus hidg, or
B — hidg, while the latter is B — h2dg. The unbalanced pressure
is, therefore,

Unbalanced pressure = (B— hidg) — (B — hzdg) = (h2 — hi) dg = h3dg

From this equation we see that the difference of pressure is
proportional to the difference in level (h3) of surfaces Si and St.
In addition to this difference in level, the factors that determine
the rate of flow are the length of the tube, the smoothness and
size of its bore, and the viscosity of the liquid.

Since atmospheric pressure cannot support a column of water
which is more than 34 ft. in height (Sec. 136), it follows that hi
(Fig. 96) must not exceed this height or the atmospheric pressure
on Si will not force the water up to C, and the siphon will fail to

198

MECHANICS AND HEAT

operate. In case mercury is the liquid used, hi must not be more
than 29 or 30 inches. If made greater than this, a vacuum will
be formed at C and no flow will take place. Since a partial
vacuum is formed at C, the siphon walls must not easily collapse.

Observe that the water flows from point a to point 6, both
points being at the same pressure, namely, atmospheric pressure.
In Sec. 143 it was shown that pressure difference is not the only
thing which may maintain a steady flow, but that in sloping
pipes a component of the weight of the liquid overcomes the fric-
tion resistance. In vertical pipes the full weight of the liquid
maintains the flow. Hence, in the case of
the siphon we may consider that it is the ex-
cess weight of the right column over that of
the left which provides the force that over-
comes the friction between the flowing column
and the tube T.

145. The Suction Pump. — The common
" suction pump " used for cisterns and shallow
wells, is shown in Fig. 97 in three stages of
operation. The "cylinder" C is open at the
top and closed at the bottom, except for a
valve a which opens upward. Within C is a
snug-fitting piston P, containing a valve 6 also
opening upward. D is a pipe extending below
the surface of the water. As P is forced down-
ward by means of the piston rod R attached to
the pump handle, valve a closes, and as soon as the air in E is
sufficiently compressed it lifts the valve 6 and escapes (left
sketch). As the piston rises again, 6 closes, and the remnant of
air in E expands to fill the greater volume, thereby having its
pressure reduced (according to Boyle's Law) below one atmos-
phere. The pressure of the air in D is, of course, one atmosphere.
Hence the pressure above the valve a is less than the pressure
below it, causing it to rise and admit some air into E from D.
As air is thus withdrawn from D the pressure of the remaining
air is reduced to below atmospheric pressure, consequently the
water in the cistern, which is exposed to full atmospheric pressure,
is forced up into the tube D (middle sketch, Fig. 97). Another
stroke of the piston still further reduces the air pressure in E and
D, and the water is forced higher, until it finally passes through
valve a into the cylinder. As P descends, valve a closes, and the

FIG. 96.

PROPERTIES OF FLUIDS IN MOTION

199

water in the cylinder is forced through 6, and finally, as P again
rises fright sketch), it is forced out through the spout d.

Atmospheric pressure will support a column of water about 34
ft. in height (Sec. 136), provided the space above the column is a
vacuum. Hence we see that the theoretical limiting vertical
distance from the cylinder to the water in the cistern, or well, is

R
H

— '

--P-

:*:

FIG. 97.

34 ft. Suppose this distance to be 40 ft. Then, even if a per-
fect vacuum could be produced in E, the water would still be 6 ft.
below the cylinder. In practice, the cylinder should not be
more than 20 or 25 ft. above the water. For this reason, pumps
for deep wells have the cylinder near the bottom, the piston rod
in some cases being several hundred feet in length.

200

MECHANICS AND HEAT

146. The Force Pump. — The force pump is used when it is
desired to pump water into a tank which is at a higher level than
the pump. The pump described in the last section is sometimes
provided with a tight-fitting top at H (right sketch, Fig. 97)

having a hole just large enough
to permit the piston rod R to
| pass through it. By connecting

spout d with a hose, the pump
may then be operated as a force
pump.

The other type of force pump
(Fig. 98)"lifts"the water from the
well on the upstroke, and forces
it up to the tank on the down-
stroke, thus making it run more
evenly, since both strokes are
working strokes. In this type
the piston has no valve. As
the piston P rises, valve b closes
and valve a opens, permitting
water to enter the cylinder. As
P descends, a closes and 6 opens,
and the water is forced up into
the tank. During the down-
stroke of P, some of the water
rushes into the air chamber A
and further compresses the en-
closed air. During the upstroke
(valve 6 being then closed) this
air expands slightly and expels
some water. Thus, by the use
of the air chamber, the flow of
water from the discharge pipe
is made more nearly uniform.
The fact that the descending
piston may force some water into

A instead of suddenly setting into motion the entire column
of water in the vertical pipe, causes the pump to run more
smoothly.

147. The Mechanical Air Pump. — The mechanical air pump
operates in exactly the same way as the suction pump (Fig. 97).

FIG. 98.

PROPERTIES OF FLUIDS IN MOTION 201

In fact, when first started, the suction pump withdraws air from
D, that is, it acts as an air pump. To withdraw the air from an
inclosure (e.g., from an incandescent lamp bulb), the tube
D would be connected to the bulb instead of to the cistern. The
process by which the air is withdrawn from the bulb is the same
as that by which it is withdrawn from D (Sec. 145), and need not
be redescribed here. As exhaustion proceeds, the air pressure
in the bulb and in D becomes too feeble to raise the valves.
Hence the practical air pump must differ from the suction water
pump in that its valves are operated mechanically. The valves
and piston must also fit much more accurately for the air pump
than is required for the water pump.

The upper end of the cylinder of an air pump has a top in
which is a small hole covered by the outlet valve. If a pipe
leads from this valve to an enclosed vessel the air will be forced
into the vessel. In such case tube D would simply be opened
to the air, and the pump would then be called an Air Compressor.
Such air compressors are used ' to furnish the compressed air
for operating pneumatic drills, the air-brakes on trains, and for
many other purposes. It will be observed that such an air
pump, like practically all pumps (see Fig. 98), produces suction
at the entrance and pressure at the exit.

Let us further consider the process of pumping air from a bulb
connected to the tube D. Assuming perfect action of the piston and
valve, and assuming that the volume of the cylinder is equal to the
combined volume of the bulb and D, we see that the first stroke would
reduce the pressure in bulb and D to 1/2 atmosphere. For as P rises
to the top of the cylinder, the air in bulb and D expands to double its
former volume, and hence the pressure, in accordance with Boyle's Law,
decreases to 1/2 its former value. A second upstroke reduces the pres-
sure in bulb and D to 1/4 atmosphere, a third to 1/8 atmosphere, a
fourth to 1/16 atmosphere, etc. Observe that each stroke removes only
1/2 of the air then remaining in the bulb.

The Geryk Pump. — In the ordinary mechanical air pump there is a
certain amount of unavoidable clearance between the piston and the
end of the cylinder. The air which always remains in this clearance
space at the end of the stroke, expands as the piston moves away, and
produces a back pressure which finally prevents the further removal of
air from the intake tube, and therefore lowers the efficiency of the pump.
In the Geryk pump, air is eliminated from the clearance space by the
use of a thin layer of oil both above and below the piston.

148. The Sprengel Mercury Pump. — The Sprengel pump exhausts

202

MECHANICS AND HEAT

very slowly, but by its use a very much better vacuum may be obtained
than with the ordinary mechanical air pump. It consists essentially
of a vertical glass tube A (Fig. 99) about one meter in length and of
rather small bore, terminating above in a funnel B into which mercury
may be poured. A short distance below the funnel a side tube leads Irom
the vertical tube to the vessel C to be exhausted. As the mercury drops,
one after another, pass down through the vertical tube into the open
dish below, each drop acts as a little piston and pushes ahead of it a
small portion of air that has entered from the side tube. Thus any
vessel connected with this side tube is exhausted.

Obviously, to obtain a good vacuum, the aggregate length of the little
mercury pistons below the side tube must be greater
than the barometric height, or the atmospheric
pressure would prevent their descent. The funnel
must always contain some mercury, or air will en-
ter and destroy the vacuum. A valve at a is ad-
justed to permit but a slow flow of mercury,
thereby causing the column to break into pistons.

149. The Windmill and the Electric Fan.—
The common Windmill consists of a wheel
I A whose axis lies in the direction of the wind and

FIG. 99.

FIG. 100.

is therefore free to rotate at right angles to the direction of the
wind. This wheel carries radial vanes which are set obliquely
to the wind and hence to the axis of the wheel. In Fig. 100,
AB is an end view of a vane which extends toward the reader
from the axis (CD) of the windmill wheel. From analogy to
the problem of the sailboat (Sec. 20), we see at once that the
reaction of the wind w against the vane AB gives rise to a thrust
F normal to the vane. This force may be resolved into the two
components F\ and F2. Fz gives only a useless end thrust on
the wheel axle; while FI gives the useful force which drives the
vane in the direction FI. When the vane comes to a position
directly below the axis of the wheel, FI is directed away from

PROPERTIES OF FLUIDS IN MOTION 203

the reader. Thus in these two positions, and indeed in all other
positions of the vane, FI gives rise to a clockwise torque as
viewed from a point from which the wind is coming. Every
vane gives rise to a similar, constant, clockwise torque.

The ordinary electric fan is very similar to the windmill in
its operation, except that the process is reversed. In the case
of" the windmill, the wind drives the wheel and generates the
power; while with the fan, the electric motor furnishes the power
to drive the fan and produce the "wind." In the former, the
reaction between the vane and the air pushes the vane; while in^.
the latter it pushes the air.

150. Rotary Blowers and Rotary Pumps. — Blowers are used
for a great variety of purposes. Important among these are
the ventilation of mines; the production of the forced draft for
forges and smelting furnaces; the production of the "wind" for
fanning mills and the wind-stackers on threshing mchines; and
for the production of "suction," as in the case of tubes that suck
up shavings from wood-working machinery, foul gases from
chemical operations, and dust as in the Vacuum Cleaner. All
blowers may be considered to be pumps, and like all pumps^
they are capable of exhausting on one side and compressing on
the other, as pointed out in Sec. 147. Hence the above blowers
that produce the "wind" do not differ essentially from those
that produce "suction." Indeed many of the large ventilating
fans used in mines may be quickly changed so as to force the
air into the shaft, instead of "drawing" it from the shaft.

Blowers commonly produce a pressure of one pound per
square inch or less (i.e., a difference from atmospheric pressure
of 1 Ib. per sq. in.), although the so-called "positive" blowers
may produce eight or ten pounds per square inch. For the
production of highly compressed air, such as used in the air-
brakes on trains, the piston air pump is used (see Air
Compressor, Sec. 147).

Rotary Blowers. — Rotary blowers are of two kinds, disc blowers
and centrifugal blowers. A disc fan has blades which are radial
and set obliquely to its axis of rotation; while the fan proper
has its blades parallel to its axis and usually about radial (like
the blades of a steamboat paddle wheel). The common electric
fan is of the former type, and the fan used in the fanning mill is
of the latter type. If a disc fan is placed at the center of a tube
with its axis parallel to the tube, it will, when revolved, force a

204

MECHANICS AND HEAT

stream of air through the tube. The diameter of the fan should
be merely enough less than that of the tube to insure " clearance."
Such a blower will develop at the intake end of the tube a slight
suction and at the other end a slight pressure. This type is
widely used for ventilation purposes.

The essential difference between the Turbine Pump and the
blower just described is that the fan is stronger and propels" a
stream of water instead of a stream of air. The turbine pump is
useful in forcing a large quantity of water up a slight grade for
a short distance. It is not a high pressure pump.

The Screw Propeller, universally used on ocean steamships and
also used on gasoline launches, is essentially a turbine pump.
The propeller forces a stream of water backward and the reacting
thrust forces the ship forward.

The Centrifugal Blower is similar in its action to the centrifugal
pump described below.

The Centrifugal Pump. — One type of centrifugal pump, shown
in section in Fig. 101, consists of a wheel W, an intake pipe A

FIG. 101.

which brings water to the center of the wheel, and an outflow
pipe B, which conveys the water from the periphery of the wheel.
The direction of flow of the water at various points is indicated
by the arrows. By means of an electric motor or other source
of power, the wheel is rotated in the direction of arrow a, and the
centrifugal force thereby developed causes the water to flow
radially outward through the curved passages in the wheel as
indicated by the arrows. In this way, it is feasible to produce
a pressure of 20 or 30 Ibs. per sq. in. in the space C, which is
sufficient pressure to force water to a vertical height of 50 or 60
ft. in a pipe connected with B.

PROPERTIES OF FLUIDS IN MOTION 205

If it is desired to raise water to a greater height than this,
several pumps can be used in " series." In such a series arrange-
ment, the lowest pump would force water through outlet pipe B
to the intake of a similar pump, say 50 ft. above. This second
pump would force the water on to the next above, and so on.

151. The Turbine Water Wheel.— The Turbine Water Wheel
operates on the same general principle as the windmill; a stream
of water driving the former, a stream of air the latter. Since
water is much more dense than air, turbine water wheels develop
a great deal more power than windmills of the same size. At
the Niagara Falls power plant, water under about 150 ft. verti-
cal head rushes into the great turbines, each of which develops
5000 H.P. Turbines of 10,000 H.P. each are used in the power
plant at Keokuk, Iowa.

There are several kinds of water turbines. In the "Axial Flow
Turbines" in which the water flows parallel to the axis, the ac-
tion of the windmill is practically duplicated; so that Fig. 100 and
the accompanying discussion would apply to a vane of such a
turbine, provided w were to represent moving water instead of
moving air. In the "Radial Flow Turbines" the water flows in a
general radial direction either toward or away from the axis.

If water under considerable pressure is forced in at pipe A
(Fig. 101) through wheel W, and out at pipe B, it will drive W
in the direction of arrow a. For, as the water flows outward
through the curved radial passages, it would, by virtue of its
inertia, produce a thrust against the concave wall of the passage.
This thrust would clearly produce a positive (left-handed)
torque. Under these circumstances, the wheel would develop
power, and would be called a radial flow turbine water wheel.

The Steam Turbine, used to obtain power from steam, is similar
to the water turbine in principle, but greatly differs from it in
detail. The development of the light, high power, high efficiency
steam turbine is among the comparatively recent achievements
of steam engineering. The steam turbine is further considered
in Sec. 235.

152. Pascal's Law. — The fact that liquids confined in tubes,,
etc., transmit pressure applied at one point to all points, has
already been pointed out (Sec. 114). This is known as Pascal's
Law. Pascal's Law holds with regard to gases as well as liquids.

This law has many important applications, among which are
the transmission of pressure by means of the water mains to all

206 MECHANICS AND HEAT

parts of the city, and the operation of the hydraulic press and
the hydraulic elevator.

153. The Hydraulic Press.— The hydraulic press (Fig. 102)
is a convenient device for securing a very great force, such as
required for example in the process of baling cotton. It con-
sists of a large piston or plunger P, fitting accurately into a hole
in the top of a strong cylindrical vessel B. As water is forced
into B by means of a force pump connected with pipe D, the
plunger P rises. As P rises, the platform C compresses the
cotton which occupies the space A. By opening a valve E, the
water is permitted to escape and P descends.

In accordance with Pascal's Law, the pressure developed by
the force pump is transmitted through D to
the plunger P. It will be observed that since
the pressure on the curved surface of the
plunger is perpendicular to that surface (Sec.
114), it will have no tendency to lift the
plunger. The lifting force is pAi, in which
AI is the area of the bottom end of the
plunger, and p, the water pressure.

If the area of the piston of the force pump is
A 2, then, since the pressure below this piston
FIG. 102. is practically the same as that acting upon

plunger P, it follows that the lift exerted by P
will be greater than the downward push upon the piston of the
force pump in the ratio of AI to Az. In other words, the theo-
retical mechanical advantage is Ai-i- A%.

Instead of using a force pump, the pipe D may be connected
to the city water system. If this pressure is 100 Ibs. per in.2,
and if ^li = 100 in.2, then P will exert a force of 10,000 Ibs. or
5 tons. With some steel forging presses a force of several
thousand tons is obtained.

154. The Hydraulic Elevator. — The simplest form of hydraulic
elevator, known as the direct-connected or direct-lift type, is
the same in construction and operation as the hydraulic press
.(Fig. 102), except that the plunger is longer. If the elevator,
built on platform C, is to have a vertical travel of 30 ft., then the
plunger P must be at least 30 ft. in length.

In another type of hydraulic elevator, the plunger and con-
taining cylinder lie in a horizontal position in the basement of
the building. The plunger is then connected with the elevator

PROPERTIES OF FLUIDS IN MOTION

207

by means of a system of gears or pulleys and cables in such a
way that the elevator travels much farther, and hence also much
faster than the plunger. This type is much better than the
direct-connected type for operating elevators in high buildings.
In both types the valves that regulate the flow of water to and
from the cylinder are controlled from the elevator.

155. The Hydraulic Ram. — The hydraulic ram (shown in
Fig. 103) depends for its action upon the high pressure developed
when a moving stream of water confined in a tube is suddenly
stopped. It is used to raise a small percentage of the water
from a spring or other source to a considerable height.

The valve C is heavy enough so that the water pressure lidg
(see figure) is not quite sufficient to keep it closed. As it sinks
slightly, the water flows rapidly past above it; while at the
same time the water below it is practically still. In the next

FIG. 103.

section it will be shown that the pressure on a fluid becomes
lower the faster it moves; accordingly, the pressure above the
valve is less than the pressure below, and the valve rises and
closes. The closing of this valve suddenly checks the motion
of the water in pipe B. But suddenly stopping any body in
motion requires a relatively large accelerating force (negative),
hence here a considerable pressure is developed in pipe B. This
"instantaneous," or better brief, pressure opens the valve D
and forces some water into the air chamber E and also into
pipe F. Valve C now sinks, and the operation is repeated, forc-
ing still more water in E, until finally the water is forced through
F into a supply tank which is on higher ground than the source
A. The action of the air chamber is explained in Sec. 146.

If the hydraulic ram had an efficiency of 100 per cent., then,
from the conservation of energy, we see that it would raise 1/n

208

MECHANICS AND HEAT

of the total amount of water to a height nh. Its efficiency, how-
ever, is only about 60 per cent.; hence it will force l/n of the
total water used to a height 0.6 nh.

156. Diminution of Pressure in Regions of High Velocity. — If
a stream of air is forced rapidly through the tube A (Fig. 104),
it will be found that the pressure at the restricted portion B is
less than elsewhere, as at C or D. If the end D is short and
open to the air, manometer F will indicate that D is practically
at atmospheric pressure. The pressure at C will be slightly
above atmospheric pressure, as indicated by manometer E.
That the pressure at B is less than one atmosphere, and hence
less than at either C or D, is evidenced by the fact that the liquid
stands higher in tube 0 than in the vessel H.

FIG. 104.

That the pressure at B should be less than at C or D is ex-
plained as follows: Since the tube has a smaller cross section
at B than at C or D, it is evident that the air must have a higher
velocity at B than at the other two points, as indicated in the
figure by the difference in the length of the arrows. As a particle
of air moves from C to B its velocity, then, increases. To cause
this increase in velocity requires an accelerating force. Conse-
quently the pressure behind this particle tending to force it to
the right must be greater than the pressure in front of it, tending
to force it to the left. As the particle moves from B to D it
slows down, showing that the backward pressure upon it must
be greater than the forward pressure. Thus B is a" region of
lower pressure than either C or D simply because it is a region
of higher velocity. The reduction in pressure at B is explained

PROPERTIES OF FLUIDS IN MOTION

209

by means of Bernoulli's theorem under "Venturi Water Meter"
(see below).

The Atomizer. — If the air rushes through B still more rapidly,
the pressure will be sufficiently reduced so that the liquid will
be " drawn" up from vessel H and thrown out at I as a fine spray.
The tube then becomes an atomizer.

The Aspirator or Filter Pump. — A similar reduction in pressure
occurs at B if water flows rapidly through the tube. Thus, if
the tube is attached in a vertical position to a faucet, the water
rushing through B produces a low pressure and consequently
"suction," so that if a vessel is connected with G the air is
withdrawn from it, producing a partial vacuum. Under these
circumstances the tube acts as a filter pump or aspirator.

The "forced draft" of locomotives is produced by a jet of
steam directed upward in the smoke stack.

The Jet Pump. — If a stream of water from a hydrant is directed
through B, a tube connected with G may be employed to "draw"

FIG. 105.

water from a cistern or flooded basement. Such an arrangement
is a jet pump. In Fig. 105 a jet pump is shown pumping water
from a basement B into the street gutter /. Pipe A is con-
nected to the hydrant.

Bernoulli's Theorem. — Bernoulli's Theorem, first enunciated
in 1738 by John Bernoulli, is of fundamental importance to some
phases of the study of hydraulics. We shall develop this theorem
from a discussion of Fig. 95.

Let water flow into A at the top as rapidly as it flows out at
B, thus maintaining a constant water level. Let us next con-
sider the energy possessed by a given volume V of water in the

210 MECHANICS AND HEAT

different stages of its passage from the surface S to the out-
flowing stream at B. Its energy (potential energy Ep) when
at S is Mghi CEq. 50, Sec. 75), and, since M=Vd (volume times
density, Sec. 101), we have

As this given volume reaches point 6 at a slight distance h above
the orifice, it has potential energy Mgh, or Vdgh, and, since it
now has appreciable velocity v, it has kinetic energy %Mv*,
or %Vdv2. In addition to this it has potential energy, because
of the pressure (p) exerted upon it by the water above, which
energy, we shall presently prove, is pV. Consequently, its total
energy when at b is

E=Vdgh+pV+$Vdv* (73)

Eq. 73 is the mathmetical statement of Bernoulli's theorem.
If C.G.S. units are used throughout (i.e., if V is given in cm.3,
p in dynes per cm.2, v in cm. per sec., etc.), then E will be the
energy in ergs. If the volume chosen is unity, the equation re-
duces to E = dgh-\-p+%dv2, a form frequently given.

Observe that when the volume V is at S, p and v are zero, hence
E= Vdghi as already shown; while when this volume reaches the
flowing stream, p and h are zero, hence E = %Vdv2 (i.e., \Mv*}.
From the law of the conservation of energy we know that these
two amounts of energy must be equal, i.e., Vdghi = %Vdv*,
which reduces to v= '\l2ghi, an equation already deduced (Sec.
143) from slightly different considerations. When the volume
is half way down in vessel A, Vdgh = pV, and the third term
\Vdvz is practically zero, since v at this point has a small value.
It should be observed that when the volume under consideration
is below the surface, then the height measured from the volume
up to the surface determines the pressure; whereas the height
measured from the volume down to the orifice, determines the
potential energy due to the elevated position. Obviously the
energy due to elevation decreases by the same amount that
the energy due to pressure increases, and vice versa, and the
sum of these two amounts of energy is constant so long as the
velocity v (last term Eq. 73) is practically zero.

We shall now prove that the potential energy of the above
volume V, when subjected to a pressure p, is pV. Let the volume
V, as it passes out at B, slowly push a snug-fitting piston in B a

PROPERTIES OF FLUIDS IN MOTION 211

distance di such that diAi = V, in which Ai is the cross section
of the orifice. The work done by the volume V on the piston
is pAiXdi (force times distance), which shows that the potential
energy of V immediately before exit was pXAidi or pV.

The Venturi Water Meter.— The Venturi water meter, used
for measuring rate of flow, differs from the apparatus sketched
in Fig. 104 in that the medium is water instead of air, and the
pressure is measured by ordinary pressure gages instead of
as shown. If pipe A were 6 ft. in diameter at C, it would taper
in a distance of 100 ft. or so to a diameter of about 2 ft. at B.

Let the pressure, area of cross section, and velocity of flow
at C and B, respectively, be pe, Ac, vc, and pb, Ab, vb. Now the
energy of a given volume V when at C must be equal to its
energy when at B; hence, from Eq. 73, we have

Vdgh + PC V + 1 Vdvc* = Vdgh + pb V + £ Vdvb*
from which we get

».f) (74)

Since in unit time equal volumes must pass B and C, we have

•n-b

Substituting in Eq. 74 this value of vb gives

or vb = -T^ve (74a)

If the pressure is reduced to poundals per square foot, the
cross section to square feet, and if the density of the water is
also expressed in the British system (i.e., 62.4 Ibs. per cu. ft.),
then vc will be expressed in feet per second. Multiplying vc
by Ac (in square feet) gives, for the rate of flow, vcAc, in cubic
feet per second.

157. The Injector. — Injectors are used for forcing water into
boilers while the steam pressure is on. Their operation depends
upon the decrease of pressure produced by the high velocity of
a jet of steam, coupled with the condensation of the steam in
the jet by contact with the water spray brought into the jet by
the atomizer action (Sec. 156). Some of the commercial forms
of the injector are quite complicated.

The injector shown diagrammatically in Fig. 106 is compara-

212 MECHANICS AND HEAT

tively simple. If valves a and d are opened, b being closed, the
steam from the boiler B rushes through D, E and e and out at
a into the outside air. The steam, especially at the restricted
portion E of the tube, has a very high velocity, and hence, from
Sec. 156, we see that a low pressure exists at E. The pressure
at E being less than one atmosphere, the atmospheric pressure
upon the water in the tank forces water up through the pipe
P into E, where it passes to the right with the steam which
quickly condenses. This stream of water, due to its momentum,
raises check valve b and passes into the boiler against the boiler
pressure. As soon as the flow through b is established, valve a
should be closed. In many injec-
tors, the suction due to the par-
tial vacuum at e automatically
closes a check valve opening down-
ward at a.

It should be pointed out that in
the action of the injector, by which
steam under a pressure p forces
supply water (and also the con-
densed steam) into the boiler against
this same steam pressure plus a
FIG. 106. slight water pressure (see figure),

there is no violation of the law of

the conservation of energy The energy involved is pressure
times volume in both cases, but the volume of water forced into
the boiler in a given time is much less than the volume of steam
used by the injector.

158. The Ball and Jet— If a stream of air B, Fig. 107, is
directed as shown against a light ball A, e.g., a ping pong ball
or tennis ball, the ball will remain in the air and rapidly revolve
in the direction indicated.

The explanation is simple. There are three forces acting
upon the ball, namely, W, F\, and F2, as shown. The force FI
arises from the impact of the stream of air B. The force Fz is
due to the fact that the air pressure at a is less than at b. The
pressure at 6 is one atmosphere, while at a it is slightly less be-
cause a is a region of high velocity. W represents the weight
of A. If it is desired to determine the magnitude of F\ and Fz,
the magnitude of W may be found by weighing A, and then,
since the ball is in equilibrium, these three forces W, FI, and

PROPERTIES OF FLUIDS IN MOTION

213

FIG. 107.

Fz, acting upon it must form a closed triangle, as explained in
Sec. 18.

Card and Spool. — If a circular card, having a pin inserted
through the center, is placed below a spool through the center of
which a rapid stream of air is blown, it will be found that the card
will be supported in spite of the downward rush of air upon it
which might be expected to blow
it away. The air above the card
is moving rapidly in all directions
away from the center; consequent-
ly the region between the spool and
card, being a region of high veloc-
ity, is also a region of low pressure
—lower, in fact, than the pressure
below the card. This difference in
pressure will not only support the
weight of the card, but also addi-
tional weight.

159. The Curving of a Baseball.
— The principle involved in the
pitching of "in curves," "out

curves, " etc., will be understood from a discussion of Fig. 108.
Let A represent a baseball rotating as indicated, and moving to
the right with a velocity v. If A were perfectly frictionless, the
air would rush past it equally fast above and below, i.e., vv
and #2 would be equal. (We are familiar with the fact that a
person running 10 mi. per hr. east through still air, faces a 10 mi.
per hr. breeze apparently going west.) If the surface of the ball

is rough, however, it will be evi-
dent that where this surface is
moving in the direction of the
rush of air past it, as on the
upper side, it will not retard that
rush so much as if it were mov-
ing in the direction opposite to
the rush of air, as it clearly is
The air, then, rushes more read-
ily, and hence more rapidly, past the upper surface than past
the lower surface of the ball; hence, as the ball moves to the
right, the air pressure above it is less than it is below, and an
"up curve" results.

FIG. 108.

on the lower side of the ball.

214 MECHANICS AND HEAT

The "drop curve" is produced by causing the ball to rotate
in a direction opposite to that shown; while the "in curve" and
"out curve" require rotation about a vertical axis.

A lath may be made to produce a very pronounced curve by
throwing it in such a way as to cause it to rotate rapidly about
its longitudinal axis, the length of the lath being perpendicular
to its path.

PROBLEMS

1. A force pump, having a 3-ft. handle with the piston rod operated by a
6-in. "arm," (i.e., with the pivot bolt 6 in. from one end of the handle), and
having a piston head 2 in. in diameter, is used to pump water into an hydrau-
lic press whose plunger is 1.5 ft. in diameter. What force will a 100-lb.
pull on the end of the pump handle exert upon the plunger of the press?

2. An hydraulic press whose plunger is 2 ft. in diameter is operated by water
at a pressure of 600 Ibs. per sq. in. How much force does it exert? Express
in tons.

3. An hydraulic elevator operated by water under a pressure of 100 Ibs.
per in.2 has a plunger 10 in. in diameter and weighs 2.5 tons. How much
freight can it carry?

4. If /n = 10 ft., and h2 = lS ft. (Fig. 96), what will be the pressure at C
(a) if the left end of the siphon is stoppered? (6) If the right end is stop-
pered? Assume the barometric pressure to be equal to that due to 34 ft.
depth of water.

6. What pressure will be required to pump water from a river into a tank
on a hill 300 ft. above the river, if 20 per cent, of the total pressure is needed
to overcome friction in the conveying pipes?

6. How long will it take a 10-H.P. pump (output 10 H.P.) to pump
1000 cu. ft. of water into the tank (Prob. 5)?

7. If the water in pipe B (Fig. 94) flows with a velocity of 4 ft. per sec.,
what will be the value of &2? Neglect friction head in the portion i to c"
(Sees. 142 and 143).

8. What would be the limiting (maximum) distance from the piston to
the water level in the cistern (Fig. 97) at such an altitude that the baro-
metric height is 20 in.?

PART III
HEAT

CHAPTER XIII
THERMOMETRY AND EXPANSION

160. The Nature of Heat. — As was pointed out in the study
of Mechanics, a portion of the power applied to any machine
is used in overcoming friction. It is a matter of everyday ob-
servation that friction develops heat. It follows, then, that
mechanical energy may be changed to heat. In the case of the
steam engine or the gas engine the ability to do work, that is
to run the machinery, ceases when the heat supply is withdrawn.
Therefore heat is transformed into mechanical energy by these
engines, which on this account are sometimes called heat engines.

Heat, then, is a form of energy, a body when hot possessing
more energy than when cold. Cold, it may be remarked, is not
a physical quantity but merely the comparative absence of heat,
just as darkness is absence of light. The heat energy of a body
is supposed to be due to a very rapid vibration of the molecules
of the body. As a body is heated to a higher temperature,
these vibrations become more violent.

It has been proved experimentally, practically beyond ques-
tion, that both radiant heat and light consist in waves in the
transmitting medium (ether). To produce a wave motion in
any medium requires a vibrating body. As a body, for example
a piece of iron, becomes hotter and hotter it radiates more heat
and light. Hence, since the iron does not vibrate as a whole,
the logical inference is that the radiant heat and light are pro-
duced by the vibrations of its molecular or atomic particles.

Until about one hundred years ago heat was supposed to be
a substance, devoid of weight or mass, called Caloric, which,
when added to a body caused it to become hotter, and when
withdrawn from a body left the body colder. In 1798, Count
Rumford showed that an almost unlimited amount of heat could
be taken from a cannon by boring it with a dull drill. The heat
was produced, of course, by friction. In the process a very small
amount of metal was removed. As the drilling proceeded and
more "caloric" was taken from the cannon, it actually became
217

218 MECHANICS AND HEAT

hotter instead of colder as the caloric theory required. Further-
more, the amount of heat developed seemed to depend upon
the amount of work done in turning the drill. The result was
the complete overthrow of the caloric theory.

In 1843, Joule showed by experiment that if 772 ft.-lbs. of
work were used in stirring 1 Ib. of water, its temperature would
be raised 1° F. This experiment showed beyond question that
heat is a form of energy, and that it can be measured in terms
of work units. Later determinations have given 778 ft.-lbs.
as the work necessary to raise the temperature of 1 Ib. of water
1° F. The amount of heat required to warm 1 Ib. of water 1° F.
is called the British Thermal Unit (B.T.U.); so that 1 B.T.U. =
778 ft.-lbs.

161. Sources of Heat. — As already stated, Friction is one
source of heat. Rubbing the hands together produces noticeable
warmth. Shafts become quite hot if not properly oiled. Primi-
tive man lighted his fires by vigorously rubbing two pieces of
wood together. The shower of sparks from a steel tool held
against a rapidly revolving emery wheel, and the train of sparks
left by a meteor or shooting star, show that high temperatures
may be produced by friction. In the latter case, the friction
between the small piece of rock forming the meteor, and the
air through which it rushes at a tremendous velocity, develops,
as a rule, sufficient heat to burn it up in less than a second.

Chemical Energy. — Chemical energy is an important source
of heat. The chemical energy of combination of the oxygen of
the air with the carbon and hydrocarbons (compounds of car-
bon and hydrogen) of coal or wood, is the source of heat when
these substances are "burned," .that is, oxidized. In almost
every chemical reaction in which new compounds are formed,
heat is produced.

The Main Source of heat is the Sun. The rate of flow of heat
energy in the sun's rays amounts to about 1/4 H.P. for every
square foot of surface at right angles to the rays. Upon a high
mountain this amount is greater, since the strata of the air below
the mountain peak absorb from 10 to 20 per cent, of the energy
of the sun's rays before they reach the earth. On the basis of
1/4 H.P. per sq. ft., the total power received by the earth from
the sun is easily shown to be about 350 million million H.P.
This enormous amount of power is only about 1/2,000,000,000
part of the total power given out by the sun in all directions.

THERMOMETRY AND EXPANSION 219

Obviously a surface receives more heat if the sun's rays strike
it normally (position AB, Fig. 109) than if aslant (position A Bi),
for in the latter case fewer rays strike it. Largely for this
reason, the ground is hotter under the noonday sun than it is
earlier. The higher temperature in summer than in winter is
due to the fact that the sun is, on an average, more nearly over-
head in summer than in winter. The hottest part of the day
is not at noon as we might at first expect, but an hour or two
later. This lagging occurs because of the time required to warm
up the ground and the air. A similar lagging occurs in the sea-
sons, so that the hottest and the coldest weather do not fall re-
spectively on the longest day (June 21) and the shortest (Dec.
22), but a month or so later as a rule.

The above-mentioned sources are the three main sources of
heat. There are other minor sources. An electric current heats
a wire or any other substance — solid, liquid, or gas — through

FIG. 109.

which it passes. This source is of great commercial importance.
The condensation of water vapor produces a large amount of
heat, and this heat is one of the greatest factors in producing
wind storms as explained in Sec. 223.

162. The Effects of Heat. — The principal effects of heat are :
(a) Rise in temperature.
(6) Increase in size.

(d) Chemical change.

(e) Physiological effect.
CD Electrical effect.

(a) With but very few exceptions a body becomes hotter,
i.e., its temperature rises, when heat is applied to it. Excep-
tions : If water containing crushed ice is placed in a vessel on a
hot stove, the water will not become perceptibly hotter until
practically all of the ice is melted. Further application of heat

220 MECHANICS AND HEAT

causes the water to become hotter until the boil'ng point is
reached, when it will be found that the temperature again ceases
to rise until all of the water boils away, whereupon the contain-
ing vessel becomes exceedingly hot. In this case, the heat energy
supplied, instead of causing a temperature rise (a), has been used
in producing a change of state (c), i.e., it has been used in changing
ice to water, or water to steam.

(6) As heat is supplied to a body, it almost invariably produces
an increase in its size. It might readily be inferred that the
more violent molecular vibrations which occur as the body be-
comes hotter, would cause it to occupy more space, just as a
crowd takes more room if the individuals are running to and fro
than if they are standing still or moving less. Exception to
(6) : If a vessel filled with ice is heated until the ice is melted,
the vessel will be only about 9/10 full. In this case heat has
caused a decrease in size. This case is decidedly exceptional,
however, in that a change of state (c) is involved. It is also
true that most substances expand upon melting instead of
contracting as ice does.

(d) To ignite wood, coal, or any other substance, it is neces-
sary to heat it to its "kindling" or ignition temperature, before
the chemical change called "burning" will take place. In the
limekiln, the excessive heat separates carbon dioxide (CO2)
from the limestone, or crude calcium carbonate (CaCO3), leaving
calcium oxide (CaO), called lime. There are other chemical re-
actions besides oxidation which take place appreciably only at
high temperatures. Slow oxidation of many substances occurs
at ordinary temperatures. All chemical reactions are much less
active at extremely low temperatures such as the temperature
produced by liquid air.

(e) Heat is essential to all forms of life. Either insufficient
heat or excessive heat is exceedingly painful.

(/) The production of electrical effect by heat will be discussed
under the head of the Thermocouple (Sec. 174).

163. Temperature. — The temperature of a body specifies its
state with respect to its ability to impart heat to other bodies.
Thus, if a body A is at a higher temperature than another body
B, it will always be found that heat will flow from A to B if they
are brought into contact, or even if brought near together. The
greater the temperature difference between A and B, other things
being equal, the more rapid will be the heat transfer. The tern-

THERMOMETRY AND EXPANSION 221

perature of a body rises as the heat vibrations of its molecules
become more violent.

The temperature of a body cannot be measured directly, but it
may be measured by some of the other effects of heat, as (6)
and (/) (Sec. 162), or it may be roughly estimated by the physi-
ological effect or temperature sense. Heat of itself always passes
from a body of higher temperature to one of lower tempera-
ture. The temperature sense serves usually as a rough guide
in determining temperature, but it is sometimes very unre-
liable and even misleading, as may be seen from the following
examples.

If the right hand is placed in hot water and the left hand in
cold water for a moment, and then both are placed in tepid water,
this tepid water will feel cold to the right hand and warm to the
left hand. Under these conditions heat flows or passes from the
right hand to the tepid water. The tepid water being warmer
than the left hand, the flow is in the opposite direction. Hence
if heat flows from the hand to a body, we consider the body to be
cold, while if the reverse is true, we consider it to be warm.
If A shakes B's hand and observes that it feels cold we may be
sure that B notices that A's hand is warm.

If the hand is touched to several articles which have been
lying in a cool room for some time, and which are therefore at
the same temperature, it will be found that the articles made of
wool do not feel noticeably cool to the touch. The cotton articles,
however, feel perceptibly cool, the wooden articles cold, and the
metal articles still colder. The metal feels colder than wood
or wool, because it takes heat from the hand more rapidly, due
to its power (called conductivity) of transmitting heat from the
layer of molecules in contact with the hand to those farther
away. Wood is a poor conductor of heat and wool is a very
poor conductor; so that in touching the latter, practically only
the particles .touching the hand are warmed, and hence very
little heat is withdrawn from the hand and no sensation of cold
results.

One of the most accurate methods of comparing and measur-
ing temperatures, and the one almost universally used, makes
use of the fact that as heat is supplied to a body, its temperature
rise, and its expansion, or increase in size, go hand in hand.
Thus if 10° rise in temperature causes a certain metal rod to be-
come 1 mm. longer, then an increase of 5 mm. in length will

222 MECHANICS AND HEAT

show that the temperature rise is almost exactly 5 times as
great, or practically 50°. This principle is employed in the use
of thermometers.

164. Thermometers. — From the preceding section it will be
seen that any substance which expands uniformly with tempera-
ture rise can be used for constructing a thermometer. Air or
almost any gas, mercury, and the other metals meet this require-
ment and are so used. Alcohol is fairly good for this purpose
and has the advantage of not freezing in the far north as mercury
does. Water is entirely unsuitable, because its expansion, as
its temperature rises, varies so greatly. When ice cold water
is slightly heated it actually decreases in volume (see Maximum
Density, Sec. 185) ; whereas further heating causes it to expand,
but not uniformly.

The fact that in the case of alcohol, the expansion per degree
becomes slightly greater as the temperature rises, makes it neces-
sary to gradually increase the length of the degree divisions
toward the top of the scale. In the case of mercury, the expan-
sion is so nearly uniform that the degree divisions are made of
equal length throughout the scale.

Mercury is the most widely used thermometric substance.
It is well adapted to this use because it expands almost uni-
formly with temperature rise; has a fairly large coefficient of
expansion; does not stick to the glass; has a low freezing point
( -38°.8 C.) and a high boiling point (357° C.) ; and, being opaque,
a thin thread of it is easily seen.

165. The Mercury Thermometer. — The mercury thermometer
consists of a glass tube T (Fig. 110) of very small bore, termi-
nating in a bulb B filled with mercury. As the bulb is heated,
the mercury expands and rises in the tube (called the stem),
thereby indicating the temperature rise of the bulb. In filling
the bulb, great care must be taken to exclude air.

Briefly, the method of introducing the mercury into the bulb
is as follows: The bulb is first heated to cause the air contained
in it to expand, in order that a portion of it may be driven out of
the open upper end of the stem. This end is then quickly placed
in mercury, so that when the bulb cools, and consequently the air
pressure within it falls below one atmosphere, some mercury is
forced up into the bulb. If, now, the bulb is again heated until
the mercury in it boils, the mercury vapor formed drives out all
of the air; so that upon again placing the end of the stem in the

THERMOMETRY AND EXPANSION

223

mercury and allowing the bulb to cool, thereby condensing the
vapor, the bulb and stem are completely filled with mercury.

Let us suppose that the highest temperature which the above
thermometer is designed to read is 120° C. The bulb is heated
to about 125°, expelling some of the mercury from
the open end of the tube which is then sealed off.
Upon cooling, the mercury contracts, so that a vacuum
is formed in the stem above the mercury. It will
be evident that as the mercury in B is heated and ex-
pands, its upper surface, called its meniscus m, will
rise; while if it is cooled its contraction will cause
the meniscus to fall. Attention is called to the fact
that if mercury and glass expanded equally upon be-
ing heated, then no motion of m would result. Mer-
cury, however, has a much larger coefficient of expan-
sion than glass (see table, Sec. 171). If heat is sud-
denly applied, for example by plunging the bulb into
hot water, the glass becomes heated first, and m
actually drops slightly, instantly to rise again as the
mercury becomes heated.

The position of the meniscus m, then, except in the
case of very sudden changes in temperature such as
just cited, indicates the temperature to which the bulb
B is subjected. In order, however, to tell definitely
what temperature corresponds to a given position of
m, it is necessary to "calibrate" the thermometer.
To do this, the thermometer is placed in steam in an
enclosed space over boiling water. This heats the
mercury in B, thereby causing it to expand, and the
meniscus m rises to a point which may be marked
a. The thermometer is next placed in moist crushed
ice which causes the mercury to contract, thereby low-
ering the meniscus to the point marked 6. We have U B
now determined two fixed points, a and b, corresponding
respectively to the boiling point of water and the F
melting point of ice. It now remains to decide what
we shall call the temperatures corresponding to a and 6, which
decision also determines how many divisions of the scale there
shall be between these two points. Several different "scales"
are used, two of which will be discussed in the next section.

Thermometers should not be calibrated until several years

224

MECHANICS AND HEAT

after filling. If calibrated immediately, it will be found after
a short time that because of the gradual contraction that has
taken place in the glass, all of the readings are slightly too high.
166. Thermometer Scales. — The two thermometer scales in
common use are the Centigrade and
Fahrenheit scales. To calibrate a
thermometer, according to the cen-
tigrade scale, the point 6 (Fig. 110)
is marked 0°, and the point a is
marked 100°, which makes it nec-
essary to divide ab into 100 equal
parts in order that each part shall
correspond to a degree. Accord-
ingly we see that ice melts at zero
degrees centigrade, written 0° C.,
and that water boils at 100° C. In-
creasing the pressure, slightly lowers
the melting point of ice (Sec. 186)
and appreciably raises the boiling
point of water (Sec. 194). To be
accurate, ice melts at 0° C. and
water boils at 100° C. when sub-
jected to standard atmospheric
pressure (76 cm. of mercury). If
the pressure differs from this, cor-
rection must be made, at least in
the case of the boiling point.

The Fahrenheit scale is in com-
mon use in the United States and
Great Britain. To calibrate the
thermometer (Fig. 110) according
to the Fahrenheit scale, the "ice
point" 6 is marked 32°, and the
boiling point a is marked 212°.
The difference between these two
points is 180° so that ab will have
to be divided into 180 equal spaces

in order that each space shall correspond to a degree change of
temperature. Using the same space for a degree, the scale may
be extended above 212 and below 32.

The Fahrenheit scale has the advantage of a low zero point

FIG. 111.

THERMOMETRY AND EXPANSION 225

which makes it seldom necessary to use negative readings, and
small enough degree division that it is commonly unnecessary
to use fractional parts of a degree in expressing temperatures.
The Reaumer scale ("ice point" 0°, "boiling point" 80°), used for
household purposes in Germany, has nothing to recommend it.

It is frequently necessary to change a temperature reading
from the Fahrenheit scale to the centigrade or vice versa. For
convenience in illustrating the method, let A and B (Fig. Ill)
represent two thermometers which are exactly alike except that
A is calibrated according to the centigrade scale, and B accord-
ing to the Fahrenheit. If both are placed in crushed ice, A will
read 0° C. and B, 32° F.; while if placed in steam, A will read
100° C. and B, 212° F. If both thermometers are placed in
warm water in which A reads 40° C., then the temperature /
that thermometer B should indicate may be found as follows:
The fact that the distance between the ice point and boiling point
is 100° on A, and 180° on B, shows that the centigrade degree
is 180/100 or 9/5 Fahrenheit degrees. From the figure it is
seen that / is 40° C. above ice point or 40X9/5 = 72° F. above
32° F., or 104° F. Next, let both thermometers be placed in
quite hot water in which B reads 140° F., and let it be required
to find the corresponding reading c of A. Since 140 — 32 = 108,
the distance be corresponds to 108° F., or 108X5/9 = 60° C.
Hence 140° F. = 60° C. In the same way any temperature
reading may be changed from one scale to the other.

167. Other Thermometers. — There are several different kinds of
thermometers, each designed for a special purpose, which we shall now
briefly consider.

Maximum Thermometer. — In the maximum thermometer of Negretti
and Zambra there is, near the bulb, a restriction in the capillary bore of
the stem. As the temperature rises, the mercury passes the restriction,
but as the temperature falls, and the mercury in the bulb contracts, the
mercury thread breaks at the restriction and thus records the maximum
temperature. To reset the instrument, the mercury is forced past the
restriction down into the bulb by the centrifugal force developed by
swinging the thermometer through an arc.

The Clinical Thermometer. — The clinical thermometer, used by phy-
sicians, differs from the one just described in that it is calibrated for but
a few degrees above and below the normal temperature of the body
(98°. 6 F.). It also has a large bulb in comparison with the size of the
bore of the stem, thus securing long degree divisions and enabling more
accurate reading.

226

MECHANICS AND HEAT

Six's Maximum and Minimum Thermometer. — In this thermometer
the expansion of the alcohol (or glycerine) in the glass bulb A (Fig. 112),
as the temperature rises, forces the mercury down in tube B and up in
the tube C. As the mercury rises in C it pushes the small index c

(shown enlarged at left) before it.
When the temperature again falls, c
is held in place by a weak spring and
thus records the maximum tempera-
ture. The contraction of the alco-
hol in A as the temperature decreases
causes the mercury to sink in C and
rise in B. As the mercury rises in B
it pushes index b before it and thus
records the minimum temperature.
This thermometer is convenient for
meteorological observations. The
instrument is reset by drawing the
indexes down to the mercury by
means of a magnet held against the
glass tube.

The Wet-and-dry-bulb Thermom-
eter, also used in meteorological
work, is discussed in Sec. 198 and
Sec. 222.

The Gas Thermometer. — There are
two kinds of gas thermometers, the
constant-pressure and the constant-
volume thermometers. A simple
form of Constant-pressure Thermom-
eter is shown in Fig. 113. As the
gas in B is heated or cooled, the
accompanying expansion and con-
traction forces the liquid index / to
the right or left. The fact that for
each degree of rise or fall in tem-
perature, the volume of a given
quantity of gas (under constant pres-
sure) changes by 1/273 of its volume
at 0° C. (Sec. 171), makes possible
the accurate marking of the degree

division on the stem, provided the volume of B and the cross section of
the bore of the stem are both known.

A simple form of the Constant-volume Gas Thermometer is shown in
Fig. 114. The stem A of the bulb B which contains the gas is connected
with the glass tube C by the rubber tube T which contains the mercury.

FIG. 112.

THERMOMETRY AND EXPANSION

227

When a quantity of gas is heated and not permitted to increase in
volume, its pressure increases 1/273 of its pressure at 0° C. for every
degree (centigrade degree) rise in temperature (Sec. 171). If, when B
is at 0° C., and meniscus mi is at mark a, the meniscus m2 is at the same
level as mi, then it is known that the pressure of the gas in B is one atmos-
phere. If, now, the temperature of .B rises, mi is pushed down; but by

raising C until m2 is at the proper height h above mi, the mercury is
forced back to mark a, thus maintaining the constancy of the volume
of air in B and A. Suppose that the required height h is 10 cm. The
excess pressure of the gas in B above atmospheric pressure will then be
10/76 or 36/273 atmospheres, and the temperature of B, according to the
gas law just stated, must be 36° above zero, that is 36° C.

The Constant-volume Hydrogen Ther-
mometer is by international agreement
the standard instrument for tempera-
ture measurements. This instrument
differs in detail, but not in principle,
from the one shown in Fig. 114.

The Dial Thermometers. — If the tube
of the Bourdon Gage (Sec. 141) is filled
with a liquid and then plugged at A,
the expansion of the liquid upon beir.g
heated will change the curvature of
the tube and actuate the index just as
explained for the case of steam pressure.

The Metallic Thermometer. — A spiral
made of two strips of metal a and
b soldered together (Fig. 115) will un-
wind slightly with temperature rise if
the metal b expands more rapidly than
a. As the spiral unwinds it causes
the index I to move over the scale and indicate the temperature.

Recording Thermometer. — If the scale in Fig. 115 were replaced by a
drum revolving about a vertical axis and covered by a suitably ruled
sheet of paper, and if, further, the left end of the index 7 were provided
with an inked tracing point resting on the ruled sheet, we would then
have represented the essentials of the recording thermometer or Thermo-
graph. The drum is driven by a clock mechanism and makes (usually)

FIG. 114.

228 MECHANICS AND HEAT

one revolution per week. If the temperature remains constant, the trac-
ing point draws a horizontal line on the drum as it rotates under it. As
the temperature rises and falls, the tracing point rises and falls and
traces on the revolving drum an irregular line which gives a permanent
and continuous record of the temperature for the week. Obviously the
days of the week, subdivided into hours, would be marked on the sheet
around the circumference of the drum; while the temperature lines,
properly spaced, would run horizontally around the drum and be num-
bered in degrees from the bottom upward.

FIG. 115.

168. Linear Expansion. — When a bar of any substance is
heated it becomes slightly longer. In some cases, especially
with the metals, allowance must be made for this change in
length, called linear expansion. Thus, a slight space is left between
the ends of the rails in railroad construction. If this were not
done, the enormous force or end thrust exerted by the rails upon
expansion during a hot day would warp the track out of shape.
The contraction and expansion of the cables supporting large
suspension bridges cause the bridge floor to rise and fall a dis-
tance of several inches as the temperature changes. A long
iron girder bridge should have one end free to move slightly
lengthwise (on rollers) on the supporting pier to permit its ex-
pansion and contraction without damage to the pier.

In the familiar process of "shrinking" hot iron tires onto
wooden wagon wheels, use is made of the contraction of the tire
that takes place when it cools. Cannons are constructed of
concentric tubes, of which the outer ones are successively heated
and "shrunk" onto the inner ones. This extremely tight fitting

THERMOMETRY AND EXPANSION 229

of the outer layers insures that they will sustain part of the stress
when the gun is fired.

169. Coefficient of Linear Expansion. — When a bar, whose
length at 0° C. is L0, has its temperature raised to 1° C., its length
increases by a certain fraction a of its original length L0. This
fraction a, which is very small for all substances, is called the
coefficient of linear expansion for the material of which the bar is
composed. The actual increase in length of the bar is then L0a.
When heated from 0° to 2°, the increase in length is found to
be almost exactly twice as great as before, or L01a; while if
heated from 0° to t°, it is very closely L0at. Consequently the
length of the bar at any temperature t, which length may be
represented by Lt, is given by the equation.

Lt=Lo+L0at=L0(l + a£) (75)

whence

in which Lt—L0 is the total increase in length for a change of t
degrees, and hence (Lt— L0) divided by t is the total change for
one degree. If this total change is divided by the length L0 of
the bar (in cms.) we have the increase in length per centimeter
of length (measured at 0° C.) per degree rise of temperature,
which by Eq. 75a is a. Thus a may also be defined as the increase
in length per centimeter (i.e., per cm. of the length of the bar
when at 0° C.) produced by 1° C. rise in temperature, or the
increase in length per centimeter per degree.

To illustrate, suppose that two scratches on a brass bar are
1 cm. apart when the bar is at 0° C. Then, since a for brass is
0.000019 (approx., see table), it follows that at 1° C. the
scratches will be 1.000019 cm. apart; at 2° C., 1.000038 cm.; at
10° C., 1.00019 cm. apart, etc. Since the length Lt of a metal
bar at a temperature t differs very little from its length at 0°,
i.e., L0, we may for most purposes consider that its increase in
length when heated from a temperature t to i+1 is Lta instead
of L0a. Consequently, when heated from a temperature t to
a still higher temperature t', the increase in length is approx-
imately Lt a (t' — t). We then have the length L/ at the higher
temperature expressed approximately in terms of Lt by the
equation

230

MECHANICS AND HEAT

This equation is accurate enough for all ordinary work and it
is also a very convenient equation to use in all problems involv-
ing two temperatures, neither of which is zero. Strictly speak-
ing, a is not constant, but increases very slightly in value with
temperature rise.

AVERAGE COEFFICIENT OF LINEAR EXPANSION OF A FEW SUBSTANCES

Substance

Coeff. of Exp. a

Substance

Coeff. of Exp. a

Brass

0 0000185

Oak, with grain.

0 000005

Copper
Glass
Ice

0.0000168
0.0000086
0 000050

Platinum
Quartz, fused ....
Silver

0.0000088
0.000005
0 000019

Iron

0.000012

Zinc

0.000029

It is perhaps well for the student to memorize a for plati-
num and note that for oak it is less than for platinum and
for most metals about twice as great. In the case of glass, a
varies considerably for the different kinds.

The French Physicist Guillaume recently made the interest-
ing discovery that the coefficient of expansion of a certain nickel-
steel alloy (36 per cent, nickel), known as Invar, is only about
one-tenth as large as that of platinum, or 0.0000009. From these
figures we see that the length of a bar of this metal increases
less than 1 part in 1,000,000 when its temperature is raised
1° C. Steel tapes and standards of length are quite commonly
made of Invar.

170. Practical Applications of Equalities and Differences in
Coefficient of Linear Expansion. — In the construction of incan-
descent lamps it is necessary to have a vacuum in the bulb, or
the carbon filament that gives off the light will quickly oxidize
or "burn out." The electric current must be led through the
glass to the filament by means of wires sealed into the glass
while hot. If the glass and wire do not expand alike upon being
heated, the glass will crack and the bulb will be ruined. Plati-
num wire is used for this purpose because its coefficient of ex-
pansion is almost exactly the same as that of glass.

The differences between the coefficients of expansion for any
two metals, for example, brass and iron, has many practical
applications. Important among the devices which utilize these
differences in expansion are the automatic fire alarm, the thermo-
stat, and the mechanism for operating the "skidoo" lamp used in
signs. Another very important application of this difference

THERMOMETRY AND EXPANSION

231

in expansion of two metals is in the temperature compensation
of clock pendulums and the balance wheels of watches. By
means of these compensation devices, timepieces are prevented
from gaining or losing time with change of temperature.

The Fire Alarm. — The operation of the fire alarm will be
understood from a study of Fig. 116. An iron bar I and a brass
bar B are riveted together at several points and attached to a
fixed support D at one end, the other end C being free. Since
the coefficient of expansion for brass is greater than for iron, it
will be evident that the above composite bar will curve upward
upon being heated, and downward upon being cooled. Conse-
quently the end C will rise when the temperature rises, and fall
when the temperature falls. If such a device is placed near the

Battery

FIG. 116.

ceiling of a room, and if by suitable wiring, electrical connections
are made between it and an electric bell, it become a fire alarm.
For if a fire breaks out in the room, both bars / and B will be
equally heated, but B will elongate more than 7, thus causing
C to rise until it makes contact with P. This contact closes the
electrical circuit and causes the electric bell to ring.

The Thermostat. — If the room above considered becomes too
cold, C descends and may be caused to touch a suitably placed
point Pi, thereby closing another electrical circuit (not shown)
connected with the mechanism that turns on more heat. As
soon as the temperature of the room rises to its normal value, C
again rises enough to break connection with Plf and the heat
supply is either cut off or reduced, depending upon the adjust-
ment and design of the apparatus. When so used, the above
bar, with its connections, is called a thermostat.

232 MECHANICS AND HEAT

In a common form of thermostat, the motion of C, when the
room becomes too cold, opens a "needle" valve to a compressed
air pipe. This pipe leads to the compressed air apparatus,
which is so arranged that when the air escapes from the above-
mentioned valve, more heat is turned on.

The "Skidoo Lamp." — This device is very much used in
operating several lamps arranged so as to spell out the words
of a sign. Such a sign is much more noticeable if the lamps
flash up for an instant every few seconds than if they shine
steadily. The arrangement (using only one lamp) is shown in
Fig. 117. The binding posts E and F are connected to the light-
ing circuit. Bars / and B are arranged just as in Fig. 116, ex-
cept that the brass bar is above the iron bar instead of below.

When these bars are not touching the point p, the electric
current passes from E to a, at which point the wire is soldered

FIG. 117.

to the bars, then on through the coil D of very many turns of
fine wire wrapped about the bars, to point P, where the wire is
again soldered, and finally through the lamp, back to the binding
post marked F.

Since coil D offers very great resistance to the passage of cur-
rent, only a small current flows, and the lamp does not glow.
This small current, however, heats coil D and therefore bars B
and /; and, since B expands more rapidly than /, point C moves
down until it touches point P as explained in connection with
Fig. 116. The instant that point C touches P, practically all
of the current flows directly from a through the heavy bars to
P and then through the lamp as before. The fact that the
current does not have to flow through coil D when C and P are
in contact produces two marked changes which are essential to
the operation of the lamp. First, since the electrical resistance
of the bars is small, the current is much greater than before and

THERMOMETRY AND EXPANSION

233

the lamp glows; and second, the coil now having practically
no current, cools down slightly, thus permitting the bars to cool
down, thereby causing C to rise. The instant C rises, the current
is obliged to go through the coil, and is therefore too weak to
make the lamp glow, but it heats the coil, causing C to descend
again and the cycle is thus repeated indefinitely. If the contact
screw S is screwed down closer to P, the lamp "winks" at
shorter intervals.

The Balance Wheel of a Watch. — The same principle discussed
above is used in the "temperature compensation" of the balance

wheel of a watch, due to which compensation its period does not
change with change of temperature. When an uncompensated
wheel is heated the resulting expansion causes its rim to be
farther from its axis, thereby increasing its moment of inertia.
As its moment of inertia increases, the hairspring (H.S., Fig.
118) does not make it vibrate so quickly and the watch loses
time. To make matters worse, the hairspring becomes weaker
upon being heated.

It will be noticed in the balance wheel, sketched in Fig. 118,
that the expansion produced by a rise in temperature causes the
masses C and D (small screws) to move from the center; while

234

MECHANICS AND HEAT

at the same time it causes E and F to move toward the center.
For the brass strip B forming the outside of the rim expands
more than the iron strip I inside. If the watch runs faster
when warmed it shows that it is overcompensated; whereas if
it runs slower when warmed it is undercompensated. Over-
compensation would be remedied by replacing screws E and F
by lighter ones, at the same time perhaps replac-
ing C and D by heavier ones.

The Gridiron Pendulum. — From the sketch of
the gridiron pendulum shown in Fig. 119, it will
be seen that the expansion of the steel strip a,
and the steel rods b, d, and/, causes the pendulum
bob B to lower, thereby increasing the period of
the pendulum; whereas the expansion of the zinc
rods c and e evidently tends to raise B, thereby
shortening the pendulum and also its period
By having the proper relation between the lengths
of the zinc and the iron rods, these two opposing
tendencies may be made to exactly counterbal-
ance each other. In this case the period of the
pendulum is unaffected by temperature changes,
that is, exact temperature compensation is ob-
tained. If rods c and e were brass, their upward
expansion would not compensate for the down-
ward expansion of the iron rods. It would then
be necessary to have four rods of brass and five
of iron.

171. Cubical Expansion and the Law of
Charles. — When a given quantity of any sub-
stance, say a metal bar, whose volume at 0° C. is
V0, has its temperature raised to 1° C., its volume increases by
a certain small fraction 0 of its original volume V0. This fraction
(8 is called the coefficient of cubical expansion of the substance in
question. The actual increase in volume is then F0/3. If the
bar is heated from 0° to t°, i.e., through t times as great a range,
the increase in volume is found to be almost exactly t times as
great, or V0ftt. Accordingly, the volume at t°, which may be
represented by Vt, is given by the equation

Vt = V0+ V0&t = V0 (1 +00 (76)

whence Vt—V0

FIG. 119.

V0t

(77)

THERMOMETRY AND EXPANSION

235

In Eq. 77, Vt— V0 is the total increase in volume; (Vt—V0)-*-
t is the total increase per degree rise in temperature; and divid-
ing the latter expression by V0 gives (Vt— F0)-5- Vj, or the in-
crease per degree per cubic centimeter. But (Vt— V0) + VJt>
is /3 from Eq. 77. Hence /3 is numerically the increase in volume
per cubic centimeter of the "original" volume per degree rise in
temperature. By "original" volume is meant the volume of the
bar when at 0° C.

Equations 76 and 77 apply to volumes of solids, liquids, or
gases. The values of 0, however, differ widely for different sub-
stances, as shown in the table below. These equations apply
to gases only if free to expand against a constant pressure when
heated.

When a solid, e.g., a metal bar, expands due to temperature rise, it
increases in each of its three dimensions — length, breadth, and thick-
ness. For this reason, it may be shown that the coefficient of cubical
expansion is 3 times the coefficient of linear expansion for the same
substance; i.e., /3=3«. For, consider a cube of metal, say, each edge
of which has a length L0 at 0° C. Then, by Eq. 74, the length of
each side at a temperature t° will be L0(l + aO. The volume at 0°, or
V0, is L03; while the volume V« at 1° is

V>L0

(78)

Expanding (1 + aO8, we have !+3aZ-(-3a2£2+a3J3. Now, since a is
very small, a2 and a3 will be negligibly small (observe that (1/1000)2 =
1/1,000,000), and the terms 3aH* and <*3t3 may be dropped. Eq. 78
then becomes

Vt=V0 (1+3 at)

(79)

By comparing Eq. 79 with Eq. 76 we see at once that /3=3a. which
was to be proved. In like manner it may be shown that the coefficient
of area expansion of a sheet of metal, for example, is 2 a.

Accordingly, the fractional parts by which the length of a bar of iron,
the area of a sheet of iron, and the volume of a chunk of iron increase
per degree, are respectively 0.000012, 0.000024, and 0.000036.

COEFFICIENT OF CUBICAL EXPANSION OF A FEW SUBSTANCES

Substance

Alcohol
Ether

0.00104
0 0017

Air, and all gases
Iron

0.00367
0 000036

Mercury
Petroleum

0.00018
0.00099

Zinc
Glass

0.000087
0.000026

236

MECHANICS AND HEAT

Gas

The Law of Charles. — If a quantity of gas which is confined
in a. vessel A (Fig. 120) by a frictionless piston P, at atmospheric
pressure and 0° C., is heated to 1° C. it will expand 1/273 (or
0.00367) of its original volume; so that its volume becomes
1.00367 times as great. The fact that this value of /3 (Eq. 77)
is practically the same for all gases was discovered by Charles
and is known as the Law of Charles.

If, now, the piston is prevented from moving, then, as the gas
is heated it cannot expand, but its pressure will increase 1/273
for each degree rise in temperature, as might be detected by the
attached manometer M; while if cooled 1°, its pressure will de-
crease 1/273. If cooled to 10° below zero its pressure will de-
crease 10/273 of its original value, etc. Hence the inference,
that if it were possible to cool a gas
to— 273° C. it would exert no pressure
whatever.

Absolute Zero and the Kinetic Theory
of Gases. — According to the Kinetic
Theory of Gases, a gas exerts pressure
because of the to-and-fro motion of
its molecules (Sec. 131). These mole-
cules are continually colliding with
each other, and also bombarding the
walls of the enclosing vessel. The impact of the molecules
in this bombardment gives rise to the pressure of gases, just
as we know that a ball, thrown against the wall and then re-
bounding from it, reacts by producing a momentary thrust
against the wall. Millions of such thrusts per second would,
however, give rise to a steady pressure. Under ordinary con-
ditions the average speed of the air molecules required to pro-
duce a pressure of 15 Ibs. per sq. in. is about 1400 ft. per sec.
But a body is supposed to have heat energy due to the motion
of its molecules. It may therefore be said : (a) that at — 273° C.
a gas would exert no pressure (see above) ; hence (6) that its molec-
ular motion must cease; and therefore (c) that it would have no
heat energy at this temperature. When a body has lost all of
its heat energy, it cannot possibly become any colder. This
temperature of —273° C. is therefore called the Absolute Zero,
It is interesting to note that extremely low temperatures,
within a few degrees of the absolute zero, have been produced
artificially. By permitting liquid helium to evaporate in a par-

FIG. 120.

THERMOMETRY AND EXPANSION 237

tial vacuum, Kammerlingh-Onnes (1908) produced a temperature
of —270° C., or within 3° of the absolute zero.

172. The Absolute Temperature Scale. — If the above absolute
zero is taken as the starting point for a temperature scale, then
on this scale, called the Absolute Centigrade Scale, ice melts at
+273°; water boils at 373° (373° A.); a temperature of 20° C.=
293° A., and -10° C. = 263° A., etc. This absolute scale is of
great value from a scientific point of view. Its use also greatly
simplifies the working of certain problems.

It will now be shown that if the pressure upon a gas is kept
constant while its temperature is increased from T\ to T2, then its
volume will be increased in the ratio of these two temperatures
expressed on the absolute scale. In other words,

in which Vi and F2 represent the volume of the gas at the lower
and higher temperatures respectively, and TI and Tz, the cor-
responding temperatures on the absolute scale.

Proof: Obviously 77i = «i+273, and 772 = <2+273; i.e., the
centigrade readings t\ and tz are changed to absolute tempera-
ture readings by adding 273, which is the difference between the
zeros of the two scales. From Eq. 76, since ft is 1/273, we have

and likewise Vz = V0(l +073

(1+ ** }
Vz IV ^273/

=~ "

V T

i.e., =? = • z (pressure being kept constant) (80)

V\ i i

Eq. 80 shows that if the absolute temperature of a certain
quantity of gas is made say 5/4 as great, its volume becomes
5/4 as great; while if the absolute temperature is doubled the
volume is doubled, etc. It must be borne in mind that Eq. 80
holds only in case the gas, when heated, is free to expand against
a constant pressure. A discussion of Fig. 121 will make clear

238 MECHANICS AND HEAT

the application of Eq. 80. Let A be a quantity of gas of volume
Fi and temperature 27° C. confined in a cylinder by a frictionless
piston of negligible weight. Let the upper surface of the piston
be exposed to atmospheric pressure. The gas in A will then also
be under atmospheric pressure regardless of temperature change.
For, as the gas is heated, it will expand and push the piston up-
ward; the pressure, however, will be unchanged thereby, i.e.,
the pressure will be constant, and therefore Eq. 80 will apply.
Next let the gas in A be heated from 27° C. to 127° C., i.e., from
300° A. to 400° A. Since the absolute temperature is 4/3 as
great as before, we see from Eq. 80 that

_t o— - *__

21° C.

P, will be raised to a position PI such
that the volume of the gas will be 4/3 of
its former volume. Experiment will
show that the new volume is 4/3 times
the old, thus verifying the equation. Let
J us again emphasize the fact that the two

? volumes are to each other as the corre-
sponding absolute temperatures, not centi-
grade temperatures.

Since, as above stated, the pressure of

I a body of gas that is not permitted to

JPIG 121 expand increases i/273 of its value when

the gas is heated from 0° to t° C., it fol-
lows that the pressures p\ and p2 corresponding to the tempera-
tures ti and £2, are given in terms of p0 (the pressure when the
temperature is zero) by the equations

pl = p0(l+2j^ and ?*"^1"hj7a)
from which (see derivation of Eq. 80) we have

= TTT I volume constant) (81)

Pi L i

This equation shows that if any body of gas, contained in a rigid
vessel to keep its volume constant, has its absolute temperature
increased in a certain ratio, then its pressure will be increased in
the same ratio. • .

Boyle's Law is expressed in Eq. 72 as pV = K. Consequently
if the pressure on the gas in question is increased to p\ the volume
will decrease to Fi, but the product will still be K; i.e., p\V\ = K.
Likewise pzVi = K, and therefore p\V\ = pzVz or Vz/V\ =

THERMOMETRY AND EXPANSION 239

Summarizing, we may write the three important gas laws,
namely Boyle's, Charles's, and the one referring to pressure
variation with temperature, thus:

(E-£), <72bis>

Observe that the subscript T indicates that Eq. 72 is true only
if the gas whose pressure and volume are varied is maintained at
a constant temperature. The subscript p of Eq. 80 indicates that
the pressure to which the gas is subjected must not vary, and V
of Eq. 81, that the volume must not vary.

Attention is called to the fact that the three important variables
of the gas, namely pressure, volume, and temperature, might
all change simultaneously. If the temperature of the gas is kept
constant, Boyle's Law (Eq. 72) states that the volume varies
inversely as the applied pressure. Eq. 80 states that if the pressure
upon the gas is kept constant, then the volume varies directly
as the absolute temperature; while Eq. 81 states that if the volume
of the gas is kept constant, then the pressure varies directly as
the absolute temperature.

The General Case. — In case both the temperature of a gas and
the pressure to which it is subjected change, then the new volume
(note that all three variables change) may easily be found by
considering the effect of each change separately; i.e., by suc-
cessively applying Boyle's Law and Charles's Law. To illus-
trate, let the volume of gas in A (Fig. 121), when at atmospheric
pressure and 20° C., be 400 cm.3, and let it be required to find its
volume if the pressure is increased to 2| atmospheres, and its
temperature is raised to 110° C. The new pressure is 5/2
times the old; hence, due to pressure alone, in accordance with
Boyle's Law, the volume will be reduced to 2/5X400 cm.3 The
original temperature of 20° C. is 293° A., and the new tempera-
ture is 383° A.; hence, due to the temperature change alone, the
volume would be 383/293X400 cm.3 Considering both effects,
the new volume would then be

2 ^8*3

7 = ^X00^X400 cm.3
o ^yo

240

MECHANICS AND HEAT

We may proceed in a similar manner if both the volume and the
temperature are changed, and the new pressure that the gas will
be under is required in terms of the old pressure.

173. The General Law of Gases. — We shall now develop the
equation expressing the relation between the old and the new
values of pressure, volume, and temperature of some confined
gas when all three of these quantities are changed. Let 1, 2,
and 3, respectively, be the initial, second, and final positions
of the piston A (Fig. 122). In the initial state, A confines a
certain quantity of gas of volume V0,
pressure p0 (say 1 atmosphere), and tem-
perature To (say 0° C. or 273° A).

The second state is produced by heat-
ing the gas from T0 to T, in which T/T0
' expressed in the absolute scale is, say,
r about 3/2. This change in temperature
causes the gas to expand against the
constant pressure p0 until A is at 2, the
new volume V being about f V0. In
this second state of the gas, its condition
V, and T as indicated in the sketch, and,

FIG. 122.

is represented by p0
from Eq. 80, we have

_ _

Vo~T0' ~2V

The third state of the gas, represented by p, V, and T, is pro-
duced by placing a weight on A, thereby increasing the pres-
sure from p0 to p (as sketched p/p0 = 5/4 approx.), and push-
ing the piston from position 2 to its final position at 3, and
consequently reducing the volume from V to V (as sketched
V/V = 4/5 approx.). From Boyle's Law (Eq. 72 bis, just given),

Substituting in this equation the value of V given above,
we have

-TT -TT '

poV' = ~°T7T=~^[

that is,

(82)

THERMOMETRY AND EXPANSION 241

PoV0

in which R is equal to ^-o°' an^ *s therefore a known constant

Zi o

if p and V0 are known. Obviously, if twice as great a mass
of the same gas, or an equal mass of some other gas half
as dense, were placed under the piston, the constant R would then
become twice as large.

Eq. 82 expresses the General Law of Gases, and is called the
General Gas Equation. From this general equation, we see (a)
that for a given mass of gas the volume varies inversely as the pres-
sure if the temperature is constant (Boyle's Law) ; (6) that the
volume varies directly as the absolute temperature T if the pres-
sure p is constant (Law of Charles); and (c) that the pressure
varies directly as the absolute temperature if the volume V is
constant. The law embodied in (c) has not received any name.

Let us now use Eq. 82 to work the problem given under the
heading "The General Case" (Sec. 172). Let us represent the
first state by ptVi = RTi and the second state by p2V2

Pi p*

Consequently

V2 RT2 . RTi

Vi~ p2 ' pi
or

7^X^,0^x11x400
Pz i i 5 293

as before found. Let us again emphasize the fact that T, T\,
and T2 represent temperatures on the absolute scale.

174. The Thermocouple and the Thermopile. — If a piece of
iron wire / (Fig. 123) has a piece of copper wire C fastened to
each end of it as shown, it will be found that if one point of con-
tact of these two dissimilar metals, say, B is kept hotter than the
other junction A, an electric current will flow in the direction
indicated by the arrows. This current might be measured by
the instrument D. If B is, say, 60° hotter than A, the electric
current will be about 6 times as large as if it is only 10° hotter.
Two such junctions so used constitute a Thermocouple. Any two
different metals may be used for a thermocouple. Antimony
and bismuth give the strongest electrical effect for a given dif-
ference of temperature between junctions.

One hundred or so thermocouples, made of heavy bars and

242 MECHANICS AND HEAT

properly connected, form a Thermobattery of considerable strength.
The greatest usefulness of thermocouples, however, is in delicate
temperature measurements by means of the thermopile.

The Thermopile. — By observing
the readings of D while the tem-
perature difference between A and
B is varied through a considerable
range (Fig. 123), in other words,
by calibrating the thermocouple, it
becomes a thermometer for meas-
uring temperature differences. A
large number of such thermocouples
properly connected constitute a
FIG. 123. Thermopile, which will detect ex-

ceedingly small differences of tem-
perature. The thermopile readily detects the heat radiated from
the hand, or from a lighted match, at a distance of several feet.

PROBLEMS

1. Express 60° C. and -30° C. on the Fahrenheit scale, and also on the
absolute scale.

2. Express 200° A. on the Fahrenheit scale, and also on the centigrade
scale.

3. An iron rail is 32 ft. long at 0° C. How long is it on a hot day when at
40° C.?

4. A certain metal bar, which is 3 meters in length at 20° C., is 0.30 cm.
longer at 100° C. Find a for this metal.

6. If the combined lengths of the iron rods a, b, and d (Fig. 119) is 100 cm.,
how long must c and e each be to secure exact temperature compensation?

6. How many H.P. does the sun expend upon one acre at noon? Assume
the sun to be directly overhead.

7. The cavity of a hollow brass sphere has a volume of 800 cm.3 at 20° C.
What is the volume of the cavity at 50° C.?

8. If 600 cm.3 of gas, at 20° C. and atmospheric pressure, is heated to
40° C., and is free to expand by pushing out a piston against the pressure of
the atmosphere, what will be its new volume?

9. If 6 cu. ft. of air, at 20° C. and atmospheric pressure, is compressed
until its volume is 2 cu. ft., and is then heated to 300° C., what will be its
new pressure?

CHAPTER XIV
HEAT MEASUREMENT, OR CALORIMETRY

175. Heat Units. — Before taking up the discussion of the
measurement of quantity of heat, it will be necessary to define
the unit in which to express quantity of heat. The unit most
commonly used is the Calorie. The calorie may be roughly
defined as the quantity of heat required to raise the temperature
of one gram of water 1° C. To be accurate, the actual tempera-
ture of the water should be stated in this definition, since the
quantity of heat required varies with the temperature. Thus,
the quantity of heat required to raise the temperature of 1 gram
of water through a range of 1° is greater at 0° than at any other
temperature, and almost 1 per cent, greater than it is at 20°, at
which point it is a minimum.

Some authors select this range from 0° C. to 1° C., others
3°.5 C. to 4°.5 C., 4° C. to 5° C., etc., which gives of course
slightly different values for the calorie. In selecting 15° C. to
16° C. as the range, we have a calorie of such magnitude that 100
calories are required to raise the temperature of one gram of
water from 0° C. to 100° C. Hence the calorie is perhaps best
defined as the amount of heat required to raise the temperature of
one gram of water from 15° C. to 16° C.

In the British system, unit quantity of heat is the quantity
required to raise the temperature of 1 Ib. of water 1° F., and is
called the British Thermal Unit, or B.T.U. Since heat is a form
of energy, it may be expressed in energy or work units. One
B.T.U. =778 ft.-lbs. This means that 778 ft.-lbs. of work
properly applied to 1 Ib. of water, for example, in stirring the water,
will raise its temperature 1° F. From the above statement,
since 1 Ib. of water in falling 778 ft. develops 778 ft.-lbs. of
energy, we see that if a 1-lb. mass of water strikes the ground after
a 778-ft. fall, and if it were possible to have all of the heat developed
by the impact used in heating the water, then this heat would
raise its temperature 1° F. In fact this temperature rise is in-
dependent of the quantity of water, and depends solely upon the
height of fall. For, while the heat energy, developed by, say 3

243

244 MECHANICS AND HEAT

Ibs. of water, due to impact after a 778-ft. fall, would be 3 times as
much as above given, the amount of water to be heated would
also be 3 times as much, and the resulting temperature rise
would therefore be 1° F. as before. The calorie is 4.187 X107
ergs. That is, if 4.187 XlO7 ergs of energy are used in stirring
one gram of water, its temperature will rise 1° C. This 4. 187 X
107 ergs is often called the Mechanical Equivalent of heat. The
mechanical equivalent in the English system is 778 ft.-lbs.

176. Thermal Capacity. — The thermal capacity of a body is
denned as the number of calories of heat required to raise the
temperature of the body 1° C., or it is the amount of heat the
body gives off in cooling 1° C. It is clear that a large mass would
have a greater thermal capacity than a small mass of the same
substance. That mass is not the only factor involved is shown by
the following experiment.

If a kilogram of lead shot at 100° C. is mixed with a kilogram of
water at 0° C., the temperature of the mixture will not be 50°,
but about 3°. The heat given up by the kilogram of lead in
cooling 97° barely suffices to warm the 1 kilogram of water 3°.
In fact the thermal capacity of the water is about 33 times as
great as that of the lead; consequently, if 33 kilos of lead had been
used in the experiment the temperature of the mixture would have
been 50°. The very suggestive and convenient term "water
equivalent" is sometimes used instead of thermal capacity.
Multiplying the mass of a calorimeter by its specific heat gives
its thermal capacity or the number of calories required to warm it
one degree. Suppose that this number is 60. Now 60 calories
would also heat 60 grams of water one degree; hence the "water
equivalent" of the calorimeter is 60; i.e., the calorimeter requires
just as much heat to raise its temperature a given amount as
would 60 gm. of water if heated through the same range.

177. Specific Heat. — -The Specific Heat (s) of a substance may
be defined as the number of calories required to heat 1 gm. of the
substance 1° C. It is therefore the thermal capacity per gram of
the substance. This, we see from the definition of the calorie, is
practically equal to the ratio of the heat required to heat a given
mass of the substance through a given ran^e of temperature, to
the heat required to heat an equal mass of water through the
same range. Thus, the specific heat of lead is 0.031. This means
that it would require 0.031 calorie to heat a gram of lead one
degree; which is only 0,031 times as much heat as would be re-

HEAT MEASUREMENT, OR CALORIMETRY 245

quired to heat a gram of water one degree. The specific heat of a
substance is sometimes defined as the ratio just given. Since the
specific heat (calorie per gram per degree) of water varies with the
temperature (Sec. 175), this definition lacks definiteness as
compared with the one we are here using.

The table below gives the specific heat of a few substances.
From the values given, we see that one calorie of heat imparted
to a gram of glass would raise its temperature 5° C., while the
same amount of heat imparted to a gram of lead would raise its
temperature 1/0.031, or about 32°.5. In popular language it
might be said that lead heats 6.5 times as easily as glass, and 32.5
times as easily as water.

The specific heat of a substance is usually expressed in calories
per gram per degree. Thus, the specific heat of lead is 0.031 cal.
per gm. per deg. It may also be written 0.031 B.T.U.'s per Ib.
per degree, the degree in this case, however, being the Fahrenheit
degree. The proof that the numeric (0.031) is the same in both
cases may be left as an exercise for the student.

The specific heat of most substances varies considerably with
the temperature. In some cases there is a decrease in its value
with temperature rise, while in others there is an increase. In
the case of water the specific heat decreases up to 20° C. and then
increases. The values given in the table for the different sub-
stances are average values, taken at ordinary temperatures (ex-
cepting in the case of ice and steam).

Substance

Sp. heat in
cal. per
gm. per deg.

Substance

Sp. heat in
cal. per
gm. per deg.

Brass
Copper
Glass

0.088
0.093
0 200

Ice
Steam
Water

0.504
0.4 approx.
1 000 (15° to 16°)

Lead .
Iron

0.031
0.11

Alcohol
Petroleum

0.60
0.51

To heat a gram of any substance of specific heat s sufficiently
to cause a temperature rise of t degrees requires st calories,
i.e., t times as much heat as to cause a rise of 1 degree. Further,
to heat M grams t degrees requires M times as much heat as to heat
one gram t degrees, or Mst calories; hence, the general expression
for the heat H required to heat a body of mass M and specific
heat s from a temperature t\ to a temperature tz, is

td (83)

246 MECHANICS AND HEAT

If the substance cools through this same range, then H is the heat
given off.

178. The Two Specific Heats of a Gas.— In general, a body
when heated, expands, and in expanding it does work in pushing
back the atmosphere. This work makes it require additional
heat energy to warm the body, and therefore makes the specific
heat of the body larger than it would have been had expansion not
occurred. In case a compressed gas is permitted to expand
into a space at lower pressure, the above heat energy is taken from
the gas itself and chills it greatly. This fact is utilized in the
manufacture of liquid air (Sec. 205).

In the case of solids and liquids, this expansion upon being heated
is inappreciable, but with gases it is very great. Consequently
the specifice heat of a gas, i.e., the number of calories required to
heat one gram one degree, is less if the gas is confined in a rigid
vessel than if it is allowed to expand against constant pressure
when heated. The latter is called the specific heat at constant
pressure, and is 0.237 for air; while the former is called the specific
heat at constant volume, and is 0.168 for air. The ratio of the two
specific heats of air is 0.237/0.168, or 1.41. This ratio differs for
the various gases.

179. Law of Dulong and Petit. — Dulong and Petit, in 1819, found by
experiment that for thirty of the elements, the product of the atomic
weight and the specific heat (in the solid state) is approximately constant.
This so-called constant varies from about 6 to 6.6. For a considerable
number of the elements it is 6.4. For gases this constant is about 3.4.
This law does not hold for liquids, and there are a few solids that do not
follow it at all closely.

Let us now utilize this law in finding the specific heat of iron and gold,
whose atomic weights are respectively 56 and 196. The mathematical
statement of the law of Dulong and Petit is:

Sp. heat X atomic weight =6.4 (approximately) (84)

Whence the specific heat of gold is 6.4/196 or 0.0326, and that of iron
6.4/56 or 0 114. These computed values of the specific heat are almost
exactly the same as those found experimentally for iron and gold.

The above law shows that it takes the same amount of heat to warm an
atom one degree whether it be a gold atom, an iron atom, or an atom of
any other substance which follows this law. For, from Eq. 84, it is
obvious that if the atomic weight of one element is three times a£ great
as that of another (compare gold with iron), then its specific heat must
be 1/3 as great in order to give the same product — 6.4. But if the

HEAT MEASUREMENT, OR CALORIMETRY 247

atomic weight is three times as great for the first metal as for the second,
then the number of atoms per gram will be 1/3 as great, which accounts
for the first having 1/3 as great specific heat as the second, provided we
assume the same thermal capacity for all atoms.

180. Specific Heat, Method of Mixtures.— A method which is
very commonly used for determining the specific heat of sub-
stances is that known as the method of mixtures. The method
can be best explained in connection with the apparatus used, one
form of which is shown in section in Fig. 124. H is a heater con-
taining some water and having a tube T passing obliquely through
it as shown. This tube contains the substance, e.g., the shot, the
specific heat of which is to be determined. D is a calorimeter,
usually of brass, containing some water E.

FIG. 124.

First, the shot, the calorimeter D, and the water E, are weighed.
Let these masses be M , M i, and M 2, respectively. Next the water
in H is heated to the boiling point and kept boiling for a few
minutes. The steam surrounding T soon warms it and the
contained shot to 100° C., which may be determined by thermom-
eter C, thrust through cork B. The cork A is now withdrawn,
and the hot shot is permitted to fall into the water E to which it
rapidly imparts its heat until D, E, and the shot are all at the
same temperature. Let this temperature be t', and let the tem-
perature of E before the shot was introduced be t. The heat
Hi, which the shot loses in cooling from 100° to t', is evidently

248 MECHANICS AND HEAT

equal to the heat H% which the calorimeter and water gain in
rising in temperature from t to t1 ', that is

#1 = HZ (85)

provided no heat passes from the calorimeter to the air or vice
versa during the mixing process.

This interchange of heat between the calorimeter and the air
cannot be totally prevented, but the error arising from this cause
is largely eliminated by having D and E a few degrees lower than
the room temperature at the beginning of the mixing process and
a few degrees higher than room temperature at the end; i.e.,
after D, E, and the shot have come to the same temperature.
During the mixing process, the contents of the calorimeter should
be stirred to insure a uniform temperature throughout.

Almost always in calorimetric work, it is assumed that the
heat given up by the hot body is equal to the heat taken up by
the cold body; so that Eq. 85 is the starting point for the deriva-
tion of the required equation in all such cases. It is much better
to learn how to apply this general equation than to try to mem-
orize special forms of it. One such application will be made here.

If s, si, and s2 represent the specific heats of the shot, calorim-
eter, and water repectively, then from Eq. 83, the heat given
up by the shot is Jlfs(100— t') ; that taken up by the calorimeter is
MiS^t' — t); and that taken up by the water is MzSz(tr — t).
Since s2 is unity it may be omitted, and we have from Eq. 85

(t'-t) (85a)

The quantities M, MI, and M% are determined by weighing, and
the three temperatures are read from thermometers, so that the
one remaining unknown, s, may be solved for.

181. Heat of Combustion. — Chemical changes are, in general,
accompanied by the evolution of heat; a few, however, absorb
heat. Most chemical salts when dissolved in water cool it, in
some cases quite markedly. In still other cases solution is at-
tended by the development of heat. A complete study of these
subjects is beyond the scope of this volume, but the particular
chemical change known as combustion is so all-important in
connection with commercial heating and power development
that a brief discussion of it will be given.

Combustion is usually denned as the violent chemical combina-
tion of a substance with oxygen or chlorine, and is accompanied

HEAT MEASUREMENT, OR CALORIMETRY 249

by heat and light. In a more restricted sense it is what is popu-
larly known as "burning" which practically amounts to the
chemical combination of oxygen with hydrogen or carbon.

In scientific work, the Heat of Combustion of any substance is
the number of calories of heat developed by the complete com-
bustion of 1 gram of that substance. In engineering practice
it is the number of B.T.U.'s developed by the complete combus-
tion of 1 pound of a substance. The latter gives 9/5 as large a
number as the former for the same substance. Hence it is neces-
sary in consulting tables to determine whether the metric, or
the British system is used. Obviously, the burning of one
gram of coal would heat just as many grams of water 1° C. as
the burning of a pound of coal would heat pounds of water 1° C.
But to heat a pound of water 1° C. takes 9/5 B.T.U.'s, since
1° C. equals 9/5° F. In the following table, in which the approxi-
mate values of the heat of combustion are given in both sys-
tems, it will be observed that the numerical values are in the above
ratio of 9 to 5.

HEAT OF COMBUSTION WITH OXYGEN

Substance

Product

Calories per
gram

B.T.U.'s per
pound

Hydrogen(H)

H2O

34,000

61,000

Carbon (C)

CO2

7,800

14,000

Marsh gas (CH4)

CO2 and H2O

13 100

23 600

Alcohol (ethyl)
Petroleum
Soft coal |

CO2andH2O..
CO2andH2O..

Mainly CO2,

7,200
11,000
7,500 to 8,500
7 800

13,000
20,000
Ave. 14,500
Ave 14 000

Wood j

H2O and ash.

4,000 to 4 500

Ave 7 600

Dynamite

1,300

Iron . ...

Fe3O4

1,600

Zinc

ZnO

1,300

Copper

CuO

600

Hydrogen, it will be seen, produces far more heat per gram
than any other substance, indeed over four times as much as its
nearest rival, carbon. Coal averages about the same as carbon.
Petroleum contains hydrogen combined with carbon (hydrocar-
bons) and gives, therefore, a higher heat of combustion than pure
carbon does. The main gases that are produced in the com-
bustion of all substances known as fuels are water vapor (H2O)
and carbon dioxide (CO2).

It would be well to memorize the values in the last column for

250 MECHANICS AND HEAT

petroleum, coal, and wood. Observe that dynamite has a sur-
prisingly low heat of combustion. Its effectiveness as an ex-
plosive depends upon the suddenness of combustion due to the
fact that the oxygen is in the dynamite itself, and not taken from
the air as in ordinary combustion.

To find how much chemical potential energy in foot-pounds
exists in 1 Ib. of coal, multiply 14,500 by 778; i.e., multiply the
number of B.T.U.'s per pound by the number of foot-pounds
in one B.T.U. To reduce this result to H.P.-hours, divide by
550X3600 (1 hr. equals 3600 sec.). Due to various losses of
energy in the furnace, boiler, and engine (Chap. XVIII), a steam
engine utilizes only about 5 or 10 per cent, of this energy, so that
the H.P.-hours above found should be multiplied by 0.05 or 0.10
(depending upon the efficiency of the engine used) to obtain the
useful work that may be derived from a pound of coal. With
a very good furnace, boiler, and engine, about 1.5 Ibs. of coal
will do 1 H.P.-hr. of work. Thus it would require about 150
Ibs. of coal to run a 100-H.P. engine for an hour.

182. Heat of Fusion and Heat of Vaporization. — As stated in
Sec. 162, considerable heat may be applied to a vessel contain-
ing ice water and crushed ice without producing perceptible
temperature rise until the ice is melted, whereupon further
application of heat causes the water to become hotter and hotter
until the boiling point is reached, when the temperature again
ceases to rise. Other substances behave in much the same way
as water. These facts show that heat energy is required to
change the substance from the solid to the liquid state, and from
the liquid to the vapor state. This heat energy is supposed to
be used partly in doing internal work against molecular forces.
In case the change of state is accompanied by an increase in
volume, part of this heat energy is used in doing external work in
causing the substance to expand against the atmospheric pressure.

The Heat of Fusion of a substance is the number of calories
required to change a gram of that substance from the solid to
the liquid state without causing a rise in temperature. The
Heat of Vaporization is the number of calories required to change
a gram of the substance from the liquid to the vapor state at a
definite temperature and at atmospheric pressure. These two
changes absorb heat while the reverse changes, that is from vapor
to liquid and from liquid to solid, evolve heat. The amounts of
heat evolved in these reverse changes are the same respectively

HEAT MEASUREMENT, OR CALORIMETRY 251

as the amounts absorbed in the former changes. This equality
should be expected, of course, from the conservation of energy.

For water, the heat of fusion is 79.25 calories per gram (also
written 79.25 cal./gm.), and the heat of vaporization is 536.5
cal. pergm.; which means that to change one gram of ice at 0°
C. to water at 0° C. requires 79.25 calories, and to change 1 gm.
of water at 100° to steam at 100° and atmospheric pressure
requires 536.5 cal. The value of the latter depends very much
upon the temperature. To change a gram of water at 20° to
vapor at 20° requires 585 cal., in other words, the heat of vapori-
zation of water at 20° C. is 585 cal. per gm.

From reasoning analogous to that used in changing the heat of
combustion from the metric to the British system (Sec. 181), we
see that the above heat of vaporization multiplied by 9/5 gives
the heat of vaporization in the British system, namely, 966
B.T.U.'s per pound. That is to say, 966 B.T.U.'s are required
to change 1 Ib. of water at 212° F. to steam at the same tempera-
ture. The heat of fusion is rarely expressed in the British
system.

HEAT OF FUSION OF VARIOUS SUBSTANCES

Substance

Melting
tem-
perature

Calories
per gram

Substance

Melting
tem-
perature

Calories
per gram

Ice

0°C.

79 25

Silver . . .

960° C.

Ice
Nitrate of soda. . .
Paraffin

-6
306
52

76
65
35

Cadmium
Sulphur
Lead

315
115
325

13.7
9.37
5.86

Zinc

415

Mercury

-38.8

2.82

HEAT OF VAPORIZATION OF VARIOUS SUBSTANCES

Substance

Tem-
perature

Calories
per gram

Substance

Tem-
perature

Calories
per gram

Water .

0°C.
100
17

78
448

595
536.5
295
206
362

Ether
Mercury

35° C.
357

61
0
30.8

90
62
58.5
56
3 72

Water
Ammonia (NH3)..
Alcohol (ethyl) . . .
Sulphur

Chloroform
Carbon dioxide.
Carbon dioxide.

183. Bunsen's Ice Calorimeter. — A very sensitive form of ice
calorimeter is that of Bunsen, in which the amount of ice melted is
determined from the accompanying change of volume. It con-

252

MECHANICS AND HEAT

sists of a bulb A (Fig. 125), with a tube B attached, and a test
tube C sealed in as shown. The space between A and C is
completely filled with water except the lower portion, which
contains mercury as does also a portion of B.

By pouring some ether into C and then evaporating it by forcing
a stream of air through it (Sec. 197), some ice E is formed about
C. As this ice forms, expansion occurs, which forces the mercury
farther along in B to, say, point a. Next, removing all traces of
ether from C, drop in a known mass of hot substance D at a
known temperature i' . The heat from D melts a portion of the
ice E, and the resulting contraction
causes the mercury to recede, say to
a'. The volume of the tube between
a and a' is evidently the difference be-
tween the volume of the ice melted by
D and that of the resulting water
formed; and hence, if known, could be
used to determine the amount of ice
melted. Multiplying this amount by
79.25 would give the number of cal-
ories of heat given off by D in cooling
to 0° C.

A simpler method, however, is to
calibrate the instrument by noting the
distance, say aa" , that the mercury col-
umn recedes when one gram of water at
100° is introduced into C. Suppose

this is two inches. Then, since the gram of water in cooling to
0° C. would impart to the ice 100 calories, we see that a motion
of one inch corresponds to 50 calories. Accordingly, the distance
aa' in inches, multiplied by 50, gives the number of calories
given off by D in cooling from t' to zero. This enables the cal-
culation of the specific heat of the substance D.

184. The Steam Calorimeter. — Dr. Joly invented a very sen-
sitive calorimeter, known as the Joly Steam Calorimeter, in which
the amount of heat imparted to a given specimen in raising its
temperature through a known range, is determined from the
amount of steam that condenses upon it in heating it. A speci-
men whose specific heat is sought, e.g., a piece of ore A (Fig. 126),
is suspended in an inclosure B by a wire W passing freely through
a small hole above, and attached to one end of the beam of a sensi-

FIG. 125.

HEAT MEASUREMENT, OR CALORIMETRY 253

tive beam balance. Weights are added to the other end of the
beam until a "balance" is secured. As steam is admitted to the
inclosure, it condenses upon the ore until the temperature of the
ore is 100°, whereupon condensation ceases. The additional
weight required to restore equilibrium, multiplied by 536.5, gives
the number of calories required to heat the ore and pan from a
temperature t (previously noted) to 100°. For it is evident that
each gram of steam that condenses upon the ore imparts to it
536.5 calories. If the mass of the ore is known, its specific heat
can readily be computed (Eqs. 83 and 85).

The pan in which the ore is placed catches the drip, if any.
Obviously the amount of steam that would condense upon the
pan in the absence of the ore must be found, either by calculation
or by experiment, and be subtracted from the total. By the use

FIG. 126.

of certain refinements and modifications which will not be
discussed here, the instrument may be employed for very delicate
work, such as the determination of the specific heat of a com-
pressed gas contained in a small metal sphere.

185. Importance of the Peculiar Heat Properties of Water. —
The fact that the specific heat, heat of fusion, and heat of
vaporization of water are all relatively large is of the utmost
importance in influencing the climate. It is also of great
importance commercially. From the conservation of energy it
follows that if it takes a large amount of heat (heat absorbed) to
warm water, to vaporize it, or to melt ice; then an equally large
amount of heat will be given off (evolved) when these respective
changes take place in the reverse sense — that is, when water
cools, vapor condenses, or water freezes,

254 MECHANICS AND HEAT

Specific Heat. — In connection with the subject of specific heat,
it is seen that the amount of heat a given mass absorbs in being
warmed through a given range of temperature depends upon its
specific heat. From this fact it is evident that a body of water
would change its temperature quickly with change of tempera-
ture of the air, if its specific heat were small. The specific
heat of water is much larger than for most other substances, as
may be seen from the table (Sec. 177). Note also that water
has about twice as large a specific heat as either ice or steam.
Because of the large specific heat of water it warms slowly and
cools slowly; so that during the heat of the day a lake cools the
air that passes over it, while in the cool of the night, it warms the
air. This same effect causes the temperature on islands in mid-
ocean to be much less subject to sudden or large changes than
it is in inland countries.

Heat of Fusion. — It requires 79.25 calories to melt 1 gram of
ice; hence, according to the conservation of energy, a gram of
water must give off approximately 80 calories of heat when it
changes to ice. If the heat of fusion were very small, say 2
calories per gram, a river would not need to give off nearly so
much heat in order to change to ice, so that it might, under those
conditions, freeze solid in a night with disastrous consequences
to the fish in it, and to the people dependent upon it for water
supply. Under these circumstances, it would also be necessary
to buy about 40 tunes as much ice to get the same cooling effect
that we now obtain.

Heat of Vaporization. — Since it requires about 600 calories
to change a gram of water at ordinary temperatures to vapor,
it follows, from the conservation of energy, that when a gram of
vapor condenses to water it gives off about 600 calories of heat.
This heat, freed by the condensation of vapor, is one of the main
causes of winds. The heat developed causes the air to become
lighter, whereupon it rises, and the surrounding air as it rushes
in is called a wind (Sec. 223).

If the heat of vaporization of water were much smaller,
evaporation and cloud formation would be much more rapid,
resulting ultimately in dried rivers and ponds, alternating
with disastrous floods.

The increase in volume which accompanies the freezing of water
is of the utmost importance in nature. If ice were more dense
than water, it would sink to the bottom when formed, and our

HEAT MEASUREMENT, OR CALORIMETRY 255

shallow ponds and our rivers would readily freeze solid. As it is,
the ice, being less dense, remains at the surface, and thus forms a
sheath that protects the water and prevents rapid cooling.

The Maximum Density of water occurs at 4° C. If water at
this temperature is either heated or cooled it expands, and con-
sequently becomes less dense. Hence in winter, as the surface
water of our lakes becomes cooler and therefore denser, it settles
to the bottom, and other water that takes its place is likewise
cooled and settles, thus establishing convection currents (Sec.
208). Through this action the temperature of the entire lake
tends to become 4° C. At least it cannot become colder than
this temperature, for as soon as any surface water becomes colder
than 4° it becomes less dense, and therefore remains on the sur-
face and finally freezes. As soon as the convection currents
cease, the chilling action practically ceases, so far as the deeper
strata of water are concerned, for water is a very poor conductor
of heat.

186. Fusion and Melting Point. — The Fusion of a substance
is the act of melting or changing from the solid to the liquid state,
and the Melting Point is the temperature at which fusion occurs.
The melting point of ice is a perfectly definite and sharply defined
temperature; for which reason it is universally used as one of the
standard temperatures in thermometry. Amorphous or non-
crystalline substances, such as glass and resin, upon being heated,
change to a soft solid or to a viscous liquid, and finally, when
considerably hotter, become perfectly liquid. Such substances
have no well-defined melting point.

Solutions of solids in liquids have a lower freezing point than the
pure solvent, and the amount of lowering of the freezing point is,
as a rule, closely proportional to the strength of the solution.
It might also be added that the dissolved substance also raises
the boiling point. For example, a 24 per cent, brine freezes at
—22° C. and boils at about 107°. Many other substances dis-
solved in water produce the same effect, differing in degree only.
Solvents other than water are affected in the same way.

Alloys, which may be looked upon as a solution of one metal in
another, behave like solutions with regard to lowering of the melt-
ing point. Thus Rose's metal, consisting by weight of bismuth
4 parts, lead 1, and tin 1, melts at 94° C. and consequently melts
readily in boiling water. Wood's metal — bismuth 4, lead 2,
tin 1, and cadmium 1 — melts at 70°. Solder, consisting of lead

256

MECHANICS AND HEAT

37 per cent., and tin 63 per cent., melts at 180° C. Using either
a greater or smaller percentage of lead raises the melting point
of the solder. In all these cases, the melting point of the alloy
is far lower than that of any of its components, as may be seen
by consulting the accompanying table.

TABLE OF MELTING POINTS

Substance

Temperature

Substance Temperature

Hydrogen
Nitrogen

-255° C.
-210

Lead
Zinc

325° C.
415

Mercury
Ice

- 38.8
0

Salt (NaCl)
Silver

800
960

Phosphorus
Tin
Bismuth
Cadmium. . .

44
233
267
315

Gold
Iron
Platinum
Iridium. . .

1064
1200 to 1600
1755
2300

Supercooling. — It is possible to cool water and other liquids
several degrees below the normal freezing point before freezing
occurs. Thus water has been cooled ten or twenty degrees below
zero, but the instant a tiny crystal of ice is dropped into the water,
freezing takes place, and the heat evolved (79.25 cal. per gm. of
ice formed) rapidly brings its temperature up to zero. Dufour
has shown that small globules of water, immersed in oil, may
remain liquid from —20° C. to 178° C. Some other substances,
e.g., acetamid and "hypo" (sodium hyposulphite), are not so
difficult to supercool as is water.

Pressure. — Some substances when subjected to great pressure
have their melting point raised, while others have it lowered.
Clearly, if a substance in melting contracts (e.g., ice, Sec. 187), we
would expect pressure to aid the melting process, and hence
cause the substance to melt at a lower temperature than normal.
It has been determined, both by theory and by experiment, that
ice melts at 0.0075° C. lower temperature for each additional
atmosphere of pressure exerted upon it. This effect is further
discussed under Regelation (Sec. 188) and Glaciers (Sec. 189).

187. Volume Change During Fusion. — Some substances
expand during fusion, while others contract. Thus, in changing
from the liquid to the solid state, water expands 9 per cent., and
bismuth 2.3 per cent.; while the following contract, silver
(10 per cent.), zinc (10 per cent.), cast iron (1 per cent.). Obvi-
ously silver and zinc do not make good, clear-cut castings for the
reason that in solidifying they shrink away fiom the mold. Silver

HEAT MEASUREMENT, OR CALORIMETRY 257

and gold coins have the impressions stamped upon them. Iron
casts well because it shrinks but slightly. The importance in
nature of the expansion of water upon freezing has already been
discussed (Sec. 185).

188. Regelation.— If a block of ice B (Fig. 127) has resting
across it a small steel wire w, to each end of which is attached a
heavy weight, it will be found that the wire slowly melts its way
through the ice. The ice immediately below the wire is subjected
to a very high pressure and therefore melts even if slightly below
zero (Sec. 186). The water thus formed is very slightly below
0° C., and flows around above the wire where it again freezes, due
to the fact that it is now at ordinary pressure, and that the sur-
rounding ice is also a trifle below 0° C. Thus

the wire passes through the ice and leaves the
block as solid as ever. The refreezing of the
water as it passes from the region of high pres-
sure is called Regelation. Since every gram of
ice melted below the wire requires about 80 cal-
ories of heat, and since this heat must come
from the surrounding ice, we see why the ice
above the wire and the water and ice below
are cooled slightly below 0° C.

If two irregular pieces of ice are pressed to-
gether, the surface of contact will be very small
and the pressure correspondingly great; as a
result of which some of the ice at this point will
melt. The water thus formed, being at ordinary atmospheric
pressure and slightly below zero as just shown, refreezes and
firmly unites the two pieces of ice. A similar phenomenon occurs
in the forming of snow balls by the pressure of the hand.

In skating, regelation probably plays an important role, as
pointed out by Dr. Joly. With a sharp skate, the skater's weight
bears upon a very small surface of ice, which may cause it to melt
even though several degrees below zero. Thus the skate melts
rather than wears a slight groove in the ice. If the ice is very
cold the skate will not "bite," i.e., it will not melt a groove,
unless very sharp. Friction is also probably much reduced by the
film of water between the skate and the ice.

189. Glaciers. — Glaciers are great rivers of ice that flow
slowly down the mountain gorges, sometimes (in the far north)
reaching the sea, where they break off in huge pieces called ice-

FIG. 127

258 MECHANICS AND HEAT

bergs, which float away to menace ocean travel. Glaciers owe
both their origin and their motion, in part, to regelation. Due
to the great pressure 'developed by the accumulated masses of
snow in the mountains or in the polar regions, part of the snow is
melted and frozen together as solid ice, forming glaciers, just as
the two pieces of ice mentioned above were frozen together.

As the glacier flows past a rocky cliff that projects into it, the
ice above, although at a temperature far below zero, melts because
of the high pressure, flows around the obstacle, and freezes again
below it. The velocity of glaciers varies from a few inches a day
to ten feet a day (Muir Glacier, Alaska), depending upon their
size and the slope of their beds. The mid-portion of a glacier
flows faster than the edge and the top faster than the bottom,
evidencing a sort of tar-like viscosity.

Glaciers in the remote past have repeatedly swept over vast
regions of the globe, profoundly modifying the soil and topogra-
phy of those regions. The northern half of the United States
shows abundant evidence of these ice invasions (see Geology).
At present, glaciers exist only in high altitudes or high latitudes.

190. The Ice Cream Freezer. — Experiments show that ice,
in the presence of common salt, may melt at a temperature far
below 0° C. (-22° C. or -7°.4 F.). This fact makes possible
the production of very low temperatures by artificial means.
The most familiar example of the practical application of this
principle is the ice cream freezer. The broken ice, mixed with
salt, surrounds an inner vessel which contains the cream. The
rotation of the inner vessel serves the two-fold purpose of agitat-
ing the cream within, and mixing the salt and ice without. The
revolving vanes within aerate the cream, thus making it light and
"velvety." The freezing would take place, however, without
revolving either vanes or container, but the process would require
more time, and the product would be inferior. As the ice melts,
the water thus formed dissolves more salt, and the resulting brine
melts more ice, and so on. One part (by weight) of salt to
three parts of crushed ice or snow gives the best results. This
is the proper proportion to form a saturated brine at that low
temperature.

The theory of the production of low temperatures by freezing
mixtures, such as salt and ice, is very simple. Every gram of ice
that melts requires 79.25 calories of heat to melt it. If this heat
is supplied, by a flame for example, the temperature remains at

HEAT MEASUREMENT, OR CALORIMETRY 259

0° C. until practically all of the ice is melted. If the melting of
the ice is caused by the presence of some salt or other chemical,
the requisite 79.25 calories of heat for each gram melted must come
from the freezing mixture itself, and from its surroundings, mainly
the inner vessel of the freezer, thus causing a fall of temperature.
Still lower temperatures may be obtained with a mixture of cal-
cium chloride and snow. The cheapness of common salt, and the
fact that —22° C. is sufficiently cold for rapid freezing, accounts
for its universal use. In fact, while being frozen, that is, while
being agitated, the cream should be but a few degrees below
zero to secure the maximum "lightness."

> PROBLEMS

1. How much heat would be required to change ^.Q^m. of ice at. —10° C
to water at 20° C.?

2. Now much heat wcyuld be^equired to change 40 gm. of water at 30° C.
to steam at 140° C.? Ae heat of vaporization at 140° C. is about 510 cal.
per gm.

3. If 40 gm. of water at 80° C. is mixed with 30 gm. of water at 20° C.,
what will be the temperature of the mixture? Neglect the heat capacity of
the calorimeter. Suggestion: Call the required temperature t, and then
solve for it.

4. Find the "water equivalent" of a brass calorimeter that weighs 150
gm.

6. Same as problem 3, except that the heat capacity of the calorimeter
containing the cold water is considered. The weight of the calorimeter is
60 gm., and the specific heat of the material of which it is composed is 0.11.

6. A certain calorimeter, whose water equivalent is 20, contains 80 gm. of
water at 40° C. When a mass of 200 gm. of a certain metal at 100° C. is
introduced, the temperature of the water and the calorimeter rises to 55° C.
Find the specific heat of the metal.

7. How many B.T.U.'s would be required to change 100 Ibs. of ice at 12° F.
to water at 80° F.? (Sees. 181 and 182.)

8. How many B.T.U.'s would be required to change 100 Ibs. of water at
80° F. to steam at 320° F.? When the water in the boiler is heated to 320° F.
the steam pressure is about 90 Ibs. per sq. in., and the heat of vaporization,
in the metric system, is about 495 cal. per gm.

9. How many pounds of soft coal would be required to change 100 Ibs. of
water at 70° F. to steam at 212° F.?

Assume that 10 per cent, of the energy is lost through incomplete combus-
tion, and that 30 per cent, of the remaining heat escapes through the smoke-
stack, or is lost by radiation, etc. See table, Sec. 181.

10. How high would the energy obtainable from burning a ton of coal
raise a ton of material, (a) assuming 12.5 per cent, efficiency for the steam
engine? (b) assuming 100 per cent, efficiency?

CHAPTER XV
^^i,..-,- - VAPORIZATION

191. Vaporization Denned. — Vaporization is the general term
applied to the process of changing from a liquid or solid to the
vapor state. Vaporization takes place in three different ways,
evaporation, ebullition (Sec. 192), and sublimation. The first two
refer to the change from liquid to vapor, the^last, from solid to
vapor. If aisolid passes directly 'iSo the vapor state without
first becoming aflTJtfid, it is said to sublime, and the process is
sublimation. Snow sublimofr— slowlyV disappearing when per-
fectly dry and far below zero. Other substances besides snow
sublime; notably camphor, iodine, and arsenic.

In whatever manner the vaporization occurs, it requires heat
energy to bring it about, and when the vapor condenses an equal
amount of heat (the heat of vaporization, Sec. 182) is evolved.
Hence a molecule must contain more energy when in the vapor
state than when in the liquid state, due, according to the kinetic
theory (Sec. 171), to the greater rapidity of its to-and-fro motion.
The above absorption and evolution of heat which accompany
vaporization and condensation, respectively, are of the utmost
importance in nature (Sec. 185) and also commercially. In
steam heating, the heat is evolved — about 540 calories for each
gram of steam condensed — at the place where the condensation
occurs, namely, in the radiator. Note the similar absorption of
heat in the melting of ice (utilized in the ice-cream freezer, Sec.
190) and the evolution of heat in the freezing of water. Thus,
vaporization and melting are heat-absorbing processes; while
the reverse changes of state, condensation and freezing, are
heat-liberating processes.

192. Evaporation and Ebullition. — The heat energy of a body
is supposed to be due to its molecular motion (Sec. 160), which,
as the body is heated, becomes more violent. The evaporation
of a liquid may be readily explained in accordance with this
theory. Let A (Fig. 128) be an air-tight cylinder containing
some water B, and provided with an air-tight piston P. Suppose

260

VAPORIZATION

261

this piston, originally in contact with the water, to be suddenly
raised, thereby producing above the water a vacuum. As the
water molecules near the surface of the water move rapidly to
and fro some of them escape into the vacuous space above, where
they travel to and fro just as do the molecules of a gas. After a
considerable number of these molecules have escaped from the
water, many of them in their to-and-fro motion will again strike
the water and be retained. Thus we see that there is a continual
passage of these molecules from the water to the vapor above,
and vice versa. The vapor above is said to be saturated when, in
this interchange, equilibrium has been reached; i.e., when the
rate at which the molecules are returning to the water is equal to
the rate at which they are escaping from it.

The saturated water vapor above the water in A exerts a pres-
sure due to the impact of its molecules against the walls, just as

-A

FIG. 128.

any gas exerts pressure. This vapor pressure is about 1/40
atmosphere when the water is at room temperature and becomes
1 atmosphere when the water and the cylinder are heated to the
boiling point.

Ebullition. — When water is placed in an open vessel (C, Fig.
128) evaporation into the air takes place from the surface, as
already described for vessel A. When heated to the boiling point
(D, Fig. 128), bubbles of vapor form at the point of application
of heat and rise to the surface, where the vapor escapes to the air.
When vaporization takes place in this manner, i.e., by the forma-
tion of bubbles within the liquid, it is called Ebullition, or boiling;
while when it takes place simply from the surface of the liquid,
it is called Evaporation.

As has already been stated, the pressure of saturated water

262

MECHANICS AND HEAT

vapor at 100° C. is one atmosphere. This will be evident from the
following considerations. In the formation of the steam bubble
E below the surface of the water in the open dish D, it is clear
that the pressure of the vapor in the bubble must be equal to the
atmospheric pressure or it would collapse. Indeed it must be a
trifle greater than atmospheric pressure, because the pressure
upon it is one atmosphere plus the slight pressure (hdg) due to
the water above it. We are now prepared to accept the general
statement that any liquid will boil in a shallow open dish when it
reaches that temperature for which the pressure of its saturated
vapor is one atmosphere. This temperature, known as the boiling
point at atmospheric pressure or simply the boiling point, differs
widely for the various substances.

193. Boiling Point.— Unless otherwise stated, the Boiling Point
is understood to be that temperature at which boiling occurs at
Standard Atmospheric Pressure (760 mm. of mercury). For pure
liqufds, this is a perfectly definite, sharply defined temperature,
so definite, indeed, that it may be used in identifying the sub-
stance. Thus if a liquid boils at 34. °9 we may be fairly sure that
it is ether; at 61°, chloroform; at 290°, glycerine. The boiling
points for a few substances are given in the following table.

BOILING POINTS AT ATMOSPHERIC PRESSURE

Substance Temperature

Helium

-267° C.

Alcohol (wood) . .

66° C.

Hydrogen

-253

Alcohol (ethyl)...

78.4

Nitrogen

— 194

Water

100

Oxygen

— 184

Glycerine

290

Carbon dioxide1. . .
Ammonia

- 80
- 38.5

Mercury
Sulphur

357

448

Ether

34.9

Zinc

about 930

Chloroform

Lead....

about 1500

Solutions of solids in liquids have a higher boiling point, as well
as a lower freezing point (Sec. 186) than the pure solvent. Thus
a 24 per cent, brine, which we have seen freezes at —22° C., boils
at about 107° C. The elevation of the boiling point is approxi-
mately proportional to the concentration for weak solutions.
A 24 per cent, sugar solution boils at about 100. °5 C.

194. Effect of Pressure on the Boiling Point. — When a change
of state is accompanied by an increase in volume, we readily see

1 Carbon dioxide (COz) sublimes at —80° C. and atmospheric pressure.
Under a pressure of 5.1 atmospheres it melts and also boils at —57° C.

VAPORIZATION

263

that subjecting the substance to a high pressure will oppose the
change; while if the change of state is accompanied by a decrease
in volume, the reverse is true, i.e., pressure will then aid the proc-
ess. Consequently, since water expands in changing to either
ice or steam, subjecting it to high pressure makes it "harder"
either to freeze or boil it; i.e., pressure lowers the freezing point,
(Sec. 186) and raises the boiling point. The latter volume change
is vastly greater than the former; accordingly the corresponding
temperature change is greater. Thus, when the pressure changes
from one atmosphere to two, the change of boiling point (21°) is
much greater than the change of freezing point (O.°0075). When
the steam gauge reads 45 Ibs. per sq. in. or 3 atmospheres, the
absolute steam pressure on the water in the boiler is 4 atmospheres
and the temperature of the water is 144° C. When the steam
gauge reads 200 Ibs., a pressure sometimes used, the temperature
of the boiler water is 194° C. On the other hand, to make water
boil in the receiver of an air pump at room temperature (20°), the
pressure must be reduced to about 1/40 atmosphere. (See
table below.)

BOILING POINT OF WATER AT VARIOUS PRESSURES »

Tempera-
ture

Pressure in
cm. of mer-
cury

Tempera-
ture

Pressure in
atmos-
pheres

Tempera-
ture

Pressure in
atmos-
pheres

0°C.

0.46

70° C.

0.3

140° C.

3.5

0.92

0.46

150

4.7

1.74

0.70

160

6.1

3.15

100

1.00

170

7.8

5.49

110

1.40

180

10.0

9.20

121°

2.00

190

12.4

14.88

130

2.67

200

15.5

Franklin's Experiment on Boiling Point. — Benjamin Franklin
discovered that if a flask partly filled with water is boiled until
the air is all expelled (Fig. 129, left sketch) , and is then tightly
stoppered and removed from the flame (right sketch), then pour-
ing cold water (the colder the better) upon the flask causes the
water to boil, even after it has cooled to about room temperature.
The explanation is simple. When the temperature of the water
is 50° C. the vapor pressure in the flask is 9.2 cm. of mercury.

1 This is also a table of the saturated vapor pressure of water at various
temperatures. (See close of Sec. 192, also Sec. 196.)

264

MECHANICS AND HEAT

(See table above.) Suppose that under these conditions cold
water is poured upon the flask. This chilling of the flask con-
denses some of the contained vapor, thereby causing a slight
drop in pressure, whereupon more water bursts into steam.
Indeed, so long as the temperature is 50°, the vapor pressure will
be maintained at 9.2 cm.; hence the colder the water which is
poured on, the more rapid the condensation, and consequently
the more violent the boiling. The flask should have a round
bottom or the atmospheric pressure will crush it when the pressure
within becomes low. Inverting the flask and placing the stopper
under water, as shown, precludes the possibility of air entering
the flask and destroying the vacuum.

This lowering of the boiling point as the air pressure decreases
is a serious drawback in cooking at high altitudes. At an altitude

FIG. 129.

of 10,000 ft. (e.g., at Leadville, Colorado), water boils at about
90° C., and at the summit of Pike's Peak (alt. 14,000 ft.), at
about 85° C. At such altitudes it is very difficult to cook (by
boiling) certain articles of food, (e.g., beans), requiring in some
cases more than a day. It will be understood that when water
has reached the boiling point, further application of heat does
not cause any further temperature rise, but is used in changing
the boiling water to steam. In sugar manufacture, the "boiling
down" is done in "vacuum pans" at reduced pressure and re-

VAPORIZATION 265

duced temperature to avoid charring the sugar. By boiling
substances in a closed vessel or boiler so that the steam is con-
fined, thereby raising the pressure, and consequently raising the
boiling point, the cooking is more quickly and more thoroughly
done. This method is used in canning factories.

Superheating, Bumping. — After pure water has boiled for some
time and the air which it contains has been expelled, it sometimes
boils intermittently with almost explosive violence known as
"bumping." A thermometer inserted in the water will show
that the temperature just previous to the "bumping" is slightly
above normal boiling point; in other words the water is Super-
heated. A few pieces of porous material or a little unboiled water
added will stop the bumping. We have seen (Sec. 186) that water
may also be supercooled without freezing. Dufour has shown
that water in fine globules immersed in oil may remain liquid
from -20° to 178° C.

195. Geysers. — The geyser may be described as a great hot
spring which, at more or less regular intervals, spouts forth a
column or jet of hot water. Geysers are found in Iceland, New
Zealand, and Yellowstone National Park. One of the Iceland
geysers throws a column of water 10 ft. in diameter to a height
of 200 ft. at intervals of about 6 hours. Grand Geyser, of the
National Park, spouts to a height of 250 ft. Old Faithful, in
the National Park, is noted for its regularity.

Geysers owe their action to the fact that water under great
pressure must be heated considerably above 100° before it boils,
and perhaps in some cases also to superheating of the lower parts
of the water column just before the eruption takes place. A deep,
irregular passage, or "well," filled with water, is heated at the
bottom by the internal heat of the earth to a temperature far
above the ordinary boiling point before the vapor pressure is
sufficient to form a bubble. When this temperature is reached
(unless superheating occurs) a vapor bubble forms and forces
the column of water upward. At first the water simply flows
away at the top. This, however, reduces the pressure on the
vapor below, whereupon it rapidly expands, and the highly
heated water below, now having less pressure upon it, bursts
into steam with explosive violence and throws upward a column
of boiling water. This water, now considerably cooled, flows
back into the " well." After a few hours the water at the bottom
of the well again becomes heated sufficiently above 100° to form

266 MECHANICS AND HEAT

steam bubbles under the high pressure to which it is subjected,
and the geyser again "spouts."

Bunsen, who first explained the action of the natural geyser,
devised an artificial geyser. It consisted of a tin tube, say 4 ft.
in length and 4 in. in diameter at the lower end, tapering to
about 1 in. in diameter at the top, with a broad flaring portion
above to catch the column when it spouts. If filled with water
and then heated at the bottom, it spouts at fairly regular intervals.
If constructed with thermometers passing through the walls of
the tube, it will be found that the thermometers just previous to
eruption read higher than 100°, and that the lowest one reads
highest.

In the case of steam boilers under high pressure, the water may
be from 50° to 80° hotter than the normal boiling point, and if
the boiler gives way, thereby reducing the pressure, part of this
water bursts into steam. This additional supply of steam no
doubt contributes greatly to the violence of boiler explosions.

196. Properties of Saturated Vapor. — If, after the space above
the water in A (Fig. 128) has become filled with saturated vapor,
the piston Pis suddenly forced down, there will then be more mole-
cules per unit volume of the space than there were before. Con-
sequently, the rate at which the molecules return to the water will
be greater than before, and therefore greater than the rate at
which they are escaping from the water. In other words, some of
the vapor condenses to water. This condensation takes place
very quickly and continues until equilibrium is restored and the
vapor is still simply saturated vapor.

If, on the other hand, the piston P had been suddenly moved
upward instead of downward, the vapor molecules in the space
above the water, having somewhat more room than before, would
not be so closely crowded together and hence would not return to
the water in such great numbers as before. In other words, the
rate of escape of molecules from the water would be greater than
their rate of return; consequently the number of molecules in
the space above the water would increase until equilibrium was
reached, i.e., until the space was again filled with saturated vapor.

In the case of a saturated vapor above its liquid, we may con-
sider that there are two opposing tendencies always at work.
As the temperature of the liquid rises, the tendency of the liquid
to change to vapor increases, i.e., more liquid vaporizes. The
effect of increasing the external pressure applied to the vapor is,

VAPORIZATION

267

on the other hand, to tend to condense it to the liquid state. At
all times, and under all circumstances, the pressure applied to
the vapor is equal to the pressure exerted by the vapor. Referring
to Fig. 128, it may readily be seen that if the vapor pressure act-
ing upward upon P is equal to, say 5 Ibs. per sq. in. at any instant,
that the downward pressure exerted by the piston upon the vapor
below it, is likewise 5 Ibs. per sq. in. Of course this would be
equally true for any other pressure.

C D E F

FIG. 130.

These characteristics of a saturated vapor above its own liquid
are beautifully illustrated in the following experiment. A ba-
rometer tube T (Fig. 130, left sketch) is filled with mercury,
stoppered, and carefully inverted in a mercury "well" A about 80
cm. deep. Upon removing the stopper, the mercury runs out of
the tube, leaving the mercury level about 76 cm. higher in the
tube than in the well, as explained in Sec. 136. Next, without
admitting any air, introduce, by means of an ink filler, sufficient
ether to make about 1 cm. depth in the tube. This ether rises

268 MECHANICS AND HEAT

and quickly evaporates, until the upper part of the tube is filled
with its saturated vapor, whose pressure at room temperature is
about 2/3 atmosphere. Consequently, the mercury drops until
it is about 25 cm. (1/3 of 76) above that in the well.

Now, as the tube is quickly moved upward more ether evapo-
rates, maintaining the pressure of the saturated vapor con-
stantly at 2/3 atmosphere, as evidenced by the fact that the level
of the mercury still remains 25 cm. above that in the well. If the
tube is suddenly forced downward, some ether vapor condenses,
and the mercury still remains at the same 25-cm. level. After the
tube has been raised high enough that all of the ether is evapor-
ated, further raising it causes the pressure of the vapor to de-
crease (in accordance with Boyle's Law), as shown by the fact
that the level of the mercury in the tube then rises.

Finally, if the tube and contents are heated to 34.9° C., the
boiling point for ether, its saturated vapor produces a pressure of
one atmosphere, and the mercury within and without the tube
comes to the same level, and remains at the same level though the
tube be again raised and lowered. If the tube is severely chilled,
the mercury rises considerably higher than 25 cm. This shows
that the pressure of the saturated vapor, or the pressure at which
boiling occurs, rises rapidly with the temperature. (See table
for Water, Sec. 194.)

Saturated Vapor Pressure of Different Liquids. — The pressure of
the saturated vapor of liquids varies greatly for the different
liquids, as shown by the experiment illustrated in Fig. 130 (right
sketch). The four tubes C, D, E, and F are filled with mercury
and are then inverted in the mercury trough G. The mercury
then stands at a height of 76 cm. in each tube. If, now, a little
alcohol is introduced into D, some chloroform into E, and some
ether into F, it will be found that the mercury level lowers by the
amounts hi, hz, and h3, respectively. The value of hi is 4.4 cm.,
which shows that at room temperature the pressure of the sat-
urated vapor of alcohol is equal to 4.4 cm. of mercury. Since
h2/hi = 4: (approx.), we see that at room temperature the pressure
of the saturated vapor is about 4 times as great for chloroform
as for alcohol.

197. Cooling Effect of Evaporation. — If the hand is mois-
tened with ether, alcohol, gasoline, or any other liquid that evapo-
rates quickly, a decided cooling effect is produced. Water pro-
duces a similar but less marked effect. We have seen that it

VAPORIZATION 269

requires 536.5 calories to change a gram of boiling water to steam.
When water is evaporated at ordinary temperatures it requires
somewhat more than this, about 600 calories. If this heat is not
supplied by a burner or some other external source, it must come
from the remaining water and the containing vessel, thereby
cooling them below room temperature.

There are two factors which determine the magnitude of the
cooling effect produced by the evaporation of a liquid. One of
these is the volatility of the liquid; the other, the value of its
heat of vaporization. From the table (Sec. 182) we see that the
heat of vaporization is about 3 times as great for water as for
alcohol. Consequently, if alcohol evaporated 3 times as fast as
water under like conditions, then alcohol and water would pro-
duce about equally pronounced cooling effects. Alcohol, how-
ever, evaporates much more than 3 times as fast as water, and
therefore gives greater cooling effect, as observed.

If three open vessels contain alcohol, chloroform, and ether,
respectively, it will be found that a thermometer placed in the
one containing alcohol shows a temperature slightly lower than
room temperature; while the one in chloroform reads still lower,
and the one in ether the lowest of all. A thermometer placed in
water would read almost exactly room temperature. The main
reason for this difference is the different rates at which these
liquids evaporate, although, as just stated, the value of the heat
of vaporization is also a determining factor. Ether, being by far
the most volatile of the three, gives the greatest cooling effect.
Observe that the more volatile liquids are those having a low
boiling point, and consequently a high vapor pressure at room
temperature. In some minor surgical operations the requisite
numbness is produced by the chilling effect of a spray of very
volatile liquid. Other practical uses of the cooling effect of
evaporation are discussed in Sees. 198, 199, and 200. The
converse, or the heating effect due to condensation, is utilized in
all heating by steam (Sec. 191), and it also plays an important role
in influencing weather conditions.

198. The Wet-and -dry-bulb Hygrometer. — The 'cooling
effect of evaporation is employed in the wet-and-dry-bulb
hygrometer, used in determining the amount of moisture in the
atmosphere. It consists of two ordinary thermometers which
are just alike except that a piece of muslin is tied about the bulb
of one. The muslin is in contact with a wick, the lower end of

270 MECHANICS AND HEAT

which is in a vessel of water. By virtue of the capillary action of
the wick and muslin, the bulb is kept moist. This moisture
evaporating from the bulb cools it, causing this thermometer to
read several degrees lower than the other one.

If the air is very dry, this evaporation will be rapid and the
difference between the readings of the two thermometers will be
large; whereas if the air is almost saturated with moisture, the
evaporation will be slow and the two thermometer readings will
differ but slightly. Consequently, if the two readings differ but
little, rain or other precipitation may be expected. The method
of finding the amount of water vapor in the air by means of these
thermometer readings, is discussed in a subsequent chapter.

As a mass of air m comes into contact with the wet (colder) bulb it
gives heat to the bulb, and as it absorbs moisture from the bulb it also
takes heat from it. A few moments after the apparatus is set up, equi-
librium is reached, as shown by the fact that the temperature of the wet
bulb is constant. It is then known that the amounts of heat "given"
and "taken" by the bulb are equal. This fact is utilized in the deriva-
tion of certain theoretical formulas for computing the amount of
moisture in the air directly from the two thermometer readings. The
practical method, however, is to use tables (Sec. 222) compiled from
experiments.

199. Cooling Effect due to Evaporation of Liquid Carbon
Dioxide. — Carbon dioxide (COa) is a gas at ordinary tempera-
tures and pressures, but if cooled to a low temperature and then
subjected to high pressure it changes to the liquid state. If
the pressure is reduced it quickly changes back to the vapor state.
We have seen that the pressure of water vapor is about 1/40
atmosphere at room temperature. Liquid carbon dioxide is so
extremely volatile, that is, it has so great a tendency to change
to the vapor state, that its vapor pressure at room temperature
has the enormous value of 60 atmospheres. It follows then, that
when an air-tight vessel is partly filled with liquid carbon dioxide
at room temperature, a portion of it quickly changes to vapor
until the pressure in the space above the liquid becomes 60 atmos-
pheres. Carbon dioxide is shipped and kept in strong sealed
iron tanks to be used for charging soda fountains, etc.

If such a tank is inverted (Fig. 131) and the valve is opened,
a stream of liquid carbon dioxide is forced out by the 60-atmos-
phere pressure of the vapor within. As soon as this liquid carbon
dioxide escapes to the air, where the pressure is only one atmos-
phere, it changes almost instantly to vapor, and takes from the

VAPORIZATION

271

air, from the nozzle, and from the remaining liquid, its heat of
vaporization, about 40 calories per gram at room temperature.
This abstraction of heat chills the nozzle to such an extent that
the moisture of the air rapidly condenses upon it as a frosty
coating. It also chills, in fact freezes, part of the liquid jet of
carbon dioxide, forming carbon dioxide "snow." This snow is
so cold ( —80° C.) that mercury surrounded by it quickly freezes.

200. Refrigeration and Ice
Manufacture by the Ammonia
Process. — There are several sys-
tems or methods of ice manufac-
ture, in all of which, however, the
chilling effect is produced by the
heat absorption (due to heat of
vaporization) that accompanies
the vaporization of a volatile
liquid. The most important of
these liquids are ammonia (NH3)
and carbon dioxide (CO2). Econ-
omy demands that the vapor be
condensed again to a liquid, in
order to use the same liquid re-
peatedly.

In the Compression System, the
vapor is compressed by means of
an air pump until it becomes a

liquid. The heat evolved in this process (heat of vaporization)
is disposed of usually by flowing water, and the cooled liquid
(e.g., ammonia) is again allowed to evaporate. Thus the cycle,
consisting of evaporation accompanied by heat absorption, and
condensation to liquid accompanied by heat evolution, is repeated
indefinitely. Since the former occurs in pipes in the ice tank
(freezing tank), we see that the heat is literally pumped from the
freezing tank to the flowing water.

Ammonia is a substance admirably adapted to use in this way.
Its heat of vaporization is fairly large (295 cal. per gm.), and it is
very volatile— that is, it evaporates very quickly, its vapor
pressure at room temperature being about 10 atmospheres. At
—38°. 5 C. its vapor pressure is one atmosphere; hence it would
boil in an open vessel at that low temperature. The liquid com-
monly called ammonia is simply water containing ammonia gas

FIG. 131.

272

MECHANICS AND HEAT

which it readily absorbs. Ammonia is a gas at ordinary tempera-
tures, but when cooled and subjected to several atmospheres'
pressure it changes to a liquid. If carbon dioxide is used instead
of ammonia, the cost of manufacturing the ice is somewhat
greater. The greater compactness of the apparatus, however,
coupled with the fact that in case of accidental bursting of the
pipes, carbon dioxide is much less dangerous than ammonia, has
resulted in its adoption on ships.

The essentials of the Ammonia Refrigerating apparatus are
shown diagrammatically in Fig. 132. A is the cooling tank which
receives a continual supply of cold water through pipe c.;B is an
air pump; C is a freezing tank filled with brine; D is a pipe filled
with liquid ammonia; and E is a pipe filled with ammonia vapor.

FIG. 132.

If valve F were slightly opened, liquid ammonia would enter E
and evaporate until the pressure in E was equal to the vapor
pressure of ammonia at room temperature or about 10 atmos-
pheres. Whereupon evaporation, and therefore all cooling action ,
would cease. If, however, the pump is operated, ammonia gas
is withdrawn from E through valve a and is then forced into pipe
D through valve 6 under sufficient pressure to liquefy it. This
constant withdrawal of ammonia gas from pipe E permits more
liquid ammonia to enter through F and evaporate. The am-

VAPORIZATION 273

monia, as it evaporates in E, withdraws from E and from the sur-
rounding brine its heat of vaporization (about 300 cal. per gm.);
while each gram of gas that is condensed to a liquid in D imparts
to D and to its surroundings about 300 calories. Thus we see
that heat is withdrawn from the very cold brine in C and imparted
to the much warmer water in A. This action continues so long
as the pump is operated. Brine is used in C because it may be
cooled far below zero without freezing.

The Refrigerator Room. — The cold brine from C may be pumped
through d into the pipes in the refrigerator room and then back
through pipe e to the tank. The brine as it returns is not so cold
as before, having abstracted some heat from the refrigerator
room. This heat it now imparts to pipe E. Thus, through the
circulation of the brine, heat is carried from the cooling room to
the tank C, and we have just seen that due to the circulation of
the ammonia, heat is carried from the brine tank C to the water
tank A.

The pipe E, instead of passing into the brine, may pass back
and forth in the refrigerator room. The stifling ammonia vapor,
which rapidly fills the room, in case of the leaking or bursting of
an ammonia pipe, makes this method dangerous.

In the Can System of ice manufacture, the cans of water to be
frozen are placed in the brine in C, and left there 40 or 50 hours
as required. In the Plate System, the pipe E passes back and forth
on one face of a large metal plate, chilling it and forming a sheet
of ice of any desired thickness upon the other face, which is in
contact with water. For every 8 or 10 tons of ice manufactured,
the engine that operates the pump uses about one ton of coal.

Observe that in "pumping" the heat, as we may say, from the
cold freezing tank to the much warmer flowing water, we are
causing the heat to flow "uphill," so to speak; for heat. of itself
always tends to flow from hotter to colder bodies, that is, "down-
hill." Observe also that it takes external applied energy of the
steam engine that operates the pump to cause this "uphill" flow
of heat.

201. Critical Temperature and Critical Pressure.— In 1869,
Dr. Andrews performed at Glasgow his classical experiments on
carbon dioxide. He found that when some of this gas, confined
in a compression cylinder at a temperature of about 32° or 33° C.,
had the pressure upon it changed from say 70 atmospheres to: 80
atmospheres, then the volume decreased, not by 1/8 as required

274

MECHANICS AND HEAT

by Boyle's Law (Sec. 139), but much more than this. He also
found that carbon dioxide gas cannot be changed to the liquid
state by pressure, however great, if its temperature is above 31° C.
This temperature (31°) is called the Critical Temperature for
carbon dioxide.

If carbon dioxide gas is at its critical temperature, it requires
73 atmospheres' pressure to change it to the liquid state. This
pressure is called the Critical Pressure for carbon dioxide. If
the temperature of any gas is several degrees lower than its
critical temperature, then the pressure required to change it to
the liquid state is considerably less than the critical pressure.
Below is given a table of critical temperatures and critical pres-
sures for a few gases.

CRITICAL TEMPERATURES AND CRITICAL PRESSURES FOR A FEW
SUBSTANCES

Substance

Critical temperature

Critical pressure in
atmospheres

Hydrogen1 (H)
Nitrogen (N)

-241° C.
-146

14
34

Air (O and N)

-140

Oxygen (O)

— 118

Ethylene (C2H4)
Carbon dioxide (CCh)
Ammonia (NHs)

10
30.92
130

52
73
115

Water vapor (H,O)

364

194.6

202. Isothermals for Carbon Dioxide. — In Fig. 134, the isothermals
which Andrews determined for carbon dioxide are shown. For the
meaning of isothermals and the method of obtaining them, the student
is referred to "Isothermals for Air" (Sec. 140).

The essential parts of the apparatus used by Andrews are shown in
section in Fig. 133. A glass tube A about 2.5 mm. in diameter, terminat-
ing in a fine capillary tube above, was filled with carbon dioxide gas and
plugged with a piston of mercury a. This tube was next slipped into
the cap C of the compression chamber D. A similar tube B, filled with
air, and likewise stoppered with mercury, was placed in the compression
chamber E.

As S was screwed into the compression chamber D, the pressure

1 The values — 234.°5 C. and 20 atmospheres, sometimes given as the
critical temperature and critical pressure, respectively, for hydrogen, are
incorrect; the first, because of extrapolation error in the readings of the resist-
ance thermometer, the second, because of manometer error in the original
determination.

VAPORIZATION

275

on the water in the two chambers, and consequently the pressure on
the mercury and gas in the two tubes A and B, could be increased as
desired. Of course, as the pressure was increased, the mercury rose
higher and higher in tubes A and B to, say, mi and m2. Knowing the
original volume of air in B and also the bore of the capillary portion of
tube B, the pressure in the chamber could be determined. Thus, if
the volume of air in tube B above m2 were 1/50 of the original volume,
then the pressure in both chambers would be approximately 50 atmos-
pheres. At such pressures there is a deviation from Boyle's law, which
was taken into account and corrected
for. Knowing the bore of A , the volume
of carbon dioxide above mi could be
found.

Plotting the values of the volumes so
found as abscissa, with the correspond-
ing pressures as ordinates, when the
temperature of the apparatus was 48.°1
C., the isothermal marked 48.°1 (Fig.
134) was obtained. The form of the
48. °1 isothermal shows that at this tem-
perature the carbon dioxide vapor fol-
lowed Boyle's law, at least roughly.

When, however, the experiment was
repeated with the apparatus at the tem-
perature 31.° 1 C., it was found that when
the pressure was somewhat above 70
atmospheres (point a on the 31.°1 isother-
mal) a slight increase in pressure caused a
very great decrease in volume, as shown by
a considerable rise in mi. As the pres-
sure was increased slightly above 75 at-
mospheres, as represented by point 6 on

the curve, a further slight reduction of volume was accompanied by a
comparatively great increase in pressure, as shown by the fact that the
portion be of the isothermal is nearly vertical. Note also that the por-
tion ab of the isothermal is nearly horizontal.

If the experiment were again repeated at, say 30° C., then as the pres-
sure reached about 70 atmospheres, liquid carbon dioxide would collect
on mi, and this liquid would be seen to have a sharply defined meniscus
separating it from the vapor above. At 31.°1 no such meniscus appears.
The limiting temperature (30.°92 C.) at which the meniscus just fails
to appear under increasing pressure, is called the Critical Temperature.

Let us now discuss the 21.°5 isothermal, which isothermal was deter-
mined by keeping the apparatus at 21.°5 while increasing the pressure.
As the volume was decreased from that represented by point A to that

FIG. 133.

276

MECHANICS AND HEAT

represented by point B, the pressure increased from about 50 atmospheres
to 60. Now as S was screwed farther into the chamber, the volume
decreased from point B to point C with practically no increase in pressure
(note that BC is practically horizontal). During this change the satu-
rated carbon dioxide vapor was changing to the liquid state, as shown
by the fact that the liquid carbon dioxide resting on mi could be seen to
be increasing. At C the gas had all been changed to liquid carbon
dioxide, and since liquids are almost incompressible, a very slight com-
pression, i.e., a very slight rising of meniscus mi, was accompanied by a
very great increase of pressure, as evidenced by the nearly vertical direc-
tion of CD.

It will be observed, that while the volume is reduced from that rep-
resented by point B to that represented by point C, the carbon dioxide
100

Volume

FIG. 134.

is changing to the liquid state, and therefore gaseous and liquid carbon
dioxide coexist in tube A. Likewise at 13.°1 the two states, or phases,
coexist from B^ to Ci, while if the temperature were, say 28° C., the two
phases would coexist for volumes between B2 and C2. Accordingly, the
region within the dotted curve through B, BI, Bz, C, Ci, C2, etc., represents
on the diagram all possible corresponding values of pressure, volume,
and temperature at which the two phases may coexist. Thus, if the
state of the carbon dioxide (temperature, pressure, and volume) is rep-
resented by a point anywhere to the right, or to the right and above
this dotted curve, only the gaseous phase exists; to the left, only the
liquid phase. We may now define the Critical Temperature of any sub-
stance as the highest temperature at which the liquid and gaseous phases
of that substance can coexist.

VAPORIZATION 277

This definition suggests the following simple method of determining
critical temperatures. A thick-walled glass tube is partly (say 1/4)
filled with the liquid, e.g. water, the space above being a vacuum, or
rather, a space containing saturated water vapor. The tube is then
heated until the meniscus disappears. The temperature at which the
meniscus disappears is the critical temperature (364° C. for water), and
the pressure then tending to burst the tube, is termed the critical pressure.
It will be noted that as the water is heated, its vapor pressure becomes
greater, finally producing the critical pressure (194.6 atmospheres)
when heated to the critical temperature.

The Distinction between a Vapor and a Gas. — When a gas is cooled
below its critical temperature it becomes a vapor. Conversely, when a
vapor is heated above its critical temperature it becomes a gas. A
vapor and its liquid often coexist; a gas and its liquid, never.

203. The Joule-Thomson Experiment. — In 1852, Joule and Thomson
(Lord Kelvin) performed their celebrated "Porous Plug" experiment.
They forced various gases under high pressure through a plug of cotton
or silk into a space at atmospheric pressure. In every case, except when

FIG. 135.

hydrogen was used, the gas was cooler after passing through the plug
than it was before. Hydrogen, on the contrary, showed a slight rise
in temperature. We may note, however, that at very low temperatures
(below— 80° C.) hydrogen also experiences a cooling effect.

The principle involved in this experiment will be explained in connec-
tion with Fig. 135. Let P be a stationary porous plug in a cylinder con-
taining two pistons C and D. Let piston C, as it moves (slowly) from
Ai to BI against a high pressure plt force the gas of volume Vi through
the plug, and let this gas push the piston D from A2 to 52, and let it
have the new volume Vz and the new pressure pz (1 atmosphere).
Now, from the proof given in Sec. 156, we see that piston C does the work
p\Vi upon the gas in forcing it through the plug; while the work done
by the gas in forcing D from A* to B2 is p2V2. Accordingly, if p2Vz =
piVi, i.e., if the work done by the gas is equal to the work done upon it,
then the gas should (on this score at least) be neither heated nor cooled
by its passage through the plug. All gases, however, deviate from
Boyle's law, and for all but hydrogen the product pV at ordinary tem-
peratures increases as p decreases. Hence here p2V2>piFi (> = is
greater than), which means that the work done by the gas (which tends
to cool it) exceeds the work done upon the gas (which tends to heat it).

278 MECHANICS AND HEAT

As a result, then, the gas is either cooled or else it abstracts heat from the
piston, or both.

Cooling Effect of Internal Work. — From the known deviation from
Boyle's law exhibited by air, it can be shown that the temperature of
the air in passing through the plug should drop about O.°l C. for each
atmosphere difference in pressure between pi and p2. Thomson and
Joule found a difference of nearly 1° C. per atmosphere. This addi-
tional cooling effect is attributed to the work done against intermolec-
ular attraction (internal work done) when a gas expands. The work
done by the gas in expanding is due, then, in part to the resulting in-
crease in pV (deviating from Boyle's law), and in part to the work done
against intermodular attraction in increasing the average distance be-
tween its molecules. Both of these effects, though small, are more
marked at low temperatures, and by an ingenious but simple arrange-
ment for securing a cumulative effect, Linde has employed this prin-
ciple in liquefying air and other gases (Sec. 206). In Linde's apparatus,
the gas passes through a small opening in a valve instead of through a
porous plug.

204. Liquefaction of Gases. — -About the beginning of the
present century, one after another of the so-called permanent
gases were liquefied, until now there is no gas known that has not
been liquefied. Indeed most of them have not only been lique-
fied, but also frozen.

In 1823, the great experimenter Faraday liquefied chlorine and
several other gases with a very simple piece of apparatus. The
chemical containing the gas to be liquefied was placed in one end
of a bent tube, the other end of which was placed in a freezing
mixture producing a temperature lower than the critical tempera-
ture of the gas. The end of the tube containing the chemical
was next heated until the gas was given off in sufficient quantity
to produce the requisite pressure to liquefy it in the cold end of
the tube.

In 1877, Pictet and Cailletet independently succeeded in
liquefying oxygen. Later Professor Dewar and others liquefied
air, and in 1893 Dewar froze some air. A few years later (1897)
he liquefied and also (1899) froze some hydrogen. Subsequently
(1903) he produced liquid helium, a substance that boils at 6°
on the absolute scale or at — 267° C. He also invented the Dewar
flask (Sec. 206), in which to keep these liquids.

In liquefying air and other gases having low critical tempera-
tures, the great difficulty encountered is in the production and
maintenance of such low temperatures. To accomplish this, the

VAPORIZATION

279

cooling effect of the evaporation of a liquid and the cooling effect
produced when a gas expands (Sec. 178) have both been utilized.

There are two distinctly different methods of liquefying air,
known as the "Cascade" or Series Method, due to Raoult Pictet
(Sec. 205), and the "Regenerative Method," due to Linde and
others (Sec. 206).

205. The Cascade Method of Liquefying Gases.— In Fig. 136 is
shown a diagrammatic sketch of the apparatus of Pictet, as modified
and used with great success in the latter part of the 19th Century by
Dewar, Olszewski, and others. It consists of three vessels A, B, and C,
the two air pumps D and E, and the carbon dioxide tube F, together
with the connecting pipes as shown.

The pump D forces ethylene through pipe K, valve G, and pipe M
into the vessel B from which vessel the ethylene (now in the vapor state)
returns to the pump through pipe N. Pump E maintains a similar

FIG 136

counterclockwise circulation of air through L, H, 0, C, and P, as is
indicated by the arrows.

The vaporization of the carbon dioxide in A produces a temperature
of —80° C. (Sec. 199). This cold gas, coming in contact with the spiral
pipe K (shown straight to avoid confusion), cools it enough that the
ethylene within it liquefies under the high pressure to which it is sub-
jected. As this cold liquid ethylene vaporizes at M, it cools the air in
L to such an extent that it in turn liquefies under the high pressure pro-
duced by pump E. As this liquid air passes through valve H and
vaporizes in C, it produces an extremely low temperature. As pointed
out in the discussion of the ammonia refrigerating apparatus, the main-
tenance of a partial vacuum into which the liquid may vaporize, as in
B and C, causes more rapid vaporization, and therefore enhances the
chilling effect. The liquid air may be withdrawn at I, and fresh air may
be admitted at J to replenish the supply.

280

MECHANICS AND HEAT

In liquefying air by this method, it is necessary to use ethylene, or
some other intermediate liquid which produces a very low temperature
when vaporized. For if L simply passed through vessel A, no pressure,
however great, would liquefy the air within it, since —80° C. is above the
critical temperature for air. Gases have, however, been liquefied when
at temperatures considerably above the critical temperatures, by sub-
jecting them to enormous pressures and then suddenly relieving the
pressure.

206. The Regenerative Method of Liquefying Gases. — The
regenerative method of liquefying gases employs the principle
(established by Thomson and Joule, Sec. 203) that a gas is chilled

200 Atmospheres Cold

as it escapes through an orifice from a region of high pressure to
a region of low pressure. This method has made possible the
liquefaction of every known gas, and also the production of liquid
air in large quantities and at a greatly reduced cost. From about
1890 to 1895 Dr. Linde, Mr. Tripler, and Dr. Hampson were all
working along much the same line, in accordance with a suggestion
made by Sir Wm. Siemens more than thirty years before; namely,
that the gas, cooled by expansion as it escapes through an orifice,
shall cool the oncoming gas about to expand, and so on, thus giving
a cumulative effect. Dr. Linde, however, was the first to produce
a practical machine.

The essential parts of Linde's apparatus are shown in Fig. 137.
A is an air pump which takes in the gas (air, e.g.) through valve

VAPORIZATION 281

E at about 16 atmospheres, and forces it under a pressure of
about 200 atmospheres through the coiled pipes in the freezing
bath B. From B, the air passes successively through the three
concentric pipes or tubes F, G, and H in the vessel C, as indicated
by the arrows. A portion of the air from G returns again through
pipe I and valve E to the pump, thus completing the cycle. The
cycle is repeated indefinitely as long as the pump is operated. It
will be understood that the freezing bath B cools the air which has
just been heated by compression. It also "freezes out" most of
the moisture from the air. The pump D supplies to the pump A,
under a pressure of 16 atmospheres, enough air to compensate for
that which escapes through J from the outer tube H, and also for
that which is liquefied and collects in the Dewar flask K.

Explanation of the Cooling Action. — The three concentric tubes
F, G, and H (which it should be stated are, with respect to the rest
of the apparatus, very much smaller than shown, and in practice
are coiled in a spiral within C), form the vital part of the appa-
ratus. The air, as it passes from the central tube F through valve
L, has its pressure reduced from 200 atmospheres to about 16
atmospheres. This process cools it considerably. The valves are
so adjusted that about 4/5 of this cooled air flows upward, as
indicated by the curved arrow, through G (thereby cooling the
downflowing stream in F) and then flows through 7 back to the
pump A. The remaining 1/5 flows directly from valve L through
valve M. As this air passes through valve M its pressure drops
from 16 atmospheres to 1 atmosphere, producing an additional
drop in temperature. At first all of the air that passes through
valve M passes up through the outer tube H and escapes through
J. We have just seen that the downflowing air in F is cooled by
the upflowing air in G, and as this downflowing air passes through
valve L it is still further cooled (by expansion), and therefore as it
passes up through G it still further cools the downflowing stream in
F, and so on. Thus both streams become colder and colder until
so low a temperature is reached that the additional cooling pro-
duced by the expansion at M causes part (about 1/4) of the air that
passes through M to liquefy and collect in the Dewar flask K.
From K, the liquid air may be withdrawn through valve N.

Quite recently liquid air has been manufactured at the rate of
about one quart per H.P.-hour expended in operating the pumps.

Properties and Effects of Liquid Air. — Liquid air is a clear, bluish
liquid, of density 0.91 gm. per cm.3. It boils at a temperature of

282 MECHANICS AND HEAT

— 191°.4 C. and its nitrogen freezes at —210°, its oxygen at
— 227°. It is attracted by a magnet, due to the oxygen which it
contains. If liquid air is poured into water it floats at first; but,
due to the fact that nitrogen (density 0.85, boiling point — 196° C.)
vaporizes faster than oxygen (density 1.13, boiling point —183°),
it soon sinks, boiling as it sinks, and rapidly disappears. Felt,
if saturated with liquid air, burns readily.

At the temperature of liquid air, mercury, alcohol, and indeed
most liquids, are quickly frozen. Iron and rubber become almost
as brittle as glass; while lead becomes elastic, i.e., more like steel.

The Dewar Flask. — If liquid air were placed in a closed metal
vessel it would vaporize, and quickly develop an enormous pres-
sure. Even if this pressure did not burst the container, the air
would soon be warmed above its critical temperature and cease to
be a liquid, so that a special form of container is required. Pro-
fessor Dewar performed a great service for low-temperature
research when he devised the double-walled flask (K, Fig. 137).
In such a container, liquid air has been kept for hours and has
been shipped to a considerable distance. The space between the
walls is a nearly perfect vacuum, which prevents, in a large meas-
ure, the passage of heat into the flask. Silvering the walls reflects
heat away from the flask and therefore improves it. These
flasks must not be tightly stoppered even for an instant or they
will explode, due to the pressure caused by the vaporization of
the liquid air. The constant but slow evaporation from the liquid
air keeps it cooled well below its critical temperature, in fact at
about — 191° C., the boiling point for air at atmospheric pressure.

The Thermal Bottle— The Thermal Bottles advertised as "Icy
hot/' etc., are simply Dewar flasks properly mounted to prevent
breakage. They will keep a liquid "warm for 12 hours," or
"cold for 24 hours." Observe that a liquid when called "warm"
differs more from room temperature than when called "cold."

CHAPTER XVI
TRANSFER OF HEAT

207. Three Methods of Transferring Heat.— Heat may be
transferred from one body to another in three ways; viz., by
Convection, by Conduction, and by Radiation.

When air comes in contact with a hot stove it becomes heated
and expands. As it expands, it becomes lighter than the sur-
rounding air and consequently rises, carrying with it heat to
other parts of the room. This is a case of transfer of heat by
convection. Obviously, only liquids and gases can transfer heat
by convection.

If one end of a metal rod is thrust into a furnace, the other end
soon becomes heated by the conduction of heat by the metal of
which the rod is composed. In general, metals are good con-
ductors, and all other substances relatively poor conductors,
especially liquids and gases.

On a cold day, the heat from a bonfire may almost blister the
face, although the air in contact with the face is quite cool. In
this case, the heat is transmitted to the face by radiation. The
earth receives an immense amount of heat from the sun, although
interplanetary space contains no material substance and is also
very cold. This heat is transmitted by radiation. These three
methods of heat transfer will be taken up in detail in subsequent
sections.

208. Convection. — Heat transfer by convection is utilized in
the hot-air, steam, and hot-water systems of heating. In these
systems the medium of heat transfer is air, steam, and water,
respectively. It will be noted in every case of heat transfer by
convection, that the heated medium moves and carries the heat with
it. Thus, in the hot-air system, an air jacket surrounding the
furnace is provided with a fresh-air inlet near the bottom; while
from the top, air pipes lead to the different rooms to be heated.
As the air between the jacket and the furnace is heated it be-
comes lighter and rises with considerable velocity through the
pipes leading to the rooms, where it mingles with the other air
of the room and thereby warms it.

283

284

MECHANICS AND HEAT

The convection currents produced by a hot stove, by means of
which all parts of the room are warmed, are indicated by arrows
in Fig. 138. As the air near the stove becomes heated, and there-
fore less dense, it rises, and the nearby air which comes in to take
its place is in turn heated and rises. As the heated air rises and
flows toward the wall, it is cooled and descends as shown.

Fig. 139 illustrates the convection currents established in a
vessel of water by a piece of ice. The water near the ice, as it is
cooled becomes more dense and sinks. Other water coming in
from all sides is in turn cooled and sinks, as indicated by the
arrows.

In steam heating, pipes lead from the steam boiler to the steam
radiators in the rooms to be heated. Through these pipes, the

1
vj

\ /

«^

" fS:

FIG. 139.

hot steam passes to the radiators, where it condenses to water.
In condensing, the steam gives up its heat of vaporization and
thereby heats the radiator. The water formed by the condensa-
tion of the steam runs back to the boiler.

In the hot-water heating system, the heated water from the boiler
(B, Fig. 140) rises through pipes leading to the radiators (C, D,
E, and F) where it gives up heat, thereby warming the radiators,
and then descends, colder and therefore denser, through other
pipes (G and H] to the boiler, where it is again heated. This
cycle is repeated indefinitely. The current of water up one pipe
and down another is evidently a convection current, established
and maintained by the difference in density of the water in the
two pipes. The rate of flow of the water through the radiators,
and hence the heating of the rooms, may be controlled by the
valves c, d, e, and/. Hot water may be obtained from the faucets
/, /, K, and L. The tank M furnishes the necessary pressure,
allows for the expansion of the water when heated, and provides

TRANSFER OF HEAT

285

a safeguard against excessive pressure should steam form in the
boiler.

If the boiler B were only partly filled with water, steam would
pass to the radiators and there condense, and the system would
become a steam-heating system. In this case it would be necessary
to provide radiators of a type in which the condensed steam would
not collect. Usually B consists of water tubes surrounded by
the flame.

In heating a vessel of water by placing it upon a hot stove, the
water becomes heated both by conduction and convection. The

FIG. 140.

heat passes through the bottom of the vessel by conduction and
heats the bottom layer of water by conduction. This heated
layer is less dense than the rest of the water and rises to the
surface, carrying with it a large quantity of heat. Other water,
taking its place, is likewise heated and rises to the surface. In
this way convection currents are established, and the entire body
of water is heated.

286 MECHANICS AND HEAT

Winds are simply convection currents produced in the air by
uneven heating. The hotter air rises, and the cooler air rushing
in to take its place is in turn heated and rises. This inrush of
air persists so long as the temperature difference is maintained,
and is called wind (Chapter XVII).

209. Conduction. — If one end A of a metal rod is heated, the
other end B is supposed to become heated by conduction in the
following manner. The violent heat vibrations of the molecules
at the end A cause the molecules near them to vibrate, and in
like manner these molecules, after having begun to vibrate,
cause the layer of molecules adjacent to them on the side toward B
to vibrate, and so on, until the molecules at the end B are vibrat-
ing violently; i.e., until B is also hot.

This vibratory motion is readily and rapidly transmitted from
layer to layer of the molecules of metals; therefore metals are
said to be good conductors.

Brick and wood are poor conductors of heat, which fact makes
them valuable for building material. Evidently it would require
a great deal of heat to keep a house warm if its walls were com-
posed of materials having high heat conductivity. Asbestos
is a very poor conductor of heat, for which reason it is much used
as a wrapping for steam pipes to prevent loss of heat, and also
as a wrapping for hot air flues to protect nearby woodwork from
the heat which might otherwise ignite it.

Clothing made of wool is much warmer than that made of
cotton, because wool is a much poorer conductor of heat than
cotton, and therefore does not conduct heat away from the body
so rapidly.

Liquids, except mercury, are very poor conductors. That
water is a poor conductor of heat may be demonstrated by the
following experiment. A gas flame is directed downward against
a shallow metal dish floating in a vessel of water. After a short
time the water in contact with the dish will boil, while the water
a short distance below experiences practically no change in tem-
perature, as may be shown by thermometers inserted. It will
be observed that convection currents are not established when
water is heated from above. A test tube containing ice cold
water, with a small piece of ice held in the bottom, may be heated
near the top until the top layers of water boil without apprecia-
bly melting the ice.

Gases are very poor conductors, pf heat — much poorer even than.

TRANSFER OF HEAT 287

liquids. The fact that air is a poor conductor is frequently made
use of in buildings by having "dead air" spaces in the walls.
It is well known that if a slight air space is left between the
plaster and the wall, a house is much warmer than if the plaster
is applied directly to the wall. If a brick wall is wet it conducts
heat much better than if dry, simply because its pores are filled
with water instead of with air. From the table of Thermal
Conductivities given below, it will be seen that water conducts
heat about 25 times as well as air. Fabrics of a loose weave are
warmer than those of a dense weave of the same material (except
in wind protection), because of the more abundant air space.
A wool-lined canvass coat protects against both wind and low
temperature.

Davy's Safety Lamp. — If a flame is directed against a cold metal
surface, it will be found that the metal cools below the combustion
point the gases of which the flame is composed, so that the flame does

FIG. 141. FIG. 141a.

not actually touch the metal. This fact may be demonstrated by pasting
one piece of paper on a block of metal, and a second piece on a block
of wood, and thrusting both into a flame. The second piece of paper
quickly ignites, the first does not. A thin paper pail quickly ignites
if exposed to a flame when empty, but not when filled with water.

If a piece of wire gauze is held above a Bunsen burner or other gas
jet, the flame will burn above the gauze only (Fig. 141), if lighted above,
and below only (Fig. 141a), if lighted below. The flame will not "strike
through" the gauze until the latter reaches red heat. Evidently, the
gas (Fig. 141a) as it passes through the wire gauze is cooled below its
ignition temperature. If a lighted match is now applied below the gauze
(Fig. 141), or above it (Fig. 141a), the flame burns both above and below
as though the gauze were absent.

The miner's Safety Lamp, invented by Sir Humphry Davy, has its
flame completely enclosed by iron gauze. The explosive fire-damp
as it passes through the gauze, burns within, but not without, and thus
gives the miner warning of its presence. After a time the gauze might
become heated sufficiently to ignite the gas and cause an explosion.

Boiler "Scale." — The incrustation of the tubes of tubular
boilers with lime, etc., deposited from the water, is one of the

288 MECHANICS AND HEAT

serious problems of steam engineering. The incrusted material
adds to the thickness of the walls of the tubes, and is also a very
poor conductor of heat in comparison with the metal of the tube.
Consequently it interferes with the transmission of the heat from
the heated furnace gases to the water, and thereby lowers the
efficiency of the boiler. Furthermore, the metal, being in con-
tact with the flame on the one side and the "scale" (instead of the
water) on the other, becomes hotter, and therefore burns out
sooner than if the scale were prevented.

210. Thermal Conductivity. — If three short rods of similar
size and length, one of copper, one of iron, and one of glass, are
held by one end in the hand while the other
^f^i^s. f° F°ld end is thrust into the gas flame, it will be found
that the copper rod quickly becomes unbear-
ably hot, the iron rod less quickly, while the
glass rod does not become uncomfortably hot,
however long it is held. This experiment
shows that copper is a better conductor than
iron, and that iron is a better conductor than
glass; but it does not enable us to tell howmany
FIG 142 times better. To do this we must compare

the thermal conductivities of the two metals,
from which (see table) we find that copper conducts about five
times as well as iron, and over 500 times as well as glass. The
fact that glass is such a very poor conductor explains why the
thin glass of windows is so great a protection against the cold.

If one face of a slab of metal (Fig. 142) is kept at a higher
temperature than the other face, it will be evident that the num-
ber of calories of heat Q which will pass through the slab in
T seconds will vary directly as the time T, as the area A of the
face, and also directly as the difference in temperature between
the two faces (i.e., ti—t2, in which the temperature of the hotter
face is t\° and the colder, £2°). It is also evident, other things
being equal, that less heat will flow through a thick slab than
through a thin one. Indeed, we readily see that the quantity Q
will vary inversely as the thickness (d) of the slab. Accordingly

we have Q cc A T(t l ~-^- - KA T^j— (85)

in which K is a constant, whose value depends upon the character
of the material of which the slab is composed, and is called the
Thermal Conductivity of the substance.

TRANSFER OF HEAT

289

Since Eq. 86 is true for all values of the variables, it is true if
we let A, T, (ti—tz), and d all be unity. This, however, would
reduce the equation to K = Q. Hence, K is numerically the num-
ber of calories of heat that will flow in unit time (the second) through
a slab of unit area and unit thickness (i.e., through a cubic centi-
meter) if its two opposite faces differ in temperature by unity (1°C.).

Temperature Gradient. — Observe that — -5 — is the fall in temperature

per centimeter in the direction of heat flow. This quantity is called
the Temperature Gradient. The heat conductivity, then, is the rate of
flow of heat (calories per sec.) through a conductor, divided by the product
of the cross-sectional area and the temperature gradient. It is better in
determining the heat conductivity for materials which are good con-
ductors, such as the metals, to use a rod instead of a slab.

The rod is conveniently heated at one end by steam circulation, and
cooled at the other end by water circulation. The temperature of the
water as it flows past the end of the rod rises from £3° to £4°. If M
grams of water flows past in T seconds, then Q = M(tt — £3), and the

Two thermometers

are inserted in the rod at a distance d apart, one near the hot end, the
other near the cold end. Let the former read ti° and the latter, t2°.

The temperature gradient is, then, -^~. The remaining quantity

A of Eq. 86, which must be known before K can be calculated, is the
cross-sectional area of the rod. If the rod is of uniform diameter and
is packed in felt throughout its length to prevent loss of heat, then the
rate of heat flow, and also the temperature gradient, will be the same at
all points in the rod.

The temperature gradient may be thought of as forcing heat along the
rod, somewhat as the pressure gradient forces water along a pipe. A
few thermal conductivities are given in the table below.

THERMAL CONDUCTIVITIES

rate of How of heat through the rod is — — IT

Substances

Thermal
conductivity
K

Substance

Thermal
conductivity
K

Silver

1 006

Marble

0 0014

Copper
Aluminum
Iron .*

0.88 to 0.96
0.34
0.16 to 0.20
0 016

Water
Hydrogen....
Paraffine. . . .
Air

0.0014
0.00033
0.00025
0 000056

Glass.. .

0.0015

Flannel. . .

0.000035

The value of the thermal conductivity varies greatly in some
cases for different specimens of the same substance. Thus, for

290 MECHANICS AND HEAT

hard steel, it is about one-half as large as for soft steel, and
about one-third as large as for hard steel. Different kinds of
copper give different results. The values given in the table are
approximate average values.

211. Wave Motion. — The kinds of wave motion most com-
monly met are three in number, typified by water waves, sound
waves, and ether waves. The beautiful waves which travel over
a field of grain on a windy day, are quite similar to water waves in
appearance, and similar to all waves in one respect; namely,
that the medium (here the swaying heads of grain) does not move
forward, but its parts, or particles simply oscillate to and fro
about their respective equilibrium positions.

Water Waves. — There are many kinds of water waves; varying
in form from the smooth ocean "swell" due to a distant storm, to
the "choppy" storm-lashed billows of the tempest; and varying
in size from the large ocean waves 20 ft. or more in height, to the
tiny ripples that speed over a still pond before a sudden gust of
wind. The Tide (Sec. 30) consists of two wave crests on opposite
sides of the earth, which travel around the earth in about 25 hrs.
Consequently, at the equator, the wave length is over 12,000 miles,
and the velocity about 1000 miles per hour.

Restoring Force. — In all cases of wave motion, at least in
material media, there must be a restoring force developed which
acts upon the displaced particle of the medium in such a direction
as to tend to bring it back to its equilibrium position. As the
head of grain sways to and fro, the supporting stem, alternately
bent this way and that, furnishes the restoring force. As the
vibrating particle reaches its equilibrium position, it has kinetic
energy which carries it to the position of maximum displacement
in the opposite direction. Thus the swaying head of grain
when the stem is erect is in equilibrium, but its velocity is then a
maximum and it moves on and again bends the stem.

In the case of large water waves, the restoring force is the gravi-
tational pull which acts downward on the "crest," and the buoy-
ant force which acts upward on the "trough" of the wave. These
waves are often called gravitational water waves. In the case
of fine ripples, the restoring force is mainly due to surface tension.
The velocity of long water waves increases with the wave length
(distance from crest to crest), while with ripples, the reverse is
true; i.e., the finer the ripples are, the faster they travel.

Sound Waves. — As the prong of a tuning fork vibrates to and

TRANSFER OF HEAT 291

fro, its motion in one direction condenses the air ahead of it;
while its return motion rarefies the air at the same point. These
condensations and rarefactions travel in all directions from the
fork with a velocity of about 1100 ft. per sec., and are called
Sound Waves. Obviously, if the tuning fork vibrated 1100 times
per sec., one condensation would be one foot from the tuning
fork when the next condensation started; while if the fork
vibrated 110 times per sec. this distance between Condensations,
called the wave length X, would be 10 ft. In other words the
relation v = n\ is true, in which v is the velocity of sound, and n,
the number of vibrations of the tuning fork per second. Sound
waves are given off by a vibrating body, and are transmitted by
any elastic medium, such as air, water, wood, and the metals.
The velocity varies greatly with the medium, but the relation
v = n\ always holds.

Ether Waves. — Ether waves consist in vibrations of the Ether
(Sec. 214), a medium which is supposed to pervade all space and
permeate all materials. These vibrations are produced, in the
case of heat or light waves, by atomic vibrations in a manner not
understood. The ether waves used in wireless telegraphy are
produced by special electrical apparatus which we cannot discuss
here.

Ether waves are usually grouped in the following manner.
Those which affect the eye (i.e., produce the sensation of light)
are called light waves, while those too long to affect the eye are
called heat waves. Those waves which are too short to affect the
eye do affect a photographic plate, and are sometimes called
actinic waves. It should not be inferred that light waves do not
produce heat or chemical (e.g., photographic) effects, for they do
produce both. Certain waves which are still longer than heat
waves, and which are produced electrically, are called Hertz
waves. These waves are the waves employed in wireless teleg-
raphy. They were discovered in 1888 by the German physicist,
H. R. Hertz (1857-94).

The longest ether waves that affect the eye are those of red
light (X = 1/35000 in. approx.). Next in order of wave length are
orange, yellow, green, blue, and violet light. The wave length
of violet light is about one-half that of red, while ultra-violet
light of wave length less than one-third that of violet has been
studied by photographic means. An occupant of a room flooded
with ultra-violet light would be in total darkness, and yet with a,

292 MECHANICS AND HEAT

camera, using a short exposure, he could take a photograph of the
objects in the room. The wave lengths longer than those of red
light, up to about 1/500 inch, have been much studied, and are
called heat waves, or infra-red. It is interesting to note that the
shortest Hertz waves that have been produced are but little longer
than the longest heat waves that have been studied. If this
small "gap" were filled, then ether waves varying in length from
several miles to 1/200000 in. would be known.

Since the velocity v for all ether waves is 186,000 miles per sec.,
the frequency of vibration n for any given wave length X is quickly
found from the relation v = n\. Thus the frequency of vibration
of violet light for which X = 1/70000 in. is about 800,000,000,-
000,000. This means that the source of such light, the vibrating
atom, or atomic particle (electron) sends out 800,000,000,000,000
vibrations per second!

Direction of Vibration. — A water wave in traveling south, let
us say, would appear to cause the water particles to vibrate up
and down. Careful examination, however, will show that there is
combined with this up-and-down motion a north-and-south
motion; so that any particular particle is seen to describe approxi-
mately a circular path. A sound wave traveling south causes
the air particles to vibrate to and fro north and south; while an
ether wave traveling south would cause the ether particles to
vibrate up and down or east and west, or in some direction in a
plane which is at right angles to the direction in which the wave is
traveling. For this reason, the ether wave is said to be a Trans-
verse Wave and the sound wave, a Longitudinal Wave. The
phenomena of polarized light seem to prove beyond question
that light is a transverse wave.

212. Interference of Wave Trains. — A succession of waves, following
each other at equal intervals, constitutes a wave train. A vibrating
tuning fork or violin string, or any other body which vibrates at a
constant frequency, gives rise to a train of sound waves. Two such
wave trains of different frequency produce interference effects, known
as beats, which are familiar to all.

Interference of Sound Waves. — Let a tuning fork A of 200 vibrations
per second be sounded. The train of waves from this fork, impinging
upon the ear of a nearby listener, will cause the tympanum of his ear
to be alternately pushed in and out 200 times per sec., thus giving
rise to the perception of a musical tone of uniform intensity. If, now,
a second fork B of, say 201 vibrations per second, is sounded, the train

TRANSFER OF HEAT 293

of waves from it, let us say the " B train," will interfere with the "A
train" and produce an alternate waxing and waning in the intensity
of the sound, known as "beats." In this case there would be 1 beat
per sec. For, consider an instant when a compressional wave from the
A train and one from the B train both strike the tympanum together.
This will cause the tympanum to vibrate through a relatively large dis-
tance, i.e., it will cause it to have a vibration of large amplitude, and a
loud note (maximum) will be heard. (The amplitude of a vibration is
half the distance through which the vibrating body or particle, as the
case may be, moves when vibrating; in other words, it is the maximum
displacement of the particle from its equilibrium position.) One-half
second later, a compressional wave from the A train and a rarefaction from
the B train will both strike the tympanum. Evidently these two dis-
turbances, which are said to be out of phase by a half period, will produce
but little effect upon the tympanum, in fact none if the two wave trains
have exactly equal amplitudes. Consequently, a minimum in the tone
is heard. Still later, by 1/2 sec., the two trains reach the ear exactly in
phase, and another maximum of intensity in the tone is noted, and so on.
Obviously, for a few waves before and after the maximum, the two trains
of waves will be nearly in phase, and a fairly loud tone will be heard.
This tone dies down gradually as the waves of the two trains get more
and more out of phase with each other, until the minimum is reached.

Had the tuning forks differed by 10 vibrations per second, there would
have been 10 beats per second. To tune a violin string to unison with
a piano, gradually increase (or decrease) the tension upon it until the
beats, which come at longer and longer intervals, finally disappear
entirely. If increasing the tension produces more beats per second,
the string is already of too high pitch.

Interference of Light Waves. — By a proper arrangement, two trains of
light waves of equal frequency and equal amplitude maybe produced.
If these two trains fall upon a photographic plate from slightly different
directions, they will reinforce each other at some points of the film,
and annul each other at other points. For certain portions of the plate,
the two trains are constantly one-half period out of phase. Such
portions are in total darkness, and therefore remain clear when the
plate is "developed," producing, with the alternate "exposed" strips, a
beautiful effect. We here have the strange anomaly of light added to
light producing darkness, for either beam alone would have affected
the entire photographic plate.

213. Reflection and Refraction of Waves.— In Fig. 143, let
AB be a stone pier, and let abc, etc., be water waves traveling in
the direction bO. Then a'b'c', etc. (dotted lines), will be the
reflected water waves, and will travel in the direction Ob', such
that bO and Ob' make equal angles 0i and 02 with the normal

294

MECHANICS AND HEAT

(NO) to the pier. This important law of reflection is stated thus:
The angle of reflection (02) is equal to the angle of incidence (0i).

If AB is a mirror and abc, etc., light waves, or heat waves,
then the construction will show accurately the reflection of light
or heat waves, as the case may be.

Proof: If the reflected wave has the same velocity as the inci-
dent wave, which is strictly true in the case of heat and light,
then, while the incident light (let us say) travels from a\ to a2,
the reflected light will travel from c\ to c2. The triangles
and c^ttzCi will be not only similar, but equal. Therefore 63
But 63 = 61 and 04 = 02, hence 0i = 02, which was to be proved.

Refraction. — Let abc (Fig. 144) represent a light wave or a
heat wave, traveling in the direction 60. Then, as the portion a
reaches a', portion c will have reached c' instead of c". The ratio
cc'/cc" is about 3/4, since light and heat radiation travel about
3/4 as fast in water as in air. The reciprocal of this ratio, i.e.,
the velocity in air divided by the velocity in water, is called the
index of refraction for water. The index of refraction for glass
varies with the kind of glass and the length of the wave, from
about 1.5 to 2. Since the ray is always normal to the wave
front, the ray Ob' deviates from the direction 60 by the angle
a, called the angle of deviation. The fact that the ray bends
sharply downward as it enters the water, accounts for the apparent
sharp upward bending of a straight stick held in a slanting
position partly beneath the surface of the water.

TRANSFER OF HEAT 295

The fact that light and heat radiation travel more slowly in
glass than in air, thus causing all rays which strike the glass
obliquely to be deviated, makes possible the focusing of a bundle
of rays at a point by means of a glass lens, and therefore makes
possible the formation of images by lenses. Since practically all
optical instruments consist essentially of a combination of lenses,
we see the great importance of the refractive power of glass and
other transparent substances. Indeed were it not for the fact
that light travels more slowly through the crystalline lens of the
eye than through air, vision itself would be ii

FIG. 144.

The production of the rainbow and prismatic colors in general
depends upon the fact that the velocity of light in glass, water,
etc., depends upon the wave length, being greatest for red and
least for violet. Consequently red light is deviated the least, the
violet the most.

214. Radiation. — If a glowing incandescent lamp is placed
under the receiver of an air pump, it will be found that it gives off
heat and heats the receiver, whether the receiver contains air or
a vacuum. It is evident, then, that the air is not the medium
of transfer of heat by radiation. Likewise, in the case of heat
and light received from the sun, the medium of transfer cannot be
air. Since the transmission of a vibratory motion from one point
to another requires an intervening medium, physicists have been
led to postulate the Ether as such a medium, and have ascribed to
it such properties as seem best to explain the observed phenomena.
The ether is supposed to fill all space and also to permeate all

296 MECHANICS AND HEAT

materials. Thus we know that the heat of the sun passes readily
through glass by radiation. This is effected, however, by the
ether in the glass and not by the glass itself. Indeed the glass
molecules prevent the ether from transmitting the radiation so
well as it would if the glass were absent.

While immense quantities of heat are transferred from the sun
to the earth by radiation, it is well to call attention to the fact
that what we call radiant heat or heat radiation, is not strictly heat,
but energy of wave motion. Radiant heat does not heat the
medium through which it passes (unless it is in part absorbed),
but heats any body which it strikes — a good reflector least, a lamp-
black surface most. Both heat radiation and light may be re-
flected, and also refracted (Sec. 213). The moon and the planets,
in the main, shine by reflected sunlight. We see all objects
which are not self-luminous, by means of irregularly (scattering)
reflected light. At South Pasadena, Cal., a 10-H.P. steam engine
is run by a boiler which is heated by means of sunlight reflected
from a great number of properly placed mirrors.

215. Factors in Heat Radiation. — It has been shown experi-
mentally that the higher the temperature of a body becomes, the
faster it radiates heat energy. Obviously, the amount of heat
radiated in a given time will also be proportional to the amount
of heated surface. It has also been found that two metal spheres,
A and B, alike as to material, size and weight, but differing in
finish of surface, have quite different radiating powers. Thus if
A is highly polished, so as to have a mirror-like surface, while B
is coated with lamp black, it will be found that B radiates heat
much faster than A. This is easily tested by simply heating A
and B to the same temperature and then suspending them to cool.
It will be found that B cools much more rapidly than A, which
shows that B parts with its heat more quickly, i.e., radiates better,
than A. A lamp-black surface is about the best radiating surface,
while a polished mirror surface is about the poorest. The radiat-
ing powers of other substances lie between those of these two.
From the above discussion, we see that the high polish of the
nickel trimmings of stoves decreases their efficiency somewhat.

Prevost's Theory of Heat Exchanges. — According to this theory,
a body radiates heat to surrounding bodies whether it is warmer
than they or colder. In the former case it radiates more heat to
the surrounding bodies than it receives from them, and its tempera-
ture falls; while in the latter case it radiates less heat than it

TRANSFER OF HEAT 297

receives, and its temperature rises. The fall in temperature
experienced by a body when placed near ice, a result which would
at first seem to indicate that cold can be radiated and that it is
not therefore merely the absence of heat, is easily explained by
this exchange theory. The body radiates heat no faster to the ice
than it would to a warmer body, but it receives less in return, and
therefore becomes colder.

Laws of Cooling. — Newton considered that the amount of heat
H, radiated from a body of temperature t, to its surroundings of
temperature t', was proportional to the difference in temperature;
i.e.,

H = K(t-t')

in which K is a constant, depending upon the size and character
of the surface. This law is very nearly true for slight differences
in temperature only. Thus a body loses heat almost exactly
twice as fast when 2° warmer than its surroundings as it does when
1° warmer. Experiment, however, shows that if this tempera-
ture difference is, say, 20°, the amount of heat radiated is more
than 20 times as great as when it is 1°.

The quite different law, expressed by the equation

is due to Stefan, and is known as Stefan's Law. In this equation
T and T' are the temperatures of the body and its surroundings,
respectively, on the absolute scale. Stefan's law, applied to
radiation by black bodies, accords with experimental results.

216. Radiation and Absorption. — It has been found experi-
mentally that surfaces which radiate heat rapidly when hot,
absorb heat rapidly when cold. Thus if the two metal spheres
mentioned in Sec. 215 were placed in the sunshine, B would be
warmed very much more quickly than A. Evidently the same
amount of solar heat radiation would strike each, but A reflects
more and consequently absorbs less than B, which has smaller
reflecting power. There is a close proportionality between radia-
tion and absorption. For example, if B, when hot, loses heat by
radiation twice as fast as A does when equally hot, then if both are
equally cold and are placed in the sunshine, B will absorb heat
practically twice as fast as A. That is, good absorbers of heat
(when cold) are good radiators of heat (when hot). If two
thermometers, one of which has its bulb smoked until black,

298 MECHANICS AND HEAT

are placed side by side in the sunshine, the one with the blackened
bulb will indicate a higher temperature than the other.

217. Measurement of Heat Radiation. — By means of the
thermopile (Sec. 174), and other sensitive devices, such as the
bolometer, many measurements of intensity of heat radiation
have been made. When white light, e.g., sunlight, passes through
a prism, the different colors of light take slightly different direc-
tions, and a "spectrum" of the colors, red, orange, yellow, green,
blue, and violet, is produced.

By exposing the bolometer successively in the violet, blue,
green, yellow, orange and the red, and then moving it still farther,
into the invisible or infra-red part of the spectrum, it is found that
the radiant energy increases with the wave length, and reaches a
maximum in the infra-red. In other words, the wave length of
the sun's radiation which contains the most energy is slightly
greater than that of the extreme red. It has been found by experi-
ment, using various sources of known temperature for producing
the light, that the wave length of maximum energy is shorter, the
hotter the source. From these considerations the temperature of
the sun is estimated to be about 6000° C. In such experiments, a
rock-salt prism must be used, since glass absorbs infra-red radia-
tions to a great extent.

218. Transmission of Heat Radiation Through Glass, Etc.—
Just as light passes readily through glass and other transparent
substances, so heat radiation passes readily through certain sub-
stances. In general, substances transparent to light are also trans-
parent to heat radiation, but there are some exceptions to this rule.

A thin pane of glass gives very little protection from the sun's
heat, but if held between the face and a hot stove it is a great
protection. It may be remarked that in the former case, the
glass is not noticeably warmed, while in the latter case it is
warmed. It is apparent, then, that the glass transmits solar
radiation better than it does the radiation from the hot stove.
This selective transmission of radiation is really due to "selective
absorption." The glass absorbs a greater percentage of the ra-
diation in the latter case than in the former, which accounts not
only for the fact that it transmits less heat in the case of the radia-
tion from the stove, but also for the fact that it is heated more.

In this connection, we may state that it has been shown by
experiment that a body, say a piece of iron, when heated to a white
heat, gives off simultaneously heat waves varying greatly in

TRANSFER OF HEAT 299

length. As it is heated more and more, it gives off more and more
energy of all wave lengths; but the energy of the shorter wave
lengths increases most rapidly. Accordingly, the wave length of
maximum energy becomes shorter the hotter the source, as stated
in Sec. 217.

Just before the iron reaches "red heat" the heat waves are all
too long to be visible. As it becomes hotter, somewhat shorter
waves, corresponding to red light, are given off, and we say that
the iron is "red hot." If heated to a still higher temperature, so
as to give off a great deal of light, in fact light of all different wave
lengths, we say that it is "white hot." A hot stove, then, gives
off, in the main, very long heat waves; while the sun, which is
intensely heated, gives off a great deal of its heat energy in the
short wave lengths.

The above-mentioned fact, that glass affords protection from
the heat radiation from a stove, and no appreciable protection in
the case of solar radiation, is explained by saying that glass trans-
mits short heat waves much better than long heat waves, i.e.,
glass is more transparent to short than to long heat waves. More
strictly, it might be said that glass does not prevent the trans-
mission of short heat waves by the ether permeating it, to so great
an extent as it does the long heat waves.

The "Hotbed. " — The rise in temperature of the soil in a Hotbed,
when the glass cover is on, above what it would be if the glass
were removed, is in part due to this behavior of glass in the trans-
mission of heat radiation. The greater part of the solar heat that
strikes the glass, being of short wave length, passes readily
through the glass to the soil, which is thereby warmed. As the
soil is warmed, it radiates heat energy, but in the form of long
heat waves which do not readily pass through the glass, and hence
the heat is largely retained. The fact that the glass prevents a
continual stream of cold air from flowing over the soil beneath it,
and still permits the sun to shine upon the soil, accounts in large
part for its effectiveness.

"Smudging" of Orchards. — Very soon after sunset, blades of
grass and other objects, through loss of heat by radiation, usually
become cool enough to precipitate part of the moisture of the air
upon them in the form of Dew (Sees. 220, 221). It is well known
that heavy dews form when the sky is clear. If the sky is over-
cast, even by fleecy clouds, a portion of the radiated heat is
reflected by the clouds back to the earth, and the cooling of

300 MECHANICS AND HEAT

objects, and consequently the formation of dew upon them, is
less marked.

Many fruit growers have placed in the orchard, a thermostat,
so adjusted that an alarm is sounded when the freezing point is
approached. As soon as the alarm is sounded, the "smudge"
fires (coal, coal oil, etc.) are started. These fires produce a thin
veil of smoke, which hovers over the orchard and protects it from
frost, somewhat as a cloud would. In addition to the protection
afforded by the smoke, the considerable amount of heat developed
by the fires is also important. If the wind blows, such protec-
tion is much less effective. Frosts, however, usually occur during
still, clear nights.

219. The General Case of Heat Radiation Striking a Body.—
Heat radiation, e.g., solar radiation, when it strikes a body, is
in general divided into three parts: the part (a) which is reflected;
the part (6) which is absorbed and therefore tends to heat the body;
and the part (c) which is transmitted, or passes through the body.
The sum of these, i.e., a+6-f-c, is of course equal to the original
energy that strikes the body. In some cases, the part reflected
is large, e.g., if the body has polished surfaces. In other cases,
the part absorbed is large, e.g., for lamp black or, in general, for
dull surfaces, and also for certain partially transparent substances.
The part transmitted is large for quartz and rock salt; much
smaller for glass, water and ice, and absent for metals unless
they are in the form of exceedingly thin foil.

PROBLEMS

1. If a piece of plate glass 80 cm. in length, 50 cm. in width, and 1.2 cm.
in thickness, is kept 20° C. hotter on one side than on the other, how many
calories of heat pass through it every minute by conduction alone?

2. A copper vessel, the bottom of which is 0.2 cm. thick, has an area of
400 cm.2, and contains 3 kilograms of water. What will be the temperature
rise of the water in it in 1 minute, if the lower side of the bottom is kept
3° C. warmer than the upper side of the bottom?

3. Assuming the sun to be directly over head, what power (in H.P.)
does it radiate in the form of heat upon an acre of land at noon. See Sec.
161.

4. A wall 10 in. thick is made of a material, the thermal conductivity of
which is 0.001 12. The wall is made "twice as warm" by rebuilding it with
an additional thickness of "dead air" space. Find the thickness of the air
space. (In practice, convection currents diminish considerably the effective-
ness of so-called "dead air" spaces.)

TRANSFER OF HEAT 301

5. How many pounds of steam at 140° C. (heat of vap. 509 cal. per gm.)
will a boiler furnish per hour if it has 1000 sq. ft. of heating surface of iron
(thermal conductivity 0.16) 0.25 in. in thickness, which is kept 5° hotter
next the flame than next the water? Note that the heat of vaporiza-
tion and the conductivity are given in C.G.S. units.

CHAPTER XVII
METEOROLOGY

220. General Discussion. — Meteorology is that science which
treats, in the main, of the variations in heat and moisture of the
atmosphere, and the production of storms by these variations.
Although the earth's atmosphere extends to a height of a great
many miles, the weather is determined almost entirely by the
condition of the lower, denser strata, extending to a height of
but a few miles.

Clouds. — Clouds have been divided into eight or ten important
classes, according to their appearance or altitude. Their altitude
varies from -1/2 mile to 8 or 10 miles, and their appearance varies
from the dense, gray, structureless rain cloud, called Nimbus, to
the interesting and beautiful "wool-pack" cloud, known as
Cumulus, which resembles the smoke and "steam" rolling up
from a locomotive. All clouds are composed either of minute
droplets of water or tiny crystals of snow, floating in the air.
Fog is merely a cloud at the surface of the earth. Thus, what is
a cloud to the people in the valley, is a fog to the party on the
mountain side enveloped by the cloud. The droplets in a fog
are easily seen. The upper clouds may travel in a direction
quite different from that of the surface wind, and at velocities
as high as 200 miles per hour.

221. Moisture in the Atmosphere. — The constant evaporation
from the ocean, from inland bodies of water, and from the ground,
provides the air with moisture, the amount of which varies
greatly from time to time. Although the water vapor seldom
forms as much as 2 per cent, of the weight of the air, never-
theless, water vapor is the most important factor in determining
the character of the weather. When air contains all of the mois-
ture it will take up, it is said to be saturated. If saturated air
is heated, it is capable of taking up more moistufe; while if it is
cooled, it precipitates a portion of its moisture as fog, cloud,
dew, or rain. If still further cooled, it loses still more of its
water vapor. Indeed the statement that the air is saturated with

302

METEOROLOGY 303

water vapor does not indicate how much water vapor it contains,
unless the temperature of the air is also given.

When unsaturated air is cooled more and more, it finally
reaches a temperature at which precipitation of its moisture
occurs. This temperature is called the Dew Point. If air is
nearly saturated, very little cooling brings it to the dew point.
After the dew point is reached, the air cools more slowly, because
every gram of water vapor precipitated, gives up nearly 600
calories of heat (its heat of vaporization) to the air. Thus, if
on a clear chilly evening in the fall, a test for the amount of
water vapor in the air shows the dew point to be several degrees
below zero, then frost may be expected before morning; while
if the dew point is well above zero, there is little probability of
frost. This might be taken as a partial guide as to whether or
not to protect delicate plants. The fact should be emphasized,
that if the moisture in the air is visible, it is in the form of drop-
lets, since water vapor, like steam, is invisible.

222. Hygrometry and Hygrometers. — Hygrometry deals with
the determination of the amount of moisture in the atmosphere.
The devices used in this determination are called hygrometers.
Only two of these, the chemical hygrometer and the wet-and-
dry-bulb hygrometer, will be discussed.

The Chemical Hygrometer consists of a glass tube containing
fused calcium chloride (CaCl2), or some other chemical having
great affinity for water. Through this tube (previously weighed)
a known volume of air is passed. This air, during its passage,
gives up its moisture to the chemical and escapes as perfectly
dry air. The tube is again weighed, and the gain in weight
gives the amount of water vapor in this known volume of air.

The Wet-and-dry-bulb Hygrometer. — From the two temperature
readings of the wet-and-dry-bulb thermometer (Sec. 198), in
connection with a table such as given below, the dew point
may be found. Having found the dew point, the amount of
moisture per cubic yard is readily found from the second table.

The manner of using these tables will be best illustrated by an
example. Suppose that when a test is made, the dry-bulb
thermometer reads 60° F., and the wet-bulb thermometer 52° F.,
or 8° lower. Running down the vertical column (first table) for
which i — t' is 8° until opposite the dry-bulb reading 60°, we find
the dew point 45°.6 F. This shows that if the temperature of the
air falls to 45°.6 F., precipitation will commence. Opposite to

304

MECHANICS AND HEAT

dew point 45° (the nearest point to 45.6) in the second table,
we find 0.299 and 0.0133; which shows that every cubic yard of
air contained approximately 0.0133 Ibs. of water vapor on the
day of this test, and that the water vapor pressure was 0.299
inches of mercury.

DEW POINTS FROM WET-AND-DRY-BULB HYGROMETER READINGS
Dry bulb temperature t. Wet bulb temperature t'. Difference t -t'.

t°-r

= 2°

8° | 10°

12°

14° F.

t°

40° F.

36.2

30.8

25.6

20.8

16.0

11.2

6.4

41.4

35.8

31.2

27.0

22.1

17.4

12.8

45.8

41.6

37.4

33.0

29.0

24.8

20.6

51.0

47.0

43.0

39.0

35.2

31.0

27.0

56.4

52.8

49.2

45.6

42.3

38.4

34.8

61.6

59.2

55.0

51.4

48.0

44.6

41.0

67.0

64.0

61.0

58.0

55.3

52.0

49.0

72.0

69.0

66.0

63.0

60.6

57.0

54.0

77.0

74.0

71.0

68.0

65.0

62.0

59.0

DEW POINTS AND THE CORRESPONDING PRESSURES AND
DENSITIES OF WATER VAPOR

Dew point

Pressure in inches of
mercury

Density in Ibs. per cu. yd

20° F.

0.102

0.0051

0.130

0.0066

0.161

0.0076

0.200

0.0091

0.252

0.0111

0.299

0.0133

0.358

0.0159

0.433

0.0196

0.512

0.0223

0.617

0.0255

0.732

0.0306

223. Winds, Trade Winds. — Winds originate in the uneven
heating of the earth's atmosphere at different points. This
heating is in part due to the direct action of the sun, and in part
to the heat of vaporization given off when a portion of the mois-
ture in the air changes to the liquid state. When air is heated
it expands, and therefore becomes lighter and rises with con-
siderable velocity. The current of colder air, rushing in to take
its place, is called Wind. This effect is easily noticed with a
large bonfire on a still day. The violent upward rush of the
heated air above the fire carries cinders to a great height. The

j METEOROLOGY 305

cool air rushinp; in to take its place produces a "wind" that
blows toward the fire from all directions.

The Trade Winds. — An effect similar to that produced by the
bonfire, as above described, is constantly being produced on a
grand scale in the tropical regions. The constant high tempera-
ture of the equatorial regions heats the air highly and causes it
to rise. The air north and south of this region, rushing in to
take the place of this rising air, constitutes the Trade Winds.

On account of the rotation of the earth, trade winds do not
blow directly toward the equator but shift to the westward.
Thus in the West Indies, the trade winds are N. E. winds, i.e.,
they blow S. W. The trade winds south of the equator are S. E.
winds, i.e., they blow N. W.

The westward deviation of the trade winds, both north and
south of the equator, may be accounted for as follows. Objects
near the equator describe each day, due to the rotation of the
earth, paths which are the full circumference of the earth;
while objects some distance either north or south of the equator
describe shorter paths in the same time, and therefore have
less velocity. Consequently, as a body of air moves toward the
equator, it comes to points of higher eastward velocity, and
therefore "falls behind," so to speak; that is, it drifts somewhat
to the westward.

Between the trade winds of the two hemispheres lies the
equatorial "zone of calms." This zone, which varies from
200 to 500 miles in width, has caused sailing vessels much trouble
with its prolonged calms, violent thunder storms, and sudden
squalls.

Since a rising column of air is cooled by expansion, it precipi-
tates its moisture; whereas a descending column, warmed as it
is by compression, is always capable of absorbing more moisture
and is, therefore, relatively dry. In the zone of calms, the air
from the two trade winds which meet in this region must rise.
As it rises, it is cooled and precipitates its moisture in torrents
of rain. Wherever the prevailing wind blows from the sea
across a mountain range near the coast, the rain will be ex-
cessive on the mountain slope toward the sea, where the air
must rise to pass over the mountain. As the air descends upon
the opposite side of the range, it is very dry and produces a region
of scant rain and, in many cases, a desert. The rainfall on por-
tions of the southern slope of the Himalaya Mountains is about

306 MECHANICS AND

30 ft. per year; while to the north of the range lie large arid or
semi-arid districts.

224. Land and Sea Breezes. — Near the seashore, especially
in warm countries, the breeze usually blows toward the shore
from about noon until shortly before sunset, and toward the
sea from about midnight until shortly before sunrise. The
former is called a Sea Breeze; the latter, a Land Breeze.

These breezes are due to the fact that the temperature of the
land changes quickly, while the temperature of the ocean is
nearly constant (Sec. 185). Consequently, by noon, the air
above the land has become considerably heated, and is therefore
less dense than the air over the ocean. This heated air, there-
fore, rises, and the air from the ocean, rushing in to take its
place, is called the sea breeze. The rising column of air becomes
cooler as it rises, and flows out to sea. Thus, air flows from
sea to land near the earth, and from land to sea in the higher
regions of the atmosphere. Toward midnight the land and the
air above it have become chilled. This chilled, and therefore
dense air flows out to sea, as a land breeze; while the air from
the ocean flows toward the land in the higher region. It will
be observed, then, that the convection circulation at night is
just the reverse of the day circulation.

It is observed that the sea breeze first originates some distance
out at sea and blows toward the land. A feasible explanation
is this: As the air over the land first becomes heated it expands
and swells up like a large blister. The air above, lifted by the
"blister," flows away out to sea in the higher regions of the
atmosphere, thereby causing an excess pressure upon the air
there. The air then flows away from this region of excess pres-
sure toward the land, where the deficit in pressure exists.

225. Cyclones. — Strictly, the term cyclone applies to the
periodical rotary storms, about 1000 miles or so in diameter,
which occur in various parts of the earth. Every few days
they pass across the central portion of the United States,
in a direction somewhat north of east. Their courses may
be followed from day to day by means of the U. S. weather
maps. The barometric pressure is usually about one-half
inch of mercury less at the center ("storm center") of a cyclone,
than at the margin. This region of low pressure, called a "low
area," is due, at least in part, to the condensation of water
vapor that occurs in cloud formation, and the consequent

METEOROLOGY 307

heating of the air by the heat of vaporization thereby evolved.
This heated air rises, and the surrounding air, rushing in,
produces wind. Due to the rotation of the earth, these winds,
instead of blowing straight toward the storm center, are, in
general, deflected to the right of the storm center. Occasionally,
due so some local disturbance, the wind may blow in a direction
nearly opposite to that which would be expected from the above
rule; but, in general, the surrounding air moves toward the storm
center in a spiral path. The rotation is counterclockwise
(viewed from above) in the Northern hemisphere, and clockwise
in the Southern. Any body in motion (e.g., a rifle ball) in the
Northern hemisphere tends to deviate to the right from its path,
and in the Southern hemisphere to the left.1 This fact accounts
for the rotatory motion of these storms, as explained below.

Cause of the Rotary Motion of Cyclones. — Let Fig. 145 represent
a top view of a level table A upon which rests a heavy ball B
loosely surrounded by a very light frame C to which is attached
a string DD\. Evidently, if the table is at rest and the lower end
of the string Z>i, which passes through a hole in the center of the
table, is pulled, ball B will roll in a straight path to the center.
If, however, the string is pulled while the table, and consequently
the ball, are rapidly revolving in the direction indicated by the
arrow a, then the ball will follow a left-handed spiral path as in-
dicated by the broken line. For, as the ball moves nearer to
the center, it reaches portions of the table of smaller and smaller
radius, and consequently portions having less tangential velocity
than its own. Therefore, the ball rolls "ahead," i.e., to the right,
of the straight line D as shown. If the table and the ball were
rotated in the opposite direction (clockwise), similar reasoning
would show that the ball would then travel toward the center in
a right-handed spiral path.

It will next be shown that any area of the globe, having a diameter
of, say, a few hundred miles, may be considered to be a flat surface,

Although this tendency of a moving body to drift to the right in the
Northern hemisphere and to the left in the Southern hemisphere is of
such great importance in determining the motion of storms, its effe.ct on
projectiles is very slight indeed. Thus, in latitude 40°, due to this cause,
an army rifle projectile veers to the right (no matter in what direction it
is fired) by only about 3 in. on a 1000-yd. range. Due to the same cause, a
heavy locomotive, when at full speed on a level track, bears only about
50 Ibs. more on the right rail than on the left.

308

MECHANICS AND HEAT

rotating about a vertical axis at its center, and that consequently, air
which tends to move toward the center of the area, as it does in cyclones,
will trace a spiral path similar to that traced by the ball (Fig. 145).
That an area about the north pole has such rotation, with the pole as
axis, is evident. Since the earth rotates from west to east, this rotation
viewed from above, is counterclockwise the same as shown in Fig.
145. The rotational velocity, say wi, is of course one revolution per
day.

Such an area at the equator would revolve once a day about a hori-
zontal (N. and S.) axis, but would obviously have no rotation about a

FIG. 145.

vertical axis. This fact accounts for the absence of cyclones near the
equator.

It can be shown that such an area, in latitude 0, has an angular
velocity o> about the vertical axis, given by the equation

w = wi sin 6

The rotation of the area is counterclockwise in the northern hemisphere
(see rotation at the north pole above) and clockwise in the southern.
Consequently, the air moves (i.e., the wind blows) toward the center of
a cyclone in a left-handed spiral path in the former case, and in a right-
handed spiral path in the latter case, as explained above for the ball.

Hurricanes and Typhoons. — The hurricanes of the West Indies, and
the Typhoons of China, might be called the "cyclones" of the tropical
and sub-tropical regions. They are more violent and of smaller diameter
than cyclones, their diameter rarely exceeding 400 miles, though they
sometimes gradually change to cyclones and travel long distances through
the temperate zones.

METEOROLOGY 309

226. Tornadoes. — Tornadoes resemble hurricanes, but are
much smaller, and usually more violent. Because of the terrific
violence, narrow path, brief duration, and still more brief warning
given, tornadoes have not been very satisfactorily studied, and
much difference of opinion exists with regard to them. The
visible part of a tornado consists of a depending, funnel-shaped
cloud, tapering to a column which frequently extends to the
ground. Due to the centrifugal force caused by the rapid
rotation of the column, the air pressure within it is considerably
reduced. Consequently as moist air enters the column it is
cooled by expansion and its moisture condenses, forming the
cloud which makes the column visible. At sea, tornadoes are
called Water Spouts. The column is not water, however, but
cloud and spray.

Origin. — Tornadoes usually develop to the southeast of the
center of a cyclone. Sometimes several may rage simultaneously
at different points in the same cyclone. Occasionally con-
ditions of the atmosphere arise which are especially favorable
to the formation of tornadoes. These conditions are a warm
layer of air saturated with moisture next to the earth, with a
layer of much cooler air above it. As, due to local disturbance,
some of this heated moist air rises to the cooler regions, it pre-
cipitates part of its moisture, thus freeing a considerable amount
of heat. This heat prevents the rising air from cooling so rapidly
as it otherwise would, and consequently helps to maintain its
tendency to rise. As this air rises, it is followed by other satu-
sated air, which in turn receives heat by condensation of its water
vapor. Thus the action, when once started, continues with
great violence. The air rushing in from the surrounding country
to take the place of the ascending air current acquires a rotary
motion, just as already explained in connection with cyclones
As the tornado advances it is constantly furnished with a new
supply of hot, damp air and it will continue just so long as this
supply is furnished, i.e., until it passes over the section of country
in which these favorable conditions exist. Tornadoes travel
across the United States in a direction which is usually about
east. A tornado may be likened to a forest fire, in that the one
requires a continuous supply of moist air, the other, a continuous
supply of fuel.

Tornadoes sometimes do not reach to the earth, which indicates
that the favorable stratum of air upon which they "feed" is, at

310 MECHANICS AND HEAT

least sometimes, at a considerable altitude. Some think it is
always at a high altitude. This, the writer doubts. The moist
stratum is probably very deep.

Extent. — The destructive paths of tornadoes vary in width
from 100 ft. to 1/2 mile, and in length from less than a mile to
200 miles. A tornado which wrecks weak buildings over a
path 1/8 mile in width may leave the ground practically bare
for a width of 100 ft. or so.

Velocity. — The velocity of tornadoes varies from 10 to 100
miles per hour. It is estimated, however, that the wind near
the center sometimes attains a velocity of 200 or 300 miles per
hour, or even greater.

Judging by the effects produced, the velocity must be very
great. An iron bed rail has been driven through a tree by a
tornado. A thin-bladed shovel has been driven several inches
into a tree. Such a shovel would not withstand driving into a
tree with a sledge hammer. Splintered boards are frequently
driven deep into the ground, and, by way of contrast, mention
may be made of a ladder which was laid down, at a considerable
distance from the path, so gently as to scarcely leave a mark on
the ground. Shingles and thin boards have been found in great
numbers 6 or 7 miles from the path, and probably 10 or 15 miles
from where they began their flight.

The rapid rotational velocity at the center, tends to produce
a vacuum, as already mentioned. It is conjectured that the
pressure at the center of the tornado may be as much as 3 or 4
Ibs. per square inch less than normal. If this be true, then, as
the tornado reaches a building filled with air at nearly normal
pressure, there will be an excess pressure within the building of
say 3 Ib. per square inch, or over 400 Ibs. per square foot, tending
to make the building explode. The position of the wreckage
sometimes indicates that this is just what has taken place.

In spite of the great violence of tornadoes, few people are killed
by them, because of their infrequency and limited extent. If a
man were to live a few hundred thousand years he might reason-
ably expect to be caught in the path of a tornado, and if immune
from death except by tornadoes he could not reasonably expect
to live more than a few million years.

CHAPTER XVIII
STEAM ENGINES AND GAS ENGINES

227. Work Obtained from Heat — Thermodynamics. — Thermo-
dynamics deals with the subject of the transformation of heat
into mechanical energy, and vice versa, and the relations that
obtain in such transformation under different conditions. No
attempt will be made to give more than a brief general treatment
of this important subject.

What is known as the First Law of Thermodynamics may be
stated as follows: Heat may be transformed into mechanical
energy, and likewise, mechanical energy may be transformed into
heat, and in all -cases, the ratio of the work done, to the heat so
transformed, is constant. Conversely, the ratio of the work
supplied, to the heat developed (in case mechanical energy is
changed to heat energy by friction, etc.), gives the same constant.
This constant, in the British system, is 778. Thus, if one B.T.U.
of heat is converted into mechanical energy, it will do 778 ft.-lbs.
of work; conversely, if 778 ft.-lbs. of work is converted into heat,
it produces one B.T.U. For example, if 778 ft.-lbs. of energy is
used in stirring 1 Ib. of water, it will warm the water 1° F. The
similar relation in the metric system is expressed by the state-
ment that 1 calorie equals 4.187X107 ergs.

Illustrations of the First Law of Thermodynamics. — By means of
the steam engine and the gas engine, heat is converted into mechan-
ical energy. In bringing a train to rest, its kinetic (mechanical)
energy is converted into heat by the brakes, where a shower of
sparks may be seen. In inflating a bicycle tire, work is done in
compressing the air, and this heated air makes the tube leading
from the pump to the tire quite warm. In the fire syringe, a
snug-fitting piston, below which some tinder is fastened, is
quickly forced into a cylinder containing air. As the air is com-
pressed it is heated sufficiently to ignite the tinder. In gas en-
gines, preignition may occur during the compression stroke, due
in part to the heat developed by the work of compression.

311

312 MECHANICS AND HEAT

The Second Law of Thermodynamics. — The second law of
thermodynamics is expressed by the statement that heat will
not flow of itself (i.e., without external work), from a colder to a
warmer body. In the operation of the ammonia refrigerating
apparatus, heat is taken continuously from the very cold brine
and given to the very much warmer cooling tank; but the work
required to cause this "uphill" flow of heat is done by the steam
engine which operates the air pump.

Lord Kelvin's statement of the second law amounts to this:
Work cannot be obtained by using up the heat in the coldest
bodies present. Carnot (Sec. 236) showed that when heat passes
from a hotter to a colder body (through an engine) the maximum

m rp

fraction of the heat which may be converted into work is — ^ — 2'

in which TI and 772 are, respectively, the temperatures of the
two bodies on the absolute scale.

228. Efficiency. — While it is possible to convert mechanical
energy, or work, entirely into heat, thereby obtaining 100 per
cent, efficiency, it is impossible in the reverse process to trans-
form more than a small percentage of heat energy into mechanical
energy. It is, indeed, a very good steam engine that changes
into work 1/5 of the heat energy of the steam furnished it by
the boiler. Considering the large amount of heat that radiates
from the furnace, and also the heat that escapes through the
smoke stack, there is a further reduction in the efficiency. The
total efficiency of a steam engine is the product of three efficien-
cies; that of the furnace, that of the boiler, and that of the engine.

The furnace wastes about 1/10 of the coal due to incomplete
combustion, through escape of unburnt gases up the smoke
stack, and unburnt coal into the ash pit. The furnace efficiency
is, therefore, about 9/10 or 90 per cent. About 4/10 of the heat
developed by the furnace escapes into the boiler room or up the
smoke stack; so that the boiler efficiency is about 6/10 or 60 per
cent. A good "condensing" engine converts into work about
1/5 of the heat energy furnished it by the boiler, in other words,
its efficiency is about 20 per cent. The total efficiency E of
the steam engine, which may be defined by the equation

work done

energy of fuel burned
has, then, the value 1/5X6/10X9/10, or about 11 per cent.

STEAM ENGINES AND GAS ENGINES 313

Calculation of Efficiency. — The efficiency of the steam engine
varies greatly with the care of the furnace, and the type and
size of the boiler and engine. Few engines have a total efficiency
above 12 per cent., and many of the smaller ones have as low as
4 or 5 per cent, efficiency, or even lower. Coal which has a heat
of combustion of 14,000 B.T.U. per lb., contains 14,000X778 ft.-
Ibs. of energy per pound. One H.P.-hr. is 3600X550 ft.-lbs.
,. , 3600X550 ., . . . t .. ., , .

Accordingly ^QQQ x 773 *"., or approximately 1/5 lb. of coal

would do 1 H.P.-hr. of work if the efficiency of the engine were
100 per cent. If an engine requires 4 Ibs. of coal per H.P.-
hr., its efficiency is approximately 1/4X1/5, or 5 per cent. In
order to make an accurate determination of the efficiency, the
heat of combustion would have to be known for the particular
grade of coal used.

Limiting, or Thermodynamic Efficiency. — Carnot (Sec. 236)
showed that the efficiency of an ideal engine, which, of course,
cannot be surpassed, is determined by the two extreme tempera-
tures of the working fluid (steam or gas) . If heat (say in steam)
is supplied to the engine at 127° C. or 400° A., and the engine
delivers it to the condenser at 27° C., or 300° A., then the maxi-

.. , ~ . . 400-300

mum theoretical efficiency is — TQ~ — , or 25 per cent. Ob-
viously, then, a gain in efficiency is obtained by using steam at
a very high pressure, and consequently at a high temperature.
The high efficiency of the gas engine is due partly to the great
temperature difference employed, and partly to the fact that
the "furnace" is in the cylinder itself, thereby reducing heat
"losses."

Some gas engines (Sec. 237) have more than 30 per cent, ef-
ficiency. Gas engines are usually more troublesome than steam
engines and also less reliable in their operation; nevertheless,
because of their greater economy of fuel, they are coming into
very general use.

The lightness of gas engines recommends them for use on
automobiles, motorcycles, and flying machines. Engines weigh-
ing about 2.5 Ibs. per H.P. have been made for use on aero-
planes. Indeed, the lightness of the gas engine has made
possible the development of the aeroplane.

229. The Steam Engine. — A modern steam engine, fully

314

MECHANICS AND HEAT

equipped with all of its essential attachments, is a very compli-
cated mechanism.

In order to bring out more clearly the fundamental principles
involved in the action of the steam engine, it seems best to omit
important details found in the modern engine, since these de-
tails are confusing to the beginner, and therefore serve to obscure
the underlying principles. In accordance with this idea, an ex-
ceedingly primitive engine is shown in Fig. 146. In Fig. 147,
an engine is shown which is essentially modern, although certain
details of construction are purposely omitted or modified, espe-
cially in the indicator mechanism, and in the valve mechanism.

In Fig. 146, A is a pipe which carries steam from the boiler to
the cylinder B, through either valve a or valve b, depending upon
which is open. P is the piston, and C is the piston rod, which

FIG. 146.

passes through the end of the cylinder (through steam-tight
packing in a "stuffing box") to the crosshead D. As the piston
is forced back and forth by the steam, as will be explained below,
the crosshead moves to and fro in "guides," indicated by the
broken lines. The crosshead, by means of the connecting rod
E attached to the crank pin F, causes the crank G to revolve as
indicated. The crank G revolves the crank shaft 0, to which is
usually attached a very heavy fly wheel H in order to "steady"
the motion.

If valves a and c are open, and b and d closed, the steam passes
from the boiler into the cylinder, and forces P to the right.
The exhaust steam to the right of P (remaining from a former
stroke) is driven out through c to the air. When P reaches the

STEAM ENGINES AND GAS ENGINES 315

right end of the cylinder, valves a and c are closed, and 6 and d
are opened, thus permitting steam to enter at 6 and force P to the
left end again; whereupon the entire operation is repeated.
These valves are automatically opened and closed at just the
right instant by a mechanism connected with the crank shaft
(Sec. 233). In practice, valve a would be closed when P had
traveled to the right about 1/3 the length of the cylinder (Sec.
231).

Speed Regulation. — A Centrifugal Governor, driven by the
engine, controls the steam supply, and hence the speed, by open-
ing wider the throttle valve (valve not shown) in A if the speed
is too low, and by partially closing it when the speed is too high,
as explained in Sec. 63. It may be mentioned that some gover-
nors control the speed by regulating the cut-off (Sec. 231); that
is, by admitting steam to the cylinder during a small fraction of
the stroke, in case the speed becomes too high.

Compound and Triple Expansion Engines. — In the Compound
Engine, the exhaust steam from cylinder B passes through pipe
/ to a second cylinder, where it drives the piston to and fro,
just as the steam from pipe A drives the piston shown in the
figure. If the exhaust steam from this second cylinder operates
a third cylinder we have a Triple Expansion Engine — so-called
because the steam expands three times. Obviously, because of
this expansion, the second cylinder must be larger than the first,
and the third larger than the second. By using steam at very
high pressure (about 200 Ibs. per sq. in.), and expanding it
successively in there different cylinders, a much higher efficiency
is obtained than with a single-cylinder engine. It will be evi-
dent, that the more the steam condenses on the walls of the
cylinder, the more rapidly its pressure drops with expansion. It
may be mentioned that the greater efficiency of the triple expan-
sion engine is due principally to a reduction of this condensation.

Superheating. — Another method of reducing condensation is
to superheat the steam. If the steam is conducted from the
boiler to the engine through coiled pipes surrounded by moder-
ately hot flame, it may thereby have its temperature raised as
much as 200° F., and is then said to be superheated 200°. Super-
heated steam does not so readily condense upon expansion in the
engine as does ordinary steam, and consequently gives a higher
efficiency.

Increasing the Efficiency. — The efficiency of the steam engine

316 MECHANICS AND HEAT

has been increased, step by step, by means of various improve-
ments, prominent among which are, the expansive use of steam
in the cylinder (Sec. 231), the expansion from cylinder to cylinder
as in triple expansion engines, and the condensation of the ex-
haust steam ahead of the piston (Sec. 230) to eliminate "back
pressure." To these may be added the use of higher steam pres-
sure, and also the use of superheated steam.

230. Condensing Engines. — It will be observed, that in the
above noncondensing engine, the steam from the boiler has to
force the piston against atmospheric pressure (15 Ibs. per sq. in).
By leading the exhaust pipe / to a "condenser," which condenses
most of the steam, this "back pressure" is largely eliminated.
The Condenser consists of an air-tight metal enclosure, kept cool
either by a water jet playing inside, or by cold water circulating
on the outside. The former is called the Jet Condenser and the
latter, the Surface Condenser. A pipe from an air pump leads to
the condenser, and by means of this pipe, the air pump removes
the water and air, maintaining in the condenser a fairly good
vacuum. Assuming that the boiler pressure is, say, 60 Ibs. per
sq. in. (i.e., 60 Ibs. per sq. in. above atmospheric pressure), and
that the condenser maintains in the cylinder, "ahead" of the
piston, a partial vacuum of 2 Ibs. per sq. in. pressure; it will be
evident that the available working pressure will be increased
to 73 Ibs. per sq. in. (15-2= 13, and 60+13 = 73), and that there-
fore the efficiency will be increased in about the same ratio.

231. Expansive Use of Steam, Cut-off Point. — If, when the
piston shown in Fig. 146 has moved to the right 1/4 the length
of the cylinder, i.e., when it is at 1/4 stroke, valve a is closed,
then only 1/4 as much steam will be used as would have been
used had valve a remained open to the end of the stroke. But the
work done by the piston during the stroke will be more than 1/4
as much in the first case as in the second, hence steam is econo-
mized. If a is closed at 1/4 stroke, the Cut-off Point is said to
be at 1/4 stroke.

The work done per stroke, if the valve a remains open during
the full stroke, is FL (or Fd, since Work=Fd), in which F is the
force exerted on the piston (its area A times the steam pressure
p), and L is the length of the stroke. Consequently, the work
done per stroke is pAL. If the cut-off is set at 1/4 stroke, then
the full pressure is applied for the first quarter stroke only, and
therefore the work done by the steam in this quarter stroke

STEAM ENGINES AND GAS ENGINES 317

is pAL/4. During the remaining 3/4 stroke, the enclosed steam
expands to four times as great volume, and because of the cool-
ing effect of expansion, it has its pressure reduced at the end of
the stroke to less than p/4, the value which Boyle's Law would
indicate. Assuming the average pressure during the last 3/4
stroke to be even as low as p/3, we have for the work of this 3/4
stroke

We see, then, that by using the expansive power of the steam
during 3/4 of the stroke, we obtain the work \ pAL, which,
added to | pAL obtained from the first 1/4 stroke, gives
\ pAL for the total work. But the work obtained per stroke by
keeping the valve a open during the full stroke was pAL. Hence
the total work per stroke, using the cut-off, is 1/2 as great as
without, and the steam consumption is only 1/4 as great; there-
fore, the Efficiency is doubled, in this instance, by the use of the
cut-off.

232. Power. — Since power is the rate of doing work, or, in
the units usually employed, the amount of work done per second
(Sec. 81), it will be at once evident that the product of the work
per stroke, or PAL (Sec. 231), and the number of strokes (to the
right) per second, or N, will give the power developed by the
steam, which enters at the left end of the cylinder. That is,
power —PALN, in which P is the average difference in pressure
upon the two sides of the piston during the entire stroke. The
average pressure is easily found from the indicator card (Sec.
234). As an aid to the memory, the symbols may be rearranged
so as to spell the word PLAN.

If P is expressed in pounds per square inch, and A in square
inches, then the average force PA exerted by the piston will be in
pounds. If L is the length of the cylinder in feet, then PAL, the
work done per stroke, will be expressed in foot-pounds. Finally,
since N, the number of revolutions per second, is also the number
of strokes to the right per second, the power, PLAN, developed
by the left end of the cylinder, is given in foot-pounds per second.
Dividing this by 550 gives the result in horse power; i.e.,

H.P. (one end) = (87)

318 MECHANICS AND HEAT

If N represents the speed of the engine in revolutions per minute
(R.P.M.), then, since 1 H. P. -33000 ft.-lbs. per min., we have

H.P. (one end) = (87a)

If the cut-off point for the stroke to the left does not occur
at exactly the same fraction of the stroke as it does for the
stroke to the right, then the average pressure pushing the pis-
ton to the left will not be the same as that pushing it to the right,
and the power developed by the right end of the cylinder will
differ from that developed by the left end. This difference usu-
ally amounts to but a few per cent, of the total power.

233. The Slide Valve Mechanism.— The slide valve V (Fig.
147) is operated by what virtually amounts to a crank of length
00', called, however, an Eccentric. The eccentric consists of a
circular disc «/. whose center is at 0', attached to the crank shaft
whose center is at 0. Over J passes the strap K connected with
the eccentric rod L. As the crank shaft revolves clockwise, 0',
which is virtually the right end of rod L, moves in the small
dotted circle as indicated. This circular motion causes L to
move to and fro, thus imparting to the slide valve V a to-and-fro
motion. By adjusting the eccentric until the angle between
00' and the crank G has the proper value, the valve opens and
closes the ports at the proper instants.

With the valve in the position shown, the steam from the
boiler, entering the steam chest S through pipe A, passes through
steam port a into the cylinder. The exhaust steam, from the
preceding stroke, escapes through steam port 6 and exhaust port
c into the exhaust pipe, which conducts the steam in the direc-
tion away from the reader to the condenser (not shown). An
instant later, port a will be closed (cut-off point), and the steam
then in the left end of the cylinder will expand (expansion period,
Sec. 231) and push, the piston to the right. As the piston ap-
proaches the right end, the valve V will close port b and at the
same time open port a into the exhaust port c. This releases the
steam in the left end of the cylinder, and is called the release point.
Since b is closed before the piston reaches the right end of its
stroke, there still remains some exhaust steam in the right end of
the cylinder. This steam acts as a "cushion" and reduces the jar-
ring. During the last part of the stroke, then, the piston is
compressing exhaust steam. This is called the compression

STEAM ENGINES AND GAS ENGINES

319

period. About the time the piston reaches the right end of its
stroke, valve V has moved far enough to the left to open port
b to the steam chest, thus admitting "live" steam to the right
end of the cylinder, and the return stroke, similar in all respects
to the one we have just described, occurs.

234. The Indicator. — The essentials of the indicator are shown
in Fig. 147 (left upper corner). / is a small vertical cylinder
containing a piston N, and is connected by pipes with the ends
of the engine cylinder, as shown. If valve e is closed and valve
d is open, it will be evident that, as the pressure in the left end
of the cylinder rises and falls, the piston N, which is held down
by the spring s, will rise and fall, and cause the pencil p at the end
of the lever Q to rise and fall.

M is a drum, to which is fastened a "card" W. This drum is
free to rotate about a vertical axis when the cord T, passing over
pulley U, is pulled to the right. As the pull on T is released, a
spring (not shown) causes the drum to rotate in the reverse
direction.

It will therefore be seen that the to-and-fro (horizontal)
motion of the crosshead D, by means of lever R and string T,
causes the drum to rotate to and fro, and consequently move
the card to and fro under the pencil p. If the pencil were station-
ary it would trace a straight horizontal line on the card.

320 MECHANICS AND HEAT

Thus we see that the change of pressure in the cylinder causes
the pencil to move up and down, while the motion of the drum
causes the card to move horizontally under the pencil. In prac-
tice, both of these motions take place simultaneously, and the
pencil traces over and over the curve shown. It will be seen
that the motion of the card under the pencil exactly reproduces,
on a reduced scale, the motion of the piston and crosshead.
That is to say, when the piston P has moved to the right, say 1/4
the length of the cylinder, or is at "quarter stroke," the pencil
p is 1/4 way across the indicator card, and so on.

The indicator card is shown, drawn to a larger scale, in the
upper, right corner of Fig. 147. At the instants that the piston,
in moving to the right, passes points 1, 2, 3, 4, the pencil p
traces respectively, the corresponding points 1, 2, 3, 4, on the
indicator card. As the piston moves back to the left from
4 to 5, pencil p traces from 4 to 5 on the curve. The indicator
card shows that full steam pressure acts on P during its motion
from 1 to 2; that at 2 the inlet valve at the left closes (i.e., cut-
off occurs, see slide valve, Sec. 233) ; and that the pressure of
the enclosed steam, as it expands and pushes the piston through
the remainder of the stroke, decreases, as indicated by the points
2, 3, and 4.

As the piston, on the return stroke, reaches the point marked
5, port a is closed and the compression period (Sec. 233) begins.
This is shown on the indicator diagram by the rounded corner
at 5. At the point marked 6, steam is again admitted through
port a, and the pencil p rises to point 1 on the diagram. The
different periods shown on the indicator diagram are, then, ad-
mission of steam from 1 to 2, expansion from 2 to 4, exhaust
from 4 to 5, and compression from 5 to 6.

If the back pressure of the exhaust steam were entirely elimi-
nated by the condenser, the pencil on the return stroke would
trace a lower line than 4-5, say, ii' . The distance j is then a
measure of the back pressure, which would be about 2 or 3
Ibs. per sq. in. when using a condenser, and about 15 Ibs. per sq.
in. without a condenser.

To obtain the indicator diagram for the other end of the
cylinder (shown in broken lines in the figure), valve d is closed
and valve e is opened. This curve should be (frequently it is
not) a duplicate of the curve just discussed, in the same sense
that the right br\nd is a duplicate of the left.

STEAM ENGINES AND GAS ENGINES 321

Use of the Indicator Card. — The indicator card enables the
operator to tell whether the engine is working properly; e.g.,
whether the admission or the cut-off are premature or delayed,
requiring valve adjustment; or whether or not the "back pres-
sure" is excessive due to fault of the condenser, and so on.

Another use of the indicator card is in determining the average
working pressure which drives the piston. By subjecting the
indicator piston to known changes of pressure as read by a steam
gauge, we may easily determine how many pounds pressure per
square inch corresponds to an inch rise of the pencil p. Having
thus calibrated the indicator, suppose we find that an increase of
40 Ibs. per sq. in. causes p to rise 1 in. Let the vertical dotted
line through 3 across the indicator curve be 1 . 5 in. in length. We
then know that at 1/2 stroke the available working pressure on
the piston, or the difference between the pressure on the left and
the exhaust pressure on the right side of the piston, is 60 Ibs.
per sq. in. Further, suppose that when we divide the total area
of the curve by its horizontal length we obtain 2 in. for its
average height. We then know that the average working pressure
P for the entire stroke is 80 Ibs. per sq. in. This average value
of p, thus found, is the P of Eq. 87, which gives the horse power
(H.P.) of the engine.

Since the average height of the indicator diagram gives the
average working pressure on the piston, and since its length is
proportional to the length of the stroke of the piston, we see that
its area is proportional to, and is therefore a measure of, the
work done per stroke, and hence a measure of the power. Ac-
cordingly, any adjustment of valves or other change which in-
creases this area without altering the speed, produces a propor-
tional increase in power. If, further, the same amount of steam
is used as before, then there is a proportional increase in efficiency.

235. The Steam Turbine. — In recent years, some large and
very efficient steam turbines have been installed. Because of
their freedom from jarring, which is so great in the reciprocating
steam engines, and also because of their high speed, they are
being used more and more for steamship power.

In the steam turbine, a stream of steam impinges against
slanting vanes and makes them move just as air makes windmill
vanes move (Sec. 149). It differs from the windmill, however,
in that the stream of steam must be confined, just as water is in
the water turbine. Note that the windmill might be called an
21

322

MECHANICS AND HEAT

air turbine. The steam turbine differs from the windmill also
in that each portion of steam must pass successively several
movable vanes alternating with fixed vanes, as indicated in
Fig. 148. The rotor vanes, attached to the rotating part called
the rotor, are indicated by heavy curved lines. The stator vanes
are stationary and are attached to the tubular shell which sur-
rounds the rotor and confines the steam. The stator vanes are
indicated in the sketch by the light curved lines. It will be
understood that the reader is looking toward the axis of the
rotor, which is indicated by the horizontal line.

As the steam passes to the right, the fixed vanes deflect
it somewhat downward, and the movable vanes, somewhat up-

FIG. 148.

ward, as indicated by the light arrows. The reaction to this up-
ward thrust exerted upon the steam by the movable vanes causes
these vanes to move downward (as explained in connection
with Fig. 100, Sec. 149, and as indicated by the heavy arrows)
with an enormous velocity, and with considerable force.

To allow for the expansion of the steam, the above-mentioned
tubular shell increases in diameter to the right, and the rotor
vanes increase in length to the right. The stator vanes are also
longer at the right.

If the steam, as it passes to the right from the turbine, enters
a condenser, the effective steam pressure and likewise the ef-
ficiency, will be increased just as is the case with the reciprocating
^team engine.

STEAM ENGINES AND GAS ENGINES

323

236. Carnot's Cycle. — Nearly a century ago, the French physicist,
Sadi Carnot, who may be said to have founded the science of thermody-
namics, showed by a line of reasoning in which he used a so-called "ideal
engine" (Fig. 149), that by taking some heat HI, from one body and giv-
ing a smaller amount H 2, to a colder body, an amount of heat Hi—H2
may be converted into work, and that the percentage of the heat that
may be so converted depends only upon the temperatures of the two
bodies.

In Fig. 149 (Sketch I-II), let a cylinder with non-conducting walls,
a non-conducting piston, and a perfect conducting base in contact with

7-77

III-IV

FIG. 149.

the perfect conducting "source" S, contain some gas at a temperature T\.
(Parts that are perfect non-conductors of heat are shown crosshatched.)
Let the gas be a perfect gas, i.e., one which obeys Boyle's law and
Charles's law. Let 7 be a perfectly non-conducting slab; R, the perfectly
conducting "refrigerator," and let S be kept constantly at the tempera-
ture TI, and R, at the temperature T2 on the absolute scale.

We shall now put the gas through four different stages, I, II, III, and
IV. In Fig. 149, we shall indicate the four processes of changing from
stages I to II, II to III, III to IV, and from IV back to I, by the four
sketches marked respectively, I-II, II-III, III-IV, and IV-I. The pis-

324

MECHANICS AND HEAT

ton, in the four stages, assumes successively the positions A, B, C, and
D, and the corresponding pressures and volumes of the gas are indicated,
respectively, by the points A, B, C, and D on the pressure- volume
diagram (Fig. 150).

Process 1: As the gas is permitted to change from stage I to II
(sketch marked I-II) by pushing the piston from A to B, it does work
on the piston (force X distance or pressure X volume, Sec. 203), and
therefore would cool itself were it not in contact with the perfect con-
ductor S. This contact maintains its temperature at TV A., i.e., the
gas takes an amount of heat, say HI, from source S, and its expansion is

L Q M N

Volume

FIG. 150.

represented in Fig. 150 by the portion AB of an isothermal. Since work
is the product of the average pressure and the change in volume LM
(Sec. 203), we see that the work done by the gas is proportional to, and
is represented by, the shaded area ABML.

Process 2: The cylinder is next placed on the non-conducting slab 7,
and the gas is permitted to expand and push the piston from B to C. In
this process (sketch II-III), since the gas is now completely surrounded
by non-conductors of heat, the work of expansion is done at the
expense of the heat of the gas itself, and its temperature is thereby lowered.
Consequently, as the volume increases, the pressure decreases more
rapidly than for the previous isothermal expansion. In case the energy
(heat) of expansion must come from the gas itself, as in this instance,
the expansion is Adiabatic. AB is an isothermal line and BC is an Adia-
latic line. The gas is now at stage III, and the work done by the gas

STEAM ENGINES AND GAS ENGINES 325

in expanding from B to C is represented by the area BCNM which lies
below the curve BC (compare Process 1).

Process 3: The cylinder is next placed upon the cold body or "refrig-
erator" R (sketch III-IV), and the gas is compressed from C to D.
Since R is a perfect conductor, this will be an isothermal compression,
and, as the volume is slowly reduced, the pressure will gradually increase
as represented by the isothermal CD. The gas is now at stage IV, and
is represented by point D on the diagram. The work done upon the
gas in this process is, by previous reasoning, represented by the area
CDQN. The work of compressing the gas develops heat in it, but this
heat, say HI, is immediately given to the refrigerator.

Process 4: Finally, the cylinder is again placed upon the non-conduct-
ing slab 7, and the piston is forced from D back to the original position
A. Since the gas is now surrounded by a perfect non-conductor, the
heat of compression raises its temperature to T\. As the volume is
gradually decreased, the pressure increases more rapidly than before, be-
cause of the accompanying temperature rise, which accounts for the fact
that DA is steeper than CD. In this case, of course, we have an Adia-
batic Compression and the line DA is an adiabatic line. The work
done upon the gas in this process is represented by the area DALQ.

Efficiency of Carnot's Cycle. — From the preceding discussion,
we see that the work done by the gas during the two expansions
(Processes 1 and 2) is represented by the area below ABC;
while the work done upon the gas during the two compressions
(Processes 3 and 4) is represented by the area below ADC.
Consequently, the net work obtained from the gas is represented
by the area A BCD.

It has just been shown (Process 1) that the gas as it ex-
pands from A to B, does work represented by the area ABML,
and since its temperature remains constant, it must take from
the source S, an amount of heat energy equal to this work.
Let us call this heat Hi. Similar reasoning shows that when com-
pressed from C to D, the gas gives to the refrigerator an amount
of heat Hz represented by the area CDQN. During the other two
processes (adiabatic processes) the gas can neither acquire nor
impart heat. Accordingly, for this cycle, the efficiency is given
by the equation

F_ work done _ ABC D _ ABML— CDQN ( >,_#!— #?

' "heat received ~ ABML " ABML Hl

(88)

Now, the heat contained by a gas, or any other substance, is

326 MECHANICS AND HEAT

proportional to the temperature of the substance (assuming that
the body has a constant specific heat). Consequently,

This equation shows (as mentioned in Sec. 228) that if the ab-
solute temperature TI of the "live" steam as it enters the
cylinder from the boiler is 400° A. and the temperature T2 of
the condenser is 300° A., then the maximum theoretical efficiency

. . 400-300 oe
of the engine is — TQX — or 25 per cent. For a rigorous, and

more extended treatment of this topic consult advanced works.

237. The Gas Engine — Fuel, Carburetor, Ignition, and Gover-
nor. — In the gas engine, the pressure which forces the piston along
the cylinder is exerted by a hot gas, instead of by steam as in the
case of the steam engine. The gas engine also differs from the
steam engine in that the fuel, commonly an explosive mixture
of gasoline vapor with air, is burned (i.e., explosion occurs)
within the cylinder itself. For this reason, no furnace or boiler
is required, which makes it much better than the steam engine
for a portable source of power. Gas engines may be made very
light in proportion to the power which they will develop. The
weight per H.P. varies from several hundred pounds for station-
ary engines, to 10 Ibs. for automobiles. As has already been
mentioned, the lightness of the gas engine (as low as 2.5 Ibs.
per H.P. for aeroplanes) has made aeroplane flight possible.

The fact that a gas engine may be started in an instant (i.e.,
usually), and that the instant it is stopped the consumption of
fuel ceases, makes it especially adapted for power for automo-
biles, or for any work requiring intermittent power. The fact
that the power can be instantly varied as required is also a point
in its favor.

Fuel. — Gasoline is the most widely used fuel for gas engines.
It is readily vaporized, and this vapor, mixed with the proper
amount of air as it is drawn into the cylinder, is very explosive
and is therefore readily ignited. Complete combustion is
easily obtained with gasoline; so that it does not foul the cylinder
as some fuels do. Kerosene is much less volatile than gasoline,
but may be used after the cylinder has first become heated by the
use of gasoline. Alcohol may also be used. Crude Petroleum
is used in some engines. Illuminating Gas, mixed with air, may

STEAM ENGINES AND GAS ENGINES 327

be used as a fuel. Natural gas, where available, forms an ideal
fuel, and is used in some large power plants. The use of "Pro-
ducer" Gas requires considerable auxiliary apparatus, but be-
cause of its cheapness, it is profitably used by stationary engines.

Briefly, producer gas is formed by heating coal while re-
stricting the air supply, so that the carbon burns to carbon
monoxide (CO) which is a combustible gas, instead of to carbon
dioxide (CO2), which is incombustible. If some steam (H2O)
is admitted with the air, the steam is decomposed into oxygen
(O) which combines with the carbon and forms more carbon
monoxide (CO). The remaining hydrogen constituent (H) of
the steam is an excellent fuel gas. All of these gases pass from
the coal through various cooling and purifying chambers, either
directly into the gas engine, or into a gas tank to be used as
required.

The Carburetor. — The carburetor is a device for mixing the
vapor of the gasoline, or other liquid fuel, with the air which
passes into the cylinder, thus forming the "charge." The ex-
plosion of this charge develops the pressure which drives the
piston. As the air being drawn into the engine rushes past a
small nozzle connected with the gasoline supply (see C, Fig.
153, left sketch), the gasoline is "drawn" out of the nozzle (see
atomizer, Sec. 156) in the form of a fine spray, which quickly
changes to vapor, and is thereby thoroughly mixed with the air
to form the "charge." This thorough mixing is essential to
complete combustion. If kerosene is used, the air must be pre-
viously heated in order to vaporize the spray. It is well to pre-
heat the air in any case.

Ignition. — The charge is usually ignited electrically, either
by what is called the "jump spark" from an induction coil, or
by the " make-and-break " method. An induction coil consists
of a bundle of iron wires, upon which is wound a layer or two of
insulated copper wire, called the primary coil. One end of this
primary coil is connected by a wire directly to one terminal of
a battery, while the other end is connected to the opposite ter-
minal of the battery through a vibrator or other device, which
opens and closes the electrical circuit a great many times per
second. On top of the primary coil, and, as a rule, carefully in-
sulated from it, are wound a great many turns of fine wire, called
the secondary coil. When the current in the primary circuit is
broken, a spark will pass between the terminals of the second-

328 MECHANICS AND HEAT

ary, provided they are not too far apart. The "spark distance"
of the secondary varies from a small fraction of an inch to several
feet, depending upon the size and kind of induction coil. For
ignition purposes, only a short spark is required. By means of a
suitable mechanism, this spark is made to take place between
two points in the " spark plug" (B, Fig. 153) within the cylinder
at the instant the explosion should occur.

In the "make-and-break" method of ignition, neither the
secondary nor the vibrator is needed. One terminal of the
primary coil, which, with its iron wire "core," is called a "spark
coil," is connected directly to the firing pin which passes through
a hole into the cylinder. The other terminal of the primary is
connected to one pole of a battery. From the other pole of the
battery a wire leads to a metal contact piece which passes into
the cylinder from which it is insulated, at a point near the firing
pin. By means of a cam, this firing pin is made to alternately
touch and then move away from the metal contact piece within
the cylinder. Consequently, by proper adjustment of the cam,
the circuit is broken by the firing pin and the gas is ignited at
the instant the explosion is desired. If the spark occurs when
the piston is past dead center it is said to be retarded, if before,
advanced. Engines running at very high speed require the spark
to be advanced, or the flame will not have time to reach all of
the gas until rather late in the stroke. The indicator card will
tell whether or not advancing the spark increases the power in
a given instance. If the spark is advanced too far "back-firing"
results, with its attendant jarring and reduction of power.

The electric current may be produced by a "magneto." The
magneto generates current only when the engine is running; so
that a battery must be used when starting the engine, after which,
by turning a switch, the magneto is thrown into the circuit and
the battery is thrown out.

Cooling. — To prevent the cylinder from becoming too hot, a
"water jacket" is provided. The cylinder walls are made
double, and the space between them is filled with water. This
water, as it is heated, passes to the "radiator" and then returns
to the water jacket again. The water circulation is maintained
either by a pump, or by convection. The radiator is so con-
structed, that it has a large radiating surface. A fan is frequently
used to cause air to circulate through the radiator more rapidly.
In some automobiles air cooling is used entirely, the cylinder

STEAM ENGINES AND GAS ENGINES 329

being deeply ribbed so as to have a large surface over which the
air is forced in a rapid stream.

The Governor. — Commonly some form of the Centrifugal Gover-
nor (Sec. 63) is used to control the speed. In the " hit-or-miss "
method no "charge" is admitted when the speed is too high.
This causes fluctuations in the speed which are readily noticeable.
In other methods of speed control, either the quantity of "rich-
ness" (proportion of gas or gasoline vapor to air) of the charge
is varied to suit the load. If the load is light, the governor re-
duces the gas or gasoline supply; or else it closes the intake
valve earlier in the stroke, thereby reducing the quantity of the
charge.

238. Multiple-cylinder Engines. — With two-cycle engines
(Sec. 240), an explosion occurs every other stroke; while in the
four-cycle engine (Sec. 239) explosions occur only once in four
strokes (i.e., in two revolutions). It will be seen that the applied
torque is quite intermittent as compared with that of the steam
engine. If an engine has six cylinders, with their connecting
rods attached to six different cranks on the same crank-shaft,
then, by having the cranks set at the proper angle apart and by
properly timing the six different explosions, a nearly uniform
torque is developed. The six-cylinder engine is characterized
by very smooth running. The four-cylinder engines, and even
the two-cylinder engines, produce a much more uniform torque
than the single-cylinder engines.

239. The Four-cycle Engine. — In the so-called four-cycle en-
gine, a complete cycle consists of four strokes, or two revolu-
tions. The four strokes are, suction or charging, compression,
working, and exhaust. The stroke, at the beginning of which
the explosion occurs, is the working stroke. With this engine,
every fourth stroke is a working stroke; whereas, in the steam
engine, every stroke is a working stroke. The operation of this
engine will be understood from a discussion of Fig. 151. In
the upper sketch, marked I (Fig. 151), valve a is open and valve
6 is closed, so that as the piston moves to the right the "suction"
draws in the charge from the carburetor. This is the charging
stroke. On the return stroke of the piston (Sketch II), both
valves are closed and the charge is highly compressed.

As the piston reaches the end of its stroke, the gas then oc-
cupying the clearance space, or "combustion chamber," is ignited
by means of either the " firing pin " c or a " spark plug," depending

330

MECHANICS AND HEAT

upon which method of ignition is used. Ignition may occur
either at, before, or after "dead center." (See Ignition, Sec.
237.) The "explosion," or the burning of the gasoline vapor,
produces a very high temperature and therefore, according to

FIG. 151.

FIG. 152.

the law of Charles, a very high pressure. This high pressure
pushes the piston to the right. This stroke is called the working
stroke (Sketch III). As the piston again returns to the left,
valve b is open, and the burned gases escape. This is the exhaust

STEAM ENGINES AND GAS ENGINES

331

stroke. The exhaust is very noisy unless the exhaust gases are
passed through a muffler.

In many engines there is no piston rod, the connecting rod
being attached directly to the piston as shown. The valves
are operated automatically by cams, or other devices connected
with the crank shaft so that by proper adjustment, exact timing
may be obtained.

Indicator Card. — An indicator mechanism may be connected
with the cylinder just as with the steam-engine cylinder (Sec. 234).
In Fig. 152, is shown the indicator card from a four-cycle engine.
The line marked 1 shows the pressure corresponding to the
charging stroke (Sketch I, Fig. 151). Line 2 shows the pressure
during the compression stroke (Sketch II, Fig. 151). At point

FIG. 153.

e, the explosion has occurred, and the pressure has reached a
maximum. Line 3 represents the pressure during the working
stroke (Sketch III), showing how it varies from the maximum
down to /. Line 4 shows the pressure during the exhaust stroke
(Sketch IV, Fig. 151).

240. The Two-cycle Engine. — The operation of the two-cycle
engine will be understood from a discussion of Fig. 153. As
the piston moves upward, compressing a previous charge, it
produces suction at port a (left sketch), and draws in the charging
gas from the carburetor C into the crank case A, which is air-
tight in this type of engine. As the piston reaches the top of its

332 MECHANICS AND HEAT

stroke, the charge is ignited by the spark plug B, and explosion
occurs. As the piston now descends it is driven, with great force,
by the high pressure of the heated gases. This is the working
stroke. As soon as the piston passes below the exhaust port
6 (right sketch) the exhaust gas escapes, in part. An instant
later, the piston is below port c, and part of the gas in the crank
case, which gas is now slightly compressed by the descent of
the piston, rushes through port c. As this charge enters port
c, it strikes the baffling plate D, which deflects it upward, thus
forcing most of the remaining exhaust gas out through port 6.
As the piston again rises, it compresses this charge preparatory
to ignition, and the cycle is completed.

PROBLEMS

1. If all of the energy developed by a mass of iron in falling 778 ft. is
used in heating it, what will be its temperature rise?

2. If the complete combustion of 1 Ib. of a certain grade of coal develops
13,000 B.T.U.'s of heat, how much work (in ft.-lbs.) would it perform if
it is used in a heat engine of 10 per cent, efficiency?

3. How many H.P.-hours of potential energy does a pound of coal
(13,500 B.T.U.'s per Ib.) possess, and how many H.P.-hours of work can a
good steam engine (say of 12.5 per cent, efficiency) obtain from it? Note
that one horse-power for one second is 550 ft.-lbs.

4. How long would a ton of coal, like that mentioned in Problem 2, run a
10-H.P. steam engine of 6 per cent, total efficiency?

6. How high would the heat energy (14,000 B.T.U.'s per Ib.) from a given
mass of coal lift an equal mass of material, if it were possible to convert all
of the heat of the coal into mechanical energy?

6. Find the H.P. of a noncondensing steam engine supplied during full
stroke with steam at 80 Ibs. per sq. in. pressure (80 Ibs. is the available
working pressure), when making 120 R.P.M. (4 strokes per sec.) ; the length
of stroke being 2 feet and the cross section of the piston being 150 sq. in.

7. How many pounds of water at 70° F. will be changed to steam at 212°
F. for each pound of coal (Prob. 2) burned in a furnace of 90 per cent,
efficiency, heating a boiler of 70 per cent, efficiency.

8. An engine whose speed is 150 R.P.M., has a piston 15 in. in diameter
which makes a 2-ft. stroke. The indicator diagram is 4 in. long and has an
area of 9 sq. in. The indicator spring is a ''50-lb. spring," i.e., a rise of
1 in. by the indicator pencil indicates a change in pressure of 50 Ibs. per sq.
in. What is the power of the engine?

9. Find the. H. P. of the engine (Prob. 6) with cut-off set at one-quarter
stroke, the average pressure during the remaining 3/4 stroke being 30 Ibs.
per sq. in.

10. Find the H.P. of the engine (Prob. 6) with cut-off at half stroke, the
pressure during the last half of the stroke being 30 Ibs. per sq. in.

STEAM ENGINES AND GAS ENGINES 333

11. How many B.T.U.'s will a 1/2-oz. bullet develop as it strikes the target
with a velocity of 1800 ft. per sec.? If this heat were all absorbed by the
bullet (lead) what would be its temperature rise?

12. What is the limiting theoretical efficiency (thermodynamic efficiency)
of a steam engine whose boiler is at 180° C., and whose condenser is at
50° C.?

INDEX

The numbers refer to pages.

Absolute temperature scale, 237

zero, 236

Absorption of heat, 297
Accelerated motion, uniform, 26, 28
Accelerating force, 26, 49, 50, 51
in circular motion, 72
in free fall, 35
in simple harmonic motion,

83,84
Accelerating torque, 66, 68

equation for, 67
Acceleration, angular, 62

with Atwood's machine, 41
of gravity, 35

variation of, 35
linear, 25, 29

and angular compared, 63
radial, 73
in simple harmonic motion, 83,

uniform and nonuniform, 26, 29
Action and reaction, 49
applications of, 51
Actual mechanical advantage, 111
Addition of vectors, 12
Adhesion and cohesion, 141
fish glue for glass, 142
Adiabatic, compression and expan-
sion, 324
line, 324

and isothermal processes, 324
Air compressor, 201
Air friction, on air, 177, 178

effect on falling bodies, 36
on meteors, 181
on projectiles, 46
Air, liquefied and frozen, 278
liquid, 278, 279, 280

properties and effects of,
281

Air pump, mechanical, 200

mercury, 201
Alloys, melting point, 255
Altitude by barometer, 187
Amalgams, 156
Ammonia, 156

refrigerating apparatus, 272
Amplitude, 87, 293
Andrews, work on critical tempera-
ture, 273

isothermals of carbon dioxide,

274

Aneroid barometer, 186
Angle of elevation, 47

of shear, 152

unit of, 62

Angular acceleration and velocity,
62

and linear velocity and accel-
eration compared, 63

measurement, 62

velocity, average, 63
Antiresultant force, 16
Aqueous vapor, pressure of, 262, 304
Archimedes' principle, 163

application to gases, 182

and floating bodies, 165

experimental proof of, 164
Army rifle, range and velocity of

projectile, 46
Artificial ice, 272
Aspirator, or filter pump, 209
Atmosphere, composition of, 180

height of, 181

moisture of, 302

pressure of, 183, 184, 197, 199

standard, 185
Atomic heat, Dulong and Petit's

law, 246
Atomizer, 209

335

336

INDEX

Attraction, gravitational, 30
Atwood's machine, 41
Avogadro's law, 180
Axis of rotation, 23

Balance wheel, of watch, tempera-
ture compensation of, 233
Balanced columns, density by, 162

forces, so-called, 51
Ball and jet, 212

bearings, 102
Ballast, use and placing of, in ships,

127
Ballistic pendulum, and velocity of

rifle bullet, 55

Balloon, lifting capacity of, 183
Barometer, aneroid, 186

mercury, 184

uses of, 187
Barometric height, 185
Baseball, curving of, 213
Beam balance, 127, 128, 129
Beams, horizontal, strength and

stiffness of, 150
Bearings, ball, 102

roller, 103

babbitt in, 101
Beats, in sound, 293
Belt speed and angular speed, 64
Bernoulli's theorem, 209, 210, 211
Black body radiation, 297
Block and tackle, 115
Blood, purification of, 158
Blowers, rotary, 203
Boiler explosions and superheating,
266

"scale," 287
Boiling, 261
Boiling point, defined, 262

at high altitudes, 264

effect of dissolved substance on,
262

effect of pressure on, 262

tables of, 262
Boyle's law, 179, 187, 192, 317

deviation from, 277

and kinetic theory, 188
Brake, Prony, 106, 107
Breaking stress, 149

British system of units, 2

thermal units, or B.T.U., 218,

243, 311
Brittleness, 144
Brownian motion, 139
Bulk or volume modulus, 152
Bullet, determination of velocity, 55

velocity at different ranges, 46
'"Bumping," due to superheating of

water, 265
Buoyancy, center of, 166

of gases, 182

of liquids, 162
Buoyant force, 162

Cailletet, liquefaction of gases, 278
Calibration of thermometer, 223
Caliper, micrometer, 7

vernier, 5

Calms, zone of, 305
Caloric theory of heat, 218
Calorie, 243
Calorimeter, Bunsen's ice, 251

Joly's steam, 252

water equivalent of, 244
Calorimetry, 243

Camphor, effect of, on surface ten-
sion, 173
Canal boat, discussion of inertia

force, 51

Cannon, "shrinking" in construc-
tion of, 228

Capacity, thermal, 244
Capillarity, 173

Capillary rise, in tubes, wicks and
soils, 174

tubes, 174

Car and hoop on incline, 98
Carbon dioxide, cooling effect of, 270

isothermals of, 274, 276

"snow," 271
Carburetor, 327

Card and spool experiment, 213
Carnot, Sadi, French physicist, 323

cycle, 323, 324

efficiency of, 325

Carnot's "ideal" engine, 313, 323
Cascade method of liquefying gases,
279

INDEX

337

Castings, when clear-cut, 256
Cavendish, gravitational experiment

of, 30

Center of buoyancy, 166
of gravity, 122

effect on levers, 123
of mass, 124
of population, 124
Centigrade scale of temperature, 224
Centimeter, denned, 4
Centimeter-gram-second (C. G. S.)

system, 4
Central force, 72

radial, 75

Centrifugal blowers, 203
cream separator, 76
dryer, 73
force, 72

effect on shape of earth, 73
practical applications of, 73,

76, 77, 79
governor, 79, 315
pump, 204
Centripetal force, 72
Chain hoist, 121
Change of state, 219, 220, 250
Charles' law, 236
Chemical hygrometer, the, 303
Choke damp, 181
Circular motion, acceleration radial

in, 75

uniform, 72

Circulation of air due to stove, 284
Clepsydra, 10
Clinical thermometer, 225
Clock, essentials of, 9
Clouds, height, character and veloc-
ity, 302
Coefficient of cubical expansion, 234

table, 235
of friction, 101

determination of, 101
limits maximum pull of loco-
motive, 102
of linear expansion, 229

differences in, and applica-
tions of, 230-234
table of, 230
Cohesion, 141

Cold denned, 219

produced by evaporation, 268
by expansion of gas, 246, 278,

280
Combustion, defined, 248

heats of, table, 249
Compensated balance wheel, 233

pendulum, 234
Components of forces and velocities,

19, 20
Compressibility of gases, 178, 179

of water, 155, 165
Compressor, air, 201
Compound lever, 130
Condenser, jet, 316

surface, 316

Condensing steam engine, 316
Conditions of equilibrium, the two,

Conduction of heat, 286
Conductivity, thermal, 288

table, 289

Cone, equilibrium of, 126
Conservation of energy, 93, 210, 251

of mass, 139

of matter, 139

momentum, three proofs of, 53,

Convection, 283, 285
Conversion of units, 4
Cooling effect of evaporation, 268,

270

of internal work, gases, 278
Cooling, Newton's law of, 297

Stefan's law of, 297
Cornsheller, fly wheel on, 69
Couple, the, 61
Crane, the, 17
Cream separator, the, 76
Crew, Henry. See Preface.
Critical temperature, and critical
pressure, 273

simple method of determining,
277

table of, 274

Cubical expansion, coefficient of, 234
Curves, plotting of and use, 48

elevation of outer rail at, 77
Curving of baseball, 213

338

INDEX

Cut-off point, steam engine, 316
controlled by governor, 315

Cyclones, 306

cause of rotary motion, 307

d'Alembert, principle of, 49, 51

Davy's safety lamp, 287

Day, the siderial and mean solar, 3

"Dead air" space in buildings, 287

Density, defined, 139
of earth, average, 30
of liquids by balanced columns,

161

of solids, liquids and gases, 140
not specific gravity, 166
of some substances, table of, 140
of water, maximum, 255

Deserts, cause of, 305

Dew, 302

point, 303
and frost, 303

Dewar flask or thermal bottle, 282
liquefaction of gases, 278

Dialysis, 158

Differential pulley, 121
Wheel and axle, 122

Diffusion of gases, 178, 180
of liquids, 156

Diminution of pressure in regions of
high velocity, 208

Disc fan, 203

Displacement, in simple harmonic

motion, 84
of a ship, 165

Dissipation of energy, 99

Distance, fallen in a given time, 40
law of inverse squares of, 31
either scalar or vector, 24
traversed in a given time, 41

Drains, flow in, 196

Driving inertia force, 51

work of, 90
torque, 69

Ductility, 144

Dufour, superheating of water, 256

Dulong and Petit's law, 246

Dynamometers, absorption and
transmission, 106

Dyne, the, 27, 36

Earth, atmosphere of, 180

attraction on the moon, 33

average density of, 30

path of, 3, 4, 32, 34

weight of, 30

Earth's rotation, effect on shape of,
73

effect on moving train, 307

and trade winds, 305
Ebullition and evaporation, 260
Eccentric, the, 318
Effects of heat, 219
Efficiency of Carnot's cycle, 325

of cream separator, 76

of gas engine, 313

of simple machines, 111, 112

of steam engine, boiler, and
furnace, 312

of steam engine, calculation of,

313

Efflux, velocity of, 196
Elastic fatigue, 145, 149

limit, 145

rebound, explanation of, 145
Elasticity, general discussion of, 142

of gases, 178

of shearing, or of torsion, 151,
152

of tension or of elongation, 146

of volume or of compression,
151, 152

perfect, 142

three kinds of, 151
Electric fan and windmill, 202

fire alarm, 231
Electrical effect of heat, 242
Elements and compounds, 138
Elevation of outer rail on a curve, 77
Elevator, hydraulic, 206
Energy, chemical, 218

conservation of, 93, 94

defined, 92

dissipation of, 99

heat, a form of, 217, 243, 244

kinetic, 92, 96

potential, 92, 95

of a rotating body, 96, 97, 98

of sun, 218

sources of, 218

INDEX

339

Energy, transformation of, 93, 94

transformation of involves work,
93, 94

units of, 95
Engineer's units of mass and force,

Equilibrant, 16

Equilibrium of rigid body, two con-
ditions of, 64

on inclined plane, 126

of rocking chair, 126

in vaporization, 266

of wagon on hillside, 127

stable, unstable, and neutral,

126

Erg, 90
Ether, the, 295

waves in, 291
Evaporation, cooling effect of, 268

and ebullition, 260
Evener, two-horse, 129
Expansibility of gases, 177, 179
Expansion, apparent, of mercury,
223

of solids, 230

and temperature rise, 221
Expansive use of steam, 316

Factor of safety, 149

Fahrenheit's thermometric scale, 224

Falling bodies, laws of, 38-48

maximum velocity in air, 36
Fan, two kinds, 203
Faraday, Michael, liquefaction of

gases, 278
''Film," width of, 172, 173

work in forming, 172
Fire alarm, electric, 231

damp, 181

syringe, 311

Fish glue, adhesion to glass, 142
Fleuss or Geryk pump, 201
Flight of aeroplane, 52

of birds when starting, 52
Floating bodies, 165

immersed, 164

Flow of liquids, gravitational, 196
Fluids, in motion, properties of, 194
Flux, use of, 141

Flywheel, bursting of, 75

calculation of, 98

design, 98

kinetic energy of, 97

speed regulation by, 98

use of, 97

Foot-pound and foot-poundal, 90
Force, accelerating, 26, 49, 50

"arm," levers, 114

buoyant, 162

central, 72

centrifugal, 72

centripetal, 72

defined, 26

driving inertia, 51

impulsive, 52

resisting, 110

resolution of into components
19, 101

of restitution in simple har-
monic motion, 83, 84, 85,
86

units of, 27, 36

working, 110

Forced draft, locomotive, 209
Forces, addition of, 11, 16

balanced, 16, 51

graphical representation of, 12

in planetary motion, 32

polygon of, 16 ,

resolution of, 19

torque due to, 60
Four-cycle gas engine, 329
Franklin, Benjamin, experiment on

boiling point, 263, 264
Freezing mixtures, 258, 259

point of solutions, 255

lowering of by pressure, 256
Friction, cause of, 100

beneficial effects of, 101

coefficient of, 101

head, 177, 194, 196

internal, 100

kinetic, 100

laws of, 100

of air on projectiles, 46

rolling, 102, 103

sliding, 99

static, 101

340

INDEX

Friction, useful, 101

work of, 90, 103, 104

produces heat, 99
Fulcrum, 113
Fundamental quantities, 1

units, 2

Furnace, efficiency of, 312
Fusion of alloys, 255

and change in volume, 256

heat of, 250

and melting point, 255

Gas, general law, 240

laws, summary of three, 239

thermometer, 226
Gases, compressibility of, 179
table of densities of, 140

diffusion of, 178, 180

general law of, 240

kinetic theory of, 179, 236

two specific heats of, 246

thermal conductivity of, 289

and vapors, distinction between,
277

average velocity of molecules,

180
Gas engine, 326

carburetor, 327

combustion chamber, 329

efficiency of, 313

four-cycle, 329

fuel, 326

governor, 329

ignition, 327

indicator card of, 331

"make-and-break" ignition, 328

multiple cylinder, 329

"richness" of charge, 329

six-cylinder, 329

spark plug, 328

two-cycle, 331

very light, for aeroplanes, 313

water jacket, 328

Gelatine film, adhesion to glass, 142
Geryk or Fleuss pump, 201
Geysers, 265

artificial, 266
Glaciers, explanation of motion, 257

location of, 258

Glaciers, origin of, 258

velocity of, 258
Gold filling of teeth, 141

foil, 144

Governor, the centrifugal, 79, 315
Gram mass, defined, 4

weight, defined, 36
Graphical method and vectors, 12

representation, of space passed
over by a falling body, 39
of force, 16
of velocity, 12, 13
Gravitation, Newton's laws of, 30

units of energy, why chosen, 95

universal, 30
Gravity, acceleration of, 35

center of, 122

flow of liquids, 196

pendulum, the simple, 86

separation of cream, 76
Gridiron pendulum, 234
Guillaume, 230
"Guinea and feather" experiment,

35
Gyroscope, 80, 81

Hardness, scale of, 144
Harmonic motion, simple, 82
Heat, absorption of, 297

a form of energy, 217, 243, 244

of combustion, 248

conduction of, 286

conductivity, 288

effects of, 219

evolution, 260

exchanges, Prevost's theory, 296

from electricity, 219

of fusion, 250

measurement of, 243

mechanical equivalent of, 244

nature of, 217

properties of water, 253

quantity, equation expressing,
245

radiation, general case of, 300
determining factors in, 296
through glass, 298

reflection, transmission, and ab-
sorption, by glass, 300

INDEX

341

Heat, sources of, 218

specific, 244

transfer, three methods of, 283

units, calorie and B.T.U., 243

uphill flow of, 273, 312

of vaporization, 250

applications, 269, 270, 271

278, 279, 284
Heating system, hot-air, 283

hot-water, 284

steam, 285

High altitudes, boiling point at, 261
" Hit-or-miss " governor, 329
"Holes" in the air, aeroplane, 52
Hooke's law, 147

Hoop, kinetic energy of translation
and rotation are equal, 98
Horizontal beams, strength of, 150
Horse power, of engines, 106, 107

French, 105

hour, 106

value of, 105
Hotbed, the, 299
Hot-water heating system, 284
Hourglass, the, 10
Hurricanes and typhoons, 308
Hydraulic elevator, 206

press, 206

ram, 207
Hydraulics, general discussion, 194,

195

Hydrogen thermometer, 227
Hydrometers, 167
Hydrostatic paradox, 161

pressure, 158
Hygrometer, chemical, 303

wet-and-dry-bulb, 269, 303
Hygrometric tables, 304
Hygrometry, 303

Icebergs, origin of, 258

flotation, 165

Ice calorimeter, Bunsen's, 251
density of, 140, 165
-cream freezer, 258
lowering of melting point by

pressure, 256

manufacture of, ammonia proc-
ess, 271

Ice, manufacture of, can system, 273

plate system, 273

"Ideal" engine, Carnot's, 313, 323
Ignition temperature, 220
Immersed floating bodies, 164
Impact of bodies, 52
Impulse equal to momentum, 52
Impulsive force, 52
Inclined plane, 117

mechanical advantage of, 118
Indicator, 319

card, gas engine, 330, 331
use of, 321

diagram or "card," 320
Induction coil, gas engines, 327
Inertia force, 49

torque, driving, 69

work done against, 89, 93
Injector, steam boilers, 211, 212
Interference of sound waves, 292

of light, 293

Intermolecular attraction, work
against, surface tension, 172
Internal work done by gas in expand-
ing, 278

Interpolation, 49
Invar, 230

Inverse square, law of, 31
Isothermal compression and expan-
sion, 324, 325

lines, Carnot's cycle, 324
Isothermals of a gas, 188, 190

of carbon dioxide, 274, 276

Jackscrew, the, 120
Jet and ball, 212

condenser, 316

pump, 209

Joly's steam calorimeter, 252
Joule, James P., 277

unit of energy, 90
Joule's determination of mechanical

equivalent of heat, 218
Joule-Thomson experiment, 277

Keokuk, water power, 205
Kilogram, 4
Kilowatt-hour, 106
Kilowatt, the, 106

342

INDEX

Kindling or ignition temperature,

220
Kinetic energy, 92, 96

and perpetual motion, 93

units of, 95
Kinetic theory of evaporation, 260,

261
of gases, 236

and Boyle's law, 188
of gas pressure, 179, 188
of heat, 217
of matter, 138
Lamp, Davy's safety, 287

the "skidoo," 232
Land and sea breezes, 306
Law, of Boyle, 187, 192
of Charles, 236
of cooling, Newton's, 297

Stefan's, 297
Dulong and Petit's, 246
of gases, general, 240
of gravitation, Newton's, 30
of inverse square of distances, 31
of Pascal, 205

Laws, of falling bodies, 38-48
of friction, 100
of gases, three, 239
Newton's three, of motion, 49
Length, measurement of, 5
standard of, 2, 4
unit of, 2, 4
Lever, "arm," 60

"resistance arm" and "force

arm," 114
three classes of, 113
the compound, 130
Light, visible, ultra-violet, and infra-
red, 291

interference of, 293
Linde's liquid air machine, 280
Linear expansion, 228
coefficient of, 229
relation to coefficients of cubical
expansion and area ex-
pansion, 235
"Line of centers," 126
Liquefaction of gases, 278-282

"cascade" or series method, 279
regenerative method, 280

Liquid air, 279, 280

properties and effects of, 281
Liquids, density of, 140, 161

elasticity of, 155

high velocity — low pressure, 208

properties of, 155

specific gravity of, 167

transmission of pressure by, 159

velocity of efflux, 196
Locomotive, maximum pull of, 102
"Loss of weight" in water, Archi-
medes' principle, 163
Low "area," in cyclones, 306

Machine, defined, 110

efficiency of, 111, 112

liquid air, 279, 280

perpetual motion, 93

simple, 112

theoretical and actual mechan-
ical advantage of, 111
Malleability, 144
Manometer, closed-tube, 191

open-tube, 191

vacuum, 193

Marriotte's or Boyle's law, 187
Mass, center of, 124

definition of, 8

and inertia, 8

measurement of, 8

and weight compared, 8
Matter, conservation of, 139

divisibility of, 138

general properties of, 139

kinetic theory of, 138

structure of, 138

three states of, 137
Maximum density of water, 255

and minimum thermometer,
Six's, 226

thermometer, 225

Mean free path, of gas molecules,
139

solar day, 3

Measuring microscope, or microm-
eter microscope, 7

Mechanical advantage, actual and
theoretical, 111

equivalent of heat, 244

INDEX

343

Melting point, 255

of alloys, 255

effect of pressure on, 256

table of, 256
Meniscus, 223, 277
Mercury, air pump, 201

barometer, the, 184

boiling point, 222

freezing point, 222

merits for therm ometric use,

222
Mercury-in-glass thermometer. 222

calibration of, 223

filling of, 222

fixed points on, 223
Metal thermometer, 227
Meteorology, 302
Meteors, cause of glowing, 181

and height of atmosphere, 181,
Method of mixtures, specific heat

determination by, 247
Metric system, the, 4
Micrometer caliper, 6

microscope, 7
Moduli, the three, 152
Modulus, of shearing or rigidity, 152

of tension, Youngs, 147

of volume or bulk, 152
Moisture in the atmosphere, 302
Molecular freedom, solids, liquids
and gases, 138

motion, kinetic theory of gases,

236

in heat, vibratory, 217
Molecules and atoms, 138

of compound, 138

"surface" and "inner," 169
Moment of inertia, defined, 66

of disc and sphere, 68

of flywheel, approximate, 68
practical applications of, 68

value and unit of, 67
Momentum, conservation of, 52, 53,
54, 55

defined, 52

equals impulse, 52
Monorail car, 82

Moon, gravitational attraction on
the earth, 33

Moon, path of, 32

production of tides by, 33
Motion, accelerated, 28

of falling bodies, 38

heat, a form of, 217

Newton's laws of, 49

non-uniformly accelerated, 29

perpetual versus the conserva-
tion of energy, 93

planetary, 32

of projectiles, 42, 43, 44

rotary, 59-71

screw, 24

of a ship in a rough sea, 24

simple harmonic, 82

translatory, 23-58

uniform, 28, 29
circular, 72

uniformly accelerated, 28, 29

wave, 290

Nature of heat, 217
Negative acceleration, 25

torque, 60
Neutral equilibrium, 126

layers, strength of beams, 150
Newton's gravitational constant, 31

law of cooling, 297
of gravitation, 30

laws of motion, 49
Nickel-steel alloy, invar, 230
Nimbus, or rain cloud, 302
Numeric and unit, 2

Olzewski, liquefaction of gases, 279
Onnes, low temperature work of, 237
Orchards, "smudging of" during

frost, 299
Osmosis, 157, 158
Osmotic pressure, 157
"Outer fiber, " strength of beams, 150

Pascal, French physicist, 185
Pascal's law, 205
Pendulum, ballistic, 55

compensated, 234

gridiron, 234

simple gravity, 86

torsion, the, 87

344

INDEX

Period of pendulum, 86

in simple harmonic motion, 85
Permanent set, elasticity, 149
Perpetual motion, 93
Physical quantity, definition of, 1
Physiological effect of heat, 219,

222

Pictet, liquefaction of gases, 278
Pitch, in music, 293

of a screw, 7
Planetary motion, 32

direction of rotation, 34
Plastic substances, 142
Platform scale, 130, 131
Platinum, why used in sealing into

glass, 230

Plotting of curves, 48
Polygon of forces, 12, 16

vector, closed, 15

Porous plug experiment, the Joule-
Thomson, 277
Potential energy, 92, 95
Pound mass, and pound weight, 2,

27,36

Poundal, 27, 36
Power, denned, 104

of engines and motors, by brake

test, 107

in linear motion, 104
in rotary motion, 106, 107
of steam engine, 317
transmitted by a shaft, 154
units of, 105, 106
Precession of equinoxes, 82

in gyroscope, 81

Precipitation, rain, snow, etc., 302
Pressure, atmospheric, 183, 199

diminution of in regions of high

velocity, 208
effect on boiling point, 262

on freezing point, 256
exerted by a gas, kinetic theory,

179

gage, Bourdon, 192
gradient, and temperature gra-
dient compared, 289
perpendicular to walls, 161,

184
steam gage, 192

Pressure of saturated vapor, 262
aqueous vapor, table, 263

transmission by liquids, 159
Prevost's theory of heat exchanges,

296
Principle of Archimedes, 163

of d'Alembert, 49, 51
Projectiles, drift due to earth's
rotation, 307

maximum height reached, 46

motion of, 42, 43, 44

range, and maximum range, 47

velocity and air friction, 46
Projection, meaning of, 83
Prony brake, 106, 107
Properties of fluids in motion, 194-
214

of gases at rest, 177-193

of liquid air, 281

of liquids at rest, 155-176

of matter, general, 139

of saturated vapor, 266, 267,
268

of solids, 144-154
Pulley, the, 114
Pulleys, "fixed" and "movable,"

115
Pump, air, 200

centrifugal, 204

force, 200

Geryk, 201

jet, 209

rotary, 203

Sprengel, 201

suction, 198

turbine, 204

Quantity of heat, measurement of,

unit of, 243
physical, defined, 1

Radial acceleration, 73
Radian, the, 62
Radiant heat, 296
Radiation, 295

and absorption, 297
Rainfall, where excessive, 305
Rain, snow and other precipitation,
302

INDEX

345

Range of projectiles, 45
Reaction, of aeroplane, 52
of birds, wings, 52
practical applications of, 51,

202

of propeller, 51
in swimming, 51
in traction, 51

Reaumer thermometric scale, 225
Receiver, the, 179
Recording thermometer, 227
Reflection and refraction of waves,

293
Refraction, 294

makes vision possible, 295
produces rainbow and prismatic

colors, 295
Refrigerating apparatus, ammonia,

271, 272

Refrigerator room, 273
Regelation, 257
Regenerative method, of liquefying

gases, 280

"Resistance arm," of levers, 114
Resisting force, F0, simple machines,

110
Resolution, forces into components,

19, 101
of vectors, 19

Restitution, force of in simple har-
monic motion, 83, 84, 85, 86
Resultant of several forces, 11, 12,

13, 16
defined, 11
torque, 61
Rifle ball, velocity at various ranges,

46
velocity by ballistic pendulum

method, 55
flight of, 44, 45, 46
Rigid body, two conditions of

equilibrium of, 64
Rigidity, modulus of, 152

of shafts

Rocking chair, equilibrium with, 126
Rolling friction, 102, 103
Rose's metal, 255
Rotary blowers and pumps, 203
motion, 59

Rotary motion, uniformly acceler-
ated, and non-uniformly
accelerated, 59
and translatory motion, for-
mulae compared, 70

Rotor and stator vanes, steam
turbine, 322

Rumford, Count, cannon-boring
experiment, 217

Safety lamp, Davy's, 287
Sailing against the wind, 20
faster than the wind, 21
Saturated solution, 156
vapor, 261

pressure, 262, 268
properties of, 266

table of, 263
Scalars and vectors, 11

addition of, compared, 12
Scale, platform, 130, 131
Screw, the, 120

propeller, 204
Sea breeze, 306
Second, defined, 2
Sensitiveness of beam balance, 128
defined, 7

of micrometer caliper, 7
of vernier caliper, 6
Shafts, rigidity of, 153

power they can transmit, 153
Shearing stress, strain and elasticity,

151, 152

Ship, motion of in a rough sea, 24
Shrinking on, or setting of wagon

tires, 228
Sidereal day, 3
Simple harmonic motion (S. H. M.)i

Simple machines, the, 112
efficiency of, 111, 112
inclined plane type and lever

type, 121

mechanical advantage of, 111
Simple gravity pendulum, 86
Siphon, the, 197

Six's maximum and minimum ther-
mometer, 226
Skate, "bite" of, 257

346

INDEX

"Skidoo"lamp, 232

Slide valve, steam engine, 318

Slug, the, 37

"Smudging" of orchards, protection

against frosts, 299
Snow, rain and other precipitation,

302
Soap films tend to contract, 171,

172
Solar day, mean, 3

variation of, 3
heat, power of per square foot,

218

motor, S. Pasadena, Cal., 296
Solids, thermal conductivity of, 288
density of, 140
elasticity of, 145-151
properties of, 145-154
Solution, boiling point of, 255
freezing point of, 262
of solids, liquids and gases, 156
of metals, amalgams, 159
saturated, 156
Sound waves, 290

interference of, 292
Sources of heat, 218
"Spark" coil, 328

Specific gravity, by balanced col-
umns, 162
defined, 166
hydrometer scale, 168
of liquids, 167
of solids, 167
Specific heat, defined, 244

method of mixtures, 247

table of, 245

the two of gases, 246

the ratio of the two of gases,
246

of water, 243
Speed, average, 24

and velocity compared, 24
Sphere of molecular attraction, 169
Spinney, L. B. See Preface.
Sprengel air pump, 201
Spring balance, 130

gun experiment, 47
Stable, unstable and neutral equi-
librium, 126

Standards of length, mass and time,

2,4

kilogram, 4
meter, 4
pound, 2
yard, 2
Steam boiler, efficiency of, 312

calorimeter, Joly's, 252
Steam engine, 311, 314, 319
compound, 315
condensing, 316
efficiency of, 312, 313
governor, 315
indicator card, 319, 321
methods of increasing effi-
ciency of, 315
power of, 317
thermodynamic efficiency of,

313

triple expansion, 315
work per stroke, 316
Steam pressures and temperatures,

table of, 274
Steam turbine, 205, 321

advantages of, 321
Steel, composition of and elastic

properties of, 149
Steelyard, the, 129
Stefan's law of cooling, 297
Stiffness of beams, 150
Strain, three kinds of, 151

tensile, 146
Strap brake, 108
Stress, tensile, 147

three kinds, 151

Stretch modulus, or Young's modu-
lus, 147

Sublimation, 260
Suction pump, 198
Supercooling, 256
Superheating, 256, 265, 266

of steam, 315

Surface condenser, steam, 316
Surface a minimum, surf ace tension,

170, 171
Surface tension, and capillarity, 168-

175

defined, 171
effects of impurities on, 173

INDEX

347

Surface tension, methods of measur-
ing, 172,173, 175
value for water, 172
Systems of measurement, British, 2
metric, 4

Table of boiling points, 262

coefficient of linear expansion,

230

of cubical expansion, 235
of critical temperatures and

critical pressures, 274
of densities, 140
of heats of combustion, 249
of heats of fusion, 251
of heats of vaporization, 251
hygrometric, 304
of melting points, 256
of saturated vapor pressure of

water, 263
of specific heat, 245
of thermal conductivity, 289
Temperature, absolute, 236

compensation, watch and clock,

233, 234
critical, 273
defined, 220
gradient, 289
sense, 221
of the sun, 298
scales, absolute, 237
centigrade, 224
Fahrenheit, 224
Reaumer, 225
sense, 221

Tensile strength, 144, 148
Theorem of Bernoulli, 209

of Torricelli, 196
Theoretical mechanical advantage,

111

Thermal capacity, 244
conductivity, 288
conductivities, table of, 289
bottle, Dewar flask, 282
Thermobattery, 242
Thermocouple, the, 241
Thermodynamic or limiting effi-
ciency, engines, 313

Thermodynamics, 311

first law, statement of, 311

illustration of first law, 311

second law, statement of, 312
Thermograph, 227
Thermometer, calibration of, 223

centigrade, 224

clinical, 225

dial, 227

gas, constant pressure, 226
constant volume, 226

hydrogen, constant volume, a
standard, 227

maximum, of Negretti and

Zambra, 225
and minimum, Six's, 226

metallic, 227

mercury-in-glass, 222

recording, 227

wet-and-dry-bulb, 269
Thermometry and expansion, 217
Thermopile, 242
Thermostat, 231, 300
Thomson, Sir Wm. (Lord Kelvin),
plug experiment, 277

statement of second law of

thermodynamics, 312
Three states of matter, 137
Tides, cause, spring and neap, 34

lagging of, 34

in Bay of Fundy, 34
Time, of flight and range, 45

measurement of, 9

measurer, essentials of, 9

spacing and spacers, 9, 10

standard of, mean solar day, 3

unit of, 2, 4
Tornadoes, 309

extent, 310

origin, 309

pressure in, 310

velocity of, 310
Torque, 59, 60, 61

accelerating, 66

driving inertia, 69

positive and negative, 60, 61

resultant, 61
Torricelli's experiment, 185

theorem, 196

348

INDEX

Torsion pendulum, 87
Trade winds, 305

Transfer of heat, three methods, 283
Transformation of energy, 93
Transmission of heat radiation
through glass, 298

of pressure, 159
Transverse wave, 292
Triple expansion engine, 315
"Tug of war," forces in, 50
Turbine pump, 204

water wheel, 205
Twilight, cause of, 181, 182
Typhoons, 308

Uniform circular motion, 72
central force of, 72
centrifugal force of, 72, 74
radial acceleration of, 73, 74
Uniform motion, linear, 28

rotary, 59
Units, absolute, or C. G. S. system, 4

of acceleration, 26

British system, 2

conversion of, 4

of force, 27, 36
and weight, 36

fundamental, 2

of heat, 243

of mass, 2, 4

of moment of inertia, 67

and numerics, 2

of power, 105

of time, 2, 4

of work, 90
Universal gravitation, 30

Vacuum, 185

cleaner, 203

gage, 193

pans, 264
Vapor and gas, distinction between,

277

Vapor pressure of water at different
temperatures, table, 263

saturated, 261
Vaporization, cooling effect of, 268

denned, 260

heat of, 250

Vaporization table, 251

two opposing tendencies in, 266
theory of, 261
Vector addition, 12
defined, 11
equilibrium, 15, 18
graphical representation of, 12
polygon, closed, represents equi-
librium, 15
resolution of into components,

scale for, 12
triangle, closed, 15
Velocity, acquired, 38, 39
angular, 62

and linear compared, 63, 70
dependent upon vertical height

of descent only, 55
average, 24, 38, 39, 40
of efflux, 196
of falling bodies, 38
head, 194, 196

initial, final and average, 38, 39
of rifle ball, at different ranges,

by ballistic pendulum, 55
versus speed, 11, 24, 25
Velocities, addition of, 13
polygon of, 15
relation of in impact, 53
resolution of into components, 19
resultant of, 13, 14
"steam," "drift," and "walk-
ing," 14, 15.

Venturi water meter, 211
Vernier caliper, 5

principle, 6
Vibration, direction of in wave

motion, 292
in simple harmonic motion, 82,

Viscosity of liquids, 155
of gases, 177

and the kinetic theory, 177
Volume, change of with change of

state, 256
elasticity of, 151
modulus, 152
strain, 152

INDEX

349

Wagon, hillside, 127

Water, compressibility of, 155, 165

critical temperature of, 273

density of in British system, 140

freezing point variation with
pressure, 256

maximum density of, 255

meter, Venturi, 211

peculiar thermal properties of,
253, 254

waves, 290

reflection of, 294
Watson, W. See Preface.
Watt, unit of power, 106
Watt-hour-meter, 106
Watt's centrifugal governor, 79, 315

indicator card or indicator dia-
gram, 320

Wave length of ether waves, 291,
292

motion, 290

direction of vibration in, 292
longitudinal and transverse
vibrations in, 292

trains, interference of, 292
Waves, actinic, 291

ether, 291

heat, 291

Hertz, 291

light, 291

reflection, 293

refraction, 294

sound, 290

water, 290
Weather bureau, service of, 187

predictions, 187
Wedge, the, 118

and sledge, 119
Weighing machines, 30, 127

the earth, 30

Weighing, process of, 127
Weight compared with mass, 8

in a mine, 30

variation of with altitude and
latitude, 9, 35

units of, 36
Welding, 141

Wet-and-dry bulb hydrometer, 296
Wheel and axle, 117
Windlass, 111
Windmill, reaction in, 202
Winds, 304
Wood's metal, 255
Work, defined, 89

done by a torque, 92

of driving inertia force, 90

in forming liquid film, 172

against friction produces heat,
99, 311

involved in all energy transfor-
mations, 93, 94

if motion is not in the direc-
tion of force, 91

obtained from heat, 311

from water under pressure,
210

per stroke of steam engine, 316

units of, 90

used in three ways, 89, 90, 93
Working force, 110

Yard, standard, 2
Yield-point, 148, 150
Young's modulus, 147

Zero, absolute, 236

change of, with age of thermom-
eter, 224
Zone of calms, 305

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