INTRODUCTION TO THE SCIENCE
OF
DYNAMICS
BY
D. H. MARSHALL, M.A, F.R.S.E,
FESSOR OF PHYSICS IN QUEEN'S UNIVERSITY, KINGSTON. ONTARIO.
KINGSTON, ONT.:
PUBLISHED BY R.UGLOW & CO.
SUCCESSORS TO J. HENDERSON 4 CO.
1898.
TABLE OF CONTENTS.
PAGE.
Introduction 1
Observation, experiment, space, matter, physics, C.G.S and
F.P.S. units.
Chapter I. Extension. Direction 5
Units of measurement, fundamental and dei'ived ; direction,
angle, radian ; body, particle, molecule, atom.
Chapter II. Motion. Velocity 12
Rest; relative, absolute; time, equal times; speed, tach,
vel ; angular motion.
Chapter III. Acceleration 19
Chapter IV. Uniformly Accelerated Motion 25
Chapter V. Inertia. Mass 34
Centrifugal force ; density, specific mass, specific volume.
Chapter VI. Momentum. Force 43
Gramtach, poundvel ; impulse ; dyne, poundal ; pressure,
tension, attraction, repulsion, friction ; dynamics, statics,
kinetics, kinematics, mechanics.
Chapter VII. Weight. Gravitation 52
Vertical, horizontal ; absolute and gravitation units of force.
Chapter VIII. Archimedes' Principle 62
Specific weights, hydrometer, specific gravity bottle.
Chapter IX. Pascal's Principle. The Barometer. . 70
Solid, fluid, liquid, gas, vapour; barad ; Bramah press; hy-
drostatic paradox ; Torricelli's experiment; suction-pump.
Chapter X. Specific Weights of Gases 81
Chapter XI. Exact Specific Weights 90
Baroscope ; dew-point, hygrometer.
IV.
PAGE.
Chapter XII. Work. Energy 99
Configuration ; molar energy, molecular energy, kinetic
energy, potential energy ; erg, foot-poundal ; dyntach, horse-
power ; coefficient of friction ; efficiency of a machine.
Chapter XIII. Action and Reaction 1.10
Problem of Attwood's machine ; coefficient of restitution.
Chapter XIV. Dimensional Equations 115
Chafter XV. Composition of Velocities 124
Relative velocity and acceleration.
Chapter XVI. Composition of Forces, whose lines
of action meet another ". 131
Rigid body, stereod}'namics ; resolution of velocities and
forces; equilibrium.
Chapter XVII. Motion and Equilibrium on an In-
clined Plane 140
Angle of friction ; lines of quickest and slowest descent.
Chapter XVIII. Composition of Forces whose lines
of action are parallel 148
Centre of any system of parallel forces.
Chapter XIX. Couples. Moments 154
Complete triangle of forces ; rotation about a fixed axis.
Chapter XX. Centres of Weight and Mass 161
Stable, unstable, and neutral equilibrium ; centroids.
Chapter XXI. Simple Machines 172
Mechanical, static, and kinetic advantages ; lever and ful-
crum, wheel and axle, toothed wheels, pulley and rope,
Chinese wheel, screw and nut, endless screw, wedge.
Miscellaneous Examples 184
PHYSICAL LAWS.
ARTICLE.
Law of Impenetrability 2
Newton's First Dynamical Law 40
Conservation of Mass 51
Newton's Second Dynamical Law 55
Newton's Third Dynamical Law 58
Conservation of Momentum 59
Hooke"s Law 66
Law of Universal Gravitation 69
Archimedes' Principle 72
Pascal's Principle 80
Boyle's Law of Gaseous Pressure 94
Avogadro's Law of Molecular Volumes of Gases 97
Charles' Law of Gaseous Expansion 101
Dalton's Law of Gaseous Pressure 103
Rankine's enunciation of Boyle's and Dalton's laws . . 103
Transformation of Energy 119
Conservation of Energy 120
Dissipation or Degradation of Energy 121
Law of Friction 122
D'Alembert's Principle 126
Principle of Work 189
TABLES OF MEASUREMENTS.
PAGE.
English and French units of measurement 3
Approximate values of n 4
Values of g at different latitudes 56
Value of G 58
Specific volumes of gases 83
Specific masses and specific weights of gases relatively
to dry air and hydrogen 84
Densities and specific weights of solids, liquids, and
gases 85
Regnault's maximum pressures of aqueous vapour,
for dew-points from 0 to 29 : 93
Coefficients of friction 105
Coefficients of restitution 112
Length, Area, Volume, Angle, Mass, Density, Time. . 120
Speed, Momentum, Force, Pressure-intensity 121
Work and Energy, Activity 122
Useful and Important Numbers.
jr=3'14159265, log -=0'4971499, 1 radian = 180o/tt = 57"3°.
Mean value of f/ = 9805 tachs per sec, or 321 ve\s per sec
Zero of the centigrade scale = 273° air thermometer scale,
and 0°A = - 273°C.
Mean sea-level atmospheric pressure = 76 cm. of mercury
at 0° in the latitude of Paris = 147 lbs.-wt. per sq. in.
= 10^ tonnes- wt. per sq. metre = 1014 megabarad.
Earth's mean radius = 6470-9 kilometres = 3958"7 miles.
Earth's mean density = 5-67, and mass = 614 XlO21 tonnes.
PREFACE.
The present text-book embraces Part I and the half of
Part II of the author's Introduction to the Science of
Dynamics, first printed in 1886, and contains as much of
that work, as experience has shewn he is able to give to the
two divisions of his pass class at the University, at the
present stage of university education in Ontario. The
present edition will, I trust, be found to be a great im-
provement on the last. It is, however, impossible to
escape all errors, and any suggestions or corrections from
students will be thankfully received.
The names tach, gramtach, and dyntach have been re-
tained for the C.G.S. units of speed, momentum, and
activity, as no other names as good as these have yet been
proposed. The Canadian ton of 2,000 lbs. has been used in
preference to the awkward English ton. Surely to call
112 lbs. a hundred-weight is unworthy of a scientific
nation. Let such absurdities disappear, like that foolish
but fast fading notion, that a knowledge of the dead
languages is necessary to a liberal education, or that
equally absurd one, that a knowledge of Hebrew should
form an essential part of the education of a modern
preacher.
It is difficult and I think pedantic for an author to
attempt to enumerate the books and authors to whom he
is indebted, but I cannot refrain from at least thankfully
acknowledging my gratitude to my old teacher and friend,
Prof. Tait, of Edinburgh University, to whose clear ex-
position of the great fundamental facts and laws under-
lying the constitution of the universe, so many thousands
of students are indebted; and also my indebtedness to my
Vlll.
friend and former colleague, Prof. R. H. Smith, of London,
emeritus professor of engineering in Mason College, Bir-
mingham, for his trenchant criticism of the methods of
dealing with some of the difficult problems in that only
sure foundation of the higher problems in all the sciences,
the science of Dynamics.
The full Table of Contents, as well as the lists of the
Tables of Measurement and Physical Laws expounded in
the text, which precede this preface, will, I trust, make
reference to the text sufficiently easy to the student.
D. H. Marshall.
Elmhurst, Kingston, Ont.
9, IV, 1898.
INTRODUCTION.
All our knowledge of the material world is derived
from experience, which can be conveniently divided into
observation and experiment. Astronomy is an example of
a science in which all our knowledge is primarily derived
from simple observation, whereas in the science of
electricity all important advances have been made by the
performance of experiments. Hence, whilst the history
of astronomy stretches over more than two thousand
years, that of electricity hardly extends over two hundred.
Observation consists in simply observing with the aid
only of our senses what is taking place in the material
world.
Experiment is the controlling to a greater or less ex-
tent what is to take place, in order to find out what will
take place under special circumstances.
What we observe and experiment with is matter. This
term, like the terms space, direction, and time, it is im-
possible to define satisfactorily.
Space is limitless extension in ail directions. It is the
abode of matter, in which all motions take place, though
itself immaterial. The term matter is applied to anything
which is perceived by our senses, and which occupies
space. A shadow can be perceived but is not matter,
since it does not occupy space. So with motion, perplex-
ity, anger, joy. The Torricellian vacuum occupies space,
but it is not matter, since (as yet) it cannot be perceived
by the senses.
All great advances in Science have been made by
measuring what is observed. Mathematics may be defin-
ed as the science of measurement. It is divided generally
into (1) Pure Mathematics, and (2) Applied Mathematics.
In the former, measurements of space and time are
principally considered. In the latter, besides space and
time, the properties and conditions of matter, such as
mass, weight, energy, temperature, potential, are measured.
In a wider sense Applied Mathematics is known as
Natural Philosophy or Physics. Natural Philosophy is
the science which investigates and measures the properties
of matter as discovered by direct observation and experi-
ment and deduces the laws connecting these properties.
So extensive, however, has our knowledge of the properties
and conditions of matter become, that different branches
of Natural Philosophy are conveniently separated from
the parent stem. Chemistry, Astronomy, Geology, Bi-
ology, &c, though originally branches of Natural Philoso-
phy, have put forth roots like the branches of the banyan
tree and become themselves trees of knowledge, sending
forth their own branches, and these in their turn new
roots. But the same vital force permeates trunk and
branches alike, and it is this vital force, under its new
name energy, which now forms the subject-matter of phy-
sical science. Natural Philosophy or Physics is thus the
science of energy, and is divided into the following prin-
cipal parts : (1) Dynamics, which treats of molar energy.
(2) Sound, (3) Heat, (4) Magnetism and Electricity, (5)
Light and other kinds of radiant energy.
Before any measurements can be made, certain units
of measurement must be fixed upon. Thus, the navigator
measures the run of his ship in knots, the surveyor his
land in acres, and states of heat are measured in ther-
mometric degrees. Now, not only in different countries,
but even in the same country, different units, bearing no
simple relations to one another, are constantly used in
measurements of the same kind. In order to avoid all
unnecessary calculations in the comparison of different
observations, scientific men have agreed to adopt a uni-
form system of units. This is founded on the French
system of units and is known as the Centimetre-Gram-
Second or C. G. S. system. With the English foot, pound
and second as fundamental units an English system of
units is formed called the F. P. S. system. In the follow-
ing pages the student is exercised in the use of the C. G. S
as well as the English units.
When for special measurements it is desirable to use
larger or smaller units than the standards, these are form-
ed in the C. G. S. system quite uniformly, except in mea-
surements of time, by prefixing the words deca, hecto, kilo,
mega, to the name of the standard to indicate multiples of
10, 102, 103, 106, times the standard unit, and by prefixing
deci. centi, milli, to indicate submultiples of 10-1, 10~'2,
10~3. Taken in connection with the decimal notation in
the writing of numbers, such a system of forming the
multiples and submultiples saves all unnecessary calcula-
tions in reducing to the standard unit. In the English
system of forming multiples and submultiples, the num-
bers seem to have been chosen with a view of containing
as many prime factors as possible, an imaginary advantage
which has occasioned a very great amount of unnecessary
calculation. In conformity with the C. G. S. system of
units, all temperatures in the following pages are given in
degrees centigrade.
Units of Length.
103 millimetres = 102 centimetres = 10 decimetres = 1
metre = 10_1 decametre = 10~2 hectometre = 10-3 kilo-
metre =10-6 megametre.
3 feet=l yard, 6 feet = l fathom, 100 links = 1 chain =
22 yards, 5280 feet=1760 yards = 80 chains = 1 mile.
Units of Surface.
1 are = 1 square decametre = 10 2 square metres = 106
square centimetres.
1 acre = 10 square chains, 640 acres = 1 square mile.
Units of Volume.
1 litre = 1 cubic decimetre = 10 3 cubic centimetres.
1 gallon = 277*274 cubic inches, and holds 10 lbs. avoir,
of water at 62° F.
Units of Mass.
1000 milligrams = 100 centigrams = 10 decigrams =
1 gram = 10_1 decagram = 10-2 hectogram = 10-3 kilogram
= 10~6 tonne.
1 pound (avoirdupois) = 7000 grains, 1 English ton =
2240 lbs., 1 Canadian ton = 2000 lbs.
The following useful formulce should be quite familiar
to the student:
7r=3i 31416, 355/113, 31415926536.
Circumference of a circle = 7td = 2nr
Area of a circle = ~r2
Surface of a sphere = Tzd2 = 4nr2
Volume of a sphere =^7ir3 =^7td3
u
INTRODUCTION TO THE SCIENCE OF DYNAMICS.
Chapter I.
Extension. Direction.
1. The student of elementary dynamics is not concern-
ed with the ultimate structure of matter, of which various
theories have been advanced by scientific men, but only
with its properties. The principal of these which we shall
consider are extension, inertia, mass, weight, and energy.
2. Any portion of matter is called a body. The grains
of sand on the sea-shore, our own bodies, houses, the whole
earth, the planets, the fixed stars, are examples of bodies.
The expression of the fact that two or more bodies cannot
at the same time occupy the same portion of space is
known as the principle of impenetrability.
3. Extension is that property of matter implied in the
statement that every body occupies a limited portion of
space. Every body has therefore form or shape. The
volume of a body is the measure of its extension. The
term bulk is often used in the same sense. The internal
volume of a body, e.g. that of a cup or of a hollow sphere,
is the amount of space enclosed by the body, and is called
its capacity.
4. Before any measurements can be made it is neces-
sary to fix upon units or definite quantities of what we
desire to measure, in terms of which all other quantities
of the same kind are expressed by means of numbers. In
measurements of extension three units are used, viz., units
of length, area, and volume. Of these the unit of length
may be taken as a fundamental or independent unit, and
the pthers made to depend upon it, and these are hence
called derived units. In any system of units a funda-
mental unit is one whose magnitude is independent of
that of any other unit, otherwise than as a mere multiple
or submultiple of a unit of the same kind. A derived
unit is one whose magnitude depends upon the magnitudes
of one or more other units, but is not a mere multiple or
submultiple of a unit of the same kind.
5. The English standard unit of length (or distance)
is the yard, which is defined by Act of Parliament as the
distance between two points on a bar of metal at a definite
temperature. The French unit, the metre, although de-
rived originally from the dimensions of the earth, is sim-
ilarly defined. The unit of length adopted in the C. G. S.
system of units is one of the submultiples of the French
unit, viz., the centimetre, and its multiples and sub-
multiples are the same as the French.
6. Whatever unit of length be used, it is found most
convenient in measurements of surface to take as the unit
of area (or surface) the area of a square of which the side
is unit of length, or a multiple or submultiple thereof.
Hence the C. G. S. unit of area is a square centimetre.
The French unit of area, the are, is a square decametre.
7. Similarly the unit of volume is immediately and
most conveniently derived from the unit of length by de-
fining it as the volume of a cube of which the edge is unit
of length or a multiple or submultiple of the unit of
length. Hence the C. G. S. unit of volume is a cubic cen-
timetre. The French unit of capacity, the litre, is a cubic
decimetre, and the unit of volume, the stere, a cubic
metre.
8. Direction is relative position irrespective of dis-
tance. It is the only property or characteristic of an in-
definite straight line. It is the characteristic property of
motion (Chap. II), and thus indicates how one must go
from one point of space to reach another. An angle is
difference of direction. The unit cf angle in common
measurements is the degree, which is the 90th part of a
right angle. It cannot be said that the unit of angle is
derivable from the unit of length, but it is most conven-
iently measured as the ratio of two lengths. This will be
understood when we remember that if a circle be describ-
ed with the vertex of an angle as centre and with any
radius, the magnitude of the angle is measured by the
ratio of the length of the arc on which it stands to the
length of the radius. We may, therefore, define the unit
angle as that angle which is subtended by an arc of unit
length at the centre of a circle of unit radius. This is
just the same as the angle which is subtended by an arc,
whose length is equal to the radius, at the centre of any
circle whatsoever, and is called a radian.
9. As mentioned above, we learn from elementary ge-
ometry that, whatever unit of angle be adopted, the fol-
lowing formula expresses the relation between the length
of any arc (a) of a circle, the length of the radius (?•),
and the magnitude of the angle (i) subtended at the
centre of the circle by the arc, a = C ri, where C is a con-
stant number, whose value depends upon the unit of angle
adopted. If we measure (i) in radians, this reduces to
the simple form
a = ri
10. To express the value of a radian in degrees. ■
Since the arc subtended by a straight angle, i.e. by two
right angles =nr, if (0 be the measure of two right angles
in radians, we get rtr = ri, .'. i = n, i.e. two right angles = n
radians, and .\ a radian = 180°/;r = 57° IT 44"'8 true to the
tenth part of a second of angle, or very nearly 57°'3.
11. In many dynamical investigations it is unnecessary
to consider in any way the dimensions of a body, or the
8
distances between the different parts of a system of bodies.
When this is the case, the body or system of bodies is
called a particle or material particle. Thus in the ex-
planation of the seasons, or of the phases of the moon, the
earth or moon is a body, as we cannot neglect its dimen-
sions, whereas in the determination of a planet's position
in the sphere of the heavens at any time, the planet is a
particle. In considering the proper motion of the solar
system amongst the fixed stars, the sun, and indeed the
whole solar system, are merely particles. The term par-
ticle is frequently defined as an indefinitely small body.
The terms small and large are merely relative, and what
is small at one time or from one point of view is large at
another or from another point of view. In looking at a
star through a large telescope we generally speak of the
star as a particle, and of the telescope as a large body, and
yet the star is immeasurably larger than the telescope. A
grain of sand or a mote of salt we generally consider a
very small body, but if it gets into one's eye, its size is
enormous.
12. The most generally accepted theory of the ultimate
structure of matter at the present day is known as the
atomic theory. According to this theory matter is not
infinitely divisible, but consists ultimately of excessively
small indivisible particles. The smallest portion of any
substance, beyond which mechanical sub-division is sup-
posed to be impossible, is called a molecule. A molecule
may, however, be chemically divided into atoms. Thus a
molecule of water (H20) may be chemically divided into
three atoms, two of Hydrogen, and one of Oxygen.
Examination I.
1. Define matter, and distinguish between the terms
body, particle, molecule, atom.
2. Define extension, space, length, and area.
9
3. What is impenetrability ? Give illustrations of the
apparent contradiction of this principle, and explain
them.
4. Distinguish between volume, bulk, and capacity.
5. What is a unit of measurement ? Give the C. G. S.
and F. P. S. units of length, area, and volume.
6. What is a metre, a litre, a stere ? State the numeri-
cal relations between the are and sq. cm., and between
the litre, stere, and cub. cm.
7. Define direction and angle. Name and define the
principal units of angle.
8. Give in radians the angles of an equilateral triangle,
a right angle, 30°, a circumangle, and express a radian and
f n radians in degrees.
9. If the unit of length be a yard, and the unit of
angle a right angle, what must be the value of C in the
formula a = C r i ? Is the value of C in this formula de-
pendent on the unit of length ? Why ?
Exercise I.
The following examples in mensuration are appended
to exercise the student in the use of the C. O. S. units and
also of logarithmic tables to which he should early accus-
to m h imself. Log n = 0-4971499.
1. The great pyramid of Gizeh is a regular pyramid on
a square base. The original length of an edge of the base
was 22042m., and of a slant edge 232-865m.; find (1) the
area of the ground on which it stands, (2) the exposed
area of the pyramid, (3) the volume.
2. Assuming the earth to be a sphere, and that the
length of an arc of a degree on a meridian is equal to
111-19 kilom., find (1) the length of the diameter, (2) the
area of the earth's surface, (3) the volume.
10
3. If the nature of the earth's crust be known to a
depth of 8 kilometres, find the ratio of the known to the
unknown volume, supposing the earth to be a sphere of
6370-9 kilometres radius.
4. On the same supposition, how much of the earth!s
surface could a person see who was at a height of 4 kilo-
metres above the sea level ?
5. If the atmosphere extend to a height of 70 kilo-
metres, what is the ratio of its volume to that of the solid
and liquid earth ?
6. Compare the earth's surface (taken as 100) with the
torrid, temperate, and frigid zones of the earth, supposing
the first to extend to an angular distance of 23c30 from
the equator, and the last to a distance of 23c30' from each
pole.
7. Two sectors of circles have equal areas, and the radii
are as 1 to 2 ; find the ratio of the angles.
8. A gravel walk of uniform breadth is made round a
rectangular grass-plot, the sides of which are 20 and 30
metres ; find the breadth of the walk, if its area be three-
tenths of that of the grass-plot.
9. Find the number of litres of air in a room whose
dimensions are 12*50 m., 545 m., and 3"70 m.
10. Find in radians the angle of a sector of a circle, the
radius of which is 20 metres, and the area a deciare.
11. The horizontal parallax of the sun (i.e. the angle
subtended by the earth's radius at the sun) is 8"-85, and
of the moon 57' 3" ; find the distances of these bodies in
terms of the earth's radius.
12. Find also, in terms of a great circle of the earth,
the areas of the moon's orbit and of the ecliptic, supposing
these to be circles.
11
13. Find the circumference and area of the circle of
latitude passing through Kingston. Ont., latitude 44° 13',
(See ex. 3).
14. A pendulum whose length is 1| metre swings
through an arc whose chord is a decimetre ; find the angle
and the length of the arc of oscillation.
15. What must be the diameter and surface of a
spherical balloon that its capacity may be 150000 litres?
Answers.
When no unit is appended to an answer, the units of
the C. G. S. or F. P. S. system arc to be understood.
When the answer cannot be expressed exactly by a num-
ber, the answer given is true to the last figure.
1. 485-850 ares; 904-313 ares; 2-80195 megasteres.
2. 12741-4 kilom. ; 5-10019X1012 ares; T08306X1012
cub. kilom. 3.1:265. 4. 160018 sq. kilom. 5.1:30.
6. 100: 40: 52: 8. 7. 4:1. 8. 168-6. 9. 2520625.
10.1/20. 11. 23307; 60-3. 12. 3631; 5-432x10*.
13. 28689-5 kilom.; 654991 XlO7 sq. kilom.
14. 3° 49' 13"-7; 10002. 15. 659; 1-365 are.
Chapter II.
Motion. Velocity.
13. Motion is change of position. Although the ideas
conveyed by the terms matter and motion are quite differ-
ent, yet it is evident that all the motions we are cognizant
of are the motions of matter directly, or are indirectly pro-
duced by motions of matter. Thus the motion we see when
a boy throws a stone is the motion of the stone directly.
A wave, on the other hand, which is motion of form, is not
directly the motion of the medium through which the
wave is passing, but is indirectly produced by the motion
of this medium. What of the motion of a shadow, or of
tlie sphere of the heavens ?
14. The opposite (or the zero) of motion of rest. All
the motion or rest of a body that we can know of is rela-
tive, i.e. with respect to some other body. In infinite
space absolute motion or rest is indeterminable, if indeed
conceivable. When we speak of rest and motion we gen-
erally mean either relatively to our own bodies, or rela-
tively to our abode in space, the Earth. Every body is
simultaneously at rest and in motion. When a person is
sitting at ease in a railway carriage, he is said to be at
rest. But this is merely relatively to the train. Rela-
tively to the earth he is moving as fast as the train is, and
when we consider that the earth is rotating about its axis,
is further revolving around the sun, and with the sun and
other members of the solar system careering through
space, it is easily seen how complex is the person's motion.
The aim of the 'physicist is to determine those co>u1itions
of matter and motion which, a port from the world of sen-
sation, thought, and consciousness, constitute the life of
the universe.
13
15. Time is continuous and limitless duration or exist-
ence, marking out the succession of events. In dynamical
science it is conceived as a uniformly increasing quantity.
It might also be denned as the immeasurable flow or con-
tinuity of instants, and is provisionally measured by the
rotation of the sphere of the heavens.
16. Velocity is time-rate of motion, i.e. rate of change of
position per unit of time. By rate is meant here degree
of quickness. When two bodies are moving, and one
moves over a greater distance in the same time than the
other, the velocity of the former is said to be the greater.
Velocity has direction as well as magnitude. The term
sj>ee<l is used for magnitude of velocity irrespective of
direction. In any motion of a body the velocity may be
uniform, i.e. the same throughout the motion, or it may be
variable, i.e. continuously or at intervals changing during
the motion. Similarly a speed may be constant or varia-
ble. The velocities of all bodies that we see moving are
really variable. The motions of the hands of a chrono-
meter, or the rotations about their axes of the different
members of the solar system, are cases of motion in which
the speeds are nearly constant. The test of constant
speed is that equal distances are moved over in equal
times, however small these times may he.
17. What, however, we naturally ask, are equal times ?
We look at our clocks or watches and say that they tell us
equal times. Some watches go slow, others go fast, and
how are we to know which go right ? It is well-known
that our clocks and watches are regulated by the apparent
motion of the sun in the sphere of the heavens. This mo-
tion is the resultant of two motions, viz., (1) the apparent
rotation of the sphere of the heavens, produced by the real
rotation of the earth, which takes place in a sidereal day :
and (2) the apparent revolution of the sun in the ecliptic
in a sidereal year, produced by the real revolution of the
14
earth in its orbit around the sun. As the sidereal year is
estimated in sidereal days, we find that ultimately the ap-
parent rotation of the sphere of the heavens is our stand-
ard measurer of time, and we define equal times as times in
which the sphere of the heavens apparently rotates
through equal angles. Whether the term equality is
rightly applied to such times or not is a legitimate en-
quiry.
The mean or average time in which the sun appar-
ently rotates about the earth is called a mean solar day,
and our unit of time, the second, is a well-known fraction
of the mean solar day. In the C. G. S. and F. P. S. sys-
tems of units of measurement, the unit of time is, like the
unit of length, one of the fundamental units.
18. Speed is measured by the number of units of length
passed over in a unit of time. The unit of speed is deriv-
ed immediately from the units of length and time ; it is
the speed in which a unit of length is passed over in a
unit of time. Hence the C. G. S. unit of speed is 1 centi-
metre per second. It will be convenient to call this a
tacit. The F. P. S. unit of speed is 1 foot per second, and
is called a vel. If s be the distance in centimetres or feet
described in / seconds by a body moving with a constant
speed of v tachs or vels, then s = vt, and v = s/t.
19. When a body is moving with variable speed it has
of course a definite speed at every instant, which is mea-
sured by the number of units of length which iron Id he
passed over in a unit of time, if for such a period from the
instant in question the speed did not change. Hence we
talk of a ship sailing at the rate of 12 knots an hour, or of
a man walking at the rate of 4 miles an hour, although
the speed of the ship or of the man may not be the same
for any two consecutive seconds. When a body is moving
with variable speed, the equation r = s/t gives the mean
speed during the time /, and, by taking t «*mall enough,
15
we can approximate in any degree of exactitude to the
speed at any instant.
20. A velocity can be completely represented by a
straight line, the direction of the line representing the
direction of the motion (the tangent to the path of the
moving particle at the instant in question), and the length
of the line representing the speed. Since a body may
move in two directions along a line, the one being opposite
to the other, it is convenient to distinguish these by the
signs + and — , as is customary in the applications of
algebra to geometry. If AB be a straight line, and a
velocity in the direction of AB be called + , a velocity in
the direction BA will be called — .
Angular Motion.
21. Angular velocity is rate of change of direction (of
one point with respect to another) per unit of time.
Example. — When a particle moves in a circle the time-
rate of change of the angle, which the radius through the
particle makes with a fixed radius, is the angular velocity
of the particle about the centre of the circle.
If the angles described by the radius through the par-
ticle be equal in equal times, however small these may be,
then the angular velocity is uniform and is measured by
the angle described in a unit of time. The unit of angu-
lar velocity is that in which a unit of angle is described in
a unit of time, i.e., 1 radian per second.
When the angular velocity is variable, the angular
velocity at any instant is measured by the angle which
would he described in a unit of time, if for such a period
from the instant in question the angular velocity did not
change.
Corresponding to the equation .<? = vt (Art. 18), we evi-
dently have the equation i=ot, in which i is the angle
described in time t with angular velocity o.
16
22. From the formula a = ri (Art. 9) it follows at once
that if v represent in tachs or vels the speed of a particle,
moving in the circumference of a circle, r the radius in
centimetres or feet, and o the angular velocity about the
centre in radians per second,
v = ro, and o — v\r.
23. If the angular velocity be like that of the hands of
a watch, it is represented by the - sign, and if unlike, by
the + sign. Let it be carefully observed, however, that
the sign given to the angular velocity of a body depends
upon the side of the plane of motion from which the mo-
tion is observed. Thus, if we could see the motion of the
hands of a watch through the back of the watch, the an-
gular velocity would be + . If we look northwards at the
rotation of the sphere of the heavens it seems to be + ,
and if we look southwards it seems to be
Just as - - linear motion is the plane image (i.e. the
image in a plane mirror) of + motion, so - angular mo-
tion is the plane image of + , and vice versa.
Angular velocity is completely represented by a num-
ber with the sign + or - prefixed to it.
Examination II.
1. Define motion, rest, velocity, speed.
2. What is a wave ? How is it produced ? Give ex-
amples.
8. Illustrate the meanings of the terms relative and
absolute with respect to extension, motion, and direction.
4. Distinguish between uniform and variable velocity s
and define the speed of a body at any instant when the
velocity is variable.
5. What is the test of constant speed ?
('). Defino time, equal times, and the unit of time.
17
7. Distinguish between fundamental and derived units,
and give examples of each.
8. Name and define the C. G. S. and F. P. S. units of
speed.
9. Give the relation between s, v, f in linear motion,
and between i, o, f in angular motion.
10. Define a sidereal and a mean solar day, and give the
numerical relations between them and a sidereal year.
11. How may velocity, speed, and angular velocity be
completely represented ?
12. Define angular velocity, and the unit thereof.
13. Give and prove the relation between the speed and
angular velocity about the centre of a circle of a body
moving in the circumference.
14. Distinguish between -f and - angular velocity.
How are they related to one another ?
Exercise II.
1. A body has a speed of 10 tachs, how long will it
take to pass over 600 metres ?
2. Which is greater a speed of 72 tachs or one of 252
metres per hour, and by how much ?
3. Express a speed of 72 kilometres per hour in deci-
metres per minute.
/jUIf a line a foot long represent a velocity of 3*75
miles per hour, what length of line would represent a
velocity of 80 yards per minute ?
5. Two bodies start from the same point, the one 10
minutes after the other, and travel in perpendicular direc-
tions with speeds of 120 tachs and 100 metres per minute.
How far apart will they be in an hour from the starting of
the first ?
6. Two travellers leave the same place at the same time
in directions inclined to one another s^ an angle of tt/3,
18
and each travels with a speed of 166 tachs. how far apart
will they be in two hours ?
7. A man two metres high walks in a straight line at
the rate of 6 kilometres an hour away from a lighted lamp
3 metres high ; find in tachs the speed of the end of his
shadow, and the rate at which his shadow lengthens.
8. If 3 minutes be the unit of time, and 50 decimetres
the unit of length, what number measures the average
rate of walking of a person who goes over 40 kilometres in
12 hours ?
9. If 7 metres per 3 minutes be the unit of speed, and
4 decimetres the unit of length, what must be the unit of
time ?
10. If 3 metres per 7 minutes be the unit of speed, and
4 seconds the unit of time, what must be the ur.it of
length ?
11.) A body moving uniformly in a circle describes the
circumference twice in 3 minutes, what is the measure of
its angular velocity about the centre ?
/12,) What is the angular velocity of any body on the
earth's surface due to the earth's rotation ?
13. The diameter of the driving wheel of a locomotive
is 2 metres, what is the angular velocity of a point on the
wheel about the centre, when the train is moving at the
rate of 80 kilometres an hour ?
14. A body moving in the circumference of a circle of
radius 10 has unit angular velocity about the centre ; find
the space described in 10 seconds, and the time taken to
complete a revolution.
ANSWERS.
1. 100 min. 2. 72 tachs ; 65 tachs. 3. 12000.
4. T«T ft. 5. 660775. 6. 11952m. 7. 500; 333-3.
8. 33-3. 9. 10 I sec. 10. 2 f cm. 11. tt/45
12. Tr/43200. J3. 22-2. 14. lm.; 2 tt sec. '
()
Chapter III.
Acceleration.
24. Just as a body's position may change, giving rise to
motion, so a body's velocity may change, giving rise to
acceleration.
Acceleration is change of velocity, not merely change
f speed. A body's velocity may change in magnitude
only, or in direction only, or in both magnitude and di-
rection. The total acceleration during any time is the
whole change of velocity during that time. The accelera-
tion at any instant is the rate of change of velocity per
unit of time at that instant.
The rate at which a body's velocity changes may be
slow or fast. Compare the accelerations of trains in a long
railway like the Canada Pacific, in which the stations are
far apart, with the accelerations of trains in large cities,
run for the convenience of passengers hurrying from one
part of the city to another, such as on the Underground
Railroad in London or on the Elevated Railroad in New
York.
25. Acceleration has direction as well as magnitude,
and may be uniform or variable. An acceleration is uni-
form when equal changes of velocity take place in equal
times, however small these times may be. and is then
measured by the velocity acquired in unit of time.
The direction of a body's acceleration may or may not
be the same as the direction of its motion. We shall first
consider acceleration, the direction of which is the same
as that of the body's motion or opposite thereto. The
effect of such an acceleration is evidently to change a
body's speed without changing its direction of motion. If
the direction of motion be considered + , then the accel-
eration will be + or - according as the speed is increasing
or decreasing.
20
26. The systematic unit of acceleration (in magnitude)
is unit of speed per unit of time. Hence the C. GK S. unit
of acceleration is 1 tach per second, and the F. P. S. unit
is 1 vel per second.
Observe carefully that the unit of acceleration, by in-
volving the units of speed and time, involves the unit of
length once and the unit of time twice. This must be
particularly attended to if in the solution of a problem the
units require to be changed. Thus a tach is represented
by t6q°q- if a metre and minute be the units of length and
time, but with the same units 1 tach per second will be
represented by 6yo%°- One of the most important cases
of the motion we are now considering is that of a body
moving vertically upwards or downwards in vacuo. Such
a body has a uniform acceleration vertically downwards.
Its value, denoted by g, depends upon position, the mean
value over the earth's surface, at the sea level, being 980*5
tachs per sec, or 32^ vels per sec, nearly.
If a, represent the acceleration of a body uniformly ac-
celerated in the direction of its motion ( + ly or - ///).
and v denote the whole change of speed in time /, then
r = at, and a = r f.
27. A body may have a uniform acceleration which is
different in direction from the direction of motion. The
resultant motion of the body in this case is very different
from that of the preceding case. Such would be the mo-
tion of a body near the earth's surface moving in vacuo in
any but a vertical direction; it is very nearly that of a
leaden bullet projected in the air in any but a vertical di-
rection, and with a small speed. Such a body's speed will
be always changing, though not at a constant rate, and the
direction of motion will be always changing, so that the
path described will be a parabola with its axis in the di-
rection of acceleration. The parabolic path is well seen in
the motion of a jet of water.
21
28. Again, a body may have an acceleration constant in
magnitude but not in direction. Any body revolving uni-
formly in a circle (which is approximately the motion of
the moon in its orbit) has such an acceleration. If o denote
the angular velocity of the body about the centre, and r
the radius of the circle, it can be shewn that the magni-
tude of acceleration is measured by ro2, but the direction
of acceleration is always towards the centre of the circle,
and therefore changing at every instant.
29. When a body's acceleration is variable, the accelera-
tion at any instant is measured by the number of units of
velocity by which the body's velocity would he changed in
a unit of time, if for such a period from the instant in
question the acceleration remained uniform. When the
acceleration is variable, the formula a — v/t gives the aver-
age acceleration during the time t, and by taking t small
enough we can approximate as closely as we please to the
acceleration at the beginning of time /.
30. A bullet shot vertically upwards with great speed is
an example of a body whose acceleration is constant in
direction, but variable in magnitude on account of the
varying resistance of the air. If the ball be shot in any
but a vertical direction we have a case of motion in which
the acceleration is always changing both in magnitude
and direction.
31. Acceleration, like velocity, is completely represented
by a straight line, the direction of the line being the di-
rection of acceleration, and the length of the line repre-
senting the magnitude of the acceleration.
Examination III.
1. Define acceleration, total acceleration, and accelera-
tion at any instant.
2. Define the unit of acceleration, What fundamental
units does it involve ?
22
3. What is the acceleration of a falling body ? What
of a body rising upwards ?
4. Under what conditions is the formula v = at true ?
What is the test of uniform acceleration ?
5. How is acceleration measured when variable ?
6. Give examples of bodies having accelerations, (a)
uniform; (6) variable, 1) in direction only, 2) in magni-
tude only, 3) in both direction and magnitude.
7. Shew that an acceleration of a metre per minute per
second is equal to an acceleration of a metre per second
per minute.
Exercise III.
In the following examples the acceleration is supposed
to be uniform a in I in the direction of motion.
1. A body has an acceleration of 20 tachs per sec; find
in decimetres per minute the speed acquired in an hour.
2. Express the acceleration of a body falling in vacuo
(980*5) in units of a metre and hour.
3. The acceleration due to the weight of a body is 32|
vels per sec. ; find the same in units of a yard and minute.
4. A body is thrown vertically upwards with a speed of
6000 tachs; what is its velocity at the end of 4 and of 8
seconds, neglecting the resistance of the air ?
5. A body uniformly accelerated starts with a speed of
6 metres per minute, and in half an hour has a speed of
36 kilometres per hour; find the acceleration in tachs per
second.
('). Compare the acceleration 2 tachs per sec. with that
in which a speed of 1800 metres per hour is acquired in
an hour.
7. Compare an acceleration 3 when a yard and minute
are the fundamental units with an acceleration 1 when a
foot and second are the fundamental units.
8. If 1 tach per 10 seconds were the unit of accelera-
tion, what would be the measure of an acceleration of 10
tachs per second ?
( 9. If 6 kilometres per second per minute were the unit
of acceleration, and 1 metre the unit of length, what would
be the unit of time ?
10. If 1 decimetre per hour per second were the unit of
acceleration, and 1 metre per minute the unit of speed,
what would be the units of length and time in a scientific
system of units ?
' 11. What is the difference between an acceleration of a
metre per hour per second and one of a metre per minute
per minute ?
12. Find in tachs per second the difference between an
acceleration of 24 metres per minute -per second and one
of 21 '6 kilometres per minute per hour.
'13 j If 216 kilometres per minute per hour be the unit of
acceleration, and a second be the unit of time, what must
be the unit of length ?
^14) If the unit of speed be 96 metres per 15 minutes,
and 10 seconds be the unit of time, express in C. G. S.
measure the unit of acceleration.
15. If the unit of speed be 5 tachs, and 3 metres be the
unit of length, express in kilometres per hour per hour
the unit of acceleration.
16. /If 7 metres be the unit of length, and 3 minutes the
unit of time, what speed in tachs will a body acquire in
half an hour with an acceleration 18 ?
17. A body starts with a speed of 120 tachs, and has an
acceleration of 6 metres per minute per minute ; another
starts at the same time from rest with an acceleration of
36 kilometres per hour per hour; when will their speeds be
equal ?
24
18. If 5 inches represent an acceleration of 10 tachs per
minute, what length of line will represent an acceleration
of 6 when a metre and minute are the units of length and
time ?
19. A body is thrown vertically upwards with a speed
of 10 kilotachs; after how many seconds will it be moving
downwards with a speed of 5 kilotachs ?
20. The values of g were represented by two different
nations by 12 and 25, and the speed of sound in air at 0°
by numbers which were as 6 : 5; find the ratios of their
units of length and time.
21. If g be represented by 1754T^r and an acre by 10 in
a system of units; find what must be the units of speed
and time.
Answers.
1. 432000. 2. 127072800. 3. 38600.
4. 2078 upwards; 1844 downwards. 5. ||. 6. 144:1.
7. 1:400. 8. 100. 9. ^ sec. 10. 10 m.; 10 min.
11. 0. 12. 30. 13. 1 m. 14. 106. 15. 10'8.
16. 700. 17. 18 min. 18. 5 inches. 19. 153.
20. 1:3; 2:5. 21. 11 vel; 1 minute.
Chapter IV.
Uniformly Accelerated Motion
32. We shall in this chapter consider more fully the
nature of the motion of a body which is uniformly accel-
erated in the direction of its motion. If u be the velocity
;at any instant and a the acceleration, the velocity at the
I end of any time t will evidently be u + at; denoting this
by v we get the first equation of motion,
v — u + at.
33. To determine the space described in th(j time t.
Since the velocity during the time / increases uniform-
'. I y from u to u + at, the average velocity is \\u + (u + at) \
or u + ^at. If a body moved uniformly during the time t
with this speed, the space described would be (u-\-\at)t,
! or ut+\at2. This will evidently be also the space de-
I scribed during the time t by a body whose velocity increases
' uniformly from u to u + at in that time. Hence if a body
has an initial velocity u, and an acceleration a in the di-
! rection of its motion, and if s denote the distance describ-
ed in the time t,
s = ut + | at2.
34. The following may be considered by the student a
more rigorous proof of the same result:
Let the time t be divided into any large number of equal
j parts. If n denote the number, the duration of each little
interval will be t/n. The velocities at the beginning of
I each of the little intervals will be
t « t , ^ t
i u , u + a — , w + 2a— , u-\-(n — l)a— .
n n' v ' n
The velocities at the end of each of the little intervals
• will be
* o * n * t
I u + a — , u + let — , it + oa — , u + na — .
n ' n * n ' n
Si =
26
Suppose now that a body A moved uniformly during
each little interval with the velocity indicated above at the
beginning of each interval; if sx be the whole distance
moved over during the n intervals, i.e. during the time t,
■-— | m+ {u+a— ) + (u + 2a— ) + . . O+w-la— j
= ut + a~Y . j 1 + 2 + 3+ n-1 I
I2 n(n-l) , ( 1
= ut+a^-- -g- ~ = nt + $al2 j 1 - —
Similarly if s2 be the whole distance moved over by a
body B, which moved uniformly during each little inter-
val with the velocity indicated above at the end of each
interval,
s2 = ut + ^af2 ] 1 + — [
Now if s be the whole space described during the time t
by the body uniformly accelerated, it is evident that s is
greater than s-y and less than s2, for the body A moved
during each little interval with the least velocity, which
the body uniformly accelerated had during that interval,
and the body B with the greatest.
.-. s >ut + %at2 j 1 - — |
<ut + iat*\ 1 +•— j
Now these two quantities between which s lies differ
only in the sign of l/>/. What is n ? n is any number
whatsoever, and may be made as large as you please. But
by taking n large enough, 1 -1/n and 1 + 1/n may be made
to differ from 1 by as small a fraction as you please.
Hence when n becomes indefinitely great, the motions of
A and B do not differ from the motion of the body uni-
formly accelerated, and the three quantities s, slt s2 be-
come id + \at2.
27
35. From the equations of uniformly accelerated motion
just determined
v =u + at (1)
8 =ut + %at2 (2)
we derive by algebraic analysis the following useful though
not independent equation :
v2 = u? + 2as (3)
Cor. 1. If the acceleration be opposite in direction to
that of motion it must be represented by— a, and the
equations become
v =u- at (4)
s = ut-\af (5)
v2=u2-2as .(6)
Cor. 2. If the body start from rest, u=0 and the equa-
tions become
v =at (7)
s =Jr«i2 (8)
v2=2as (9)
Comparing (2) or (5) with (8) we might say that ut is
the distance described in virtue of the speed u, and \at 2
that described in virtue of the acceleration a.
36. As already stated in art. 26, the motion of a body
moving freely in a vertical direction is of the character we
have been considering. Strictly speaking, this applies
only to bodies moving in vacuo, but unless the velocity be
great we may often neglect the action of the atmosphere.
Let us consider the motion of a body thrown vertically
upwards with a velocity u.
1). How long will it rise ?
It rises until its velocity is zero. Hence from equation
(4) we get 0 = u - gt, .. t = u/g .
2). What is the greatest height readied ?
28
In equation (5) putting t = u/g we get
u \ u ) 2 u2
s=u7~i9u\ =27'
We might get the same result more simply from equa-
tion (6). When the body ceases to rise, v = 0
.-. 0 = u2-2gs. .-. s = u2/2g.
3). When will the body return to the point of projec-
tion 9
The distance described from the point of projection in
the required time is zero ; hence from equation (5),
0=ut-%gt2, .-. t=0or2u/g.
Comparing this result with 1), we see that the time
taken for a body to fall from the greatest height reached,
back again to the point of projection, is just the same as
that taken by the body to reach its greatest height.
4). What is the velocity of the body after returning to
the point of projection?
From equation (4), v — u — g \ — J- = — u,
< 9 >
that is, the velocity is the same in magnitude as that at
starting, but opposite in direction. Now, since any point
in the path might be considered a point of projection, we
infer from this result that the return or downward motion
of the body is a plane image of the upward motion.
37. The distances described in successive seconds (or
other equal inter rah of time) by a body, which starts from
rest and is uniformly accelerated, are as the odd numbers.
The distances described in 1, 2, 3, (w-1), n seconds
are £a(l)2, £a(2)2, £a(3)2 \a (n-\)\\an\
.-. the distances described in the 1st, 2nd, 3rd wth
seconds are \a, fa, fa, |a(2ra-l). Thus the
distance described in the nth second, where n is any num-
ber whatsoever, is equal to $a(2n-l), which varies as
(2m -1) the.Mth odd number.
29
Ex. A body is thrown vertically upwards with a ve-
locity of 3922 tachs; find 1) the time taken to describe
58*83 metres, 2) the velocity at that height, 3) the greatest
height reached, 4) the time of ascent, 5) the distance de-
scribed in the half second following the fifth second from
the instant of starting, 6) the distance described in 10
seconds.
1). Let /sec. = time required. From equation (5),
5883 = 3922 *-J(980'5)*2, .-. t = 2 or 6.
2). From equation (4), velocity at tliQ end of 2 seconds
= 3922-2 (980-5) = 1961 tachs; velocity at the end of 6
seconds = 3922 - 6 (980-5) = - 1961 tachs.
We thus see from 1) and 2) that the body in its ascent
has risen 58'83 metres in 2 seconds; that after 6 seconds it
is at the same height, but is then descending; that at both
times the speed is the same.
3). From 2), art. 36, the greatest height reached
=■_; „_.: „ = 7844 centimetres.
2 X 980-5
4). From 1), art. 36, the time of ascent = =4 sec.
980 'o
5). Distance described in 5^ seconds
= 3922 ( V ) - 1(980-5) x ( V ) 2
Distance described in 5 seconds = 3922 X 5-^(9805) X52
.-. the required distance
= 3922x|- 1(^80-5) XV= -612if cm.
The -sign tells us that the body has descended down
this distance in the 11th half second of its motion.
6). From equation (5) the distance required is
3922 X 10 - 1(980-5) X 102 = - 9805 cm.
The -sign tells us that the body is below the point of
projection.
30
Examination IV.
1. Determine the equations of motion of a body uni-
formly accelerated in the direction of its motion.
2. Deduce the formula v2 = u2 -2as.
3. A body starts from a given point with a velocity u.
and has an acceleration a opposite in direction to n; de-
termine 1) after what time will the velocity be zero? 2)
After what time will the body return to the point of pro-
jection ? 3) What is the velocity on returning to the
point of projection ? 4) What is the greatest distance
travelled over '?
4. Give the three equations of motion of a body let fall
to the ground, neglecting the resistance of the air.
5. Prove that the distances described in successive equal
intervals of time by a body, which starts from rest and is
uniformly accelerated, are as the odd numbers.
6. Trace the motion of a body projected vertically up-
wards, and shew that the downward return motion is a
plane image of the upward.
Exercise IV.
In the following crumples the directions of velocity
nml deceleration are the same, and in the ease of bodies
moving vertically the resistance of the air is neglected.
In all examples in the text-book take (/ = 980'5 or 32\ mi-
les* otherwise stated.
Log 980-5 = 2-9914476, log 321 = 15074061.
1. A stone is observed to fall to the bottom of a pre-
cipice in 9 seconds; what is the depth? Given f/ = 980.
2. The height of the piers of Brooklyn Bridge is 277
feet: how long will a stone let fall from the top take to fall
into the water?
31
3. A body is projected vertically upwards with a velo-
city of 320 vels. 1). How long will it rise ? 2). How far
will it rise ? 3). When and where will its speed be 150
miles per hour ? 4). How long will it take to rise 1000
ft.? 5). What will its speed be at that height? 6). How
far will it travel in the seventh second? Given g = 32.
1. A body starts with a speed of 1 metre per second,
and has an acceleration of 10 tachs per second; what will
its speed be after traversing 6^ metres?
5. How long would a body which is projected with a
downward velocity of 150 tachs take to fall through 15
kilometres, if there were no atmospheric resistance?
6. The speed of sound in air is constant, and at 10= C.
is equal to 33833 tachs. The depth of the well in the fort-
ress of Konigstein in Saxony is 195 metres. In what time
should the splash of a stone dropped into the well be
heard, if there were no atmospheric resistance ?
^7JWhen a bucket of water is poured into this well, the
splash is heard in 15 seconds; what is the average accel-
eration produced in the water by the resistance of the air?
8. A body whose acceleration is 10, traverses 6 metres
in 10 seconds; what is the initial speed?
9. A body uniformly accelerated moves over 31'3 metres
in the fourth second of its motion from rest; find the ac-
celeration.
10. A person, starting with a velocity of 1 metre per
second, and accelerating his velocity uniformly, traverses
960 metres in a minute; find his acceleration.
11. A body starts from a given point with a uniform
velocity of 9 kilometres per hour; in an hoar afterwards
another body starts in pursuit of the first with a velocity
of 2 metres per second, and an acceleration of 5 decatachs
per hour; when and where will the second body overtake
the first?
32
12. A body projected vertically upwards passes a point
10 metres above the point of projection with a velocity of
9805 tachs; how high will it still rise, and what will be its
speed on returning to the point of projection?
13. A body uniformly accelerated describes 65 metres
and 4*5 metres in the fourth and sixths seconds of its mo-
tion; find the initial speed and acceleration.
14. Two bodies uniformly accelerated in passing over
the same distance have their speeds increased from a to bf
and from c to <l respectively; compare their accelerations.
15. Find the acceleration when in one-tenth of a second
a speed is produced, which would carry a body over 10
metres every tenth of a second.
16. A particle is projected vertically upwards, and the
time between its leaving a point 21 feet above the point of
projection and returning to it again is observed to be 10
seconds; find the initial velocity. Given r/ = 32.
17. Two bodies are let fall from the same place at an
interval of two seconds; find their distance from one an-
other at the end of five seconds from the instant at which
the first was allowed to fall.
18. Two bodies let fall from heights of 40 metres and
169 decimetres reach the ground simultaneously; find the
interval between their starting. Given g = 980.
19. Two bodies start from rest and from the same point
on the circumference of a circle; the one body moves along
the circumference with uniform angular velocity about
the centre, and the other, starting at the same time, moves
along a diameter with uniform acceleration; they meet at
the other extremity of the diameter; compare their speeds
at that point.
20. A body, starting from rest with an acceleration of
20 tachs per second, moves over 10 metres; find the whole
time of motion, and the distance passed over in the last
second.
33
21. A body moves over 9 ft. whilst its velocity increases
uniformly from 8 to 10 vels; how much farther will the
body move before it acquires a velocity of 12 vels?
22. The path of a body uniformly accelerated is divided
into a number of equal spaces. Shew that, if the times of
describing these spaces be in A.P., the mean speeds for
each of the spaces are in H.P.
23. A body falling freely is observed to describe 24^
metres in a certain second; how long previously to this has
it been falling? Given g = 980.
24. A body is dropped from a height of 80 metres; at
the same instant another body is started from the ground
upwards so as to meet the former half way; find the initial
velocity of the latter body, and the speeds of the two
bodies when they meet.
25. A body has a uniform acceleration a. If p be the
mean speed, and q the change of speed in passing over any
portion s of the path, shew that pq = as.
26. A body uniformly accelerated is observed to move
over a and b feet respectively in two consecutive seconds;
find the acceleration.
Answers.
1. 3969m. 2. 415. 3. 10; 1600; 3& or 16J,
843|; 39 or 161; 19596; 112. 4. 150.
5. 54-9. 6. 6-9. 7. 793. 8. 10. 9. 980.
10. 50. 11. 4h. 18min. 59-8sec; 47849-6m.
12. 49025; 9904-5. 13. 1000,-100. 14. b^a2:d^c2.
15. 105. 16. 164-15. 17. 7844. 18. 1.
19. ?r:4. 20. 10; 190. 21. 11. 23. 2.
24. 2800: 2800,0. 26. b— a.
Chaptek V.
Inertia. Mass.
38. Inertia is the inability of a body to alter its own
condition of motion or rest. If a body be at rest, it re-
mains so; if it be in motion, it goes on moving with the
same velocity, i.e., with constant speed in a straight line;
and if it be rotating, it goes on rotating with the same
angular velocity, about the same axis, which maintains a
constant direction ; unless some other body interfere with
it. To change the state of rest or motion of a. body re-
quires the presence of another body. The term force is
applied to the action of a body in altering the status quo
of another body.
39. Inertia may be called a negative property, and yet
it is one of the most obtrusive properties of matter. It
is lucidly illustrated in railway and horse-riding accidents,
in vaulting, jumping, and circus-riding, in shaking the
dust from off a book, in the difficulty of driving over
smooth ice, and in the action of a fly-wheel, which is used
to regulate either an irregular driving-power, as in a foot-
lathe, or an irregular resistance, as in a circular saw cut-
ting wood. The tendency of bodies moving in circles, to
fly off at every instant along the tangent, commonly but
misleadingly called centrifugal force, is just inertia.
Herein we have an explanation of the spheroidal form of
the earth, and of the decrease of a body's weight, as we
approach the equator. On letting a bullet fall from the
top of a high tower or down a deep mine, it will, on account
of its inertia, be found to fall somewhat to the east of the
point vertically below that from which it fell, thus afford-
ing an ocular demonstration of the earth's rotation from
west to east. The rotations of the earth and other mem-
35
bers of the solar system afford beautiful examples of
[inertia as regards rotation. The constancy of direction of
the earth's axis, (except in so far as it is interfered with
by the sun and moon) furnishes the most important step
in the explanation of the changes of the seasons. By its
inertia that interesting physical toy, the gyroscope, will
prove that it is the earth and not the sphere of the heavens
which daily rotates. The same principle, applied to the
j plane of oscillation of a pendulum, enabled Foucault to
|give one of the most convincing experimental proofs of the
earth's rotation from west to east.
40. Sir Isaac Newton clearly enunciated the inertia of
[matter in his First Dynamical Law;
Every body remain* in its state of rest or of uniform
[motion, except in so far as it may he compelled by im-
| pressed force to change that state.
In a scholium he referred to inertia as regards rotation.
Here indeed there is a difficulty, for evidently the individ-
ual small particles of the rotating body move in circles,
and must therefore be acted on by forces amongst them-
selves: else, on account of inertia, they would move in
straight lines. However, when by internal forces the
relative positions of the particles are fixed, the body will
be as inert in its rotation as in its motion of translation.
41. Every body offers resistance to any change of its
state of rest or motion. When the same force acts on
different bodies it is found that the changes from the pre-
vious states of rest or motion are different, and this fact
is expressed by saying that the bodies differ in mass.
Mass, thus, is a property in which bodies may differ, just
as they may differ in colour, volume, or weight. It might
be defined as the dynamical measure of a body's inertia.
or as the capacity of a body to resist change of state of
rest or motion.
36
42. The difference in mass of different bodies {e.g. of
balls of wood, ivory, lead, and iron, of different radii) may
be lucidly illustrated by suspending the bodies by strings,
and allowing the same spring, bent through the same
angle, to act upon them in succession so as to give the
bodies a horizontal motion. It will be found that the
accelerations imparted will be very different.
The accelerations so produced would be the same if one
of the bodies were at the surface of the moon, another at
the sun's surface, and a third at the surface of Jupiter,
where their weights would be respectively \th, 28 times,
and 2| times as great as at the earth's surface.
43. How is mass measured? When the same force is
applied to different bodies, the masses of the bodies are
defined as inversely proportional to the accelerations pro-
duced.
Hence if the same force acts upon two bodies, and pro-
duces equal accelerations, the masses of the bodies are
defined as equal to one another ; but if the accelerations
be in the ratio of m:n, the masses are defined to be in the
ratio of n:m.
The C. Gr. S. unit of mass is called a gram. It is the
mass of a cubic centimetre of water at 4C C (under the
mean atmospheric pressure). The French unit of mass is
the kilogramme, and is the mass of a litre of water at 4°C.
The English and the F. P. S. unit of mass is a pound
(avoirdupois). The pound was chosen perfectly arbitrar-
ily. The present standard pound is a cylinder of platin-
um with a groove near one end. It is denoted as the P.
S. or Parliamentary Standard, and is carefully preserved
in London. The unit of mass is the third of the funda-
mental units in the C. G. S. and F. P. S. systems.
44. Let it be observed that by means of one force the
masses of all bodies can be theoretically determined.
When the same force acts upon bodies of the same mate-
37
rial, e.g. two pieces of iron at the same temperature, it is
found that the accelerations are inversely as their vol-
umes, (take as an illustration the opening of doors of the
same kind of wood but of different sizes) ; but not so for
bodies of different material, (take as an illustration the
opening of a wooden and of an iron door). Hence it
follows that the masses of bodies of the same materia]
{and at the same temperature and press are) are directly
proportional to their volumes, but not so for bodies of
different material.
45. These facts lead to the consideration of density and
specific mass. The density of a substance is the mass per
unit of volume. Hence in using C. Gr. S. units the density
of water at 4° will be represented by 1. A cub. cm. of
gold at 0C is found to be 193 grams, of rock-crystal 266
grams, of mercury 136 grams, of sea-water 1027 grams, of
dry air (under the mean atmospheric pressure) 0'0012932
gram. These facts are expressed by saying that the den-
sity of gold is 19*3, of rock-crystal 266, of mercury 136,
of sea-water 1-027, of dry air 00012932.
46. The density of water, as of all other substances,
varies with temperature, and (under the mean atmospheric
pressure) is a maximum at 4C C. Hence it is that in de-
fining unit of mass, the water is taken at this tempera-
ture. The density of water, as of all liquids, is very little
changed by ordinary changes of pressure, so that it is
hardly necessary to state, in defining unit of mass, that
the water is supposed to be under the mean atmospheric
pressure, the changes of atmospheric pressure making only
immeasurably small changes of density.
47. The density of a body may be uniform, i.e. every
part having the same density, or it may be variable. In
the latter case we may speak of the density at any point
of the body, or of the mean density of the whole body.
38
48. The specific mass of a substance is the ratio of the
mass of any volume of the substance to the mass of an
equal volume of water at 4°C. Whatever units be used,
the specific masses of substances will evidently be repre-
sented by the same numbers, and with the C. G. S. units
the density and specific mass of any substance will be
represented by the same number. Hence the terms density
and specific mass are frequently used indiscriminately, in
the sense of specific mass, and almost always so when
F. P. S. units are used.
The specific volume of a substance is the ratio of the
volume of any mass of that substance to the volume of an
equal mass of water at 4°C, and is evidently the reciprocal
of the specific mass. The rarity of any substance or
medium is the volume per unit of mass. Like density and
specific mass the terms rarity and specific volume are
generally used indiscriminately. Thus (Art. 45) the rarity
or specific volume of dry air at 0° is 773*3.
49. Using C.G. S. units, the relation between the mass
(m), the volume (F), and the density (d) of a body is given
by the equation m= Vd. Using F.P.S. units, ra=62'4 Vd
expresses the same relation, since a cubic foot of water at
4° is 62'4 pounds nearly. In the above equations we see
what an immense advantage the C. G. S. system has over
the F. P. S. system of units.
50. The mass of a body is sometimes defined as the
measure of the quantity of matter in it, or as the dynam-
ical measure of the quantity of matter in it. Since we
do not know the ultimate nature of matter this can hardly
be scientifically correct. We only know the properties of
matter, and can only measure its properties. Why then
should quantity of matter be measured by one of these
properties, mass, rather than by any other. We might
reason thus: when the same quantity of heat is applied to
bodies of the same substance, the changes of temperature
39
produced are inversely proportional to their volumes; but
when applied to bodies of different substances, the changes
of temperature are not inversely proportional to their
volumes. We express this fact by saying that bodies
differ in thermal capacity, and we define the thermal
capacities as inversely proportional to the changes of tem-
perature produced. Just then as with mass we might
define the thermal capacity of a body as the thermal
measure of the quantity of matter in it. We should then
find the thermal measure and measurement by mass were
quite different.
So long as we are dealing with bodies of one substance
there are many ways in which we may measure quantity
of matter quite intelligibly, e.g. by volume, by weighing
in the same place, in the case of food by the length of
time it will supply nourishment, in the case of fuel by the
amount of water it will boil away, or by the amount of oxy-
gen gas necessary for its complete combustion, and all
these measurements would be found to agree with one an-
other as well as with the measurement by mass. But
when we come to deal with bodies of different substances,
none of these measurements will be found to give results
consistent with one another.
51. The reason doubtless why mass is stated to measure
the quantity of matter in a body, is that this is the most
familiar. and most easily measured of the few properties
of matter which remain measurably invariable through
whatever changes the body may pass. Thus whilst by
pressure, motion, heat, chemical action, or other agencies,
we can easily alter the other measurable properties of a
body, such as its volume, weight, elasticity, or thermal capa-
city, its mass, through whatever changes the body may
pass, remains unchanged. This may be clearly illustrated
by many experiments, e.g. by dissolving a piece of sugar
in tea, by freezing a body of water, by mixing alcohol and
40
water, and, generally, in all chemical combinations.
Whence the great law which forms the foundation of
chemical science, the Conservation of Mass : — Through
whatever changes matter may pass, the total mass remains
unchanged. Hence the total mass of the universe is in-
variable.
Examination V.
1. Define inertia, and state the different forms thereof.
2. Give various illustrative examples of inertia.
3. What is centrifugal force ? Suggest a better name
for it, and give illustrations thereof.
4. How may the earth's rotation from west to east be
proved by ocular demonstration ?
5. Enunciate Newton's First Dynamical Law.
6. Define mass. How is it measured ?
7. Describe a simple experiment to shew difference of
mass in different bodies.
8. Name and define the units of mass in the C. G. S.
and F. P. S. systems of units.
9. What relation exists between the volumes and masses
of bodies of the same material ? How is this proved ?
10. Define density, specific mass, specific volume, and
rarity. Give the densities of a few common substances,
and the rarity of air.
11. Why is water at 4° taken as the standard substance
in measuring mass and density?.
12. Give algebraical equations connecting the mass,
volume, and density of a body. In the case of a body of
variable density how do you express the relation?
13. Criticise the usual definition of mass as the measure
of the quantity of matter in a body. /
14. How did the above definition probably arise?
15. Enunciate the principle of the Conservation of Mass.
41
Exercise V.
1. A rectangular block of limestone is 2 metres long,
15 metre broad, and 1 metre thick. If 2-7 be its density,
find its mass.
2. The sides of a canal shelve regularly from top to bot-
tom. The width of a section at the top is 10 metres, at the
bottom 5 metres, and the depth is 3 metres. If the canal
be filled with water to a depth of 25 metres, find the mass
of water per kilometre of length.
3. If the density of sea-salt is 2-2 and of sea-water
r027, find the mass and volume of salt obtained in evap-
orating 100 litres of sea-water, if no contraction took place
in solution.
4. The density of copper is 8*8, of zinc 7, and of brass
formed from these 8'4; find the quantity of copper in 100
grams of brass.
5. The mass of a sphere of rock-crystal is 400'5, and
its radius 3*3; find its density.
6. Find the mass of the earth, supposing it to be a
sphere of radius 6371 kilometres, and of mean density 567.
/7J Equal masses of copper and tin, whose densities are
88 and 7-3, are melted together; what would be the density
of the alloy if no contraction or expansion took place?
'8. When 63 litres of sulphuric acid, whose density is
1'84 is mixed with 24 litres of water, the volume of the
mixture is 86 litres; find the mass and density of the
mixture.
9. The mass of a nugget of gold-quartz is 350, and its
density it 7'4; find the mass of gold in it. See art. 45.
10. The density of sea-water is 1027; 100 litres of sea-
water are frozen, and 20 kilograms of ice free from salt
formed therefrom; what is the density of the residue?
11. What is the density of mercury, if 9 cubic inches
have a mass of 4'42 lbs?
42
12. The density of milk is 103; how much water must
have been added to 10 gallons of milk to reduce its density
to 102.
13. From the summit of the Eiffel tower at Paris (lat. 48:
50'. r/ = 981) a bullet is let fall 300 metres ; neglecting the
resistance of the air, find how far to the east of the point,
which is vertically under the point from which the bullet
was dropped, it reaches the ground. Will the atmosphere
increase or diminish the eastward deflection '? How ?
14. When a vessel is filled with equal volumes of two
liquids, the density of the mixture is 9/8 of what it is
when the vessel is filled with equal masses of the same
liquids ; find the ratio of the densities of the two liquids.
15. Two liquids whose densities are as 1:2 are mixed to-
gether, (1) by masses in the ratio of the volumes of equal
masses, (2) by volumes in the ratio of the masses of equal
volumes ; find the ratio of the densities of the mixtures.
Answers.
1. 8100 kilogrs. 2. 17708-3 tonnes. 3. 4950; 2250.
4. 815. 5. 266. 6. 61418 XlO17 tonnes. 7. 7"98
8. 139920; 1-63. 9. 260. 10. 1-034. 11. 136.
12. 5 gals. 13. 11-23. 14. 1:2. 15. 18:25.
Chapter VI.
Moment urn. Force.
52. The momentum of a particle is a property depend-
ing upon its velocity and mass, the direction of momentum
being the direction of motion, and the magnitude of
momentum being defined as proportional to the mass and
speed conjointly. The term quantity of motion was used
by Newton for momentum.
To vividly realize momentum let a person bathe close
to a waterfall, say 200 ft. high, when he will feel the <lroj)s
of water, which separate from the main mass, strike his
body as if they were sharp stones. If he attempted to
enter the main mass of falling water he would be roughly
thrown on the ground.
53. In a scientific system of units, the unit of momen-
tum is best defined as the momentum of a particle of unit
mass moving with unit speed. Hence the C. G. S. unit of
momentum is a gromUieh, which is the momentum of 1
gram moving with a speed of 1 tach. Similarly the F.P.S.
unit of momentum is a pound 'ret, the momentum of 1
pound moving with a speed of 1 vel. The unit of momen-
tum evidently involves each of the three fundamental
units of length, mass, and time, the first two directly and
the third inversely.
The equation M — mr expresses the numerical relation
between the momentum, mass, and speed of a moving
particle.
54. The time-rate of change of momentum at any in-
stant will evidently be measured by the acceleration of the
moving particle at that instant, and its mass conjointly,
and is called the acceleration of momentum.
44
(
Force is that aspect of any external influence exerted
on a body which is manifested by change of momentum.
Whenever the momentum of a body changes, a force is
said to act on the body.
If a definite change of momentum takes place in an
immeasurably short time, {e.g., when a cricket ball is
struck by a bat), the action is called an impulse, the term
force being usually applied when a finite time is required
to produce a finite change of momentum, {e.g. when a
body falls to the ground). All forces can be conceived to
be made up of immeasurably small impulses, just as a
curved line can be conceived to be made up of immeasur-
ably short straight lines.
55. An impulse is measured in magnitude and direc-
tion by the whole change of momentum produced. A
force is measured in magnitude by the acceleration of
momentum, and its direction is the direction of the change
of momentum. This is what Newtou taught in his Second
Dynamical Law:
Change of momentum is proportional to the impressed
force and takes place in the direction of the straight line
in which the force acts.
By impressed force Newton meant external to the body
concerned. It is well in defining force to avoid the word
cause. All that we are aware of is a change of momentum
and the word force is conveniently used as a measure of
the rate of this change. Under energy, one of the most
important properties of matter, the student will learn that
force may be defined as the space-rate of transference of
energy, i.e. the rate of expenditure of energy per unit of
length.
56. The unit of force is that force which produces unit
of momentum per unit of time. Hence the C G. S. unit
of force is that force which produces 1 gramtach per
second, or that force which acting upon a particle whose
45
mass is a gram gives it an acceleration of 1 tach per
second. This is called a dune. Similarly the F.P.S. unit
of force is that force which produces 1 poundvel per
second, or the force which acting upon a particle whose
mass is a pound gives it an acceleration of 1 vel per
second, and this is called a poundal.
When a force of / dynes or poundals acts upon a
particle whose mass is m grams or pounds, and produces
an acceleration of a tachs or vels per second, the equation
which expresses the numerical relation between/, m and a is
/= ma
Force like velocity, acceleration, and momentum, is
completely represented by a straight line.
57. What does Newton's second dynamical law really
teach ?
1). It defines the measurement of mass and force.
Just as it is theoretically possible to measure all masses
by means of one force, so is it possible to measure the
magnitude of all forces theoretically by one mass.
When different forces act upon the same body, the
magnitudes of the forces are by definition directly
proportional to the accelerations produced. It is indeed
evident that the mass of any body is measured in grams
by the reciprocal of the acceleration in tachs per second
produced, when a dyne acts upon it ; and any force is
measured in dynes by the acceleration in tachs per second
produced, when it acts upon a body whose mass is a gram.
2)" It enunciates the important experimental fact: With
whatever force different masses be measured, and with
whatever mass different forces be measured, the measure-
ments will always be alike. 3). It asserts that the effect
of a force depends in no way upon the motion of the body,
and that when more than one force is acting on the body,
each force produces its effect quite independently of the
others.
46
58. It has been pointed out (Art 38). that whenever a
force acts, there are always two bodies concerned. We
generally speak of one of the two as receiving a change of
momentum, and of the other as being concerned in the
production of this change. Newton clearly pointed out in
his Third Dynamical Law that the action was a mutual
one ; that change of momentum was received by both
bodies, equal in magnitude but of opposite direction:
To every action there is always an equal and contrary
reaction ; or the mutual actions of any two bodies are
always equal (in magnitude) and opposite in direction.
The word force is properly used when we consider the
effect of the action between any two bodies in changing the
momentum of one of them only. Stress is a term applied
to the mutual action between any two bodies, when there
is special reference to the dual character of that action,
as enunciated by Newton. This third law tells us that all
dynamical actions between bodies are of the nature of
stresses. When a body falls to the ground under the
action of its weight, the earth rises to meet it with an
equal momentum. Since, however, the mass of the earth
is so very much greater than that of any body on its sur-
face, the motion of the earth is so small that it may be
neglected. When two like magnetic poles, free to move.
arc brought near one another, it will be found that the
repulsion is mutual. WThen the loadstone attracts a piece
of iron, the iron attracts the loadstone with an exactly
equal force. When the table is pressed by the hand, we
fee! that the hand is likewise pressed by the table. When
a horse draws a canal boat by means of a stretched rope,
the horse is drawn backwards with as great a force as the
boat is drawn forwards. This may easily be proved by
cutting the rope, when immediately the horse falls for-
wards. This is further seen, when we reflect that relatively
to the boat the horse does not move at all. If two boats
47
are floating and one is drawn towards the other by means
of a rope, the latter is also drawn towards the former with
an equal momentum ; i.e. the same rope is pulling both
boats at the same time, and with equal force but in
opposite directions. When two railroad trains or other
bodies collide, the change of momentum in the one is just
equal, and opposite indirection, to the change of momentum
in the other whatever be the original direction or rate
of motion of either train.
59. Since momentum has direction as well as magni-
tude, it at once follows from the above law, that the total
momentum of two bodies is not altered by their mutual
action. From this the important principle called the
Conservation of Momentum is at once deduced:
The total momentum of any body, or system of bodies,
cannot be altered by the mutual actions of its several parts.
As an illustration of this principle let us consider the
kick of a gun. Here we have a system consisting of 3
bodies, the gun, the gas formed from the gunpowder, and
the ball ; it will be at once seen that the backward momen-
tum of the gun is just the equivalent of the forward
momentum of the ball.
The total momentum of the universe is a constant
quantity is an immediate deduction from the same
principle.
60. Different names are given to different aspects of
force, such as pressure, tension, attraction, weight, repul-
sion, friction.
Pressure calls up the idea of pushing. This term is
applied to a stress between particles close together, when
the direction of each force is towards the particle acted
upon, eg. the pressure of a fluid on the containing vessel.
Tension calls up the idea of pulling. This term is
applied to a stress between particles close together when
48
the direction of each force is away from the particle acted
upon, e.g. the tension of a stretched cord.
Attraction is a term applied to forces exerted between
bodies, when there is no sensible material medium through
which the force is exerted, and in consequence of which
the bodies approach one another. The force between two
unlike magnetic poles is a familiar case of attraction.
Weight, a well known form of attraction, is applied to
the force exerted by the earth on any body at its surface.
Forces are often conveniently measured by the weights of
bodies of known mass. Thus, when a force of p grams is
spoken of, a force equal to the weight of a body whose
mass is p grams is meant. It would be better to speak of
a force of p grams-weight.
Repulsion is a term generally applied to forces between
bodies, when there is no sensible material medium through
which the force is exerted, and in consequence of which
the bodies recede from one another. The force between
two like magnetic poles is a familiar example of repulsion.
Resistance is a term frequently applied to any force
opposing the motion of a body, i.e. producing a negative
acceleration. One of the most familiar and important of
such resistances is the ubiquitous force of friction, a term
applied to that force which is called into play, when one
body moves or tends to move over the surface of another
body. It is principally the force of friction which a loco-
motive works against in pulling a trai?! along. The resist-
ance which bodies experience in falling through the air is
largely the force of friction between the bodies and the
aerial particles they rub against.
61. In modern nomenclature the science of force is
called Dynamics. It is divided into Statics and Kinetics.
Statics treats of equilibrium or the balancing of forces.
It is chiefly concerned in determining the relations which
must exist amongst a set of forces which keep a body at rest.
49
Kinetics investigates the forces acting on bodies having
varying motions. The exact determination of the motions
of the Solar System is the grandest problem in Kinetics,
and is commonly known as Physical Astronomy.
Kinematics is the science of motion, when studied with-
out any reference to mass. It forms an appropriate intro-
duction to Kinetics. Chapters II., III., IV. belong to
Kinematics.
Mechanics treats of the construction and uses of
machines, and the relations of the forces applied to them.
It forms the practical side of Dynamics.
Examination VI.
1. Define momentum ; name and define the unit of
momentum in the C.G.S. and F.P.S. systems of units ; and
write down the numerical relation between momentum,
mass, and speed.
2. Define force and impulse, and give the measures and
units of these in the two systems of units.
3. Enunciate Newton's Second Dynamical Law, and
state fully all that it teaches. Express the law in
algebraical form.
4. Enunciate Newton's Third Dynamical Law and
give several illustrations thereof.
5. Enunciate and illustrate the Conservation of
Momentum.
6. Define the terms pressure, tension, attraction, weight,
repulsion, resistance, friction.
7. What is meant by a force of 10 pounds, or a force of
10 kilograms ? Give better expressions for these.
8. Define the terms Dynamics, Statics, Kinetics, Kine-
matics, Mechanics.
9. How can momentum be strikingly felt ?
50
Exercise VI.
1. A kilodyne acts upon a body at rest whose mass is
50 grams ; find the speed and distance passed over at the
end of 10 seconds.
2 A body whose mass is 5, has an acceleration 2. At
one instant the speed is 10 ; what is the momentum a
minute afterwards ?
3. Find the acceleration produced by a megadyne act-
ing on a solid sphere whose diameter is a decimetre and
density 10.
4. A force of 50 kilodynes acts upon a body which
acquires in 10 seconds a speed of a kilotach ; find the mass
of the body.
5. The mean radius of the earth is 20,902,000 feet, its
mean density 567, its mean distance from the sun 927
million miles, and the time of its revolution around the
sun 365j days ; compare its momentum with that of a train
of 10,000 tons, rushing along at 60 miles an hour.
6. The distance of Jupiter from the sun is 5*2 times
thai of the earth, its period 1X'!21, days, its mass 310 times
that of the earth ; compare the momenta of Jupiter and
of the earth.
7. A body of 10 grams has a uniform acceleration of
10 m. per min. per min.; what force is acting upon it ?
8. A body acted on by a uniform force is found to be
moving, at the end of the first minute from rest, with a
speed which would carry it through 20 kilometres in the
next hour ; compare the force with the weight of the
body which would give it an acceleration g — 9805.
9. A kilogram is supported by a string 20 m. long.
The string's mass is 2 grams per metre. Find the tension
at the middle point, and at 5 decim. from the upper end.
10. Compare the momentum of a man, whose mass is
150 lbs. and latitude that of Kingston, Ont. (44s 13'),
arising from the earth's rotation, with that of a steamship
51
pf 10,000 tons, going at the rate of 15 miles an hour. A
(sidereal day = 86.164'1 sec. See also ex. 5.
11. A body, acted on by a force of 100 kilodynes, has
}!its speed increased from 6 to 8 kilometres per hour in
Jpassing over 84 metres ; find the mass of the body.
12. A body of 1 kilogram is acted on by a uniform
[force in the direction of its motion, and is found to pass
lover 905 and 805 cm. in the 10th and 20th second of its
Imotion from rest ; find the force acting upon it and its
•initial speed.
13". Two bodies, acted on by equal forces, describe the
pame distance from rest, the one in half the time the other
does ; compare their final speeds and momenta.
• 14. Two bodies of equal mass uniformly accelerated
ijfrom rest, describe the same distance, the one in half the
Itime the other does; compare the forces acting on the bodies.
^15J Two balls, one of silver and the other of ivory,
iwhose diameters are as 1 to 2, are subjected to equal
fimpulses ; the speeds produced are as 22 to 15 : compare
[the densities of silver and ivory.
16. If a ship be sailing with uniform velocity, what
Srelation must exist between the driving force and the
iresistances of the air and water ?
17. The density of lead is 11*4 and of cork 024. Two
| balls of these substances, whose diameters are as 1 to 10.
jare acted upon by equal forces during the same time ;
1 compare their momenta and speeds.
Answers.
1. 200; 10m. 2. 650. 3. 191. 4. 500.
5. 74937 X101H. 6. 1359:10. 7. 25/9.
8. 1 : 105-894. 9. 1020 grs. - wt. ; 1039 grs. - wt.
10. 1:2685. 11. 77760. 12. 10 kilodynes; 1 kilotach.
13. 2:1; 1:2. 14. 4:1. 15. 60:11. 16. Equal.
17. Equal: 400:19.
Chapter VII.
Weight. Gravitation.
62. Weight is the attraction of the earth for every body
on its surface, in virtue of which any body, unless it is
supported, falls to the ground. It is also called the force
of gravity. According to their weights bodies are called
heavy or light. All bodies at the same place are found to
fall in the same direction relatively to the surface of the
earth, and bodies falling in contiguous places (practically.
within a decametre of one another) fall in parallel straight
lines. The direction in which a body falls at any place is
called the vertical direction at that place, and is easily
found by means of a plumb-line. Any direction at right
angles to the vertical is called horizontal. The surface of
any liquid (practically, within an are of area), at rest
relatively to the earth, is a horizontal plane, except just
where it meets the vessel containing it.
63. When different particles fall at the same place and
in vacuo, the vertical acceleration is found to be the same
for all. This is proved by the guinea and feather experi-
ment. A simple but very instructive experiment which
illustrates this fact, may be made by cutting out a piece of
paper just sufficient to cover the mouth of a tin lid of a
small box, and then allowing the lid and piece of paper to
fall simultaneously from the same level, 1) when they are
apart. 2) when the paper covers the mouth of the lid.
Hence (Art. 56) we deduce the very important fact:
The masses of bodies are directly proportional to their
weights at the same place.
w=mg, g is constant, .'. moc ic
64. The following extract from Lucretius, De rerum
natura, shows that the fact, that all bodies would fall
53
equally fast in vacuo, was believed in, though not proved,
2000 years ago:
In water or in air when weights descend,
The heavier weights more swiftly downwards tend;
The limpid waves, the gales that gently play,
Yield to the weightier mass a readier way;
But if the weights in empty space should fall,
One common swiftness we should find in all.
65. Weight is measured like any other force in dynes
or poundals. Thus the weight of a body whose mass is 1
gram is g or 980'5 dynes, and the weight of a pound of
matter is g or 32^ poundals. Forces are often conveniently
measured in terms of the weights of known masses. Thus
we read of a force of a kilogram-weight or a force of 10
lbs. wt., and these expressions are generally abbreviated
into a force of a kilogram or a force of 10 lbs. The
measure of a force in terms of weight is called its gravita-
tion measure, that in dynes or poundals being called in
contradistinction its absolute measure. Since g varies
with latitude, it is evident that the gravitation measure of
a force has not a definite value, unless the place be stated.
The dyne or poundal on the other hand is independent of
place and time, and is hence called an absolute unit.
66. The simple relation between the weights of bodies
at the same place and their masses gives the best practical
method of measuring the masses of bodies, as is done in a
common balance. Observe that in a common balance, by
comparing the weights of bodies with those of standard
masses, we really measure mass; whereas in a spring bal-
ance we directly measure weight.
The law which explains to us the measurement of
weight by means of a spring balance is known as Hooke's
law: The extension, compression, or distortion of a solid
body, icithin the limits of elasticity, or the compression of
54
a liquid, is directly proportional to the force which pro-
duces it.
67. The direct proportionality between the masses of
bodies and their weights explains why mass and weight
are constantly confounded with one another. The follow-
ing illustrations in which these two properties of matter
are contrasted, will assist the student to apprehend their
difference:
1. a). The mass of a body is the same at whatever part
of the earth's surface it be.
6). The weight changes with change of place, and is
different at the Equator, at either Pole, or at the summit
of the Rocky Mountains, from what it is in the class-room.
2. a.) The opening of a room door is essentially a
question of mass ; and, however heavy the door may be, if
the hinges are truly vertical and well oiled, a small child
may open it, though slowly.
6). If the same door formed the lid of a box, and swung
on horizontal hinges well oiled, the child could not open
it, unless he had strength enough to exert muscular force
equal to at least half the weight of the door.
In either case the child has to overcome the force of
friction, which, though greater in the first than in the sec-
ond case, is in either case small.
3. a). In moving a cart along a level road the horse
has to exert a greater force at starting than afterwards,
because he has to exert force to give the mass a given
velocity, i.e. to produce momentum. After having started
he has only to balance the force of friction.
b). When, however, he comes to a hill he has again to
put forth his strength, for now he has, in addition to the
force of friction, to overcome part of the weight oi the cart,
4. «). The action of a fly-wheel, or of a small hammer,
depends entirely upon its mass.
55
6). The action of a large steel hammer, such e.g. as the
125 ton hammer at Bethlehem, Pennsylvania, worked by
steam, and used in shaping large and massive bodies,
depends essentially upon its weight.
5. a) . In athletic sports the " long jump " is essentially
a question of mass.
b). In the " high jump " weight in addition has to be
considered. Hence the actual distance of the high jump
is never so great as that of the long jump.
6. a). In an undershot water-wheel the miller depends
upon the momentum (and hence also the mass) of the
running water to drive the wheel.
b). In an overshot water-wheel he depends upon the
weight of the water which enters the buckets of the wheel.
68. How is g, the acceleration due to the force of gravity,
at any place measured ? The most accurate method of
finding this important physical quantity is by means of
pendulum experiments. There is, however, one method of
finding a fairly accurate value of g, which at this stage the
student can understand. This is by means of the well-
known physical instrument called Attwood's machine.
The essential part of the apparatus is a grooved wheel
which turns upon an axle, each end of which rests on two
other wheels called the friction wheels, so that the -friction
on the axle of the first wheel is reduced to a minimum ;
over this wheel passes a fine thread connecting two bodies
of different weights. If m and m! be their masses, and m
be the greater, the bodies will move on account of the
greater weight of m with an acceleration equal to (m - m') g
-T- (m-\-m'), if we neglect friction and the masses of the
wheels and thread. This acceleration can evidently be
made as small as the experimenter pleases, by making the
difference between m and m' small enough. By a clock and
suitable adjuncts the acceleration of the moving bodies
can be accurately measured, and therefore g approximately
determined.
56
The following values of g at the sea-level have been
determined by experiment and calculation :
Latitude. Value of g.
Equator 0= 0' .978-1
Sydney, N.S.W 33°51' 979-6
Tokyo 35°40' 979-8
Washington 38°54' 980-1
Rome 41=51' 9803
Kingston, Out 44c13' 980-5
paris 18°50 980-9
London 51c30' 981-2
Berlin 52°30 981-3
Edinburgh 55°57' 981 -5
St. Petersburgh 59°55' .981-9
Pole 90= 0' 9831
From the above values of g it is seen that the maxi-
mum variation over the earth's surface is about \ p.c. of
the mean value.
69. The profound investigations of Sir Isaac Newton
into the attractions between a few of the larger particles
of matter in the universe, in order to explain the motions
of the planets and their satellites, and especially the
motion of the moon, the nearest neighbour in the universe
to our own abode the earth, led this remarkable philos-
opher to the inevitable conclusion that it is the weight of
the moon which keeps her revolving around the earth, and
that weight is but a particular case of gravitation, which
pervades the whole universe, and which acts according to
the following law.
Law of Universal Gravitation: Every particle of.
matter in the universe attracts every other with a force,
whose direction is that of the line joining the particles, and
whose magnitude is directly as the product of their masses
au<l inversely as the square of their distance.
57
Weight is thus only a particular case of a force which
has been found to govern the motions of every body in the
universe. Since Newton published his law to the world,
its truth has been confirmed by every astronomer who has
lived after him. By means of it, not only have the mi-
nutest perturbations in the motions of the heavenly bodies
been rationally explained, but eclipses, transits, return of
comets, and other heavenly phenomena have been pre-
dicted years before they took place, and actually did take
place a few seconds within the times predicted. The
nautical almanac, which guides our ships across trackless
oceans, is but a book of predictions depending on the
truth of this law. A great triumph in its application was
the discovery of Neptune, the most distant planet of our
solar system, which, though invisible to the naked eye,
was discovered by mathematical calculations, which in-
structed the astronomer how to direct his telescope. Nor
is this law confined to our own small solar system, but
extends to systems millions of millions of millions of miles
beyond our own, where not only does satellite revolve
around planet, and planet around sun, but where one sun
revolving around another has its motions governed by this
same grand law.
70. We may express the law algebraically by the for-
mula f—G. — ;— , where m and m' are the masses of two
particles, r their distance apart, and / the gravitation be-
tween them. G is a constant, and measures the gravitation
between two particles, each of unit mass, and unit distance
apart. To determine G, in C. Gr. S. measure, we may
observe that, by comparing the attraction of the whole
earth with that of a large leaden ball, and by other means,
the mean density of the earth has been calculated to be
5"67. Hence, taking the earth's mean radius r to be 6,371
kilometres, and the mean attraction of the earth for a gram
58
of matter at the sea-level (allowing for centrifugal force)
to be 982"3 dynes, we get
982-3 = G. i^L?!*?"?7 = G | n X 637,100,000 x 5-67
.-. G=6-5xl0-8
Hence two particles, one centimetre apart, and each of one
gram mass, attract one another with a force of 6*5 X 10-8
dyne.
Examination VII.
1. Define weight. By what other name is it known ?
How is the direction of weight practically found ?
2. How is it proved that g is the same for all bodies at
the same place?
3. Define the terms vertical and horizontal, and give an
illustration of each.
4. Prove that the weights of bodies at the same place
are directly proportional to their masses.
5. Explain wh-at is meant by an absolute unit of force,
and express, in absolute units, forces of a pound-weight
and of a kilogram-weight.
6. How would you prove to a person that the weight of
a body is less, the nearer it is to the equator?
7. How are mass and weight practically measured?
8. (iive illustrations of weight and mass, which contrast
with one another, to shew the difference between these two
properties of matter.
9. How is the value of g experimentally determined?
Describe the essential parts of Attwood's machine.
10. Enunciate the law of universal gravitation, and
shew how to determine approximately the gravitation
between two grams of matter at the distance of a centi-
metre from one another.
59
11. Give the values of (j at the Equator, Kingston Out.,
Paris, and the North Pole, true to a decitach per second.
12. Enunciate Hooke's Law, and apply it to the spring
balance.
13. If a merchant buys goods in London by means of
a spring balance, and with the same balance sells in King-
ton Ont., will he gain or lose in the transaction ? Why ?
By how much p.c. ?
14. Show that a poundal is nearly half an ounce-weight;
and that a dyne is nearly a milligram-weight.
Exercise VII.
1. A body whose mass is 10 grams is falling in vacuo;
what is the force acting on it, and its momentum at the
end of 10 seconds from rest?
(%) A force of 50 grams weight acts upon a body which
acquires in 10 seconds a speed of 3922 tachs; find the
mass of the body.
3. A force produces in a sphere of radius 10 cm. and
density 10 an acceleration of 100 tachs per second ; find
what weight the force could balance.
4. If 250 lbs. be hung to the lower end of a rope 80 ft.
long, find the tensions at the ends, the middle point, and
20 ft. from the upper end, the mass of the rope being 4 oz.
per foot.
5. Jin Attwood's machine, if 10 kilograms be the mass of
one Taody, and 15 kilograms that of the other ; find the ac-
celeration of momentum, and the speed at the end of two
seconds.
(5. A force of 1^ pounds-weight acts upon a mass <>f '2
pounds ; what is the speed after traversing a mile ?
7. A mass of 10 pounds is acted on by a uniform force,
and in 4 seconds passes over 200 feet ; express in gravita-
tion measure the force acting.
60
8. If the earth's mean radius be 20,900,000 feet and
the mean attraction of the earth for a pound of matter at
the sea level be 32'23 poundals, find the gravitation
between two pounds of matter one foot apart.
9. In Attwood's machine one mass is known to be 10
lbs., and the distance described in 2 sec. is found to be 16
ft. 1 in.; find the other mass.
10. Find the diameters of two equal spheres of gold,
such that the gravitation between them, when they just
touch, is a dyne.
11. Find the unit of time, if a metre be the unit of
length, a gram the unit of mass, and a gram-weight the
unit of force, in a scientific system of units.
12. Find the unit of length, if a second, gram, and
gram-weight be the units of time, mass, and force.
13. Find the unit of mass, if a second, centimetre, and
gram-weight be the units of time, length and force.
14. A sphere of rock-crystal of density 2*66 has a diam-
eter 65 cm.; find its volume, mass, and weight at Rome.
15. A body of 6 lbs. pulls by its weight another body
of 4 lbs. along a smooth horizontal table ; find the time
taken to move through 965 ft. from rest, and the distance
passed over in the last second.
16. Find the tensions of the three parts of a string,
which supports at different heights bodies of 12, 6, and 4
lbs. respectively.
17. Oxygen combines chemically with hydrogen to form
steam in the proportion of 8 parts by mass of oxygen to 1
of hydrogen. If the gases be weighed by means of a
spring balance graduated at Edinburgh, how many milli-
grams-weight of oxygen at the Ecpaator will combine with
100 milligrams-weight of hydrogen at Edinburgh to form
steam?
61
18. Determine the mass of steam so formed, and its
weight at Kingston Ont., as indicated on the above spring
balance.
19. Answer the above (17 and 18) when the gases are
weighed in a common balance, and explain your answers.
/20J If a kilogram be placed on a horizontal plane, which
is made to descend vertically with an acceleration of 100
tachs per second; find in gravitation measure the pressure
on the plane.
21. If 10 lbs. be placed on a horizontal plane, which is
made to ascend vertically with an acceleration of 20 vels
per sec; find in lbs.-wt. the pressure on the plane.
22. If the speed of each of the bodies in Attwood's
machine be 20 vels, when they are at the same height
above the ground, and if at that instant the string be cut,
find how far apart the bodies will be in 5 seconds.
23. If in ex. 20 and 21 the motions be vertical velocities
of 100 tachs and 20 vels respectively, instead of accelera-
tions, find the pressures on the planes.
24. One spring is stretched 2 cm. for every kilogram
appended to it; another. 4 cm. ; if 4 kilograms be appended
to both, how far will they both be stretched?
Answeks.
1. 9805; 98050. 2. 12500. 3. 4272 grams.
4. 250,270,260, and 265 lbs. wt. 5. 4902500; 392-2.
6. 1427. 7. 7-772 lbs.-wt. 8. 104X10-° poundal.
9. 6 or 16| lbs. 10. 197 cm. 11. 032 sec.
12. 980-5 cm. 13. 9805 grams.
14. 1438; 382-5; 374955. 15. 10; 18335.
16. 22,10, and 4 lbs.-wt. 17. 797-2.
18. 9decigrs.: 899-1 milligrs. 20. 898-01 grs.-wt.
21. 16-22. 22. 200 ft. 24. 5^ cm.
Chapter VIII.
A rch imedes" Principle.
71. Since the weights of bodies at the same place are
directly proportional to their masses, and since different
bodies differ in their specific masses, therefore they will
also have different specific weights, or, as they are also
called, specific gravities.
The specific weight of a body is the ratio of its weight
to the weight of an equal volume of water at 4° C (its
maximum density point) at the same place.
Specific weight, being a ratio of quantities of the same
kind, viz. weight, is like angle and specific mass merely a
number, and is independent of all units.
The specific weight of water at 4° C will evidently be
represented by unity, and it is evident that the numbers
which represent the specific mass and specific weight of
the same substance are the same.
In the case of a body whose specific weight is not uni-
form throughout, the above definition gives the mean
specific weight of the body.
72. The most convenient methods of determining the
specific weights of liquid and solid bodies depend upon
the Principle of Archimedes :
Every hod// immersed in a fluid is subjected to a
vertically upward pressure equal to the weight of the fin id
displaced.
The truth of this principle is at once seen when we
think that, if the body were replaced with a portion of
fluid of the same kind without any other change, the
weight of the fluid would be supported. Its truth is
sensibly felt in bathing on a shingly beach, when it is
found that, the deeper one enters the water, the less are
63
the soles of the feet hurt by the pressure of the stones on
[them. It can be proved directly by immersing in a liquid,
i body, whose volume can be measured exactly (such as a
3ube, cylinder, or sphere), observing the apparent loss of
weight of the body, and then weighing the amount of
[liquid displaced. In the case of a floating body, the
weight of fluid displaced will be found to be equal to the
antire weight of the floating body. Convenient experi-
ments to show these facts are given in books on Experi-
mental Physics.
According to Newton's Third Law (Art. 58), the fluid,
on the other hand, is subjected to a vertically downward
[pressure equal to the weight of the fluid displaced. This
3an easily be shewn experimentally by balancing a vessel
containing water in a common balance, and immersing in
the water a body held by a cord. The equilibrium will be
immediately destroyed, and the force necessary to restore
equilibrium will be found to be equal to the weight of
water displaced.
73. The occasion which led Archimedes to the dis-
covery of this principle was the giving to him by King
Hiero of Syracuse the problem: — to discover the amount
of alloy which, the king suspected, had been fraudulently
put into a crown, which he had ordered to be made of pure
gold. It is said that Archimedes saw the solution of the
problem one day on entering the bath, and probably it was
by his observation of the buoyancy of the water. It may
have been, however, by his noticing that the volume of the
water which he displaced would just be equal, by the
principle of impenetrability (Art 2). to the volume of the
immersed part of the body. Indeed, one of the most im-
iportant applications of the principle of impenetrability is
to determine the volume of any irregularly shaped body,
by immersing it in a liquid contained in a measuring glass.
and noting the change of level which takes place.
64
74. Archimedes' principle is applied practically in many
ways; e.g. in finding the volumes of irregularly shaped
bodies like King Hiero's crown, in floating balloons in the
air and iron ships in the sen, in lifting ships over bars
formed at the mouths of rivers, in removing piles used in
the construction of docks, as well as in determining specific
weights, as explained in the following article.
75. There are three instruments used in finding specific
weights accurately, viz. the balance, hydrometer, and
specific gravity bottle. For less accurate values a measur-
ing glass may be used, and for liquids, also specific gravity
beads.
I. Liquids, to an approximation of the first degree:
a). By means of a balance.
Weigh a body which is not attacked either by water or
the liquid, e.g. a piece of agate or a platinum ball, firstly
in air, secondly in water, thirdly in the liquid whose
specific weight is required:
Let w-i = weight of the body in air,
w2 = water,
w3 = the liquid.
then s. w. of the liquid
b). By means of hydrometers.
'f'i - w2
These instruments, also called areometers, are essentially
closed tubes, weighted at one end, for determining specific
weights by observing how far they sink in water and other
liquids, or by observing what weight will make them sink
to a certain depth. The latter are called hydrometers of
constant immersion, the former hydrometers of variable
immersion.
1. By means of a hydrometer of constant immersion.
e.g. Nicholson's.
65
Let Wx= weight of the hydrometer in air,
w2 = weight required to sink the hydrometer to the
marked depth in water,
M?3= ditto, ditto, ditto, in the liquid.
then s. w. of the liquid = , 3
wx-\-w2
2. By means of hydrometers of variable immersion.
These are called salimeters or alcoholimeters, according
as they are used for liquids of greater or less specific
weight than that of water. Both kinds have scales attach-
ed to them, which tell either the specific weight directly for
any immersion, or the volume immersed, in which case the
specific weight must be calculated. A thermometer is fre-
quently attached to tell the temperature of the liquid.
c). By means of a specific gravity bottle.
Let wx = weight of the s. g. bottle empty,
w2 = full of water,
w3 = full of the liquid,
then s. w. of the liquid = —5 x
w2- wx
II. Solids, to on approximation of the first degree:
o ). By means of a balance.
Let u\ = weight of the body in the air,
. w% = water,
then s. w. of the body = - —
wx - w2
h). By means of a hydrometer of constant immersion.
Let w-i = weight of the body in air.
w2 = weight required to make the hydrometer
itself sink to the marked depth in water,
?r;. = weight required to make the hydrometer,
with the body attached to the lower part of
it, sink to the marked depth in water,
then s. w. of the body = - -1-
6o
for evidently if w denote the weight of the hydrometer in
air, then w + w% will be the weight of water displaced by
the hydrometer, and w-\- "i + ^'h the weight of water dis-
placed by the hydrometer and body together: therefore the
difference io1 + w3 - w2 will be the weight of water dis-
placed by the body alone : u^ can easily be determined by
the hydrometer, although it is simpler to measure it by
means of a common balance. This method is useful in the
case of bodies like cork which float in water,
c). By means of a specific gravity bottle.
This method is particularly convenient for finding the
specific weights of powders. -j
Let w1 = weight of the powder, ^ v>
w2 = weight of the specific gravity bottle, full of
water, .v>
w3 = weight of specific gravity bottle, after the
powder has been inserted, and the bottle
thereafter filled up with water,
then s.w. of the powder =
wx
<7). When a solid body is soluble in water, we find its
specific weight relatively to a liquid in which it is insoluble,
and multiplying this by the specific weight of the liquid,
we get the specific weight of the body relatively to water.
As an example let us take common salt, and adopt
method a).
Let Wi = weight of salt in air,
«\j = weight in petroleum or turpentine, of a vessel
to hold the salt.
i/-3 = weight in petroleum or turpentine of the vessel
containing the salt,
s = the s. w. of petroleum or turpentine,
then s. w. of the salt = -1 "
/r, -f >r2 - w3
76. The following propositions follow immediately from
Archimedes' principle :
67
1). The mass of a floating body is equal to the mass of
the displaced fluid.
2) . When a body floats in a liquid, the volume immersed
is to the whole volume as the specific weight of the body is
to the specific weight of the liquid.
Examination VIII.
1. Define the specific weight of a body, and prove that
the numbers which measure the density and specific
weight of any body are the same.
2. Enunciate and prove Archimedes' principle.
3. Describe several illustrative experiments which prove
the same principle, and state several practical applications
thereof.
4. Explain why a boat built of iron can float in water.
What is the s. w. of iron?
5. Give the history of the discovery by Archimedes of
his principle.
6. What are the three chief practical methods of
determining the specific weights of solid and liquid
bodies?
7. Give formulae for all the methods in the case of both
solid and liquid bodies.
8. What is a hydrometer? Give the names of the dif-
ferent kinds, and their respective uses.
9. When a solid body is soluble in water, how is its
specific weight found?
10. How would you find the s. w. of sulphuric acid,
sulphate of copper, sand, cork, paper, snow, ice, mercury?
11. Given a common balance with a hook to weigh
bodies in water, a piece of cork, and a piece of lead suf-
ficient to sink the cork in water; shew (giving a formula )
how to determine the specific weight of the cork.
12. How can the volume of an irregularly shaped stone
be accurately determined?
68
Exercise VIII.
1. A piece of limestone weighs 2021 grams in air, and
12'82 in water; find its specific weight.
2. The s. w. of ice is 092, and of sea-water 1027; find
what fraction of an ice-berg is below the surface of the sea.
3. Compare 1) the mass of an ice-berg below the sur-
face of the sea with that above, 2) the average depth of
the ice-berg below the surface with the average height
above.
•^i^A block of pine, the volume of which is 4 litres, 340
cub. cm., floats in water with a volume of 2 litres, 240 cub.
cm. above the surface; find the s.w. of the pine.
5. A person whose mass is 150 lbs. enters the sea to
bathe. If the s. w. of sea-water be 1-027, and of the human
body 0-9, find the pressure on his feet when 5/6 of his body
is immersed.
(6) A ball of platinum whose mass is a kilogram, when
in water weighs 955 grams; what will it weigh when in
mercury (s. w. 13'6)?
^V)A piece of iron (s.w. 7'5) floats in mercury; find what
part of the iron is above the surface of the mercury.
8. When 1 lb. of cork is attached to 21 lbs. of silver, the
whole is found to weigh 16 lbs. in water. If the s. w. of
silver be 105, find that of cork.
9. A vessel, containing water, weighs 2034 grams; a
kilogram of bronze (s. w. 8'4) is held in the water by a
string ; find what will now be the apparent weight of the
vessel and water.
10. A piece of cork has s. w. \, and mass 534 grams;
find the pressure necessary to keep the cork under sea-
water, whose s. w. is T027.
11. A kilogram of lead, whose s. w. is 114, is suspended
in water by a string; find the tension of the string.
12. Neglecting friction, with what acceleration will a
silver ball (s. w. 105) sink, and an elm ball (s.w. 0'7) rise
in sea-water (s. w. 1"027)?
69
13. Find what force would be necessary to immerse a
kilogram of oak (s. w. 097) in mercury (s.w. 13-6.)
U.4?) A body of 58 grams floats in water with two-thirds
of its bulk submerged; find its volume.
15. A man whose mass is 68 kilograms can just float in
fresh water; find the maximum weight he could bear up
clear of the water, when floating in the sea (s. w. T027.)
/16^How much lead (s. w. 11 '4) will a kilogram of cork
(s. wT^|) keep from sinking in the sea (s. w. l-027)?
(V?) A piece of hard wood of mass 7*6 grams is attached
to the lower part of Nicholson's hydrometer, and it is then
found that the force required to sink the hydrometer in
salt water is just the same as before the wood was attached,
viz., 12*6 grams-weight. If 1*03 be the s. w. of the salt
water, find that of the wood.
•fl^A vessel quite full of mercury weighs 725 kilo-
gram^; a kilogram of iron is put into the vessel and held
completely immersed in the mercury ; what will now be
the apparent weight of the vessel and contents?
(l9.)lf in ex. 18 the iron be fixed to a hook at the bottom
of a vessel, what will be the weight of the vessel and con-
tents? Explain the difference.
20. Answer 18. when the vessel is only partially tilled
witnwmercury, and no mercury is spilled in completely
immersing the iron.
Answers.
1. 2-735. 2. 0-90. 3. 86; 205. 4. 15/31.
5. 72 lbs.-wt. 6. 388 grs. 7. 61/136. 8. \.
9. 2153 grs. 10. 1660 grs.-wt. 11. 912 grs.-wt.
12. 884-6 and 458-0 tachs per sec.
13. 13 kilogrs.-wt. 14. 87. 15. 1836 grs.
16. 3415-7 grs. 17. 1*03. 18. 72-5 kilograms.
19. 71686-6 grs. 20. 743133 grs.
Chapter IX.
Pascal's Principle. The Barometer.
77. Matter is divided into solid and fluid. Asolide.f/.
agate, is distinguished from a fluid in offering more or less
resistance to change of form, a fluid offering little or none.
Hence fluids under the action of weight must be kept in
solid vessels. Fluids again are divided into liquids and
gases. A liquid is a very incompressible fluid, and can be
kept in an open vessel. Water, petroleum, and mercury
at ordinary temperatures are liquids. A gas is a very
compressible fluid, and must be kept in a closed vessel,
inasmuch as it will fill any space into which it is admitted,
even if that space be already occupied by gas. Air is a
mixture of several gases, though principally of two, Nitro-
gen and Oxygen. When a gas is near its point of con-
densation, it is called a vapour. Aqueous vapor or steam
is one of the components of air or the atmosphere, though
the quantity is comparatively small and varies with time
and place.
In elementary dynamics a solid is assumed to be rigid.
i.e. that its parts maintain the same relative positions,
whatever forces may be acting on it; a liquid is supposed
to be incompressible; and a gas as obeying Boyle's law.
78. A fluid may be defined as a body which will change
its shape, more or less quickly, under the action of any
force however slight, until all force applied to it is normal
to its surface at every point.
It is evident that a fluid cannot remain at rest under
the action of any external pressure, unless pressure is
applied normally at every part of the surface. Hence in
measuring the action between a fluid and a surrounding
body (which may itself be fluid of the same or of a differ-
ent kind), it is necessary to get the pressure per unit of
71
area at each point of contact. Again when we think of
the equilibrium of a small spherical particle of a fluid,
whose centre is at a given point; and make the particle
smaller and smaller, we see that there is pressure in every
direction af any point of a fluid.
That the pressure at any point of a fluid is the same
in all directions may be accepted as an experimental fact.
It is deduced later on as a necessary consequence of the
fundamental property of a fluid, viz. that hydrostatic
pressure on any surface is normal to that surface.
79. The pressure at any point of a fluid is the pressure
per unit of area on any plane surface containing the point,
when the area of the plane is indefinitely diminished. Or,
The pressure at any point of a fluid is the pressure per
unit of area which would be on any plane containing the
point, if the pressure at every point of the plane were the
same as at the point in question.
The systematic unit of fluid pressure (or of hydro-
static pressure, or of pressure-intensity in general) is
unit of force per unit of area. Hence the C. G. S. unit of
fluid pressure is 1 dyne persq. cm. This is called a barad.
The F. P. S. unit is 1 poundal per sq. ft.
80. The following important property of a fluid is gen-
erally known as Pascal's principle:
Pressure applied at the surface of a fluid contained in
a closed vessel is transmitted without change to every
particle of the fluid.
The student may compare this principle with the
equality of tension at every particle of a stretched cord,
due to forces applied at the ends of the cord. Var-
ious experimental illustrations can be found in books
on experimental physics. It may also be deduced from the
fundamental property of a fluid at rest (Art. 78). Hence,
if a fluid were at rest, and subject only to forces applied
/
72
at its surface, the pressure would be the same at every
point of the fluid.
81. Pascal's principle is admirably illustrated and most
usefully applied in the hydrostatic or Bramah press, a
machine by means of which great mechanical advantage
is acquired. It consists essentially of two hollow cylinders
connected by a tube. Water-tight pistons fit into these
cylinders, whilst they and the connecting tube are filled
with water or oil. Pressure applied to the smaller piston
is transmitted through the liquid to the larger, with a
mechanical advantage measured by the ratio of the areas
of the bases of the pistons. The moving force is generally
applied to the smaller piston through a lever which further
increases the mechanical advantage gained. It was by
means of this machine that the heavy tubes used in the
construction of the Britannia bridge, which crosses the
Menai Strait, were lifted into their places. The student
will easily satisfy himself that the principle of work
applies in this as in all other machines.
Let it be observed that Pascal's principle applies to
any fluid whether homogeneous or not. Thus in the piez-
ometer the principle is applied to measure the compressi-
bilities of liquids, when in general the compressed fluids
consist of two different liquids and a gas, The Cartesian
divers is an amusing physical toy which illustrates in a
unique manner the principles both of Pascal and Archi-
medes; also Boyle's law (Art. 94), and Art. 83.
82. Weight is a force which acts on every particle of a
fluid and not merely at its surface. Hence, applying Pas-
cal's principle, the following four propositions can be de-
duced:
In a fluid at rest under the action of no external I force
except Had of weighty surfaces of equal pressure are
horizontal planes.
73
In large fluid bodies, like one of the oceans or the
atmosphere, surfaces of equal pressure are at each place
perpendicular to the direction of weight at that place, and
are nearly spherical surfaces having small curvature.
83. In a liquid of uniform temperature under the
action of no external force except that of weight, the pres-
sure increases uniformly with the depth.
Take a vertical right cylinder of the liquid, the area of
each horizontal base being s, its length It. and the density
of the liquid d. It is evident that the pressure on the
lower base must be greater than that on the upper by the
weight of the cylinder, i.e. in C.G.S. units by s h dg dynes.
Hence the increase of fluid pressure per unit of depth is
gd barads, or d grams-wt. per sq. cm. In F.P.S. units the
increase of fluid pressure per unit of depth will be 62'4#d
poundals per sq. ft., or 62'4 d pounds-wt. per sq. ft.
This proposition was experimentally illustrated by
Pascal by bursting a large barrel by means of a long fine
column of water. It shows us that in the supply of water
from a reservoir to the houses of a town, the pipes far be-
low the level of the reservoir need to be much stronger
than those near the level of the reservoir. In Barker's
mill and many applications of turbine-wheels the principle
is taken advantage of to produce motion in machinery.
84. When two or more fluids of different specific
weights, which do not mix with one another, are j)ut into
the same vessel, they arrange themselves in the order of
their specific weights, and their surfaces of separation,
when the fluids are at rest, are horizontal planes.
This is an immediate corollary of the two preceding-
articles. It may be illustrated by putting mercury, water,
benzine, and air into the same bottle. In this, as well as
in the following article, let it be observed that just where
the surfaces of separation meet the containing vessel, they
74
are not horizontal on account of the action of the external
molecular force of adhesion.
If a globule of oil be put into a mixture of water and
alcohol, having the same s.w. as the oil, it will assume the
form of a sphere under the action of internal molecular
force.
85. The free surface of a liquid at rest, under the
action only of weight, an I the jrressure of the atmosphere,
is a horizontal plane.
This is just a particular case of last article, and is
lucidly illustrated by putting a liquid into a series of com-
municating vessels of different shapes. The adage "water
will always find its level " is a popular way of expressing
the truth of the proposition. The principle is most use-
fully applied in supplying the houses of cities with water,
in water and spirit levels, and in the construction of foun-
tains. It explains the action of Artesian wells.
If two liquids of different specific weights be contained
in different vessels, whose bottoms communicate by means
of a tube completely filled with the denser of the liquids
(art. 84), the free surfaces cannot be in one plane, but
will be at heights above the common surface of separation
which are inversely proportional to the specific weights of
the liquids (art. 83.)
86. If A denote the pressure of the atmosphere, and d
the density of a liquid, the pressure on any horizontal
plane of area S at a vertical distance z below the free sur-
face of the liquid is (A+gdz) S, by arts. 80 and 83.
The pressure does not depend in any way upon the
form of the vessel, but only upon z and S.
This is beautifully illustrated by the famous experi-
ment of Pascal's vases.
87. The student must not confound the pressure on tne
bottom of any vessel containing a liquid, with the pressure
75
resulting from the presence of the liquid on the body sup-
porting the vessel. The latter is just the weight of the
liquid, whilst the former may be greater or less according
to the shape of the vessel. This fact is commonly called
the hydrostatic paradox. It is easily understood when it
is noticed that the pressure of the vessel on the body sup-
porting it is the weight of the vessel together with the
resultant of the pressures of the liquid on the whole
interior surface of the vessel, and not merely the resultant
of the pressures on the bottom.
88. The pressure of the atmosphere at any place is
measured by the length of a vertical column of mercury
which it can support, as shown by Torricelli in 1643 in his
famous experiment: — Fill with mercury a glass tube which
is closed at one end and is about 90 centimetres long and 1
centimetre in diameter; prevent air from entering the tube
by placing a finger over the open end; put the open end
into a vessel of mercury and remove the finger; it will
then be found that the mercury will sink in the tube till
the level inside is about 76 centimetres above the level of
the mercury outside. This column of mercury Torricelli
conclusively proved was supported by the pressure of the
atmosphere. It was the first barometer, and at the present
day is the most perfect barometer which can be made.
89. The pressure of the atmosphere changes both with
time and place. The mean value over the earth's surface
at the sea level is the same as would be produced by the
weight of a vertical column of mercury 76 cm. long, at 0°( '.,
in the latitude of Paris. The pressure per sq. cm. is
therefore equal to the weight at Paris of 76 cub. cm. of
mercury at 0°, i.e. (since 136 is the density of mercury at
0°) 13-6x76, or 1033-6 grams-wt. at Paris, i.e. (since 980-9
is the value of g at Paris) 9809 X 136x76 barads, or 1014
megabarad.
76
90. The pressure of the atmosphere is produced by its
ucight. This was conclusively proved in 1648 by Pascal.
Amongst other experiments he sent a Torricellian barom-
eter to the top of the Puy de Dome, and found that there,
the length of the mercurial column was considerably
shorter than it was at the bottom of the mountain, just as
he had predicted. It was fully 9 centimetres shorter. As
in the case of liquids (art. 83), it is evident that at any
altitude the length of Torricelli's column can be affected
only by the weights of the aerial particles at higher levels.
Hence as we ascend the mercury falls.
The very great pressure of the atmosphere (at the sea-
level, nearly 14'7 lbs.-wt. per sq. inch, over 1 ton-wt. per
sq. ft., and over 10 tonnes-wt. per sq. metre) may be strik-
ingly shewn by means of the Magdeburg hemispheres. It
is taken advantage of in a useful manner in the suction-
pump, siphons, pipettes, and other appliances. The limit-
ing height (about 33 feet or 10 metres) to which water
can be raised by means of a suction-pump was accidently
discovered in 1640 by some Florentine workmen, and it
was this discovery which first led Galileo to suspect that
the pressure of the atmosphere was due to its weight. The
very name suction-pump recalls the old explanation of the
rise of the water, by means of the long abandoned axiom
" Nature abhors a vacuum."
91. Hydrodynamics is the dynamics of fluids, and is
divided into hydrostatics and hydrokinetics. Pneumatics
is a term frequently used to denote the dynamics of gases,
and hydraulics the science which treats of machines for
the conveyance of water or other liquids. Archimedes
screw is one of the oldest of hydraulic machines. It is
used for raising water or any other liquid from one level
to another, in a most ingenious manner, by taking advan-
tage only of the weight and fluidity of the liquid particles.
It consists of a tube wound round a cylinder into a helix
77
or screw. If the axis of the cylinder be inclined to the
vertical at a greater angle than the angle of the screw, and
the lower end of the tube be immersed in water, the water
can be raised by rotating the screw-tube about the axis of
the cylinder. Such machines were used in ancient Egypt
for draining the land after inundations of the Nile.
Examination IX.
1. Define the terms solid, fluid, liquid, gas. vapor,
rigid, and point of condensation.
2. Give a full definition of fluid, sufficient for the study
of hydrostatics.
3. Explain the statement that the pressure at any point
of a fluid is the same in all directions.
4. Define the pressure at any point of a fluid, and give
the C.G.S. and F.P.S. units of fluid pressure.
5. Enunciate and explain Pascal's principle. Give an
important practical application thereof.
6. Enunciate and prove four important propositions
regarding the action of weight on a fluid at rest.
7. Explain the construction of a Cartesian diver. What
important principles does it illustrate?
8. Explain the bursting of Pascal's barrel. Give im-
portant applications of the same principle.
9. Calculate the mechanical advantage of an hydro-
static press. What is meant by saying that the principle
of work applies to it?
10. What is the form a fluid would take under the
action of internal molecular forces alone? How would
you shew this experimentally?
78
11. How would you illustrate experimentally the adage,
" water will always find its level." State several important
practical applications.
12. What does the experiment of Pascal's vases prove?
What is the hydrostatic paradox? Explain it.
13. Describe Torricelli's experiment whereb)r he first
measured the pressure of the atmosphere.
14. State in gravitation and in absolute measure the
mean sea-level atmospheric pressure over the earth's sur-
face.
15. How did Pascal prove that the pressure of the
atmosphere was due to weight?
16. Describe the experiment of the Magdeburg hemi-
spheres.
17. Explain the actions of the suction-pump, siphon,
and pipette.
18. If the barometer be inclined at an angle i to the
vertical, what is the length of the mercurial column?
19. A siphon is filled and held with its legs pointing
downwards and the ends closed; what will take place, when
a) one end is opened, b) both ends are opened, 1) when
the ends are in the same horizontal plane, 2) when they are
not in the same horizontal plane.
20. How does a change of atmospheric pressure affect
the pressure between a liquid and the containing vessel?
21. Does a change of atmospheric pressure affect the
action of a siphon, or of a suction-pump?
22. Examine the effects of making a small aperture in
different parts of the barometer-tube.
23. A small piece of glass gets into a barometer and
floats; is the reading vitiated thereby?
79
Exercise IX.
1. Find in lbs.-wt. per sq. in. at what rate the pressure
increases per 10 ft. of vertical distance from the reservoir
in the water-pipes of a town.
"~2) The deepest sounding taken on the " Challenger "
was 8184 metres between the Carolines and Ladrones in
the N. Pacific ocean. Find at that depth the pressures in
atmospheres, and in tonnes-wt. per sq.(im., taking 1-027 as
the mean s. w. of the water.
' 3?) The legs of a siphon are equal in length and inclined
at^fi angle /; how should the siphon be placed so as to
remove the most liquid?
4. Find the greatest height over which sulphuric acid
(s.w. 1*84) can be carried by a siphon when the barometer
is at 30 inches.
5. If a mercurial barometer 1 sq. in. in section stand at
30 inches, what will be the height of a sulphuric acid
barometer of section 1/1 84 sq. inch?
6JIf the coefficient of cubical dilatation of a barometer
tube were the same as that of mercury, would the height
of the mercurial column be affected by change of tempera-
ture? Explain your answer.
7. The scale of a barometer is etched on the glass tube
and is true at 17°. The readings of the barometer and
thermometer are 7567 and 10°. Find the reduced barom-
etic pressure, i.e. the length of the Torricellian column
when the temperature of the mercury is 0°; given A's
(coefficient of cubical dilatation) for mercury to be
1-8 XlO"4, and />-, (coefficient of linear dilatation) for glass
tobe8xl0"6.
8. If the temperature at Edinburgh be 15°, and the
scale of the barometer (A'^l^X 10"5) be true at 0°. find
the reading of the barometer when the atmospheric pres-
sure is a megabarad.
80
9. A siphon-barometer is held suspended in a vessel of
water by a string attached to its upper end. If h denote
the depth of the upper mercurial surface, and A gram-wt.
per sq. cm. the atmospheric pressure, and .s the internal
section of the tube, find the difference of level of the two
mercurial surfaces. At what rate does the tension of the
string change as the barometer is lowered?
10. A string can bear a tension of 10 kilograms wt.;
determine how much cork (s.w. \) it can keep below the
surface of mercury (s.w. 13*6).
11. Two bodies of 1 and 2 kilograms are attached to a
string passing over a smooth pulley ; the bodies rest in
equilibrium when they are completely immersed in water;
if the specific weight of the first body be 2, find that of
the second.
12. A specific gravity bottle when entirely filled with
distilled water has a mass of 530 grains; 26 grains of sand
are put into the vessel, and the whole then weighs 546
grains; find the specific weight of the sand.
13. Two liquids which cannot mix are poured into a
circular tube so as to occupy a quadrant each; the diameter
joining the free surfaces is inclined at \ti to the vertical;
find the ratio of the densities of the liquids.
Answers.
1. U. 2. 813; 8405.
3. Legs equally inclined to the vertical.
4. 221-7 inches. 5. 18 ft. 5'7 inches. 6. Yes.
7. 7554. 8. 7510.
(.). ( A-{- h ) H- 126 ; s, 252 grams-wt. per cm.
10. 187-3. 11. 4/3. ' 12. 2-6. 13. 373.
Chapter X.
Specific Weights of Gases.
92. We are aware from the effects of wind in driving
windmills, ships, &c, that air has mass. That it has weight
like solid and liquid bodies can easily be proved by the
following experiments :
Exp. 1. Weigh a globe with a tightly fitting stop-cock,
firstly full of air, secondly after the air has been extracted
from it by means of an air-pump.
Exp. 2. Boil water in a flask until all the air is driven
out, cork it up tightly, weigh when cool, admit air and
weigh again.
Exp. 3. Instead of extracting air from the globe in
exp. 1, compress air into it, when it will be found to be-
come heavier.
Exp. 4. Weigh the globe in exp. 1 when filled, firstly
with air, secondly with hydrogen, thirdly with carbonic
acid.
The second and third experiments are due to Galileo;
from the third the specific weight of air may be approxi-
mately measured by collecting the compressed air in a
pneumatic trough. The fourth experiment proves that
gases like liquid and solid bodies, differ in specific weight.
93. The fact that gases have weight, and even flame,
which is essentially incandescent gas, was known to the
Epicureans, if we take Lucretius as their mouth piece. In
his great poem, "De rerum natura," written about 56 B.C.,
he says:
See with what force yon river's crystal stream
Resists the weight of many a massy beam ;
To sink the wood, the more we vainly toil,
The higher it rebounds with swift recoil :
82
Yet that the beam would of itself ascend,
No man will rashly venture to contend :
Thus too the flame has weight, though highly rare
Nor mounts but when compelled by heavier air.
94. Before considering how the specific weights of
gases have been determined, it will be necessary to know
how the density of a gas depends upon its pressure. The
physical law which tells us this is called Boyle's law, and
may be enunciated in either of the following ways :
The density of a gas is directly proportional to its
pressure, if the temperature be far above the point of con-
densation, and remain constant, d oc p.
The volume of a gaseous body is inversely propor-
tional to the pressure, if the temperature be far above the
point of condensation, and remain constant. pV = c.
The truth of Boyle's law depends of course entirely
upon experimental evidence. Dry air is an example of a
gas far above its point of condensation, and for dry air at
all ordinary pressures and temperatures the law may be
said to be exact. When the temperature approaches the
point of condensation, the product pV gradually de-
creases. In the case of solid and liquid bodies it is not
necessary to consider the pressure to which they are sub-
jected, for the changes of density, arising from the
changes of pressure to which bodies are in general exposed,
would be immeasurably small. The case of gases is very
different.
95. It was the great French physicist, Regnault, who
first overcame the experimental difficulties necessary to an
exact determination of the specific weights of gases. The
secret of his success depended upon counterbalancing the
globe containing the gas he was weighing, with another
globe of equal volume and weight as nearly as possible, so
that it was unnecessary for him to make any corrections
for barometric, thermometric, or hygrometric changes in
83
the state of the atmosphere during the time of experiment-
ation. Having several times exhausted one of these globes
and filled it with a dried gas until he was satisfied that the
globe was thoroughly dry, he put the globe into a mixture
of ice and water (0° C), and filled it once again with the
dried gas at the pressure of the atmosphere, say P cm. of
mercury. He then partially exhausted the globe, to
pressure p say, keeping it at the same temperature 0° C,
and noted the change of weight. This change of weight
(w) will, by Boyle's law, be the weight of the gas at 0°
which would fill the globe at pressure P-p\ therefore the
weight of gas at 0° required to fill the globe at the mean
atmospheric pressure will be 76 w-t- (P -p).
In this way Regnault found the weights of equal vol-
umes of dry air and other gases. It remained for him to
determine the specific weight of dry air with respect to
the standard substance, water at 4° C. If w' denote the
difference of weight between the globe when filled with
water at 0°, and when filled with dry air at 0C and pressure
P, then w,Jr{w P-t-(P-p) \ will denote the weight of the
water which the globe would hold at 0°. If therefore s
denote the s. w. of water at 0°, the s.w. of dry air at 0° and
under the mean atmospheric pressure will be
76 w s . r , . tvP ,
If it be necessary to take into consideration the buoy-
ancy of the air in determining w and w', the methods indi-
cated in the next chapter will explain how this can be
done. The following table gives the results of some of
Regnault's experiments:
Mass of 1 litre at 0° and 76 cm. Specific volume.
Air (dry) 1293187 7733
Oxygen 1429802 6991
Nitrogen 1-256167 7961
Carbonic Acid 1977414 5057
Hydrogen 0'089578 11.163
84
By 76 cm., or dpressure of 76 cm. the mean atmospheric
pressure is always meant, (Art 89).
96. On account of the small densities of gases it is gen-
erally more convenient to measure their specific weights
with respect to dry air or hydrogen as a standard, than
with respect to water at 4° C.
The specific weight of a gas with respect to dry air (or
hydrogen), is defined as the ratio of the weight of any vol-
ume of the gas at 0° C and under the mean atmospheric
pressure, to the weight at the same place of an equal vol-
ume of dry air (or hydrogen) at the same temperature and
pressure.
The following table gives the specific weights of the
above gases with respect to dry air and hydrogen:
Hydrogen 00693 1000
Nitrogen 09714 14023
Air (dry) 10000 14436
Oxygen 11056 15962
Carbonic Acid 1-5291 22075
97. The numbers in the last column and similar results
have enabled chemists to establish a most important law,
called Avogadro's law, which may be enunciated in the
three following ways:
The specific weights of gases, at the same temperature
and pressure, are directly proportional to their molecular
weights, if the temperature be far above the points of con-
densation.
The molecular volumes of gases, at the same tempera-
ture and pressure, are all equal to one another, if the
temperature be far above the points of condensation.
The number of molecules in a gaseous body, at a given
pressure and temperature, the temperature being far
above the point of condensation, is directly proportional to
the volume, and is independent of the nature of the gas.
85
In the following table for gases the specific weights
have been calculated from the molecular weights, with the
exception of those of hydrogen, nitrogen, air, and oxygen,
which are Regnault's experimental measurements.
Table of Densities and Specific Weights.
I. Solids at 0° C.
Platinum, stamped 22*10
rolled 22-07
cast 20-86
Gold, stamped 19-36
" cast 19-26
Lead, cast 1 1 '35
Silver, cast 10*47
Copper, hammered 8*88
cast 8-79
Bronze, i An
R , > Average 8 -40
Steel 7-82
Iron, wrought 7 '79
" cast 7"21
Tin, cast 729
Zinc, cast 7'00
Aluminium 267
Magnesium 1 '74
Sapphire 4
Diamond 3
01
52
Glass 2-5 to 3-3
Kingston Limestone 2
Rock-crystal (Quartz) 2
Ice 0
Ivory 1
Anthracite 1
Ebony, American 1
Mahogany, Spanish 1
Box, French 1
Oak, English 0
Maple, Canadian 0
Elm 0
Willow 0
Poplar 0
Cork 0
70
66
92
92
80
33
06
03
97
75
70
58
38
24
Pith, of sun-flower 0-028
II. Liquids at 0° C.
Mercury. 13 596
Sulphuric Acid 1 84
Human Blood 1 '05
Milk, of Cow 1'03
Sea-water 1 '027
Water at 4° 1 '000000
" at 0° 0-999873
Olive Oil 0-92
Alcohol 0.80
Ether 0"72
III. Oases at 0°C and 76 cm. pressure.
Hydrogen 0-0000896
Ammonia 0'0007619
Aqueous Vapour 0 -0008044
Steam at 100° 0-0005887
Nitroa-en 0 001 2562
Air (dry) 0-0012932
Oxygen 00014298
Carbonic Acid 0 00 19658
Carbonic Oxide . .00012510 Chlorine 0-0031684
86
98. By a good air-pump hydrogen can be rarified
to a density 104 times less than what it has under the
atmospheric pressure. We thus see, by comparing the
density of platinum with that of rarified hydrogen, the
great range of density there is, even in substances which
can be easily obtained. It will be seen from the above
table that the density of a solid depends to a certain
extent on the way in which it has been prepared. Even
in the case of natural bodies like sapphires and diamonds,
different specimens from different places are found to vary
slightly in density. In mixtures the density is not always
the mean of the densities of the component parts. Thus
bronze has a greater density than the mean of the densities
of the component metals; so with a mixture of alcohol and
water. In the case of woods different parts of the same
tree vary in density, as well as specimens from different
trees of the same species. Liquids can be obtained more
easily in a state of purity, but in such liquids as blood,
milk, and sea-water, differences of density are found in
different specimens.
Examination X.
1. How do we become aware that air possesses the
property of mass? Describe three experiments to prove
that air has weight.
2. Enunciate Boyle's law in two ways, and show that
the one follows from the other.
3. What was the secret of Regnault's success in weigh-
ing gases? Describe fully his method of finding the spec-
ific weight of dry air.
4. Define the specific weight of a gas with respect to
dry air, and also with respect to hydrogen, and give the
s.w. of dry air, 1) with respect to water at 4°, 2) with re-
spect to hydrogen.
5. Enunciate Avogadro's law in three ways.
6. What is the range of density as found by experiment.
87
Exercise X.
1. Determine the mass and weight at the equator of 10
litres of oxygen at 0°, and at a pressure of 74 cm. of mer-
cury at 0° at the equator.
2. Determine the pressure in barads under which
chlorine has a density 3 with respect to air at 0° and 76 cm.
3. A balloon is filled with hydrogen, the pressure and
temperature being 76 cm. and 0°. If the capacity of the
balloon be a megalitre, and the non-gaseous material have
a mass of 500 kilograms, and a mean density of 1, find
with what acceleration the balloon will begin to ascend.
4. A flask of 2 litres capacity was found to weigh 1"6
grams more, when filled with carbonic acid, than when
filled with air at 0° ; find the pressure of the atmosphere.
5. Find what the volume and density of a litre of air at
0° and 76 cm. will become at the bottom of the deepest
known part of the ocean (s.w. 1*027), viz. 8184 metres, if
the air obeyed Boyle's law to that depth.
6. A uniform tube, 1 m. long, closed at one end, is 3/4
full of mercury, and is then inverted as in the Torricellian
experiment into a vessel of mercury; if the barometic
column be 75 cm., find what will be the height of the mer-
cury in the tube.
7. A uniform tube, 1 m. long, open at both ends, is
immersed in a vessel of mercury to a depth of 90 cm. If
the top be now closed, and the tube raised, until the length
out of the mercury be 90 cm., find the height of the mer-
cury within the tube, 75 cm. being the length of the baro-
ometric column.
8. What will be the length of tube out of the mercury
in last example, when the air within the tube is 30 cm.
long?
9. A barometer was carefully calibrated, and it was
found that on account of a small bubble of air getting into
88
the tube, the reading was 75 cm., when the true pressure
was 76 cm., and that the space occupied by the air was
equivalent to 10 cm. of tube; what would be the true
pressure, at the same temperature, when the reading was
76 cm., and what would be the reading when the true
pressure was 75*4 cm.?
10. If 75 cm. be the height of the mercurial barometer,
find how far a conical wine-glass must be immersed mouth
downwards in water, so that the water may rise half-way
up in it, 7 cm. being the length of the axis of the cone,
and 13#5 the density of the mercury relatively to the water.
11. Prove that the atmosphere must be at least 5 miles
high. Is this true from whatever level you measure?
12. A cylindrical diving-bell, 3m. high, is lowered to
the bed of a river 12m. deep. If 75 cm. be the baromet-
rical pressure, find the height of water in the bell, and
compare the mass of air in the bell with what must be
forced in to keep the water out.
13. A diving-bell is suspended at a fixed depth ; a man
who has been seated in the bell falls into the water and
floats ; find the effect on 1 ) the level of the water in the
bell, 2) the amount of water in the bell, 3) the tension of
the chain holding the bell.
14. The receiver of an air-pump is 4 times as capacious
as the barrel. Shew that after 3 strokes the pressure
of the enclosed air is reduced to nearly f , and that it
takes another 3 to reduce the pressure to |. Given
log. 2=0-3010300.
15. The gauge of a condensing pump consists of a
glsss tube containing air, whose volume is determined by
the position of a drop of mercury C in the tube. If A be
the position of the mercury when the air in the condenser
is uncompressed, and B the end of the tube, prove that if
air is forced uniformly into the condenser, the ratio
AC: CB increases uniformly.
89
16. A condenser and suction-pump have the same
barrel and receiver, the capacity of the barrel being y1^ of
that of the receiver. Ten strokes are given to the con-
denser; how many strokes must be given to the suction-
pump, so that the pressure of the air in the receiver may
be that of the atmosphere. Given log. 11=0-0413927.
17. Two barometers of the same length and the same
section are immersed in the same reservoir, and each con-
tains a small quantity of air; their readings at one time
are d, e, and at another time h. k; if / denote the length
of each tube above the surface of mercury in the reservoir,
shew the quantities of enclosed air are as
d-e_h—k. d-e h-k
i^k~i^e: T^rri^d
18. A gas contained in a cubical vessel is compressed
into the sphere which can be inscribed in the cube, shew
that the total pressures on the two confining surfaces are
equal. If the gas be allowed to expand until it fills the
sphere circumscribing the cube, shew that the total pressure
on the confining surface is lessened in the ratio j/3: 1.
19. A cylinder with a closely fitting piston is full of gas;
shew that, if the piston be pressed into the cylinder at a
constant speed, the total pressure on the base increases
harmonically, whilst the total pressure on the cylindrical
surface remains constant.
20. A water-tap is connected with a mercurial siphon-
manometer, and on opening the tap the difference of level
of the surfaces of mercury was found to be 4 ft. 3 in. ; find
the head of water, and the available water-pressure.
Answers.
1. 12556; 12281. 2. 1-2414x10°. 3. 11716.
4. 90-4 cm. 5. 123, 105. 6. 42*4. 7. 54-1.
8. 80. 9. 77-S-; 74'45. 10. 7091.
12. 152; 51/60. 13. 1) raised, 2) lessened, 3) lessened.
16. About 7J. 20. 57-8 ft.; 39-7 lbs.-wt. per sq. in.
Chapter XI.
Exact Specific Weights.
99. Since the principle of Archimedes evidently applies
to gases as well as to liquids, all bodies in the atmosphere
are subjected to a vertically upward pressure equal to the
weight of the air displaced by them. This may be illus-
trated experimentally by the baroscope and balloons. In
determining the specific weights of solid and liquid bodies
to an approximation of the first degree, we neglected the
buoyancy of the surrounding atmosphere. Let us now
determine these specific weights to an approximation of
the second degree. This is done by taking into consider-
ation the buoyancy of the air, but, without noting what
may be its barometric, thermometric, and hygrometric
states, taking as its s.w. the mean s.w. of the atmosphere
at the place where the weighings are made. A good aver-
age value is 0,0012. The following problem will illustrate
the process.
100. Given wx, w2, w-s, the number of grams which bal-
ance a solid body in air, water, and another liquid respec-
tively; to determine the specific weights of the solid body
and liquid to an approximation of the second degree.
Let r denote 0'0012 gram-weight. The approximate
volume of the solid body will be (w1-w2) cub. cm., and
therefore the weight of air displaced by the solid body will
be (tv1-w2) r gram-weight approximately.
Let s denote the s. w. of the standard masses against
which the body is weighed. This should be determined
by the maker of the standard masses. Then wx/s is the
volume of the standard masses which balance the body in
air, and therefore (w>i/s) r the approximate weight of air
91
in grams displaced by them. Therefore the approximate
weight of the solid body in vacuo
=Wi - (wx/s) r-\-(ivl - w2) r grams-weight = Wx
The approximate weight of the solid body in water
= w2 - (w2/s) r grams-weight = W2
The approximate weight of the solid body in the liquid
= w3- (w3/s) r grams-weight = W?>
Then the specific weight of the solid bodv= -== — \FrF-
Wx - W 2
and liquid body = — J — — ?-
w i — w 2
each to an approximation of the second degree.
101. To get the specific weight of a body to an
approximation of the third degree, we require to calculate
the density of the air at the pressure, temperature, and
hygrometric state in which it is at the time the weighings
are performed, as well as to allow for the temperature of
the water in which the body is weighed. We have already
learned from Boyle's law (art. 94) how the density of a gas
depends upon its pressure. The law of change of density
of a gas, arising from change of temperature, was first dis-
covered by Charles. It may be enunciated thus:
The dilatation of a gas, at temperatures far above its
point of condensation, and at a constant pressure, is
directly proportioned to the increase of temperature; and
the coefficients of dilatation are the same for all gases.
If the dilatation be reckoned from 0° C, the coefficient
of dilatation is very approximately 0'003665 or 1/273.
Hence if Vt be the volume at temperature t°, and V0 the
volume at 0C, we may express the law algebraically thus:
Vt =V0 (1+0-003665 t), or Vt = V0 (l+f/273)
If now temperature be reckoned from the zero of the
air thermometer, i.e. from — 273°C, Charles' law may be
expressed thus:
92
The volume of a gas at temperatures far above its
point of condensation, is, at a constant pressure, directly
proportional to its temperature reckoned from the zero of
the air thermometer. Or thus:
The density of a gas at temperatures far above its
point of condensation, is, at a constant pressure, inversely
proportioned to its temperature reckoned from the zero of
the air thermometer. Or thus:
The pressure of a gas at temperatures far above its
point of condensation, is, at a constant volume, directly
proportioned to its temperature reckoned from the zero of
the air thermometer.
102. Boyle's and Charles' laws can be conjointly ex-
pressed algebraically by either of the following equations :
PV_ P 7.
in which p, v, d, and / denote the pressure, volume, den-
sity, and temperature reckoned from the zero of the air
thermometer. The quantities c and k are constant, so long
as we keep to the same gaseous body. If v', d! denote its
volume and density at 0°C and 76 cm. pressure, then
' 76 v' J ,. 76
c=
and k =
273 273 d'
103. To allow for the hygrometric state of the air, we
require first to know the law of Dalton relating to the pres-
sure of a mixture of gases. It may be enunciated thus :
When two or more gases, which do not act chemically
on one another, are enclosed in a, vessel, the resultant
pressure is the sum of the pressures of the gases when
placed singly in the vessel.
The physical principle underlying Boyle's and Dalton's
laws has been beautifully expressed by Rankine thus:
When one, or more gases, which do not act chemically on
one another, is confined in a vessel, each portion of gas,
however small, exerts its pressure quite independently of
the presence of the rest of the gas in the vessel.
93
104. Boyle's, Charles', and Dalton's laws can be con-
jointly expressed algebraically by either of the following
equations:
PV_ y\pv\ _76 V' P_ = y jjj_\ _ 76
T"wi li '213' DT Xdti' 273Z)'
in which p, v, d, t denote the pressure, volume, density and
temperature of any one of a number of gases to be mixed
together; P, V, D, T, denote the pressure, volume, density,
and temperature, of the mixture; and V' , D denote the
volume and density of the mixture at the standard tem-
perature and pressure, 0° C and 76 cm. T and t must be
reckoned from the zero of the air thermometer.
105. The amount of aqueous vapor in the atmosphere
varies with time and place, whilst the other constituents
are found in almost unvarying proportions. Dalton's law
tells us that the pressure of the moist air is just the sum
of the pressures of the dry air and of the aqueous vapor
mixed with it. From the classical experiments of Reg-
nault the pressure of the aqueous vapour in the atmos-
phere can be determined, as soon as the dew-point is
known. The dew-point is the temperature at which the
atmosphere at any place would be saturated with the
aqueous vapour which it contains. It is found experi-
mentally by means of a hygrometer.
The following is a part of Regnault's table of the max-
imum pressures (or pressures of saturation, or pressures of
condensation) of aqueous vapour at different temperatures.
It gives the pressure of the aqueous vapour in the atmos-
phere, in millimetres of mercury at 0° at the latitude of
Paris, for dew-points from 0°C to 29°C.
mm.
23-6
250
265
28-1
29-8
°c
mm.
°c
mm.
°C
mm.
°c
mm.
°c
mm.
°c
0
4-6
5
65
10
92
15
127
20
174
25
l
4-9
6
70
11
9-8
16
135
21
185
26
2
53
7
75
12
10-5
17
144
22
197
27
3
57
8
8-0
13
11-2
18
154
23
20-9
28
4
61
9
8-6
14
11-9
19
163 |
|24
22-2 J
29 |
94
Observe, that whilst the pressure of a gas at a tempera-
ture far above its point of condensation depends upon its
temperature and volume, the pressure of the same gas at
its point of condensation (or, in contact with its own liquid)
depends upon its temperature alone.
106. The following example will illustrate how the
density of the atmospheric air can be calculated when its
barometric, thermometric and hygrometric states are
known.
Ex. The reading on the barometer is 76*4, the temper-
ature 20°, the dew-point 8°, and the latitude 44° 13' (King-
ston, Ont.) ; to determine the density of the air, given the
coefficient of dilatation of the barometer scale, which is true
at 0°, to be 0-000008, and the mean coefficient of dilatation
of mercury between 0° and 20° to be 000018.
The barometric pressure in centimetres of mercury at
0° in the latitude of Paris will be
76-4 (1+20x0-000008) 9805 = 76>n
~ 1+20x0-00018 980-9 "
This pressure is due, according to Dalton's law, partly
to dry air, and partly to aqueous vapor in the air. Accord-
ing to Regnault's tables the pressure of the aqueous vapor
for the dew-point 8° is 0'80 cm. Therefore the pressure of
the dry air in the atmosphere is 7531 cm. Hence apply-
ing Boyle's and Charles' laws, the density of the dry air in
the atmosphere
= i^932- X I*?* x273 =0001194,
1000 76 293
the density of the aqueous vapour in the atmosphere
= 8^44 x 0-80 x 273 =0>0000()8
10000 76 293
.-. the density of the atmospheric air =l-202xl0-3.
107. To determine the specific weight of a solid and
of a liquid, body to an approximation of the third degree.
95
Let wlf iv2, w3, denote the number of grams which bal-
ance the solid body in air, distilled water and the liquid
respectively ; the reading of the barometer 764, the tem-
perature 20°, the dew-point 8°, and the latitude 44° 13' ;
the coefficient of dilatation of the barometer scale 0*000008,
the mean coefficient of dilatation of mercury between 0°, and
20°. according to Regnault. 0'00018, and the density of
distilled water at 20°, according to Despretz, 0-998213.
Find, as in art. 100, Wl the approximate weight of the
solid body in vacuo, S the s. w. of the solid body of an ap-
proximation of the second degree, and, as in last article, R
the density of the air. Denote by s the s. w. of the stand-
ard masses against which the body is weighed, and by S'
the s. w. of distilled water at 20°.
The weight of the solid body in vacuo (in grams) is
very nearly = wx+ ( Wx/S - w1/s) R-W1
The weight of the body in distilled water at 20°
= W2-(w2/s) R= W"
The weight of the body in the liquid at 20°
= w3-(w3/s) R= W">
Then the s. w. of the solid body at 20° = == ===- 6'
W - W
and the s. w. of the liquid at 20° = — — ttttt^'
each to an approximation of the third degree.
If the coefficients of dilatation of the solid body and
liquid be known, the specific weights at any other temper-
ature may be determined. By taking the specific weight
of the solid body just determined in place of S, and W' in
place of Wx, and repeating the method above, we could
find the specific weights to an approximation of the fourth
degree and so on to higher degrees. This would however
be useless, as the errors of experimentation would certainly
be greater than any errors, from the exact values, of the
specific weights to an approximation of the third degree.
96
Examination XI.
1. How can it be proved experimentally that Archi-
medes' principle applies to gases? Explain the rise of
smoke in the air.
2. Given the weights of a solid body in the air, water,
and another liquid, to determine the specific weights of the
solid body and liquid to an approximation of the second
degree.
3. Enunciate in four ways the law of Charles, and de-
duce each one from the others.
4. Enunciate Dalton's law relating to the pressure of a
mixture of gases. Give Rankine's statement of the physical
principle underlying Boyle's and Dalton's laws.
5. What are the various corrections to be made in de-
termining the specific weight of a body to an approxima-
tion of the third degree? What are the physical instru-
ments used for this purpose?
6. Define the dew-point. What does it tell us?
7. Write down an algebraical equation which expresses
conjointly the gaseous laws of Boyle, Charles, and Dalton.
8. A merchant buys against lead standard masses, and
sells against aluminium standard masses. Does he gain or
lose thereby? By how much p.c?
9. Which is more favourable for purchasers of goods, a
high or a low barometer?
Exercise XI.
1. The reading of the barometer in a room is 77*34, the
thermometer 15°, the dew-point 10:, the latitude 44 13' ;
the coefficient of expansion of the barometer scale is
0000008, and the mean coefficient of expansion of mercury
between 0^ and 15°, according to Regnault, 000018; the
room is 125m. long, 545 m. broad, and 3'7 m. high; find
the volume, mass, and weight of the air in the room.
97
2. A lump of gold weighs 437,008 grams in air, 414357
in distilled water, and 420699 in ether; the reading of the
barometer is 77*3 cm., of the thermometer 9°, the dew-
point 4°; the latitude, that of London; the s. w. of the
standard masses against which the body is weighed is 8*4,
the coefficient of linear expansion of the barometer scale
O'OOOOIS, the mean coefficient of expansion of mercury
between 0° and 9C, according to Regnault, 000018, and
the density of distilled water at 9°, according to Despretz,
0'999812; to determine the specific weights of gold and
ether to approximations of the first, second, and third
degrees.
3. A cubic decimetre of aluminium just balances a
lump of lead when both are in water; which will weigh
the heavier in air? Why? Find their difference in
weight, 1) in vacuo, 2) in air (s. w. 00012), 3) in air as
indicated by a common balance, the s. w. of the standard
masses being 8*4.
4. Find the force necessary to hold down a balloon, of
which the capacity is 150000 litres, when filled with hydro-
gen, the pressure and temperature being 77 cm. and 15 C,
and the weight in the air (s. w. 0'0012) of the solid
material of the balloon being 14'5 kilograms.
5. Two hollow spheres (radii 1:2) contain equal
masses of air at 10 and 20 : respectively; compare 1) the
pressures of the gases, 2) the total pressures on the
spheres.
6. Find at what temperature the density of dry air is
0'01 at pressure 76 cm., and at what pressure the density
is 1 at temperature 17°C?
7. Compare the densities of the air at the top and bot-
tom of a mine shaft, the temperatures being respectively
11° and 18°, and the pressures 74 cm. and 77 cm.
8. The temperatures at the bottom and top of the
mountain Fuji in Japan were respectively 20° and 4°, and
98
the reduced barometric pressures 76*2 cm. and 48 cm.
Shew that the density of the air at the top of the moun-
tain was just about f of that at the bottom.
9. Ten litres of oxygen at 74 cm. and 18 c are mixed
with 5 litres of hydrogen at 75 cm. and 15° ; find the pres-
sure of the mixture when the volume is 10 litres, and tem-
perature 0°.
10. Compare the volumes of hydrogen at 0° and 77
cm., and of oxygen at 20° and 74 cm., which will be in
proper proportion to form steam (Ha0) which consists of
8 parts by mass of oxygen to 1 part of hydrogen.
11 A vessel filled with hydrogen contains some water,
and the pressure at 0° is found to be 76 cm.: find what
the pressure will be at 20°, when the volume of the gas is
reduced one-half.
12. The Torricellian vacuum is 40 cm. long, and 1 sq.
cm. in section, the temperature 0°, and the atmospheric
pressure 76 cm. Find what would be the height of the
mercurial column if there was admitted into the vacuum 1)
a centigram of dry air, 2) a milligram of hydrogen, 3) a
decigram of water, 4) a gram of ether, all at 0°. Given
the pressure of condensation of ether at 0° to be 18"4 cm.
Answers.
1. 252,062,500; 312,084-6; 306 megadynes.
2. 19293, 0-72001; 19271, 072035; 19267, 072023.
3. 838-65 grs.-wt. ; 837'64 grs.-wt. ; 837'76 grs.-wt.
4. 152,592-4 grs.-wt. 5. 2264:293; 566:293.
6. - 238°C or 35°A; 821 atmospheres.
7. 197:200 nearly. 9. 105 cm. 10. 1-786:1.
11 163-89 cm. 12. 6457; 6067; 7553; 5759.
Chapter XII.
Work. Energy.
108. Work is the production of motion against resist-
ance. Energy is the power to do work. Work is phys-
ically manifested either in accelerating the motions of
bodies, or in changing the configuration of a material
system. Thus when a man lifts a body up, he does work
against the body's weight, and by the work done produces
a change of configuration of the earth. Again, when he
throws a cricket ball, he does work in giving the ball
motion.
We have firstly defined work, then energy. The order
might have been reversed, thus: Energy is the power to
overcome resistance through space; work, the expenditure
of energy, or the transference of energy from one body to
another.
By the configuration of a material system at any in-
stant is meant the condition of the system as regards the
relative position of its several parts. The term form refers
particularly to the bounding surface of any body or system
of bodies. Configuration refers to all parts of the body or
system, whether internal or on the bounding surface.
109. A body in motion has energy in virtue of its mass
and speed (art. 112), and this is called kinetic energy.
Such is the energy of a cannon ball, which enables it to
tear down a rampart against the resisting molecular forces.
Similarly, in virtue of its mass and speed, a running stream
can drive a water-wheel and thus grind our corn. These
are examples of molar kinetic energy.
110. A body may also possess energy in virtue of its
mass, and of its position with respect to other bodies, and
this is called potential energy. It is found by experience
100
that there are forces which act between every pair of par-
ticles in the universe. The force of gravitation, the mol-
ecular forces (cohesion, elasticity, crystalline force, &c),
the atomic force or chemical affinity, are different aspects
of such force. When work is done against such force
upon a body which forms a part of a material system, so
as to alter the configuration of that system, the body in
virtue of its new position has energy which it did not pre-
viously possess. Thus a head of water has energy in
virtue of its position with respect to the earth. The
wound up spring of a clock can keep it going for a week
or longer. Compressed air, such as is used for the con-
veyance of letters in large cities, is a store of energy in
virtue of the configuration of the aerial particles. These
are examples of molar potential energy.
A material system, such e.g. as the solar system, pos-
sesses molar energy; firstly, on account of the motions of
its component parts; this is its kinetic energy; secondly,
on account of its configuration; this is its potential energy.
An oscillating pendulum, or a vibrating spiral spring, is a
beautiful and simple example of a body whose molar
energy is constantly passing from the one form into the
other. At the extremities of the line of vibration, the
energy is wholly potential; at the middle point, it is
wholly kinetic; and at intermediate positions it is partly
kinetic and partly potential. In an undershot water-
wheel the miller depends upon the kinetic energy of the
water to grind his corn; in an overshot water-wheel, upon
the potential energy of the water to drive the wheel.
111. Work is measured by the force overcome and the
distance through which it is overcome conjointly. Thus
in measuring the work done in raising bricks to the top of
a house, the builder multiplies the weight of the bricks by
the vertical height through which they are raised. To
raise double the number of bricks through double the
height will evidently require four times as much work.
101
The unit of work is that in which unit of force is overcome
through unit of distance. In the C.G.S. system the unit
of work is the work of overcoming a dyne through a centi-
metre, and is called an erg. The equation w =fs evi- ""T
dently gives the relation between the work done in ergs,
the force overcome in dynes, and the distance in centi-
metres through which the force is overcome.
112, To determine the kinetic energy of a body whose
mass is m and speed r.
Let the body move against a uniform resistance /.
This will give the body an acceleration f/m, opposite in
direction to the body's motion. If s be the whole distance
through which the body can act against this resistance, so
that after passing through the distance s the speed is
zero, by equation (6) art. 35,
0 = v2 - 2 (f/m)s, .-. fs = \mv2
but/s is evidently the total work done by the body against
the resistance/, and is all that it can do, since, after doing
this work, the speed is zero. Hence ^mv2 measures the
body's kinetic energy, or the amount of work the body can
do, in virtue of having mass m and speed v. If m be
measured in grams, and v in tachs, \mv2 measures the
body's kinetic energy in ergs. Since the kinetic energy
of a body varies as the square of its speed, it is evident
that it is independent of direction; in this respect it is
well to note, energy differs from momentum and force.
113. As an illustration of the preceding article let us
consider the case of a body of weight w and mass m,
thrown vertically upwards in vacuo with speed u. In vir-
tue of its kinetic energy it raises itself against its own
weight. If h be the greatest height reached, the work
done is wh. Now tv = mg (art. 63), and h = u2+2g (art. 36),
therefore wh = ^mir, which, as might be expected, is the
same result as we got in last article.
7s the energy of the body in its elevated position de-
stroyed? No, it is merely in a latent form; for, without
102
imparting any more energy to the body, we can get out of
it, in virtue of its new position, the same amount of work
as it was capable of doing at starting. This will be at
once understood when we remember that by letting the
body fall to its point of starting, it acquires the same speed
which it had at starting (art. 36), and has therefore again
the original kinetic energy imparted to it. In its elevated
position the energy of the body is potential. Such is the
energy of a head of water used to drive machinery, or of
the elevated heavy bodies whose energy is used to drive
piles into the ground.
We see from the above that the potential energy of an
elevated body, relatively to the earth, is measured by wh,
where w is the body's weight, and h its height above the
ground. If iv be measured in dynes and h in centimetres,
then wh measures the potential energy in ergs. Also from
art 35, equations (3) and (6), it is easily seen, that in any
intermediate position of the body between the ground and
height h, the energy of the body is partly kinetic and
partly potential, and that the total energy is constant and
equal to wh or \mu2.
114. Just as it is convenient in many practical ques-
tions to have a gravitation as well as an absolute unit of
force, so in the practical measurement of work it is often
convenient to use a gravitation unit. Such a unit is the
kilogrammetre, or work done in raising a body of 1 kilo-
gram vertically upwards against its weight through the
height of 1 metre. Evidently 1 kilogrammetre = 105(/ ergs.
115. The F.P.S. unit of work is that required to over-
come a poundal through the distance of a foot, and may be
called a foot-poundal. English engineers use as a gravi-
tation unit of work a foot-pound, i.e. the work done in
raising 1 pound vertically upwards through the distance of
1 foot. The foot-pound is evidently equal to g or nearly
32^ foot-poundals.
103
If m be the mass of a body in pounds and v its speed
in vels, then \mv2 measures its kinetic energy in foot;
p_oundals (art. 112), and therefore |rat>2-i-(/, its energy in
foot-pounds. Similarly if a body, whose mass is m pounds
and weight w poundals, be at a height of h feet above the
earth's surface, it has potential energy measured by wh
foot-poundals, i.e. mgh foot-poundals, or mk foot-pounds.
116. The unit rate of working, or the unit of activity
in the C.G.S. system is 1 erg per second. If H denote the
rate of working in ergs per second, / the resistance in
dynes, and v the speed in tachs of the body moved against
the resistance, then H—fv. This formula suggests the
name dyntach for the unit of activity. Watt's horse-
power is a convenient gravitation unit adopted by English
engineers, and is equal to 550 foot-pounds per second.
The French force-de-cheval is a similar gravitation unit
equal to 75 kilogram metres per second, or nearly 7'36xl09
dyntachs. These were supposed to be rates at which a
good horse works, but are now allowed to be too high.
117. The examples of energy we have hitherto taken as
illustrations are energies of systems, the motions and con-
figurations of whose parts are manifest. Our grandest
sources of energy are, however, derived from systems, the
motions and configurations of whose parts are impercept-
ible. Whence the energy which enables the labourer to
dig the ground, the student to pursue his studies, or the
horse to draw his load? These are examples of vital
energy which the man and horse derive from the food they
eat and drink, and the air they breathe. The energy of
gunpowder, of steam, and of a voltaic battery are other
examples of what is called molecular energy.
118. Food and fuel are our principal immediate sources
of energy. Thus coal and the oxygen of the air form a
system which, before combustion, in virtue of the separa-
tion of the atoms of coal and the atoms of oxygen, pos-
sesses potential energy of atomic separation. During
104
combustion the energy becomes kinetic, and may be com-
municated to the water in a boiler so as to heat the water
and form steam, and through this be used to drive an
engine, and by means of the engine do all sorts of me-
chanical work. Similarly food and air form a great store
of potential molecular energy, which is transformed dur-
ing digestion into the vital energy by means of which we
do our daily work. Winds and running water, including
waterfalls, such as Niagara, and the ocean tides, are other
considerable sources of energy made use of by man.
119. Heat, light, and electricity, in their physical
aspects are well denned as forms of molecular energy.
Sound forms a sort of connecting link between molar and
molecular energy. The Transformation of Energy is the
enunciation of the fact:
Any one form of energy may be transformed, directly
or indirectly, into cm equivalent of any other form.
120. Amongst the most important of the modern
advances in Physical Science is the measurement of the
different forms of molecular energy in dynamical units.
Thus the energy of a unit of heat, (the heat required t<>
raise the temperature of 1 gram of water from 4°C to 5°C)\
has been determined experimentally to be nearly equal
to 42 million ergs. From such measurements the very
important generalization, known as the Conservation of
Energy, has been deduced:
Through whatever forms energy may pass, it cannot
be changed in quantity, and hence the toted energy in the
universe remains constant.
As the Conservation of Mass forms the foundation of
modern chemistry, the Conservation of Energy may be
said to form the foundation of modern physics.
121. Although the total energy in the universe remains
constant, it is gradually being transformed into lower
forms so as to be less useful to man. This is the principle
105
enunciated by Lord Kelvin, and known as the Dissipation
or Degradation of Energy:
The energy of the universe is gradually being trans-
formed into a form in which it cannot be made use of by
man, viz., that of uniformly diffused heat.
122. Perhaps the principal force through which energy
is being constantly dissipated, or degraded into the useless
form of diffused heat, is friction (art. 60). The direction
of this force is always diametrically opposite to the direc-
tion of motion, or to that in which motion would take
place under the influence of the other acting forces.
When the surfaces between which friction is called into
play are plane, and sliding motion does, or is just about to
take place, the law of friction, determined by experiment
to an approximation of the first degree, and sufficiently
accurate for practical purposes, may be thus expressed:
For like surfaces the friction varies directly (is the
K^normal pressure between the surfaces, and is independent
jof the areas of the surfaces in contact, arid of the relative
/ speed between the surfaces: F=kE. Or thus:
K For like surfaces the friction per unit of area depends
\rnly upon, and varies directly as, the normal pressure
per unit of area between the surfaces : f= kr.
When there is no relative motion between the surfaces,
the friction may have any value from 0 to the maximum
value, which is reached when motion is just about to take
place. The constant k which measures the ratio of the
maximum friction to the normal pressure is called the co-
efficient of friction for the two surfaces in question.
Rankine has shown that the value of k lies between 0'2
and 0*5 for wood on wood, 02 and 0'6 for wood on metals,
0'3 and 07 for metals on stone, and 015 and 025 for
metals on metals.
On account of the dissipation of energy through friction
and other causes, a machine does not do as much useful
106
work as the equivalent of the energy imparted to it. The
ratio of the useful work done to the energy supplied is
called the efficiency or modulus of the machine. The duty
of a steam-engine is the amount of useful work performed
per unit mass of fuel consumed.
Examination XII.
1. Define energy, work, and the configuration of a ma-
terial system. How is work physically manifested?
2. Define kinetic and potential energy, and give three
good examples of each.
3. How is work measured? Give examples. Name
and define the unit of energy and work.
4. Determine the kinetic energy of a body whose mass
is m, and speed v.
5. Prove that the potential energy of a body whose
mass is m and height above the earth's surface h, is mgh.
6. Prove that when a body is moving vertically, under
no other force than its weight, its total energy, relatively
to the earth, is independent of its position.
7. Define a kilogrammetre and foot-pound, and deter-
mine their values in absolute measure.
8. If a body of m pounds be moving with a speed of v
vels; find its kinetic energy in foot-pounds.
9. Name and define the unit of activity in absolute and
gravitation measures, according to both the C.G.S. and
F. P. S. systems.
10. Give various examples of molecular energy, both
kinetic and potential.
■11. What are our principal sources of energy?
12. Define. the unit of heat, and give its measurement
in ergs and kilogrammetres.
13. Enunciate the principles known as the Transforma-
tion, Conservation, and Degradation of Energy.
107
14. Enunciate the law of friction for plane surfaces in
two ways, and define the coefficient of friction.
15. Define the modulus of a machine, and the duty of
a steam-engine.
16. Prove that when a body is projected upwards in
the atmosphere, the time of ascent is less than the time of
descent.
Exercise XII.
1. How much work must be done to pump 1000 cub. ft.
of water from a mine 150 fathoms deep?
2. In pile-driving 30 men raised a rammer of 500 kilo-
grams through a height of 40 metres 12 times in an hour;
find the average rate of working per man?
3. How many ergs of potential energy are there in a
mill-pond near Kingston, Ont., which is 40 in. long, 20 m.
broad, and 1 m. deep, and has an average fall of 5 metres?
'4. A ball of 40 lbs is moving at the rate of 300 miles
per hour; find its kinetic energy in ft.-lbs.
5. A machine (modulus |) for raising coals is worked
by two horses; how much coal will be raised in a day of 8
working hours from a pit 90 metres deep?
6. An engine is found to raise 6 tons of material per
hour from a mine 110 fathoms deep; find the horse-power
of the engine, supposing \ of its energy to be lost in una-
voidable resistances.
7. A railway train of 300 tons, in passing over a certain
mile, has its speed increased from 40 to 50 miles per hour.
If the average friction be 10 lbs.-wt. per ton, find the work
done by the engine in passing over the mile.
8. What must be the horse-power of an engine whose
modulus is f, working 8 hours per day,'which supplies 3000
families with 100 gallons of water each per day, the mean
height to which the water is raised being 60 feet?
108
'.). How many bricks will a labourer raise to the mean
height of 20 ft., working 8 hours per day; given that the
mass of 17 bricks is 125 lbs., and that the average rate of
doing such work is 1200 ft. -lbs. per minute?
10. If a load be 10 bricks (Ex. 9), and the man's own
mass 140 lbs., what is the rate at which he expends his
vital energy when working?
11. What would be the cost per ton to raise coals from
a pit 25 fathoms deep, allowing $3 per day for a horse and
driver, and that the horse performs 24000 ft-lbs. of work
per minute, working 8 hours per day?
\(l2)At what rate will a train of 100 tons be drawn by a
locomotive-engine of 70 H.P., the frictional resistance be-
ing 10 lbs.wt.-per ton; and how far, after steam is shut off,
will it go before being brought to rest?
13. If 8 lbs-wt. per ton (Ex. 12) be the average fric-
tional resistance until full speed is attained, how long will
it take for the train to attain its maximum speed after
starting; and how far will it have travelled in this time?
14. In what time will a locomotive of 100 force-de-
cheval, drawing a train of 100 tonnes, complete a jour-
ney of 100 kilometres, supposing that the frictional resist-
ance until full speed is attained, and after steam is shut
off until it stops, be on the average 3 kilograms-weight
per tonne, and after full speed is attained, 4 kilograms-
weight per tonne. (1 tonne = 106 grams.)
15. Determine the H. P. of the river Niagara which
has a total descent of 334 feet, and discharges about
4xl07 tons of water per hour.
16. A body rest on a rough horizontal board, which is
moving horizontally; determine the maximum acceleration
the board can have without the body slipping.
17. Shew that it requires as much work to increase the
speed of a ship from 24 to 25 miles per hour as to give it
the first 7 miles per hour.
109
18. A ball of 10 kilograms is fired from the mouth of a
cannon 3 metres long with the speed of 15 kilotachs; find
the mean pressure of the gaseous products on the ball.
19. There were 4000 cub. ft. of water in a mine of depth
60 fathoms, when an engine of 70 H.P. began to work the
pump; the engine worked for 5 hours before the mine was
cleared of the water; if the modulus of the engine were §,
find the rate at which water was entering the mine.
20. Find in dyntachs the rate at which a fire-engine
works, which discharges 10 kilograms of water per second
with a speed of 1500 tachs.
21. A railway carriage of 5 tons mass is started on a
level railroad with a speed of 8 vels, and moves over 200
ft. before it stops; determine the coefficient of frictional
resistance.
22. A cistern is 10 ft. long, 7 ft. broad, and 8 ft. deep.
The height of the top of the cistern from the water in the
well is 56 ft. If a man can work with a pump at the rate
of 2600 ft.-lbs. per minute, and the modulus of the pump
is 066, how long will he take to fill the cistern?
23. A spring tide raises the level of the river Thames,
between London and Battersea bridges, on an average 15
feet. If 5 miles be the distance between the bridges and
900 feet the mean breadth of the river, find the potential
energy of the spring tide when full.
Answers.
1. 5-616 XlO7 ft.-lbs. 2. 20/9 kilogrammetres per sec.
3. 3622 XlO14. 4. 120,373. 5. 32 tonnes.
6. 5. 7. 16,948 ft.-tons. 8. 14-2. 9. 3916-8.
10. 3484-8 ft.-lbs per min. 11. 7lf cents.
12. 26^ miles per hr.: 4608 ft.
13. 3 min. 19| sec. ; 1280 yds. 14. 1 hr. 37 min. 233 sec.
15. 13| millions nearly. 16. kg. 18. 3-75 XlO9.
19. 5522 cub. ft. per min. 20. 1-125 XlO10.
21. 1/201 nearly. 22. 20^- hrs. ; 1668 X 108 ft.-lbs.
Chapter XIII.
Action and Reaction.
123. Two heavy bodies are connected by an inextensible
string which passes over a fixed smooth peg, (or pully, as
in Attwood's machine); required to determine the tension
of the string.
Let T denote the tension of the string, m and m' the
masses of the bodies, m being the greater. Since the ten-
sion of the string is the same throughout, if the weight of
the string may be neglected, by Newton's third law (art.
58); the acceleration of the heavier body will be (mg — T)
-4-w downwards, and of the lighter body (T —m/g)-7-mf
upwards; since these must be equal,
9-—=— -ff> •• T= , ,9
m m m-\-m
~ mi ^ 4.- mq-T T-m'q m-m'
Cor. The acceleration =_^_ _=_ _» = _ — g
m m' m-\-m'
as already proved (art. 68). If m=m' , the tension of the
string is mg, and there is no acceleration, so that the
bodies must either be at rest or moving with uniform
speed.
The above completes the solution of the problem of
Attwood's machine (art. 68), when the weight and mass of
the string, the pully's mass, and friction may be neglected.
124. As an additional illustration of Newton's third
law let us consider one of the very simplest cases of im-
pulse (art. 54), viz., the direct impact of two spherical parti-
cles. If the centres of two spheres move in the straight
line joining them, and the spheres impinge on one
another, the impact is called direct; otherwise, the impact
is called oblique.
Denote by m^, m2, the masses of the particles, and by
ult u2, their velocities before impact. If the direction of
Ui be called + , w2, will be -+- or - according as m2 is or-
Ill
iginally moving in the same or opposite direction to r»i«
The action which takes place during impact may be ex-
plained thus:
a). Alterations of form and volume take place by work
being done against the molecular forces, until the relative
velocity of the two bodies is destroyed. If i? denote the
stress during this first stage of the impact, and v the com-
mon velocity, we get from Newton's dynamical laws,
i? = m1(i/1- v) =m2(v- u2). . . . ... .... (a)
, mvvb\ 4- m2u2 / ■, x
whence v= 1 1 =-^- .... (1)
mx -} m2
R= mim'2 (»x-u2) (2)
m1-\-m2
These equations contain the complete solution of the
problem, if the bodies do not separate again after impact.
This will be the case, when the force of adhesion between
the bodies counterbalances the force of elasticity, which
tends to separate them.
b). If the bodies be sufficiently elastic, they have the
common velocity v only for an instant, for an amount of
molecular potential energy has been stored up in conse-
quence of the change of configuration of each sphere, and
in the transformation of this energy into the kinetic form
through the force of elasticity, the original forms and vol-
umes are as much as possible restored. Daring this sec-
ond stage of the impact it is evident that the bodies re-
ceive accelerations of momentum in the same directions
as during the first stage, and if R denote the stress called
into play, and V\, v2 the velocities after impact,
R' = mi(v-v{) = m2(v2-v) (b)
Now it has been proved by experiment, that if the im-
pact do not make any sensible permanent alteration of
form, the relative velocity of the bodies after impact bears
a constant ratio to the relative velocity before impact, i.e.
vx - v2 = - e(Ux -u 2) (c)
112
where e is a proper fraction, whose value depends only
upon the material natures of the spheres. From (a), (b),
and (c), by algebraical analysis, R = eR. Also
v1 = u1- — 17^—{\+e){u1-u2) (3)
mx-\-m2
v2 = M2+ — (1 + f) ( '*i - u2) (4)
m1-\-m2
R+R' =mim2(1+e)(u1 + u2) (5)
The value of e was found by Newton to be -§ for balls
of compressed wool and steel, § for balls of ivory, and -j-|
for balls of glass. It is called by most writers the co-
efficient of elasticity, a name strongly objected to by Tait
and Thomson, who call it the coefficient of restitution.
Cor. 1. If m2=oo, and u2=0, the case is that of a
sphere impinging normally on a fixed plane. The equa-
tions (3), (4), (5), become then
Vi= —eu-y , v2=0 , R-\-R' = m1(l+e)n1.
Cor. 2. If m1 = m2, and e = l, then v1 = u2, and v2 = u1,
i.e. the bodies interchange velocities. This may be shown
to be nearly the case for balls of ivory or glass. Also, if
u2 = 0, and m1=em2 , then vx = 0 , and v2 — eux.
125. The following results are at once deduced from
the preceding investigation:
1. Whether the bodies be elastic or not, the total momen-
tum is not affected by the impact. (Art. 59.)
From (1), (mx + m2)v = m1Ui-{-m2u2
From (3) and (4), m1v1 + m2v2 =m1u1-\-m2u2
2. The total molar kinetic energy after impact is less
than before impact.
£(ro1 + m2>?2=im1w12 + £ra2M22-| r^-(u1 - u2f ,
mx -f m2
and \mxVi2 + \m2v22 =
|wii«i2 +£m2w22 - ^ mimz (1 - e2) (Mi - u2)~.
nil + m2
113
What becomes of the molar kinetic energy lost? It is
transformed into the molecular kinetic energy of heat, so
that the bodies after impact are warmer than before impact.
126. The Conservation of Momentum (art. 59) teaches
that change of momentum in a body or system of bodies
must be produced by forces external to the body or sys-
tem. Let any forces act upon a body of mass m and pro-
duce in it an acceleration a, then ma is the measure of
the single force which would produce the same dynamical
effect on the body. If, after Newton, we call a force
measured by - ma the resistance to acceleration, which
the body offers in virtue of its mass and inertia, then
WAlembert's Principle at once follows as a corollary to
Newton's third law:
The external forces acting upon a body (or system, of
bodies), together with the resistance (or resistances) to ac-
celeration, form a, system of forces in equilibrium.
This principle evidently amounts to saying that the
molecular or internal forces acting within a body or sys-
tem of bodies are themselves in equilibrium.
127. Newton published his axioms or laws of motion in
his celebrated work " Philosophiae Naturalis Principia
Mathematical At the end of the scholium appended to
his laws he points out that another meaning may be at-
tached to the words action and reaction besides that of
force :
If the action of an agent be measured by the product
of Us force into its velocity; and if, similarly, the reaction
of the resistance be measured by the velocities of its sev-
ered parts into their several forces, whether these arise
from friction, cohesion, weight, or acceleration; action and
reaction, in all combinations of machines, will be equal
and opposite.
As pointed out by Tait and Thomson, this remarkable
passage contains in it the foundation of that great modern
generalization, the Conservation of Energy.
114
Examination XIII.
1. A string passing over a smooth peg connects two
heavy bodies; determine its tension, 1) when the bodies
have different weights, 2) when the weights are the same.
2. Two spherical particles impinge directly; describe
the nature of the impact, and determine the equations of
motion. Define stress.
3. What is denoted by e in the theory of impact? How
can it be experimentally determined? Give its value for
a few substances.
4. How do we deduce the equations of impact of a
sphere on a fixed plane. Give the equations.
5. Determine under what conditions will two spheres,
impinging directly, interchange velocities?
6. Prove that the momentum of a system of spherical
particles is not altered by direct impacts of its component
parts.
7. Determine the change of molar kinetic energy in both
stages of impact, when two spheres impinge directly.
What becomes of it?
8. Enunciate and explain D'Alembert's principle.
9. How can it be said that Newton in his third law laid
the foundation of the science of energy?
Exercise XIII.
1. A boulder of 2 tonnes is rolled from the summit of
El Capitan in the Yosemite valley, a rock rising vertically
3000 feet; find the speed of, and distance travelled by, the
earth when the boulder strikes the ground. (See Ex. V, 6.)
2. A ball is let fall from a height h above a fixed smooth
table, and rebounds to a height h', prove that e for the
ball and table =1 '(h'/h).
3. Find the tensions of the strings in Ex. VII, 5, 9, 15.
115
4. A chain 20 ft. long and mass 2| lbs. per ft. is hang-
ing vertically, and is connected by a fine wire of insig-
nificant mass, which passes over a smooth pully to a body
of 56 lbs.; find the tensions 1) of the wire, 2) of the chain
at its middle point, 3) of the chain 2 ft. from the free end.
5 Prove that in Attwood's machine, if the total mass
of the moving bodies be constant, the greater the tension
of the string is, the less is the acceleration.
6. A body of 5 kilograms, moving with a speed of 3
kilotachs, impinges on a body of 3 kilograms moving with
a speed of 1 kilotach ; e. = § , find the speeds after impact.
7. Two bodies of unequal masses, moving in opposite
directions with momenta equal in magnitude, meet; shew
that the momenta are equal in magnitude after impact.
8. The largest gun in the United States in 1891, with
a charge of 440 lbs. of prism powder, sent a projectile of
1000 lbs. with a speed of 1865 vels: if the mass of the gun
and carriage were 100 tons, find the speed of recoil of the
gun, and the potential energy in a lb. of powder.
9. The result of an impact between two bodies moving
with equal speeds in opposite directions, is that one of
them turns back with its original speed, and the other
follows it with half that speed; find e and the ratio of the
masses.
10. A bomb-shell moving with a speed of 50 vels bursts
into two parts whose masses are 70 and 40 lbs. After
bursting, the larger part turns back with a speed of 10
vels; find the speed of the smaller part.
11. A and B are two uniform spheres of the same ma-
terial and of given masses. If A impinges directly upon a
third sphere Cat rest, and then C,on B at rest, find the
mass of C in order that the velocity of B may be the great-
est possible for a given initial velocity of A.
12. Find the necessary and sufficient condition that one
body moves after direct impact with the original velocity
of the other.
116
13. Two balls, each \ cub. decim., one of elm and the
other of silver, are connected by an inextensible cord and
immersed in Lake Ontario; find the tension of the cord in
grs.-wt., and the acceleration, neglecting friction and the
cord's weight.
14. Two particles of 1 and 2 kilograms are connected
by a cord which passes over a smooth pully; this pully and
a particle of 3 kilograms are connected by another cord
which passes over a smooth fixed pully; neglecting the
masses and weights of the pullies and cords, find the ten-
sions of the cords and the accelerations of the three par-
ticles.
15. A jet of water is projected against an embankment
so as to strike it normally. If the speed of the jet be
2500 tachs, and 50 kilograms of water strike the embank-
ment per second, find the pressure of water against the
embankment, 1) when the water does not rebound, 2) when
it rebounds with a speed of 500 tachs.
16. A strikes B which is at rest, and after impact re-
bounds with a speed equal to that of B; shew that B's
mass is at least 3 times A'a mass.
17. If the sum of the masses of two impinging spherical
masses be 2m, find the greatest loss of molar kinetic energy
for given values of e, ulf and u2.
Answers.
1. 1,376 X 10"1* cm. per year; 29,776 X 10~21 cm.
3. 12 kilogrs.-wt.; 75 or 125 lbs.-wt.; 2.4 Ibs.-wt.
4. 52-8, 26-4, 5-28 lbs.-wt. 6. 1750, 30831.
8. 9-325; 123,491 ft.-lbs. 9. J; 1:4. 10. 155.
11. C= j/(A B). 12. Ratio of masses e: 1.
13. (0-82)0; 437^ grs.-wt. nearly.
14. 24/17 and 48/17 kilogrs.-wt.; 70/17, 5#/17, g/11.
15. 127,486 and 152,983 grs.-wt. 17. imil-'e2)^-^)2.
Chapter XIV.
Dim ens iona I Eq uatioit s .
128. In the previous pages the student has been intro-
duced to two distinct scientific systems of units, called the
C. Gr. S. and F. P. S. systems respectively. In both sys-
tems three independent or fundamental units are chosen,
and from these all others are derived. It is not necessary
that any three special units be taken as the fundamental
ones. The three, however, which are most easily fixed
upon; and with standards of which, comparisons are most
easily and directly made, at all times and at all places; and
in relation to which the derived units are most easily de-
fined, and are of the simplest dimensions, in virtue of the
established relations between the different units; are the
units of length, mass, and time.
Dimensional equations are such as express in algebrai-
cal form the relations between dynamical units, and are
used more particularly to express how a derived unit de-
pends upon the fundamental units.
129. Whatever units of length, time, and speed be used,
V oc L/T; where V measures the speed of a body moving
with constant speed, and L is the distance passed over by
the body in the time T. Now if we take the unit of speed
as that in which unit of length is passed over in unit of
time, the relation is expressed thus, V = L/T. Hence if
v, I, t, denote the units of speed, length, and time in a
scientific system, v — l/t. This is called a dimensional
equation. It tells us that the unit of speed depends upon
the unit of length to the first power directly, and the unit
of time to the first power inversely. Hence if the unit of
length be increased or diminished n times, so will the unit
of speed be increased or diminished n times; and if the
unit of time be increased or diminished n times, the unit
of speed will be diminished or increased n times.
118
Similarly if m, a, M, f, w, h, denote respectively the
units of mass, acceleration, momentum, force, work, and
activity, in a scientific system of units.
v I iK/r ml n M ml ni mP
«=T=^ M=mv=T,f=T=T,w=fl = -r,
Hence if the unit of length be increased or diminished
x times, the unit of mass y times, and the unit of time z
times, the unit of activity will thereby be increased or
diminished x2y/z3 times.
If i, o, denote the units of angle and angular velocity,
i= arc/radius = l/l = 1°, i.e. the unit of angle is independent
of the fundamental units; and o = i/t=t~1.
If A, p denote the units of area and pressure -intensity,
A = l2, and p =f/A = m/ ( It2 )
If V, d denote the units of volume and density,
V= I3, and d = m/V= m/ls .
130. In whatsoever way the dimensions of a derived
unit be deduced, they must of necessity always be the
same. Thus (art. 22) o=v/r =(l/t)/l =t~1. Similarly
(art. 112) the dimensions of energy and work are mv2, i.e.
(ml2)/t2 as above.
When an equation occurs in which different units are
involved, it is evident that the dimensions of each term
relatively to the fundamental units must be the same;
otherwise, by simply changing the values of the funda-
mental units, the equation becomes untrue. For examples
see articles 35, and 123 to 125.
131. An important use of dimensional equations is to
facilitate the calculations of the numerical relations be-
tween the derived units of different systems, when the
numerical relations between the fundamental units are
known. Thus if /', m\ t' denote the fundamental units in
119
the F. P. S. system, and p' the derived unit of pressure-
intensity, and I, m, t, p the corresponding C. G. S. units,
, m' _ m . p' _ m' I ( t y
'' JF2' p '"W'" y ~"1^' T'{ ~F
= 453-593x0-0328087 = 14-8818 (see tables art. 132), i.e. 1
poundal per square foot = 14'8818 barads.
Ex. Find the units of length, mass, and time in a
scientific system in which a mile per hour is the unit of
speed, a pound-weight the unit of force, and a foot-pound
the unit of work.
Let L, M, T, denote the fundamental units:
JL_5280 J^ = 22_ V_
~f ~3600" /' "15 " t
(1)
ML Q01 ^m'V _ 193 m'V ,9.
~W~ ¥ XP 6~' ~F~ " {)
ML2 _ 193 nvT2
T2 ~6 t'<
2
(3)
.-. L = l'=l foot, T=lh' = ~ second, and
M=™ ( *>•. « = 14||pounds.
132. The following tables give the numerical relations
between the C G. S., F. P. S., and a few other frequently
occurring units. The numbers in the tables of length and
mass give the results of the most accurate observations
made in comparisons of the French and English standards.
Those in the other tables are calculated from the dimen-
sional equations of the units, as explained in last article.
Each number is true to the last decimal place given, and
the mantissae of the logarithms of the true ratios are
added,
120
1 foot
1 mile (statute)
1 metre
1 square foot
1
1
1
acre
are
square kilometre
I. Length or Distance.
= 30*4797 centimetres
= 1609*33 metres
= 3-28087 feet
II. Area or Sin-face.
= 929*014 sq. centimetres
= 40-4678 ares
= 1076*41 square feet
= 247-110 acres
cubic foot
gallon
litre
III. Volume, Bulk, or Capacity.
= 28-3161 litres
= 4-54102 "
= 61-0254 cubic inches
right angle
radian
IV. Angle.
= 1-5707963268 radian
= 57-295779513 degrees
V. Mass.
ounce avoirdupois -- 28-34954 grains
pound " = 453-5927 "
1 gram
1 kilogram
Mantissae.
4840111
2066451
5159889
9680222
6071101
0319778
3928899
4520332
6571531
7855105
1961199
7581226
4525461
6566661
1884321
= 15*43235 grains
= 2-204621 lbs. avoirdupois 3433339
VI. Density.
1 gram per cub. cm. = 624262 lbs. av. per cub. ft. 7953672
VII. Time.
1 day (mean solar) = 86400 seconds 9365137
1 sidereal day = 86164-1 " 9353264
1 mean sidereal month = 27*321661 days 4365071
1 mean synodic " = 29*530589 " 4702721
1 sidereal year = 31558149*6 seconds 4991116
= 365*2564 days 5625978
1 mean tropical year = 365*2422 " 5625809
121
A solar day is the time in which the sun apparently
revolves around the earth. A sidereal day is the time of
the apparent rotation of the sphere of the heavens. A
sidereal month is the time in which the moon makes a
complete revolution in the sphere of the heavens amongst
the fixed stars. A synodic month is the time between two
consecutive full moons. A sidereal year is the time in
which the sun apparently makes a complete revolution in
the sphere of the heavens amongst the fixed stars. A
tropical year is the time between two consecutive appear-
ances of the sun on the vernal equinox, one of the points
in which the equinoctial cuts the ecliptic; it governs the
return of the seasons, and varies slowly through a maxi-
mum range of about a minute on each side of the mean
value. The student will do well to satisfy himself that a
positive rotation of the earth would produce an apparent
negative rotation of the sphere of the heavens, and a pos-
itive revolution of the earth around the sun would produce
an apparent positive revolution of the sun around the
earth. It follows from this that the number of sidereal
days in the sidereal year exceeds the number of mean solar
days by unity; whence the relation between these days.
VIII. Speed.
1 poundvel
vel = 30-4797 tachs
mile per hr., or 22/15 vels = 44*7036 "
IX. Momentum.
= 138254 gramtachs
X. Force (taking (j = $8Q-o).
= 13825-4 dynes
= 1'0199 gram-weight
= 63'5354 dynes
XI. Pressure-intensity.
poundal per sq. foot = 14-8818 barads
mean atmosphere = T0136 megabarad
poundal
kilodyne
grain-weight
Mantissae.
4840111
6503425
1406772
1406772
0085524
8030157
1726550
0058495
122
XII. Work and Energy.
1 foot-poundal = 421394 ergs 6246883
1 kilogrammetre = 7-23307 foot-pounds 8593228
XIII. Activity.
1 foot-poundal per second = 421394 dyntachs 6246883
1 force de cheval =735405 megadyntachs 8665266
Examination XIV.
1. What determines the choice of fundamental units?
2. Why is the French method of forming multiples and
submultiples of units the best?
3. Define a dimensional equation. Write down the
dimensional equations between angular velocity, momen-
tum, energy, angle, pressure-intensity, and density, and
the fundamental units.
4. Determine the ratios of the units of acceleration,
angular velocity, density, and the gravitation units of the
rates of doing work, in the F. P. S. and C. G. S. systems.
5. Define the following terms: mean solar day, sidereal
day, sidereal month, synodic month, sidereal year, tropical
year, equinox, equinoctial, ecliptic.
6. How is the ratio of the sidereal day to the mean
solar day determined? Calculate the ratio.
7. Check all the ratios in tables VIII to XIII. art. 132.
Exercise XIV.
1. If a kilometre and hour be the units of length and
time, what number will express the mean value of g?
2. If a metre, kilogram, and minute were the funda-
mental units, what number would express the mean atmos-
pheric pressure at the sea-level (T014 megabarad).
123
3. If a metre per second, a kilogram-weight, and a kil-
ogrammetre were the units of speed, force, and work, find
the units of length, mass, and time, and the number which
expresses the pressure at a kilometre-depth of ocean (s.w.
1-027).
4. If a metre, kilogram, and 10-4 of a day be the funda-
mental units, find in dyntachs and barads the derived
units of activity and hydrostatic pressure.
5. If g, a kilogram-weight, and a force-de-cheval be the
units of acceleration, force, and activity, find the units of
momentum and pressure-intensity.
6. The units of speed, acceleration, and force are 1 kil-
ometre per hour, g, and the weight of a kilogram ; find the
units of length, mass, time, and density in terms of the
C. G. S. units.
7. The relation between g, and the time of oscillation
(f) and the length (I) of a pendulum is t- n \ {l/g). If 1
second be the unit of time, and the length of the second's
pendulum at the latitude of 45 :, where </ = 9805 tachs per
second, be 100 units of length, find the unit of length in
centimetres, and the number which measures the mean
value of g.
Answers.
1. 127,072-8. 2. 36504X104. 3. lm.;9805 grs.;
1 sec. ; 1-037 X 106. 4. 1013/8643; 105/8642.
5. 7-5 megagramtachs; 980-53x754^105.
6. 106 -T- ( 362X 980-5 );103; 103- (36x9805);
366x980-53--1015. 7. 0-993454 cm. ; 986-96.
Chapter XV.
Composition of Velocities.
133. If a body have simultaneously two velocities, re-
presented by lines drawn from a point, the resultant veloc-
ity will be represented by the diagonal, drawn from that
point, of the parallelogram described on the two lines as
adjacent sides.
Let a body have a velocity along the line AX, repre-
sented by AB, and at the same time let the line AX move
parallel to itself, the end A always keeping on A Y, with a
velocity represented by AC. It is evident that every
point in AX, and also the body moving along AX, has
this velocity represented by AC, and therefore the body
under consideration has simultaneously velocities repre-
sented by AB and AC. It is required to prove that AD,
the diagonal through A of the parallelogram ABDC, re-
presents the resultant velocity of the body.
1). In any time t let the body move along AX through
the distance AP, and in the same time suppose that AX
in virtue of its motion moves parallel to itself through the
distance A A', so that at the end of time t the position of
AX is A'X. In AX' take A'P' equal to AP, then evi-
dently P' is the position of the moving body at the end of
time t. Now AP: AA' = AB: AC, .: A'P' : AA = CD-.AC,
.'. A,D, and P' must be in one straight line. Hence in any
time t the body i^ found to be in AD or AD produced,
and therefore the diagonal AD represents the direction of
the resultant velocity.
2). Because the ratio AP' : AP is equal to the constant
ratio AD : AB, and since AP varies as t, therefore also
AP' varies as t, i.e. the velocity along AD is uniform.
125
3). Because AP' : AP: AA = AD: AB: AC, therefore
AD represents the resultant speed on the same scale that
AB and AC represent the component speeds in the direc-
tions of AX and A Y.
The above important theorem is called the Parallelo-
gram of Velocities. It may be lucidly illustrated by the
motion of a boat which is propelled directly across a
stream whilst carried down by the current. The triangle
of velocities (art. 134) is another way of expressing the
same truth.
Cor. If a body have simultaneously two velocities re-
presented by lines AB.. AC, and if E be the middle point
of BC, the resultant velocity is in the direction of AEand
is measured in magnitude by twice the length of AE.
The Triangle of Velocities.
134. If a body hare simultaneously two velocities re-
presented by lines AB, BC, the Hue AC represents the re-
sultant velocity.
Cor. 1. If a body have simultaneously three velocities
represented by the sides of a triangle taken in order {e.g.
AB, BC, CA: or AC. CB, BA) the body will be at rest.
Cor. 2. If AB, AC represent the velocities of a body
at the beginning and end of any interval, BC represents
the total acceleration during that interval.
The Polygon of Velocities.
135. If a body have simultaneously velocities repre-
sented by lines AB, BC, CD, . . .DM, MN, the line AN
represents the resultant velocity.
Cor. If a body have simultaneously velocities repre-
sented by the sides of any polygon taken in order, the
body will be at rest.
The polygon of velocities is immediately deduced by
repeated applications of the triangle of velocities. Note
that this theorem is true whether the lines representing
126
the velocities be all in one plane or not. The following
may be taken as an important particular case.
The Parallelepiped of Velocities.
136. If a body have simultaneously three velocities re-
presented by the edges of a parallelepiped which meet at
a point the diagonal of the parallelepiped through the
point will represent the resultant velocity.
137. Since acceleration is measured by the change of
velocity per unit of time (art. 24), it is evident that there
are propositions relating to the composition of simultaneous
accelerations exactly similar to those of the preceding
articles for velocities. Hence
1. The Parallelogram, and Parallelepiped of Accelerations.
If a body have simultaneously two or three accelera-
tions represented by lines drawn from a point, the result-
ant acceleration will be represented by the diagonal,
drawn from that point, of the parallelogram or parallele-
piped described on the lines as adjacent sides or edges.
2. The Triangle and Polygon of Accelerations.
If a body have simultaneously accelerations represented
by the sides of any triangle or polygon taken in order, the
resultant acceleration will be zero.
Relative Velocity.
138. By the term relative velocity we denote the veloc-
ity of one body with respect to or relatively to another
body. All velocity is relative (art. 14).
Ex. 1. If a body A be at rest, and another body B is
moving eastwards with a speed 12 ; or, if B be at rest, and
A is moving westwards with a speed 12; or, if A is moving
westwards with a speed 4, and B is moving eastwards with
a speed 8; in all three cases the velocity of B relatively to
A is 12 eastwards, and the velocity of A relatively to B is 12
westwards; for evidently B is separating from A eastwardly
with a speed 12, and A is separating from B westwardly at
the same rate.
127
Ex. 2. Let two persons A and B start from the same
point with equal speeds v, and let Pls P>, Ps> • • • • be
the positions in successive units of time of A who is
travelling northwards, and Qy, Q2, Q3, • • • • the correspond-
ing positions of B who is travelling eastwards. At the
end of the first unit of time. A is at a distance Qx Px or
v |/2 in a N.W. direction from B; at the end of the second
unit of time, the distance is Q2 P2 or 2 v 4/2 in a N.W.
direction; at the end of the third unit of time, the distance
is Qs P3 or 3 v \/2 in a N.W. direction; and so on. Hence
we learn that A is moving relatively to B with a velocity
v |/2 in a N.W. direction. Similarly, the velocity of B
with respect to A is v \/2 in a S.E. direction.
From these examples it is evident that the velocities of
two bodies with respect to one another are equal in mag-
nitude and opposite in direction.
139. Having given the velocities of two bodies, (i.e.
with respect to the earth), to determine their velocities
relatively to one another.
Let two bodies P and Q have velocities represented by
AB. AC respectively. Give to each of them a velocity
represented by BA : P is then brought to rest, and Q has
a velocity represented by BC (art. 134). Since the rela-
tive velocities of the two bodies cannot be altered by giv-
ing to each the same velocity, BC must represent the
velocity of Q relatively to P. Similarly by giving to each
a velocity represented by CA, it is manifest that CB rep-
resents P's velocity relatively to Q. Hence
If two particles have velocities represented by lines
drawn from a point, the line joining the other extremities
of the two lines represents the relative velocities of the
particles. Or thus:
Iftuio particles P and Q have velocities represented
by lines AB, AC, then BC represents the velocity of Q
relatively to P, and CB the velocity of P relatively to Q.
128
The direction of relative velocity is not necessarily the
direction of relative position at any time. The latter de-
pends upon the relative positions of the bodies at the be-
ginning of motion as well as on their relative velocities.
Cor. If two particles have accelerations represented by
lines drawn from a point, the line joining the other ex-
tremities of the two lines represents the relative acceler-
ations of the particles.
140. If a body have simultaneously two velocities or
accelerations denoted by vx and v2, and if i denote the
angle between the directions of vx and v2, v the resultant
velocity or acceleration, t\ the angle between the directions
of v and v1} and i2, the angle between the directions of v
and v2, the following formulae are easily deduced:
v2=v12-\-v22 + 2v1 v2 cos i .... .... ....(1)
sin ix = - sin i, and sin i2 = — - sin i .... .... (2)
v v
v2 sin i i , vx sin i , ., x
tan ix — — . , and tan i2 - — ^ . .... (a)
V\-\-v2 cos / » v2-\-vx cos i
Examination XV.
1. Enunciate and prove the parallelogram of velocities.
Give three good physical illustrations thereof.
2. Enunciate the triangle, polygon, and parallelepiped
of accelerations.
3. Given the velocities of two bodies, determine their
relative velocities. Apply your result, and illustrate by a
figure, when the bodies are moving in the same or in
opposite directions.
4. Explain the directions of the trade winds, the anti-
trade winds, and the Gulf stream.
5. Simplify the equations in art. 140 when i = 0, ^n,
and n\ also when v2 = vv
6. Given the velocities and accelerations of two bodies,
express their relative velocities and accelerations by means
of algebraical equations similar to those in art. 140.
129
Exercise XV.
1. Prove that the resultant of two equal velocities
bisects the angle between them; and conversely.
2. The resultant of two equal velocities is equal to
either of them; find their inclination.
3. A river is 12 miles broad; a boat is rowed directly
across at the rate of 3 miles per hour; the current is 2
miles per hour; how far does the boat travel in crossing,
and how long does it take her to cross?
4. Find the resultant of velocities 2, 2, 2, 3, 3, 3, which
a particle receives simultaneously in directions parallel to
the sides of a regular hexagon taken in order.
5. Two persons start from the same place, the first an
hour before the second; they travel along roads inclined
to one another at an angle \ n, each with a speed of 150
tachs; find their relative velocity, and their distance apart
at the end of 1 hours from the starting of the first.
6. The speed of light is 3 X 1010 tachs, and that of the
earth in its orbit 3 megatachs; what is the maximum
angular displacement of a star owing to the aberration of
light.
7. Two straight railroads cross each other; a train on
each line is approaching the junction with constant speed;
what is the necessary and sufficient condition that the
trains collide?
8. Two equal circles in the same plane touch each
other, and from the point wL contact two persons move
along the circumferences in opposite directions with equal
speeds; shew that each will appear to the other to move in
the circumference of a circle of double the diameter of the
real circles of motion, the observer being in the circum-
ference of the circle of apparent motion of the other.
What will be the apparent motions, if the two persons start
from the point of contact in the same direction?
130
9. Two trains, 200 and 150 ft. long respectively, are
travelling on a double track railroad, with speeds of 20 and
25 miles per hour respectively. How long do they take to
pass one another, 1) when going in the same direction, 2)
when in opposite directions.
10. If a body have simultaneously velocities repre-
sented by p. OA and q. OB, its resultant velocity is repre-
sented by (p+q) OC, where C is a point in AB, such that
p. AQ=q. BC.
pLL) A man can row a boat at 4 miles per hr. If the
current of a river be 2 miles per hr., in what direction
must he row relatively to either bank so as to cross 1) at
right angles to the current, 2) in the shortest time?
fl-2-jA- ship is steaming due east across a strong south-
ward current. At the end of 4 hours the ship is found to
have gone 40 miles 30° south of east. Find the current.
13. To a man walking at 2 miles per hour the rain ap-
pears to fall vertically; when he increases his speed to 4
miles per hour, it appears to meet him at an angle of \n;
find the velocity of the rain.
14. The wind blows along a railroad, and two trains
moving with equal speeds, have the aqueous cloud-track of
the one double that of the other; find the ratio of the speed
of each train to that of the wind.
Answers.
2. §tt. 3. 7614-9 ft.; 24 min.
4. 2. in the direction of the middle velocity 3.
5. 150, \k with either road; 1,946,998. 6. 20"-6.
8. Each will seem to move from and to the other in a
line perpendicular to the common tangent.
9. 47r8T; m 11. 1) trr, 2) fcr to either bank.
12. 5 miles per hr. 13. 2 |/2 miles per hr. at \n to
vertical. 14, 3:1.
Chapter XVI.
Composition of Forces.
A. Forces whose lines of action meet one another.
141. A single force which would produce the same
effect as two or more forces is called their resultant. The
term component is correlative to resultant. The student
should familiarize himself with the term line of action of
a force by imagining the force to act on a body either
through a stretched inextensible cord or through a rigid
straight rod, when the line of the cord or rod will be the
line of action of the force, and the point at which the cord
or rod meets the body may be said to be the point at which
the force acts on the body. A rigid body is one whose
configuration remains constant, whatever forces act upon
it. In the dynamics of solids (stereodynamics) the bodies
are assumed to be rigid, unless otherwise stated. This
gives only a first approximation to the complete solution
of problems, but it is an approximation sufficiently accur-
ate for most practical purposes. We speak of the tension
of a stretched cord, assuming thereby that the stress be-
tween every pair of contiguous particles is the same. If we
may neglect the weight of the cord, this may be taken as
an immediate result of Newton's third law (art. 58), and if
the cord passes round any smooth surface, e.g. a peg, the
tension remains unaltered, as any smooth surface can only
exert force normal to itself, and this will not affect the
stress along the cord, as will appear in what follows. A
force is completely represented by a straight line, when
the line is the actual line of action of the force, and its
length measures the magnitude of the force. Any parallel
line of equal length would represent the force in magni-
tude and direction.
132
142. The resultant of any number of forces which hove
the same line of action is their algebraical sum.
Let/l5 /2, —fz,—f±, /5, . . . denote any forces (in dynes
or poundals) acting on a particle of mass m (in grams or
pounds). If alt a2, —a3, - «4, a5, . . .denote the accelerations
(in tachs or vels per second) which the forces acting
separately would respectively produce, then when all the
forces act simultaneously, the resultant acceleration will
be Oi+a2 _ «3~ #-4+«5 •■• •> an(i therefore the resultant
force m (a^+a® - «s - «4+«5 ■•• >)• If therefore / denote
the resultant force, f=m («! + a2 - a3 - aA + a5 . . .)
= ma1+ma2-ma3 - ma^+ma5 . . . =fi-\-f2 -/s -ft+fs ■ • •
143. From the parallelogram of velocities we passed at
once to the parallelogram of accelerations (art. 137). When
now we take into consideration the mass of the moving
body, we at once deduce the parallelogram of accelerations
of momentum, or as it is more commonly called,
The Parallelogram of Forces.
If two forces acting on a particle be represented in
magnitude and direction by lines drawn from a point, the
resultant force will be represented in magnitude and
direction by the diagonal, drawn from that point, of the
parallelogram described on the two lines as adjacent si<les.
Denote by/!,/2, and m, the forces and the mass of the
particle, and by / the resultant force. If a1} a2, a denote
the accelerations produced by flf f2, f respectively, then
a1=fi/m, a2—f2/m, a=f/m, (art. 56). Let the lines AB,
AC represent /^T^ and complete the parallelogram ABDC.
Since a^ : a2— f\ '-f>, AB, AC may be taken to represent
the accelerations alf a2, and then AD will represent the
resultant acceleration a (art. 137.) Therefore AD must
be the direction of the resultant force/, and since ci\ : a2 : a
-fi '• fi :f AD must also represent the resultant force in
magnitude on the same scale as AB, AC represent the
components.
133
Cor. 1. If AB, AC represent two forces acting upon a
particle A, and if E be the middle point of BC, the re-
sultant will be completely represented by 2 AE.
Cor. 2. If two equal forces (/, /) act upon a particle
at an inclination i, the resultant is 2/ cos \ i, and is equally
inclined to the components.
Cor. 3. When / is substituted for v in art. 140, the
formulae apply equally well to forces acting upon a particle.
144. The parallelogram of forces can be proved experi-
mentally by simple mechanical contrivances, as explained
in treatises on Experimental Physics.
The triangle of forces is another way of stating the
same truth. Observe the use of the term particle in these
enunciations. If body were used, it would be necessary to
add that the lines of action of the forces were concurrent.
The Triangle of Forces:
If three forces acting upon a particle can be repre-
sented in magnitude and direction by the sides of a
triangle taken in order, they will keep the 'particle in
equilibrium. Or thus:
If two forces acting on a particle be represented in
magnitude and direction by two sides of a triangle taken
in order, the resultant force will be represented by the
third side taken in the reverse order.
Conversely: If three forces acting on a particle keep
it in equilibrium, an<l a triangle be drawn having its sides
parallel to the lines of action of the forces, the magnitudes
of the forces will be proportional to the lengths of the
sides of the triangle respectively parallel to thou.
Cor. 1, If three forces keep a particle in equilibrium,
and a triangle be drawn having its sides perpendicular to
the lines of action of the forces, the magnitudes of the
forces will be proportional to the lengths of the sides of
the triangle respectively perpendicular to them.
134
Cor. 2. If P, Q, R represent three forces which keep
a particle in equilibrium, and A, B, C denote the angles
between the directions of Q and R, R and P, P and Q re-
spectively, then P: Q:R=sin J.: sin B-.sin C.
The Polygon of Forces:
145. If (my number of forces, whose lines of action are
concurrent, can be represented in magnitude and direction
by the sides of a polygon taken in order, they will be in
equilibrium. Or thus:
If any number of forces, whose lines of action are con-
current, be represented in magnitude and direction by the
sides of a polygon but one, taken in order, the remaining
side taken in the reverse order will represent the resultant
force in magnitude and direction.
The polygon of forces is immediately deduced by re-
peated applications of the triangle of forces. The converse
is not true. This will be understood at once when it is re-
membered that equiangular polygons are not necessarily
similar. Note further that the polygon of forces is true
whether the forces all act in one plane or not. It contains
a geometrical solution of the problem: To determine the
resultant of any number of forces acting upon a rigid body
wltose lines of action are concurrent. The parallelepiped
of forces may be taken as a particular case.
Resolution of Velocities and Forces.
146. Just as two or more velocities or forces can be
compounded into one resultant, so can any velocity or force
be resolved into two or more components. Thus if AD
represent a velocity or force, and it be desired to resolve it
into components in the directions of A X and A Y, draw
DB, DC parallel to AY and AX respectively, to meet
these lines in B and D, then the components will be repre-
sented by AB and AC. When the components are to be
in one plane, the solution is determinate if there be only two
components, but indeterminate if more than two (art. 145).
135
The only important case of the resolution of force is
that in directions at right angles to one another. The
component in any direction is then called the resolved
force, or the resolute, or the principal component in that
direction, or better, simply the component in that direction;
it measures the effectiveness of the force in that direction.
If i denote the angle between the line of action of a force
ff and its principal component in any direction, the com-
ponent will be measured by /cos i.
147. The algebraical sum of the principal components in
any direction of two forces, which act upon a particle, is
equal to the principal component in the same direction of
the resultant of the two forces.
Let AB, AC represent the forces, and AD their result-
ant. If XAY be the direction in which the forces are to
be resolved, draw BE, CF, DG perpendicular to XAY.
Since AC and BD are equal and parallel, their projections
AF, EG on XY are equal. Now AG = AE + EG =
AE-\-AF; which proves the proposition.
If AE be reckoned + , AF and AG will be + or -
according as F and G lie on the same or opposite side of
A as E does.
The proposition can evidently be extended to any num-
ber of forces whatsoever, and we therefore deduce : The
effectiveness in any direction of any number of forces act-
inn upon a particle is measured by the principal compon-
ent in that direction of the resultant of the forces.
148. To determine algebraically the resultant of any
number of forces acting on a particle.
For simplicity we shall confine ourselves to forces in
one plane. Let 0 denote the particle, and/i,-/a./3i • • • tne
forces. Draw through O any axes OX, OY, at right
angles to one another, and let alf a2, a3}. . . denote the
angles which the forces make with OX. Let R denote the
resultant force, and A its inclination to OX. Then (art. 147)
136
R cos A =/i cos «! 4-/2 cos o2 +/s c°s a3 + . . . = 2 (f cos a)
i? sin ^4 =/i sin ax +/2 sin 0-2+/3 sin «3 + • • • = 2 (/ sin a)
.-. i22={2'(/cosa)P+ii'(/sina)i2
tan J. = -T(/sin a)/2(f cos a).
Cor. If the forces^are in equilibrium, B = 0; then
^(/cos a)= 0, ^'(/si11 «) =0
which express in algebraical language the necessary and
sufficient conditions of equilibrium of any system of forces
acting on a particle in one plane.
Examination XVI.
1. Define the terms resultant, rigid, the component in
any direction, stereodynamics, equilibrium.
2. Enunciate and prove the parallelogram, triangle,
and polygon of forces. Enunciate these conversely, and
state which then remain true.
3. Deduce the algebraical equations between any two
forces and their resultant, and write down the resultant of
two equal forces P, P acting at an angle 2 A.
4. State and prove the relations between 3 forces in
equilibrium, and the angles between the lines of action.
5. Determine, both geometrically and algebraically,
the resultant of any number of forces acting upon a
particle.
6. Prove that the algebraical sum of the components in
any direction of any number of forces, which act upon a
particle, is equal to the component in the same direction
of the resultant of the forces.
7. Express, both geometrically and algebraically, the
necessary and sufficient conditions of equilibrium of any
number of forces, which act upon a particle.
137
Exercise XVI.
1. Prove that three equal forces, whose lines of action
pass through one point, and are inclined to one another at
an angle of §7r, will be in equilibrium.
2. A cricket ball of 200 grams is moving eastward with
a speed of 45 metres per second; find the impulse necessary
to make it move northwards with an equal speed.
3. Prove that forces represented by lines drawn from
the angular points of a triangle to the middle points of the
opposite sides are in equilibrium.
/ 4. A body of 10 kilograms is supported by two strings
whose lengths are 1*2 and 0*9 metre; the other ends of the
strings are fastened at two points in a horizontal line 1^
metre apart; find the tensions of the strings.
5. Two forces acting at M are represented by MA and
MB, and two others acting at N by NC and ND; shew
that the four forces cannot be in equilibrium unless MX
bisects both AB and CD.
6. Find tlie northward and eastward velocities of a ship
which is sailing in a direction N. 30 E. with a speed of
12 miles per hour, and is carried in a S.W. direction by a
current which flows at the rate of 2 miles per hour.
7. Shew that if the angle at which two given forces are
inclined to one another be increased, their resultant is di-
minished.
8. The circumference of a circle is divided into any
number of equal parts; shew that equal forces acting at
the centre towards the points of division are in equilibrium.
9. A picture is suspended by a cord passed round a
smooth pin and fastened to two rings on the picture frame;
if 10 kilograms be the mass of the picture, and 30: the in-
clination of the two parts of the cord, find the tension of
the cord.
138
10 A sphere of 2 tonnes rests on two smooth planes
inclined to the horizon at angles of 60° and 30c; find the
pressure on each plane.
11. Three pegs A, B, C are stuck in a vertical wall so
as to form an equilateral triangle, having A highest and
BC horizontal ; a string passed over the pegs supports a
kilogram at each end; find the pressure on each peg.
12. If the ends of the string (Ex. 11) be attached to a
body of 2 kilograms, which is then supported at a point D,
so that DBC is an equilateral triangle below ABC; find
the pressures on the pegs.
13. A boat is tied by a rope to the right bank of a river
flowing N.E., and is acted upon by a pressure S from the
river and a pressure T^from a S.E. wind; find the tension
and direction of the rope.
14. A body of 12 lbs. is suspended from a point by a
string 6 ft. long, and is acted on by a horizontal force of
9 lbs.-wt.; find how far the body is displaced and the ten-
sion of the string.
15. Two bodies of 5 kilograms each are connected by a
string which is passed over 2 smooth pegs in a horizontal
line 1 m. apart; a body of 3 kilograms is then hooked on
to the string between the pegs; how far will it descend?
16. Forces are represented by 4 lines OA, OB, OC, OD;
shew that their resultant is represented by 4 00, O being
the middle point of the line which joins the points of
bisection of AC and BD.
17. Forces act at the middle points of the sides of a
rigid polygon, in the plane of the polygon and at right
angles to the sides; if the forces be proportional to the
sides to which they are respectively perpendicular, shew
that if they all act outwards or all act inwards, they will
be in equilibrium.
139
18. A string is wrapped around a regular smooth poly-
gon of n sides and pulled with a tension T; find the total
crushing force at the angular points of the polygon.
Hence determine the total pressure, and the pressure per
unit of length, on the circumference of a smooth circular
hoop in like circumstances.
19. The circumference of a circle (radius r) is divided
into any number (n) of equal parts; find the resultant of
a system of forces acting at one of the points of division
and represented by the straight lines drawn from that
point to the other points of division. See ex. 8.
20. Forces acting on a particle are represented by lines
drawn to the angular points of a triangle from the centre
of the circumscribing circle ; prove that the resultant is
represented by the line drawn from the same point to the
intersection of the perpendiculars on the sides from the
angular points.
21. Three forces P, Q, R represented by AB, AC, AD
act on a particle A and keep it at rest; if the direction of
P be fixed, but that of Q vary, find the locus of D which
determines the direction of R.
Answers.
2. 1,272,792 gramtachs N.W. 4. 6 and 8 kilogrs-wt.
6. 8-9781 and 4-5858 miles per hr. 9. 5176-4 grs.-wt.
10. 1 tonne-wt.; 1732 kilogrs-wt. 11. 1732, 517-6, and
517-6 grs.-wt. 12. 2000, 1154-7, and 1154-7 grs.-wt.
13. \{S2+ W2) at tan-1 ' IV /S to the bank.
14. 3-8 ft.; 15 lbs.-wt. 15. 157.
18. 2 nT sin n/n; 2 tt T, T/r.
19. A force measured by nr towards the centre.
21. A sphere with radius equal to AC, and centre at E
in BA produced so that AE-AB.
Chapter XVII.
Motion and Equilibrium on an Inclined Plane.
149. As practical applications of the principles of the
preceding chapter let us consider (1) the motion of a heavy
body sliding on an inclined plane, (2) the conditions of
equilibrium of such a body.
A body slides down an inclined plane, to determine the
nature of the motion.
The body is acted upon by two forces; (1) its weight or
the attraction of the earth, which acts vertically down-
wards, (2) the pressure of the plane. The latter is gener-
ally divided into the two principal components, 1) the
normal pressure of the plane, 2) the tangential action or
friction, which being always opposite to the direction of
motion, acts along the plane upwards. Let m denote the
mass of the body, mg its weight (art. 63), R the normal
pressure of the plane, F the friction, and a the inclination
of the plane to the horizon. Resolving the forces along
and perpendicularly to the plane the components are
mg sin a — F, and R — mg cos a
Since there is no motion perpendicular to the plane
R-mg cos « = 0, .-. R = mg cos a.
If k denote the coefficient of friction between the body
and the plane, F=kR = kmg cos a (art. 122); therefore the
resultant force acting along the plane which moves the
body is mg sin a — kmg cos a. The acceleration of the
body is therefore g (sin a - k cos a) which is constant.
From this we see that the motion of a body sliding down
an inclined plane is exactly similar to that of a body fall-
ing freely, the only difference being that the acceleration
of the body is less.
141
Cor. 1. If a body be projected up an inclined plane,
the motion is uniformly accelerated, the acceleration being
g (sin a + k cos a) down the plane.
Cor 2. If the plane be smooth, the, acceleration is g sin a
downwards, whether the body be moving up or down.
150. If there be no acceleration, but the body either
moving or just about to move, g (sin a-k cos a)=0, and
therefore fc=tan a.
a is then called the angle of friction or angle of repose.
It is the greatest inclination a plane can be made to take
without the body, when laid on the plane, actually sliding
down. At any less inclination the body will not slide,
but then the maximum friction is not called into play. The
angle of repose is beautifully illustrated in the slopes of
moving sand-dunes.
The above value of k points out one of the best and
simplest methods of practically determining the coeffi-
cient of friction between two surfaces.
151. The following propositions are of interest in illus-
trating principles and affording intellectual exercise.
1. The time of sliding from rest down any smooth chord
of a sphere drawn from the highest or lowest point is
constant.
Let d denote the diameter, and i the inclination of the
chord to the vertical diameter. The length of the chord
will be d cos i, and the acceleration of any body sliding
down the chord g cos i (art. 149, cor. 2), therefore the
time of descent down the chord will be j/ (2d/g.) This is
a constant quantity, and is the time of falling freely
through a diameter of the sphere, which might be expected
as the vertical diameter is one of the chords.
2. If two spheres touch at their highest or lowest points,
the time of sliding from rest down any smooth straight line,
intercepted between the surfaces and passing through the
point of contact, is constant.
142
Let A be the point of contact, AB and ^4Cthe vertical
diameters of the spheres, and DE any line through A
meeting the spheres in D and E. On BC as diameter, let
a sphere be described, touching the other spheres in B and
C. Join DB and EC. Let EC cut the sphere BC in F,
and join BE. Evidently DEEB is a rectangle, so that
DE is parallel and equal to BF, and tlierefore the time of
descent down DE is equal to that down BF, which is
equal to the time of descent down BC (prop. 1), i.e. the
time any body would take to fall freely through a distance
equal to the difference or sum, according as the spheres
touch internally or externally, of the diameters of the
spheres.
152. In these propositions we have the keys to the solu-
tion of a set of interesting problems relating to lines of
quickest or slowest descent. The following will serve as
illustrations. The lines are supposed to be smooth.
1. To find the lines of quickest and slowest descent
from a given sphere to a given point without it.
Let 0 be the centre of the sphere and A the point.
Draw the vertical radius OB. Join AB and let C be the
other point in which AB meets the sphere. CA will be
the line of quickest or slowest descent according as B is
the highest or lowest point of the sphere. Join OC and
produce it to meet the vertical line through A in P. F is
the centre of a sphere having A for its lowest point and
touching the sphere 0 in C. Take any other point E in
the sphere O. Join EA and let F be the other point in
which EA meets the sphere P. Now the time of descent
down CA is equal to the time of descent down FA (art.
151, 1). and therefore less or greater than the time of de-
scent down EA.
2. To find the lines of quickest and slowest descent
from the higher of two given spheres, without one (mother,
to the lower.
143
Let O and P be the centres of the two spheres, A the
highest or lowest point of the former, and B the lowest or
highest point of the latter. Join AB and let C and D be
the other points in which AB meets the spheres. CD is
the line required.
Join PD and produce it to meet the vertical radius OA
in Q. Q is the centre of a sphere which will touch the
spheres O and P in A and D respectively. By last prob-
lem CD is the line of quickest or slowest descent from the
sphere 0 to the point D. Take any other point E in the
sphere P. Join AE and let F and G be the other points
in which AE meets the spheres 0 and Q. FE is the line
of quickest or slowest descent from the sphere 0 to the
point E. But the time of descent down CD is equal to the
time of descent down FG (art. 151, 2), and therefore less
or greater than the time of descent down FE.
153. Let us now consider the conditions of equilibrium
of a heavy body resting on an inclined plane and acted on
by some force P in addition to those already considered.
Let a denote the inclination of the plane to the horizon,
b the inclination of P's direction to the plane, W the
body's weight, R the normal pressure of the plane, and k
the coefficient of friction. P is supposed to act in the
same vertical plane as W and R.
1. If P just prevents the body from sliding down, or
the body is sliding down uniformly, the friction equals kR
and acts along the plane upwards. Resolving the forces
along and at right angles to the plane,
Pcos b -j-kR -W sin a = 0, P sin b + R - W cos a =0,
i D Jxr sin a . — k cos a D JIr cos (a -\-b)
whence P= W— — — : , R= W — -^ — : — r
cos b — k sin b cos 6 - A' sin b
2. If the body is just about to move up the plane, or
moves up uniformly, the friction will be along the plane
downwards, and we get
p_ rar8"1 (i + k cos a z> _ tit cos (a + ^)
cos b + k sin 6' cos 6 + k sin b
144
If/ denote the angle of friction (art. 150), fc = tan/;
substituting this value of k in the above, we find that for
equilibrium P must lie between
wsm_(a-f) and Trsin (a+f)
cos (b+f) cos (& — /")
Cor. 1. If there be no friction A; = 0, and we get
P : W : R = sin a : cos b : cos (a + b).
Cor. 2. If P act along the plane, b = 0, and we get
P : W : P = sin «=p/>" cos a : 1 : cos «.
Cor. 3. If the force P act along the plane and there be
no friction, k=0 and 6=0, and we get
P : W : i? = sin a : 1 : cos a, i.e.
= height of plane : length : horizontal base.
Cor. 4. For different values of b, P in dragging up is
least when 6=/, it value being then W sin (a +/); and this
becomes IF sin /"when a =0, i.e. when the body is dragged
along a horizontal plane.
Examination XVII.
1. Write down the equations of motion of a body sliding
down a rough inclined plane, and of a body projected up
1) a smooth, 2) a rough inclined plane.
2. A body is projected up a rough inclined plane, find
the time taken to return to the point of projection, and the
speed on reaching it. What becomes of the molar energy
lost?
3. Define the angle of friction, and prove the relation
between it and the coefficient of friction.
4. Shew that the kinetic energy acquired by a body
sliding down a smooth plane is the same as it would have-
acquired in falling freely through the same vertical height.
5. Find the limiting values of the force which will
keep a body in equilibrium on an inclined plane. Explain
what takes place for other values of the force.
145
6. Find the direction and magnitude of the least force
required to drag a body 1) up an inclined plane, 2) along
a horizontal plane.
7. If a body is dragged up an inclined plane by a force
acting along the plane, shew that the work done is the
same as in dragging it along the base supposed to be of
the same material as the plane itself, and then raising it
vertically through the height of the plane.
Exercise XVII.
( 1/)A body lies on a horizontal slab 10 feet long ; if the
coefficient of friction be |, bow high may one end of the
slab be raised before the body will begin to slide down.
2. Find the speed with which a body must be projected
up a rough plane, inclined to the horizon at an angle of
30°, so as to travel just 10 metres up the plane, the co-
efficient of friction being tan 30. Find also the time it
will take to descend 10 metres, if projected downwards
with a speed of 100 tachs.
3. A body of 10 kilograms hanging freely is connected
by a cord passing over a small smooth pully with another
body of 4 kilograms resting on a plane inclined to the
horizon at \tz; if | be the coefficient of friction between
the latter body and the plane, find the acceleration of
motion and the tension of the cord.
4. A plane is inclined to the horizon at an angle \ic\
find into what two parts a body of LOO Lbs. must be divided,
so that one part, connected by a string with the other and
hanging over the plane, may balance the other part resting
on the plane, fc = tan 30-.
5. At what rate can an engine of 30 horse-power draw
a train of 50 tons up an incline 1 in 250, the resistance
from friction being 7 lbs.-wt. per- ton?
146
6. It is found that it requires double the force acting
along an inclined plane just to drag a body up, as it does
just to keep the body from sliding down; find the relation
between a and k.
7. A railway carriage, detached from a train when going
up an incline of 1 in 280, is found to move over 1500 yds.
before it begins to descend; if the friction be 6| lbs.-wt.
per ton, find the speed of the train.
8. If the train (ex. 7) were going at the rate of 30 miles
an hour on a level piece of road when the carriage was de-
tached, how long and how far would the carriage move be-
fore stopping?
9. Find the lines of quickest and slowest descent from
a point without a sphere to the sphere.
10. Find the lines of quickest descent between a sphere
and a point within it.
11. Find the lines of quickest descent (1) from a
straight line without a circle to the circle, (2) from a circle
to a straight line without it, the circle and line being in
the same vertical plane.
12. Find the lines of quickest descent between two
spheres, one being within the other.
13. If I be the distance of a point from a plane and a
the inclination of the plane to the horizon, find the short-
est time in which a body can fall from the one to the other.
14. Find the locus of a point without a sphere of radius
r, such that the shortest time in which a body can fall be-
tween the point and the sphere is equal to /.
15. Find the same (ex. 14) when / is the longest time a
body can take to fall.
16. A body of 30 lbs. descending under the action of its
weight draws another body of 30 lbs. up a plane 50 ft. long
inclined at ^tt to the horizon, by means of a cord passing
over a small smooth pully ; find when the cord must be cut,
147
in order that the ascending body may just reach the top of
the plane, 1) when the plane is smooth, 2) when A* = ^.
17. Two bodies support one another on a rough double
inclined plane by means of a fine string passing over the
vertex, and no friction is called into play; shew that the
plane may be tilted about either extremity of the base
through an angle 2/ without disturbing the equilibrium,/
being the angle of friction and both angles of the plane
being less than \it -f.
18. A body is kept in equilibrium on an inclined plane
by a force in a given direction; prove that the pressure of
the plane, if the plane be smooth, is an harmonic mean
between the greatest and least normal pressures, if it be
rough.
Answers.
1. 6 ft. 2. 1400-4; 10. 3. 387-75; 6045-8 grs.-wt.
4. The hanging body may be any mass not greater than
50. 5. 15 miles per hr. 6. tan a—Sk.
7. 30 miles per hr. 8. 7 min. 17-7 sec; 3210 yds.
9 and 10. The lines through the lowest and highest
points of the sphere.
11. Through A the lowest or highest point of the circle
draw the tangent AB meeting the line in B: take
BC up or down the line equal to BA: join CA
cutting the circle in D; CD is the required line.
12. The lines between the two lowest and the two highest
points of the spheres. 13. (sep. ^a)i/(2l/g).
14. Spheres of radius r-\-\gt2, which touch the given
sphere internally at its highest and lowest points.
15. Spheres of radius \gt%-r, which touch the given
sphere externally at its highest and lowest points.
16. After going 1) 33| ft., 2) 441 ft.
Chapter XVIII.
Composition of Forces.
B. Forces whose lines of action are parallel.
154. In Chapter XVI. we considered the composition
of forces whose lines of action passed through one point,
and could then neglect the dimensions of the body, or
speak of the forces as acting on a particle. From the very
nature of the case we cannot neglect the dimensions of a
body in determining the resultant of parallel forces acting
upon it. Forces whose lines of action are parallel and act
in the same direction are called like parallel forces; if they
act in opposite directions, they are called unlike parallel
forces. A pair of like parallel forces may be illustrated
by two men supporting a heavy bar, one man at each end.
The vertically upward forces applied by the men balance
the weight of the bar. A pair of unlike parallel forces is
illustrated in breaking a nut by means of a pair of nut-
crackers, the pressure of the nut being opposite to that of
the hand on either arm of the crackers. Unlike parallel
forces are conveniently distinguished by the signs -f- and -.
155. To determine the magnitude, direction, and line
of action of the resultant of two like parallel forces.
Let P and Q denote the forces, and let A and B be
their points of application. Since P and Q have the same
direction, the direction of the resultant acceleration will
be the same as that of the forces, and the magnitude equal
to the sum of the accelerations produced by each force
separately. Hence the resultant force R equals P+Q,
and is parallel to P and Q. This may also be deduced
from art. 143, cor. 3 by, making i=0.
To determine the line of action of R, we assume the
following axiom: The line of action of the resultant of
149
any two concurrent forces passes through their point of
intersection. If now a force Si act at A along AB, and an
equal force S2 &t B along BA, R will evidently be the re-
sultant of the four forces P, Q, Si, S2. Let X denote the
resultant of P and Si, and Y that of Q and S2, and let the
lines of action of X and Y intersect in D, then D must be
a point in the line of action of R. If G be the point in
which R cuts AB, CD is the line of action of R and is
parallel to P and Q. To determine C, v P: 8 = CD: AC,
and S: Q=BC: CD (art. 144), .-. P: Q=BC:AC
156. It is evident that the position of D depends upon
the magnitude of S and the direction of P and Q, whilst
that of C is independent of both. The position of C de-
pends only upon the magnitudes of P and Q, and the posi-
tions of their points of application, and is hence called the
centre of the two parallel forces.
157. To find the magnitude, direction, and line of action
of the resultant of two unlike parallel forces.
We can deduce this directly as in art. 155, or thus:
Let P and Q denote the forces, Q being the greater,
and A and B their points of application. Join ^4P» and
produce it to C, so that AB: BC—Q - P: P. If now a
force Q-P parallel and like to P act at C, the three
forces P, Q, and Q — P are in equilibrium (art. 155), and
therefore the resultant of P and Q must be a force Q-P
parallel and like to Q, and having C for centre.
158. From arts. 155 and 157 it follows that if (P, Q) or
(P, — Q) be a pair of parallel forces acting at A and B re-
spectively, and C be their centre, and if AB be denoted
by -\-l in magnitude and direction, then AC is denoted in
magnitude and direction by Ql/(P+Q) or -Ql/(P-Q),
and CB by Pl/{P+Q) or Pl/{P-Q).
159. By repeating the above processes it is evident that
we can determine the magnitude and line of action of the
resultant of any number of parallel forces whatsoever.
150
The magnitude is simply the algebraical sum of the com-
ponents. The sign of this resultant indicates the direction:
if it is + , the direction is the same as the + components ;
if — , the same as the - components. The resultant may
be supposed to act at the centre of the system.
Def. The centre of a system of parallel forces is a
point, fixed relatively to the points of application of the
component forces, and through which the resultant of the
system must pass, whatever be the direction of the com-
ponent forces, provided their directions relatively to one
another remain unchanged.
By repeated applications of arts. 155 to 157 the centre
of any system can easily be found geometrically. In the
following articles it is determined algebraically.
160. Given the distances of the points of application of
two like parallel forces from any plane, to determine the
distance of their centre from the plane.
Let two like parallel forces P and Q act at A and B,
and let the distances AD and BE from a plane be denoted
by p and q. Let C be the centre of P and Q, and B the re-
sultant. Join AB, draw CF perpendicular to the plane,
and through C draw a line parallel to DE to meet AD
in G and BE in H. Denoting CF by>, we get
P: Q = BC:AC (art. lbb), = BH : AG, = q-r: r-p,
••• r={Pp+Qq)/(P + Q), or Rr=Pp + Qq.
161. Given the distances of the points of application of
two unlike parallel forces from any plane, to determine
the distance of their centre from the plane.
Let two unlike parallel forces P, and - Q act at dis-
tances p, and- q (A and B being on opposite sides of the
plane in this case) from the plane. Suppose £>P. Con-
struct a figure as in last article. Then
P:Q = BC: AC (art. 157), = BH : AG, = r -q : p + r,
.: -7-= (Pp + Qq)/(P-Q), or R{-r) = Pp + Qq
the distance of C from the plane in this case being -r.
151
Hence we get the following rule for determining the
distance of the centre of any two parallel forces from a
plane:
Multiply each force by the distance of its point of ap-
plication from the plane, take the algebraical sum of the
products, and divide by the algebraical sum of the forces.
162. Given the distances of the points of application
of force, in any system of parallel forces, from any plane,
to determine the distance of the centre from the plane.
Let the forces be denoted by P, - Q,- R, S, T, . . . .
and the distances of their points of application from the
plane by p, q,-r,-s, t,
By the previous articles the forces P, and - Q, are equal
to a single force P-Q acting at a distance (Pp-Qq)
-^(P-Q) from the plane; compound this with the parallel
force - R, and we get a resultant P- Q - R acting at a
distance (Pp - Qq + Rr)/(P -Q-R) from the plane; com-
pound this with the parallel force S, and we get a" resultant
P-Q-R + S acting at a distance (Pp-Qq + Rr-Ss)
+-{P-Q- R + S) from the plane. Thus we see that exactly
the same rule to determine the centre of two parallel
forces (art. 161) applies to any number of parallel forces.
It may be concisely expressed thus: d2\P) = 2\Pp),
where P denotes any force of a system of parallel forces,
p the distance of its point of application from a plane, and
d the distance of the centre of the system from the plane.
Examination XVIII.
1. Define unlike parallel forces; find directly the re-
sultant of a pair of unlike parallel forces, and thence de-
duce the resultant of a pair of like parallel forces.
2. Shew how to find the centre of a pair of parallel
forces 1) geometrically, 2) algebraically.
3. Define the centre of any system of parallel forces,
and deduce the rule for finding its distance from any plane.
152
Exercise XVIII.
/l./K two bodies balance each other on a straight lever
in any one position inclined to the vertical, they will bal-
ance each other in any other position of the lever.
2. A shopkeeper uses a balance having arms 10 and
11 inches in length, and sells from the longer arm; what
percentage of money drawn does he gain dishonestly?
3. Find the true mass of a body which balances a grams
when placed in one scale of a false balance, and b grams
when placed in the other; find also the ratio of the lengths
of the arms, and how much the fulcrum should be shifted,
21 being the length of the beam.
4. A shopkeeper possessing a balance, whose arms are
a foot and 12^ inches long respectively, sells from each
arm alternately; will he gain or lose in the long run? by
how much p.c. of the money drawn?
5. If the arm of a cork-squeezer be 30 cm., and a cork
be placed 5 cm. from the fulcrum, find the pressure on the
cork, when 50 lbs.-wt. is applied by the hand.
6. Explain the boast of Archimedes, " Give me a lever
and whereon to rest it and I shall move the world." What
much easier way is there of moving the world?
7. A rod, whose weight may be neglected, rests between
two pegs which are 1 ft. apart and in a horizontal line; a
body of 10 lbs. is hung from one end of the rod, 1^ ft. from
the nearer peg; find the pressures on the pegs.
8. A man carries a bundle at the end of a stick over
his shoulder; if the piece of stick between his hand and
shoulder be shorteued, is the pressure on the shoulder in-
creased or diminished? Is his pressure on the ground
altered thereby? Explain your answers.
9. Two bodies of P and Q lbs. balance at the ends of a
lever whose weight is insignificant; if the bodies be inter-
changed, so that the greater P now hangs where Q was,
153
and Q where P was, find what additional weight must be
added to Q to maintain equilibrium.
10. 0 is any point within a triangle ABC; like parallel
forces act at A, B, and C, proportional to the areas BOC,
CO A, and A OB respectively; prove that O is the centre.
11. If O be outside of the triangle, and the forces in
the same proportion as in ex. 10, under what condition
may O be still the centre of the parallel forces?
12. Find a single body whose weight will produce the
same effect as the weights of bodies of 1, 2. 3, 4. and 5 kil-
ograms hanging on a rod at distances of 1, 2, 3, 4, and 5
decimetres from one end of the rod.
13. A square board (side 3 decim.) is kept horizontal
by an attached string, when bodies of 1. 2, 3, and 4 lbs.
respectively hang at the corners. Find the point where
the string is fastened to the board.
14. Parallel forces K, L, M , ^V act at E, F, G, H, and
K: L: M: iV=area FGH : area GHE : area HEF : area
EFG, shew that the centre is at the intersection of EG
and FH.
15. Like parallel forces of 3, 5, 7, 5 lbs.-Wt. act at the
angular points A, B, C, D respectively of a quadilateral,
taken in order; shew that parallel forces of F, 10 - F,
4 + F, 6- i*1 lbs.-wt., where F may have any value, acting
at the middle points of AB, BC, CD, DA respectively,
have the same centre and resultant.
Answers.
2. 9TV 3. ^(ab); \/a: \/b; 7( | « - ^b)j{ v/«+ xb).
4. Lose, TV. 5. 300 lbs.-wt. 7. 25 and 15 lbs.-wt.
8. Increased; no. 9. (P2 -Q*)IQ. 11. The force
at the vertex of the double angle, within which O
lies, must be unlike to the other two.
12. 15 kilogrs. at 3| decim. from the same end of the rod.
13. 21 cm. from 12, and 15 cm. from 23.
Chapter XIX.
Couples. Moments.
163. There is one case of a pair of parallel forces for
which the foregoing articles fail to give a single resultant,
viz., the case of a pair of equal unlike parallel forces.
According to art. 157 the resultant would be a force of in-
definitely small magnitude, having a line of action at an
indefinitely great distance; the effect of which it would be
impossible to foresee. We must, therefore, as in all cases
in which reasoning from established principles fail us,
appeal to experiment to ascertain what is the dynamical
effect of such a pair of forces. The answer is: the pro-
duction, not of translation, but of rotation of the affected
body about an axis normal to the plane of the couple.
A pair of equal unlike parallel forces is called a couple.
The distance between the lines of action of the forces is
called the arm, of the couple. The moment of a couple is
measured by the product of the numbers which measure
the magnitude of either force and the length of the arm,
and is + or - according as the couple tends to produce -f
rotation {i.e. opposite to the apparent rotation of the sphere
of the heavens, when looking southwards), or - rotation.
It can easily be proved that two unlike couples of
equal moments, in the same or parallel planes, balance one
another. Hence, as is also proved by experience, the
dynamical action of a couple is measured by its moment.
164. If three forces can be represented in magnitude
and line of action by the sides of a triangle taken in order,
they are equivalent to a couple, whose moment is measured
by twice the area of the triangle. {The complete Triangle
of Forces).
Let 8, T, R denote the forces acting along the sides
BC, CA, AB of the triangle ABC. By the triangle of
155
forces (art. 144) the resultant of S and T is a force R,
whose line of action passes through C, and is parallel to
BA, and therefore the system of forces is equal to a couple
RR whose arm is the distance of 0 from AB, and whose
moment is therefore measured by twice the triangle ABC,
since R is represented by AB.
165. The moment of a force about a point is measured
by the product of the numbers which represent the mag-
nitude of the force and the perpendicular on its line of
action from the point, and is 4- or - according as the force
tends to produce + or - rotation about the point.
The moment of a force about any point measures the
effect of the force in producing rotation about the point.
This may be taken as an experimental fact illustrated
in the use of a lever, or it is easily seen that the force is
equivalent to a couple, whose moment is the same as that
of the force about the point, and an equal parallel force
through the point, which evidently cannot produce rota-
tion about the point.
Cor. Just as a force can be resolved into a couple and
an equal and parallel force in the plane of the couple, so a
force and couple in the same plane can be compounded
into a single force equal and parallel to the original force.
166. The moment of a force about any point will evi-
dently be measured by twice the area of the triangle
formed by drawing lines from the point to the extremities
of the line representing the force. The distance of the
point from the line of action of the force is called the arm
of the force about the point.
When will the moment of a force about a point vanish?
Either 1) when the force itself vanishes, or 2) when the
arm vanishes, i.e. when the point lies in the line of action
of the force. In either case there can evidently be no
tendency to rotation about an axis through the point.
156
The algebraical sum of the moments of the forces form-
ing a couple, about any point in the plane of the couple,
is evidently equal to the moment of the couple.
167. The algebraical sum of the moments of two
coplanar forces about any point in their plane is equal to
the moment of their resultant about the point.
1) When the forces are not parallel. Let S and T de-
note the forces, R the resultant. Let AB, AC, AD repre-
sent these forces. If 0 be the point about which moments
are taken, join OA, OB, OC, OD. Suppose O lies between
AD produced and CD produced, then
moment of S : moment of R— /\ OAB : - £\ OAD
moment of T : moment of R= - /\ OAC : - /\ OAD
.-. mo. of S+mo. of T: mo. of R= OAB -OAC: -OAD
now OAB = OCD + DAB = OCD+ACD = OAC -OAD
:. moment of S~\- moment of T= moment of R.
2) When the forces are parallel. Take S and T unlike
forces, S the greater, and let R denote their resultant.
Draw through 0 a line cutting the lines of actiou of the
forces in A, B, and C. Suppose O lies between S and R,
moment of S : moment of R— - S.OA : + R.OC
moment of T: moment of R= + T.OB : -\-R.OC
.'. mo. of S+mo. of T : mo. of R=T.OB - S.OA : R.OC
now T.OB- S.OA = T.CB - T.OC- S.CA + S.OC
= (S-T) OC=R.OC
.'. moment of >S'+ moment of T= moment of R.
The student will find it very instructive to verify the
proposition for all possible positions of the point O.
Cor. 1. When any number of forces act upon a body in
one plane, the moment of the resultant force (or couple),
about any point in the plane, is the algebraical sum of the
moments of the component forces about the same point.
Cor. 2. If, therefore, the forces be in equilibrium, the
algebraical sum of the moments is zero; if the system be
157
equal to a single resultant, the sum depends upon the
position of the point, and vanishes only when the point
lies on the line of action of the resultant: if the system
reduces to a couple, the sum is a constant quantity, but
not zero, whatever be the position of the point.
Cor. 3. Conversely, if the algebraical sum of the mo-
ments of any number of forces acting in one plane, about
three points not in a straight line, be zero for all three
points, the forces are in equilibrium ; if the sum be not of
the same value for all three points, the forces have a single
resultant; if the sum be of the same value but not zero for
all three points, the forces are equivalent to a couple.
168. When a body can move in any manner whatever,
it is said to be free; if its motion be restricted in any man-
ner or by any condition, it is said to be constrained. We
have already considered a case of motion of a constrained
body in Chap. XVII. A oscillating pendulum, a sliding
window, a swinging door, a ring moving on a retort stand
will serve as other illustrations of constrained bodies.
If a body can only rotate about a fixed axis, and is
acted upon by a system of forces whose lines of action are
all at rigid angles to the axis, it is required to find the
necessary and sufficient condition of equilibrium.
The body will be in equilibrium if the forces do not
produce rotation about the axis. The effect of any one
force to produce rotation being measured by the product
of the force into the distance of its line of action from the
axis, i.e. by the moment of the force about the axis, the
necessary and sufficient condition of equilibrium is, that
the algebraical sum of the moments of the forces about
the axis vanish.
In the wheel and axle, toothed wheels, and other forms
of the lever we have practical examples of bodies con-
strained in the manner just considered.
158
169. If the line of action of any force Ph not at right
angles to the axis, resolve i°into two components, one par-
allel to the axis, and the other at right angles to the axis.
The moment of the latter about the axis, will evidently
measure the effect of P in producing rotation about the
fixed axis. This effect will vanish, (1) when /'vanishes,
(2) when the line of action P is parallel to the axis, (3)
when the line of action of P meets the axis.
The other component will produce motion parallel to
the axis, and may be neglected if the body can only rotate;
if, however, the body can also slide parallel to the axis, as
a ring on a retort-stand or a screw in its nut, there cannot
be equilibrium, unless 1) the sum of the moments about
the axis, of the components at right angles to the axis,
vanish, and 2) the sum of the components parallel to the
axis vanish. If, as in a sliding window, rotation is impos-
sible, equilibrium is established, if 2) alone is satisfied.
170. When three forces keep a body in equilibrium,
their lines of action, must all lie in one plane, and must be
all concurrent or all parallel.
Let R, S, T denote the forces. Take A, B, and C points
in the lines of action of R, S, and T, such that BC, CA,
and AB are not parallel to the lines of action of R, S,
and T respectively. Since the body is in equilibrium, we
may suppose BC s. fixed axis. The forces S and T whose
lines of action meet this axis, can have no effect in pro-
ducing rotation about it, nor therefore, since the body is
in equilibrium, can R produce rotation about it. The line
of action of R must therefore cut BC (art. 169), and must
therefore lie in the plane ABC Similarly it may be shewn
that the lines of action of S and T lie in the plane ABC.
If the lines of action be not all parallel, let two of them
K and S meet in O, then the resultant of R and S passes
through (), and, since this resultant is balanced by T, the
line of action of T must also pass through O.
159
Examination XIX.
1. Define a couple, the arm and moment of a couple,
and write down the dynamical dimensions of the moment
of a couple or force.
2. Prove that two unlike couples of equal moments in
the same plane balance one another, and shew that when
three parallel forces are in equilibrium they are really a
pair of such balancing couples.
3. Enunciate and prove the complete triangle of forces.
Hence prove the complete polygon of forces: If a system
of forces can be represented in magnitude and line of
action by the sides of a plane polygon taken in order, it is
equivalent to a couple whose moment is measured by twice
the area of the polygon.
4. Define the moment of a force about a point, and
prove from theory what it measures.
5. Find the magnitude and line of action of the result-
ant of a force I' and a couple Qq in the same plane.
6. Prove that the algebraical sum of the moments of
two coplanar forces about any point in their plane is equal
to the moment of their resultant, and extend the proposi-
tion to any number of forces in one plane.
7. Any number of forces act upon a body in one plane;
find the conditions that the forces can be reduced to 1) a
single resultant, 2) a couple, 3) equilibrium.
8. Define the moment of a force about an axis; find the
condition of equilibrium of a body which can only rotate
about a fixed axis; and give examples of such bodies.
9. Prove that when three forces keep a body in equili-
brium, their lines of action must all lie in one plane.
Exercise XIX.
1. If the sum of the moments of forces in one plane be
of the same value, but not zero, for two points in the plane,
160
the straight line which join these two points is parallel to
the resultant force, or the forces reduce to a couple.
2. In a wheel and axle the radii are as 8 to 3; two
bodies of 6 and 15 lbs. are suspended from ropes wound
round the wheel and the axle respectively; one is sup-
ported by a prop; find the pressures on the prop, and on
the fixed supports of the wheel and axle.
3. Bodies of 1 and 4 lbs. are suspended from the ends
of a straight lever of insignificant weight; the fulcrum and
a point at which another body is suspended divide the
lever into three equal parts; find the mass of the third
body in order that the lever may be in equilibrium.
4. A lever of insignificant weight is 5 ft. long; two
strings, 3 and 4 ft. long, attached to the extremities of the
lever, support a body of 10 kilograms; if the lever be kept
in equilibrium in a horizontal position, find the tensions
of the strings and the position of the fulcrum.
5. Two coplanar forces S and T act at the ends of a
straight lever AB, whose weight may be neglected; find
the position of the fulcrum in order that there may be
equilibrium, the inclinations of S and T to AB being a
and b respectively; find also the pressure on the fulcrum.
6. Forces are represented by the perpendiculars drawn
from the angular points of a triangle on to the opposite
sides; find under what condition they are in equilibrium.
7. 0 is any point within a triangle ABC; AO, BO,
CO cut the sides in D, E, F; find under what conditions
forces represented by AD, BE, and CF are in equilibrium.
Answers.
2. | and 20g lbs.-wt, 3. 2 lbs.
4. 8 and 6 kilogrs.-wt. ; AC : CB=9 : 16.
5. AC: CB = T sin. h : S sin. a;
tflSt+T2 -2 ST cos (a + b)\.
6. That the triangle be equilateral. 7. D, E, F must
be the middle points of the sides.
Chapter XX.
Centres of Weight and Mass.
171. We may consider any body or system of bodies to
be made up of small particles, whose positions can be de-
fined by geometrical points, and the weights of the par-
ticles may be supposed to act at these points, and to be
parallel (art. 62). The centre of such a system of parallel
weights is called the centre of weight or centre of gravity
of the body or system of bodies. To save circumlocution
in what follows, we shall use the term body either for a
single body, or for a system of bodies whose configuration
does not change.
It follows from art. 159 that the position of the centre
of weight, relatively to the particles which compose the
body, is constant, whatever be the position of the body
relatively to the earth. Hence, if the different parts of a
body, acted on only by weight, be rigidly connected ivith
the centre of weight, and if this point be supported, the
body will balance in all positions. This important pro-
perty is sometimes used as a definition of the centre of
weight. The following facts follow immediately from it:
1. If a body balances on a straight line (or axis) in all
positions, the centre of weight must lie in that line.
2. If a body can turn freely round an axis which is
not vertical, it cannot be at rest unless the centre of weight
lies in the vertical plane through the axis.
3. If a body hang from a point round which it can
turn freely, it cannot be at rest unless the centre of weight
is in the vertical line through the point of suspension.
Hence when a body is suspended by strings at-
tached to different points of the body, the lines of the
strings, when the body is at rest, all pass through the
162
centre of weight. This gives a practical method of finding
the centre of weight of any body, however irregular its
configuration may be.
172. When a body is suspended by a string, the centre of
weight will necessarily be below the point of suspension
on account of the non-rigidity of the string. When it is
rigidly connected with a point, the body can be supported
at this point, provided the centre of weight and point of
support be in the same vertical line. Should, however,
the centre of weight be above the point of support,
and any slight displacement take place, the weight of the
body will cause it to rotate about the point, until the cen-
tre is below the point of support. If the centre of weight
be below the point of support, and any slight displacement
take place, the weight will bring the body back again to
to its old position. If the centre of weight coincide with
the point of support, and any slight displacement take
place the body will remain displaced without any tendency
either to recede further from, or to return to its former
position. In these respective relative positions of the
point of support and centre of weight, the body is said to
be in unstable, stable, or neutral equilibrium.
173. A body placed on a plane will stand or fall, ac-
cording as the vertical line through its centre of weight
passes within or without the base of support.
By base in this statement is meant the polygon of
greatest area which can be formed by joining points of
contact of the body and plane.
174. Since weight is a vertically downward force, its
effect on a body is to bring down the centre of weight as
far as possible. Hence a body supported in any way from
falling will be in stable equilibrium, for a displacement
in any direction, if such a displacement raises the centre
of weight ; in unstable equilibrium, if the displacement
lowers the centre of weight ; and in neutral equilibrium,
163
if the displacement does not alter the vertical height of
the centre of weight. A right circular cone of uniform
density resting on a horizontal plane illustrates the three
kinds of equilibrium according as it rests on its base,
apex, or curved surface.
A body is practically in unstable equilibrium if it be
in unstable equilibrium for a displacement in any direction
whatsoever. A sphere of uniform density, resting on a
saddle-back surface, presents the three kinds of equilib-
rium according to the direction of displacement, but prac-
tically it is in unstable equilibrium.
175. Given the weights and Cartesian coordinates of
the particles which composed body, to determine the co-
ordinates of the centre of weight.
Take three axes OX, OY, OZ mutually at right angles
to one another, and let x, y, z denote the coordinates of
any one of the particles of the body having weight w, and
a, b, c the coordinates of the c. of w., then (art. 162)
a—I{icx)^-I {to), b = I (10 y)^-2 (w), c—S{wz)^rI{w).
Cor. When a number of bodies are raised through
various heights, the work done is the same as that of rais-
ing a body, whose weight is the sum of the weights of the
bodies, through the height that the c. of w. of the bodies
is raised.
176. If in the above we write mg for w (art. (YS) we get
a— 2 (mx)-±-Z(m),b=2 (m y)^-S(m),c = Z (mz)^-2 (m),
the coordinates of a point, which, although it coincides
with the c. of w., is quite independent of weight. It is
called the centre of mass or centre of inertia of the body.
It may be also thus defined:
The centre of mass of a body is a point, which coin-
cides with the centre of a system of parallel forces, acting
at the particles which compose the body, and proportional
to the masses of the particles.
164
For a body of uniform density the position of the c. of
m. depends only upon its configuration, and is independent
of its mass. The term centroid is then used for c. of m.;
also centre of length, area, or volume according as the
body is practically a line, surface, or solid.
Let it be observed that we can speak of the centre of
mass of any body whatsoever, whether belonging to the
earth or external to it, whilst, properly speaking, the term
centre of weight is applicable only to bodies near the
earth's surface, and whose dimensions are small.
177. The centre of mass of a very thin straight rod of
uniform density and section is its middle point.
Cor. If the rod be thick, the centre of mass will lie in
the middle section, for any such rod may be supposed to
be made up of indefinitely thin rods.
178. To find the centre of mass of a thin triangular
plate of uniform density and thickness.
Such a plate can be represented by a plane triangle
ABC. Let D and E be the middle points of BC and CA.
AD is the locus of the middle points of all lines parallel
to BC. Now we may conceive the plate to be made up of
an indefinitely large number of thin rods parallel to BC,
and the centre of mass of each of these rods will lie in AD,
therefore the centre of mass of the whole plate will lie in
AD. Similarly it may be shewn that it will lie in BE.
Therefore it is C the point of intersection of AD and BE.
Since D and E are the middle points of BC and CA, DE
is parallel to AB and equal to one half of AB; and the
triangle DGE is similar to the triangle AGB; therefore
DG is one-half of GA or one-third of DA.
Cor. 1. The medians of a triangle meet in the centroid
of the triangle and trisect one another.
Cor. 2. The centre of mass of a thin triangular plate
coincides with that of three equally massive particles
165
situated at the angular points of the triangle, or at the
middle points of the sides.
Cor. 3. The centroid of a plate in the form of a paral-
lelogram is the intersection of the diagonals.
Cor. 4. The centroid of a triangular prism is the
middle point of the line joining the centroids of the oppos-
ite triangular faces.
Cor. 5. The centroid of a plate in the form of any
plane rectilineal polygon may be determined by dividing
the polygon into triangles and applying art 155.
179. To determine the centroid of a thin plate in the
form of any plane rectilineal quadrilateral.
Let ABCD represent the plate. Bisect BD in E; take
EF one-third of AE, and EH one-third of CE; join FH,
cutting BD in K; take HG equal to FK (or FO equal to
HK); G is the centroid required.
180. To find the centroid of a triangular pyramid.
Let ABCD represent the pyramid. Bisect CD in E;
take EF one-third of BE, and EH one-third of AE; let
AF and BH intersect in G; G is the centroid required.
For we may imagine the pyramid to be made up of in-
definitely thin triangular plates, all parallel to BCD. Let
bed represent one of these plates, cutting the plane ABFE
in bfe. Since dec is parallel to DEC, de : DE, = Ae : AE,
- ec : EC; therefore de — ec. Again, because bfe is
parallel to BFE, ef : EF=Af : AF, = bf : BF; therefore
e/is one-third of be, and /is the centroid of the plate bed.
Thus AF is the locus of the centroids of all the plates
parallel to BCD, and therefore contains the centroid of
the pyramid. Similarly it may be shewn that BH con-
tains the centroid, and therefore G is the centroid required.
Because EF is one-third of EB, and EH one-third of
EA, FH is parallel AB and is one-third of AB; and the
triangle HFG is similar to the triangle ABG. Therefore
FG is one-third of GA, or one-fourth of FA.
166
Cor. 1. The lines drawn from the vertices of a tetrahe-
dron to the centroids of the opposite faces meet in the
centroid of the tetrahedron, and quadrisect one another.
Cor. 2. The centre of mass of a triangular pyramid of
uniform density coincides with that of four equally mas-
sive particles placed at the vertices of the pyramid.
Cor. 3. A pyramid whose base is any plane rectilineal
polygon can easily be divided into triangular pyramids.
The centroid of the pyramid will lie in the line joining
the vertex with the centroid of the base, and be three-
quarters of this line from the vertex.
Cor. 4 A cone may be considered to be a pyramid
having a base with an indefinitely large number of sides,
and therefore the rule for finding the centroid of a pyra-
mid (Cor. 3) applies to a cone having any plane base.
181. A body is symmetrical with respect to a point,
line, or plane, when the body may be conceived to be made
up of pairs of equally massive particles, the two which
form a pair being on opposite sides of the point, line, or
plane, equidistant from it, and in the same perpendicular
to it. The point, line, or plane will contain the centre of
mass of every pair of particles, and therefore also the
centre of mass of the whole body. From this principle
of symmetry we can frequently find the centre of mass
with great facility. Thus, the centroids of a circular or
elliptic ring, of a circular or elliptic plate, of a sphere,
spheroid, or cuboid are apparent.
182. Having given the speeds of any number of par-
ticles in any direction, to determine the speed of their
centre of mass in the same direction.
Let m denote the mass of any one of the particles, and
d its distance from a fixed plane at right angles to the
direction in question, at any instant; then the distance of
the centre of mass from the plane at the same instant will
be I(md)^-lXm). Let v denote the speed of the particle
167
m in the direction in question, then at the end of time t
the distance of m from the plane will be d-\-vt, and there-
fore the distance of the centre of mass from the plane, at
the end of time /, will be 2\m(d + vf)\-7-2(m), that is
\l\md)^r-(m)\-\-\2\7nv)-h2(m)}t which shews that the
speed of the centre of mass is 2(mv) -±-2(m).
If the speeds of the particles in the given direction be
not all constant, t must be taken indefinitely small.
183. From arts. 59 and 182 we deduce the important
fact, that the velocity of the centre of mass of any system
of bodies cannot be altered by the mutual actions (e.g.
direct impact, art. 125) of its several parts. Hence the
centre of mass of the universe, or of any body not acted on
by external force, is either at rest or in uniform motion.
184. Having given the accelerations of any number of
particles in any direction, to determine the acceleration of
the centre of mass in the same direction.
Let v denote the speed, in the given direction, of any
particle having mass m, at any instant, then -(mv)^--(m)
is the speed of the centre of mass at the same instant (art.
182). If « denote the acceleration of m, then at the end
of time / the speed of m will be v + at, and therefore the
speed of the centre of mass will be I{m(v-\-at)\^2\m), or
{ I(mv) -4- 2\m) } + j I(ma) + -(m) \ t, which proves that the
acceleration of the centre of mass is 2 (ma) -*- 2'(ra) .
If the accelerations of the particles in the given direc-
tion be not all constant, t must be taken indefinitely small.
185. Articles 182 and 184 shew us, that at any instant,
the total momentum and acceleration of momentum of any
system of particles, is the same as that of the total mass of
the system concentrated at the centre of mass. Hence,
when any forces act upon a system of bodies, the motion of
the centre of mass is the same as that of a particle, whose
mass is the total mass of the system, and which is acted
upon by the same forces. Hence, so far as translation is
168
concerned, the motion of any body is represented by the
motion of its centre of mass.
186. The kinetic energy of any system of particles is
equal to the kinetic energy of the ivhole mass of the system
moving with the speed of the centre of mass, together with
the kinetic energies of the different parts of the system
relatively to the centre of mass.
Let O J represent the velocity of the c. of m. and IP
that of any particle (mass m) of the system relatively to the
c. of m., then OP represents the velocity of m. Draw PQ
at right angles to 01; then I(m. IQ) =0, (art. 182). Now
OP2 = OI2+IP2±2 01. IQ /. I(^mOP2)=^(7n). OF2
-{-I (\ m. IP2), which proves the proposition.
Ex. A body of mass M hanging vertically, draws an-
other body of mass m along a horizontal plane, by means
of a string passing over a smooth pulley. If v denote the
speed at any given instant, and k the coefficient of friction
on the plane, find the motion of the centre of mass, the
masses of the string and pulley being insignificant.
At the given instant the centre of mass has a horizontal
velocity mv-^(M-\-m), anda vertical velocity Mv^-(MJr7n)
downwards; therefore its total velocity is \/(M 2 + ra2)v-H
{M-\-m), in a direction which makes an angle with the
horizon, whose tangent is the ratio M/m.
Let T denote the tension of the string; then the accel-
eration of m is ( T- kmg)^m, and that of M is ( Mg - T)
-i-M. These are numerically equal, therefore T is equal
to Mm(l+k)g-i-(M+m), and the acceleration of either
body is (M-km)g^-(M-\-m). The centre of mass has
therefore a horizontal acceleration m (M - km)g-t-(M+m)2,
a vertical acceleration M(M -km)g^-(M + m)2, and a
total acceleration^ (M-kni) y (M 2 + m?)g-i-(M+m)2 in
a direction which makes an angle — tan_1(il//wi) with the
horizon. Hence the acceleration is constant, and its
169
direction is the same as that of the velocity at the given
instant, and therefore the centre of mass moves in a
straight line inclined at an angle— tan-1 (M/m) to the
horizon with uniformly accelerated motion.
Examination XX.
1. Define the centre of weight, and prove propositions
1, 2, 3 of art. 171.
2. How may the c. of w. of any irregularly shaped body
be experimentally determined?
3. What are the different kinds of equilibrium? Illus-
trate these by bodies supported 1) at a point, 2) on a plane,
3) on a surface of double curvature.
4. When a body is placed on a plane, state and prove
the condition of equilibrium.
5. Define the centre of mass of a body. When does it
coincide with the centre of weight? Define centroid;
what other names may be used for this term?
6. Find the controids of 1) a triangular plate, 2) a
quadrilateral plate, 3) a cuboid, 4) a triangular pyramid,
5) a right circular cone, 6) a spheroid.
7. Determine algebraical expressions for the position,
velocity, and acceleration of the c. of m. of any material
system.
8. Prove the corollary to art. 175.
9. What is meant by saying that the motion of a body
is represented by the motion of its c. of m.? Enunciate
and prove the corresponding proposition for the kinetic
energy of a material system.
10. What do we know about the c. of m. of the solar
system, and of that of the whole universe?
170
Exercise XX.
1. Three men support a heavy triangular board, of uni-
form density and thickness, at its corners; shew that all
three exert the same force. Would this be the case if they
supported it at the middle points of the sides?
2. Prove that a rhomboidal lamina of uniform density,
when placed on a horizontal plane, will rest in equilibrium
on any one of its sides, if its plane be vertical.
3. A round table stands on three legs placed on the
circumference at equal distances; shew that a body, whose
weight is not greater than that of the table, may be placed
on any part of it without upsetting it.
4. A uniform rod 1 metre long, and 500 grams mass, is
supported horizontally by means of a ringer below the rod,
5 cm. from one end, and the thumb at the end over the
rod; find the pressures on the linger and thumb.
5. Two strings have each one of their ends fixed to a
peg, and the others to the ends of a uniform rod; when the
rod is hanging in equilibrium, shew that the tensions of
the strings are proportional to their lengths.
6. A heavy bar, of uniform section and density, 3
metres long, is to be carried by two men, one of whom is
half as strong again as the other; if the weaker man sup-
ports the bar at one end, where should the stronger man
support it?
7. The base of a solid right circular cone is in contact
with a plane which can be gradually inclined; find the
ratio of the altitude of the cone to the radius of the base,
in order that the cone may be just on the point of toppling
over as it begins to slide down.
8. A uniform triangular plate is suspended from a
point by strings attached to its angular points; shew that
the tensions of the strings are proportional to their lengths.
171
9. The lower end of a rigid ladder is fixed, whilst the
upper rests against a vertical wall; compare the heights a
man and boy can ascend respectively, so that the pressure
against the wall may be the same.
10. It is observed that a rod AB, 12 feet long, will
balance at a point 2 feet from the end A; but when a body
of 100 lbs. is suspended at the end B, the rod balances at a
point 2 feet from that end; find the mass of the rod.
11. In Ex. XIII, 6 and 10, find the speeds of the
centres of mass after impact and bursting respectively.
12. Find the motions of the centres of mass in the sys-
tems described in Ex. VII, 5 and 15.
13. Three particles of equal mass are moving along the
sides of a triangle taken in order, with speeds propor-
tional to the sides along which they move respectively;
find the velocity of their centre of mass.
14. Find the amount of work required to dig a cylin-
drical well to a depth of 20 feet, the diameter being 4 feet,
and the density of the material raised 2*3.
15. A shaft 100 feet deep is full of water; find the
depth of the surface of the water when j, and also \ of the
work required to empty the shaft has been done.
16. Find the centroid of a thin shell in the shape of a
right circular cone.
Answers.
1. Yes. 4, 5 and 4^ kilogrs.-wt. 6. 50 cm. from
the other end. 7. 4 : k. 9. Inversely as their
weights. 10. 25 lbs. 11. 2250; 50.'
12. Uniform vertical ace. of ysQ; uniform ace. of 13-9
at - tan-1 f to the horizon. 13. 0.
14. 360,705 ft.-lbs. 15. 50, 705. 16. On the axis,
at two-thirds of the axis from the vertex.
Chapter XXI.
Simple Machines.
187. This chapter might be called an introduction to
mechanics (art. 61). Machines are used for various pur-
poses: for transmitting force or motion, as in many uses
of flexible cords and straps, rigid rods, or toothed wheels;
for changing the direction of force or motion, as when a
stretched rope is passed around a pin or smooth pulley, or
in the use of bevelled toothed wheels; in measuring force,
as by a balance or dynamometer; but the use principally
aimed at, in the larger number of machines, is to enable
man to balance, or do work against great forces by the
application of small forces. This is the use, for example,
of a crowbar to balance the weight of or to move a heavy
beam, or of a screw to apply great pressure, as in an ordi-
nary book-binder's press. This may be called appropriately
the dynamical advantage, of the machine.
If W denotes the resistance which is balanced or
worked against by the aid of the machine, and P the force
applied to balance or work against the resistance, the
ratio W/P measures the dynamical advantage.
In the present chapter the dynamical advantages of a
few of the simplest machines will be calculated. More
complicated machines will be found to be combinations of
such simple machines, and if the dynamical advantages of
the latter are known, it needs but simple multiplication to
find those of the former.
188. When there is no appreciable friction called into
play in the use of a machine, the dynamical advantage
arises entirely from the combination of parts or the me-
chanism of the machine, and is then called the mechanical
advantage of the machine. Thus in sliding a heavy body
173
up an inclined plane, which may be considered a simple
machine, the mechanical advantage is (cos 6)/(sin a), art.
153 cor 1, and this becomes cosec a, when Pacts along the
plane, and cotan a, when P acts horizontally.
When friction is called into play, and the machine is
used to aid in merely balancing the resistance W, then
friction can always be taken advantage of in favour of the
balancing force P. When friction is fully taken advan-
tage of, so that the resistance W is just kept in check, or
motion is just about to take place against P, the dynami-
cal advantage may then be called the static advantage of
the machine. It is the greatest advantage the machine
can offer in balancing any resistance. Thus, in the inclined
plane, the static advantage is cos (6+/)/sin(«— /), art 153,
and this becomes l/(sin a — k cos a) when P acts along
the plane, and cot (a -/) when P acts horizontally.
When it is desired to do work, i.e. move against the re-
sistance W, with the aid of the machine, the friction called
into play always acts in opposition to the moving force P. In
this case the dynamical advantage may be called the kinetic
advantage of the machine. It is evidently always less than
the mechanical advantage. Its value, like that of the
static advantage, depends upon the friction called into play
as well as on the mechanism of the machine. In the inclined
plane the kinetic advantage is cos (b -/)-i-sin (a+f),
art. 153, which becomes cot (a +f) when P acts horizon-
tally, and l/(sin a + k cos a) when P acts along the plane.
189. Less tvork is never done in overcoming any resist-
ance through a given distance with the aid of a machine,
than would be done in overcoming the resistance through
the same distance directly, i.e., without the machine.
Thus, although a man may be able to roll a heavy body
which he could not lift up an inclined plane, yet the
amount of work he does is not less than the work which
would be done in raising the heavy body vertically through
174
the height of the plane. In .reality more work is done with
the aid of the machine, for some work must always be done
against friction, and this work is entirely lost. It is a case
of the dissipation of energy (art. 122).
If there were not any friction, the ivork done with the
aid of a machine by the moving force would be the exact
equivalent of the ivork done against the resistance.
This is an immediate result of the great law of the
conservation of energy (art. 120), and is generally known
as the principle of ivork. Thus, in sliding a body up an
inclined plane, P cos b= W sin a, (art. 153, cor. 1), there-
fore PI cos b = Wl sin a; but I cos b is the distance through
which P works in its own direction in raising the body
from the bottom to the top of the plane, I denoting the
length of the plane, and I sin a is the vertical distance
through which W \s overcome; hence the work done by P
is equal to the work done against W. If P act along the
plane, Px length of plane = Wx height, (art. 153, cor. 3).
190. It is evident that from the principle of work the
mechanical advantage of a machine can be determined
with great facility by studying the kinematics of the ma-
chine. It is not always vital energy which is used as the
motive power in machines. In a steam-engine e.g. it is the
potential energy of atomic separation of coal and the oxy-
gen of the air, which is transformed into mechanical work,
and if we do not consume animal energy in doing work
with a steam-engine, we exhaust an equivalent amount of
our store of potential energy in the form of coal.
The Lever and Fulcrum.
191. A lever is a rigid rod moveable in one plane about
an axis called the fulcrum. The condition of equilibrium
can be deduced from art. 168. When, as is generally the
case, friction may be neglected, and the weight of the lever
itself is balanced at the fulcrum or may be neglected, the
condition can be stated thus: The moment of the balanc-
175
ing or moving force about the fulcrum must be equal and
unlike to the moment of the resistance.
Hence the mechanical advantage of a lever is measured
by the ratio of the arm of the moving or balancing force
about the fulcrum to the arm of the resistance. This can
be deduced easily from the principle of work by making a
small displacement about the fulcrum. To find the pres-
sure on the fulcrum, apply art. 143 or 159.
192. Levers are sometimes divided into 3 classes; 1)
those in which the fulcrum lies between the places of ap-
plication of the balancing or moving force and resistance,
as a crow-bar, a claw-hammer, a common balance, or com-
mon scissors; 2) those in which the place of application of
the resistance lies between the fulcrum and place of appli-
cation of the balancing or moving force, as in a wheel-
barrow, an oar, nut-crackers, or cork-squeezers; 3) those in
which the place of application of the moving force lies
between the fulcrum and the place of application of the
resistance, as in many parts of the animal frame- work,
such as the forearm, in the shells of bivalve molluscs, and
some forms of shears. In the first and second classes the
mechanical advantage will generally be greater than unity,
and in the third less than unity. The object sought in the
third class of levers is therefore not the acquisition of me-
chanical power, but the production of motion over a con-
siderable range by means of motion through a small
distance.
Many very powerful combinations of levers are used in
the mechanical arts, such e.g. as may be seen in the hand
printing presses, and the large paper cutting machines
used by printers and bookbinders.
Wheel and Axle.
193. A wheel and axle consists of two cylinders capable
of rotating about a common axis. The larger cylinder is
called the wheel, and the smaller one the axle. To the
176
latter a resistance to be worked against is applied by means
of a rope in tension or otherwise, to the former a moving
force to overcome this resistance. Familiar examples of
such machines are found at wells to draw water up in buck-
ets, and in the windlass and capstan which are commonly
used on board ship, the former to raise merchandise from
the hold, the latter to weigh anchor. In these the wheel
generally consists of spokes, at the extremities of which
the moving force is applied. The axis of the capstan is
vertical, and the applied forces horizontal.
The mechanical advantage can be easily deduced from
art. 168, or from the principle of work, thus: Let R and
r denote the radii of the wheel and axle respectively; if
the machine be rotated through any angle i, the work done
by the moving force P is PRi, and that done against
the resistance W is Wri; since these must be equal,
W/P = R/r. The pressure on the axis of rotation will be
the resultant of P and IV, and the weight of the machine.
Toothed Wheels.
194. Toothed wheels are principally used to communi-
cate motion from one wheel to another, as e.g. in clock-
work. In cranes and other machines, however, mechanical
advantage is sought by their aid. We shall illustrate
their use in this respect by rinding the mechanical advan-
tage of a machine consisting of one wheel and axle driving
another wheel and axle, the axle of the first and the wheel
of the second having teeth which work into one another.
Let P denote the force acting on the first wheel, which
balances, without friction being called into play, a resist-
ance W acting on the second axle. Let Q denote the
mutual pressure between the teeth in contact; this will be
normal to the surfaces in contact. Denote by R and / the
arms of P and Q about the first axis, and by V and r' the
arms of Q and IF about the second axis. Then (art. 168),
PR = Ql, QV = Wr, :. W/P = Rl'/lr'.
177
If r and R' denote the radii of the first axle and the
second wheel, and the teeth be small compared with these
radii, l/V = r/R', and therefore W/P = RR/rr', a result
easily deduced from the principle of work, thus: Let the
first wheel and axle be rotated through any angle i, then
the second will be rotated through an angle i. (r/R'),
and therefore PRi = Wr'i. (r/R), or W/P = RR'lrr'.
Endless bands are much used in machinery for the
same purposes as toothed wheels, when it is inconvenient
to bring the wheels close together, friction preventing the
bands from slipping.
Pulley and Rope.
195. A pulley is a grooved wheel which rotates about
an axis, fixed generally in a framework called a block. A
stretched rope or cord passes around the wheel within the
groove. The pulley may be used 1) merely to change the
direction of the rope, so as to apply a force in a convenient
direction, or 2) to get mechanical advantage. In the first
case the pulley is fixed, in the second it is moveable.
To understand more clearly what follows, the student
may imagine the part of the cord, which is in immediate
contact with the pulley at any instant, as forming part of
the pulley to which the forces are applied. Neglecting
friction, the tension of the rope wound round the pulley
is the same at both sides (art. 141). To balance these ten-
sions, there must be another force acting on the pulley
equal to 2 T cos i, where T denotes the tension of the rope,
and 2 i the inclination of its two branches as it leaves the
pulley. If 1) the pulley be fixed, 2 T cos i denotes the
tension of the beam which holds the pulley, necessitated
by the tension of the rope; if we add to this the weight of
the pulley and rope, we get the total tension of the beam.
If 2) the pulley be moveable, 2 T cos i denotes the resist-
ance which the tensions of the rope balance, and this re-
sistance includes the weight of the pulley and rope, which,
however, are often neglected.
178
Cor. If the two branches of the rope be parallel in a
moveable pulley, and W denote the resistance, W=2 T, a
result easily arrived at from the principle of work.
196. Pulleys in various combinations are commonly used
in practice to lift bodies against their weights. The stud-
ent will find the above results sufficient to enable him to
determine the mechanical advantage, as well as the tension
of the supporting beam, in any combination whatsoever.
In any system of pulleys the kinetic advantage is consider-
ably less than the mechanical on account of the friction
called into play, arising principally from the rigidity or
imperfect flexibility of the ropes.
Double Axle and Pulley.
197. A combination, in which the mechanical advantage
can be made as great as required with great facility, is
known as the differential axle or Chinese wheel. It con-
sists of two cylinders having a common axis: round the
larger cylinder a rope is wound a few times, then passed
under a moveable pulley, and thereafter round the smaller
cylinder. The direction of coiling the rope on the latter
is opposite to that on the former, so that as the rope is
wound on to the larger cylinder it is wound off the smaller.
The moving force is generally applied by means of a winch
which rotates on the same axis as the double axle, and the
resistance acts on the block of the pulley. Let a denote
the arm of the moving force applied to the winch, b and
c the radii of the larger and smaller cylinders respectively.
If the two lines of rope passing round the pulley be paral-
lel, then for one complete turn we get by the principle of
work, P.27ia= IV.n(b-c), and therefore W/P = 2a/(b - c).
Hence, by making the difference between b and c small
enough, the mechanical advantage can be made as great
as required without sacrificing the strength of the machine
or making it unduly bulky.
179
Screw and Nut.
198. A screw may be described as a right circular cylin-
drical bolt, on the surface of which runs a uniform pro-
jecting thread, which makes a constant angle with the
base. The nut of a screw is a hollow cylinder in which is
cut a spiral groove, the exact counterpart of the thread of
the screw.
When the screw enters its nut, it is evident that they
can only move relatively to one another by one of them ro-
tating, and then there is sliding motion parallel to the axis
of the cylinder, proportional in amount to the angle of ro-
tation. Either the screw or nut can both slide and rotate,
or one can rotate and the other slide. Generally the nut
is fixed, as in a book-press, or the screw only rotates and
the nut slides, as in a dividing engine.
The distance, measured parallel to the axis, between
two adjacent coils, is called the pitch of the screw. It is
evident that for every complete rotation the sliding motion
is equal to the pitch of the screw. The inclination of the
thread to the base of the cylinder is called the angle of the
screw.
The principal uses of a screw and nut are 1) to measure
small distances, as in a dividing engine or micrometer; and
2) to exert great pressure in the direction of the axis, as in
a book-press. For the first of these uses, the head of the
screw is provided with a carefully divided circle to enable
the experimenter to measure any small rotation; for the
second, the moving force is generally applied to a rigid bar
fixed into the head of the screw, and acts at right angles
to the axis. If a denote the arm of the moving force and
p the pitch of the screw, it is evident that the mechanical
advantage is measured by ^Inajp. '
Screw and Toothed Wheel.
199. In this machine a toothed wheel takes the place
of a screw's nut. The thread of a short screw fits into the
180
spaces between the teeth of a wheel (or. the teeth of the
wheel fit into the groove between the coils of the screw).
The axis of the screw is fixed, and so long as the screw is
turned, the wheel is made to rotate about its own axis.
Hence the machine is generally called the endless screw.
The wheel generally forms part of a wheel and axle, and
to the axle the resistance is applied. The moving force is
applied by means of a crank or winch fitted on to the axis
of the screw. The mechanical advantage is evidently
measured by the product of the numbers, which measure
the mechanical advantages of the screw, and wheel and
axle respectively.
The Wedge,
200. The wedge may be described as a rigid right tri-
angular prism having two of its faces inclined generally at
a very acute angle. The line in which those faces meet is
called the edge of the wedge, and their inclination the
angle of the wedge. The face opposite the edge is called
the head of the wedge. The wedge is practically used to
separate two bodies, as in lifting a heavy body through a
small distance against its weight, or to divide a body into
two parts against molecular force, as in splitting wood.
Friction plays an important part in the practical use of
the wedge. The moving force is frequently impulsive, as
when the wedge is driven by the sharp blows of a hammer.
Axes, knives, and chisels are different forms of the wedge,
which may be considered a double inclined plane, and as
the angle between the two faces is very small the mechan-
ical advantage is very great.
In the above machines there are in reality only three
primary principles, viz., the principles of the inclined
plane, the lever, and the moveable pulley. The wheel and
axle and toothed wheels are just special levers, whilst the
screw and nut and the wedge are special forms of the in-
clined plane.
181
Examination XXI.
1. Define the mechanical, static, and kinetic advan-
tages of a machine, and find those of an inclined plant',
when a body is slid up the plane.
2. Enunciate the principle of work and apply it to find
the mechanical advantages of the lever, Chinese wheel,
and endless screw.
3. Give examples of the different classes of levers, and
state the object of each class.
4. What is the effect of changing the direction of P or of
W in a wheel and axle? Is the mechanical advantage
thereby changed?
5. In the first system of pulleys each pulley is sup-
ported by a separate rope, one end of which is fixed to the
beam, and the other to the block of the pulley above. If
the lines of rope be all parallel, and the free end of the
rope of the highest moveable pulley pass over a fixed
pulley, find the mechanical advantage, 1) when the weights
may be neglected, 2) when the weight of each pulley is w.
Find also the tension of the supporting beam.
6. In the second system of pulleys there are two blocks,
one fixed and the other moveable; each block contains a
number of pulleys, and the same rope passes round all the
pulleys, the lines of rope being parallel or very nearly so;
find the mechanical advantage, 1) when the weight of the
lower block and pulleys may be neglected, 2) when the
weight of the lower block and pulleys is w. Find also the
tension of the beam supporting the upper block.
7. The third system of pulleys is just the first system
reversed, the beam and resistance changing places; find in
it the mechanical advantage and tension of the beam, 1)
when the weights of the pulleys are insignificant, 2) when
the weight of each pulley is w.
8. When will the static and kinetic advantages of an
inclined plane be maxima for given values of a and fe?
182
Exercise XXI.
1. Prove that the efficiency of a machine (art. 122) is
measured by the ratio of the kinetic advantage to the
mechanical advantage, and hence find the efficiency of an
inclined plane.
2. Shew how to graduate a common steelyard. What
change is produced on the graduations by increasing 1)
the moveable counterpoise, 2) the density of the rod.
3. If a counterpoise of one pound be used in a steelyard
which was graduated for a counterpoise of one kilogram,
shew that the merchant will defraud himself, or defraud
his customers, or deal justly, according as the centre of
weight of the steelyard is in the longer arm, in the shorter
arm, or just below the point of suspension.
4. A lever of insignificant weight is a metre long; a
body of 10 kilograms is supported by two strings, 6 and 8
decimetres long respectively, attached to its extremities; if
the lever be in equilibrium when horizontal, shew that the
fulcrum divides it into two parts in the ratio of 9 to 16.
What is the pressure on the fulcrum?
5. In the Danish steelyard the beam is heavy at one
end and the fulcrum moveable; the masses to be measured
are suspended at the light end; shew that the distances of
the graduations from the light end form an harmonical
progression.
6. Under what condition may there be no mechanical
advantage in the first system of pulleys, w being the weight
of each pulley? Find the force required just to balance
the weights of the pulleys.
7 The mass of a uniform straight lever, which turns
about a fulcrum at one end, is 6 lbs.; in what direction
must a force of 5 lbs.-wt. be applied at the other end, so as
to keep the lever at rest in a horizontal position ? What
will be the pressure of the fulcrum ?
183
8. Sixteen sailors, each exerting a force of 30 lbs.-wt.,
push a capstan, each with a length of lever equal to 8 ft.;
calculate the weight they are capable of sustaining, the
radius of the cylinder of the capstan being 1 ft. 4 in.
9. A body of 100 kilograms is kept from sliding down
an inclined plane of inclination \ti, by a string in tension
parallel to the plane ; the string passes round an axle of
diameter 3 decimetres; find what body must hang from a
wheel, of diameter 1 metre, having a common axis with
the axle, in order to keep the 100 kilograms at rest without
friction being called into play.
10. One body is suspended from a single moveable
pulley, and is supported by another body hanging freely
over a fixed pulley, the three lines of rope being parallel;
prove that, whatever be the vertical height of each body,
the height of their centre of weight is constant.
11. A man stands in a scale attached to a moveable
pulley; the free end of a rope passes over a fixed pulley;
find with what force the man must hold the free end, in
order to support himself, the lines of rope being parallel.
12. One tonne is to be raised by means of a third sys-
tem of 6 pulleys; if the mass of each pulley be a kilogram,
find the mechanical advantage, the tension of each rope
in kilograms-weight, and the tension of the beam.
Answers.
1. | Cos (b -/) sin a\ + j cos b sin (a+f) \ . 2. The zero
is moved 1) nearer to, 2) further from the fulcrum.
4. 10 kilograms-wt. 6. W=w\ w -w/2n.
7. sin-1 f- to the horizontal; 5 lbs.-wt.
8. 2880 lbs.-wt. 9. 15 kilogrs. 11. One-third
of the weight of the man, scale, and pulley.
12. 668; 15, 31, 63, 127, 254, 510; 1021.
Miscellaneous Examples.
l.Jlt requires a grams-wt. to sink Nicholson's hydro-
meter by itself to the mark on the stem, b grams-wt. when
a piece of amber is placed in the upper pan. and c grams-wt.
when the amber is placed in the lower pan; find the s.w.
of amber.
2. /A man whose mass is 68 kilograms can just float in
fresh water; find the greatest quantity of gold (s.w. 19'3)
he could keep from sinking, when floating in the sea
(s.w. 1-027).
3. If in Ex. XIII, 13, the balls be painted alike so that
the frictional resistance / between the water and each ball
be the same, will the resistance increase or diminish the
tension of the cord? By how much'?
4. Two bodies of 4 and 5 kilograms together pull one
of 6 kilograms over a smooth peg by means of a connecting
string; after descending through 10 metres, the 5 kilo-
grams mass is detached without interrupting the motion;
find through what distance, and for what time, the remain-
ing 4 kilograms will continue to descend.
5. To a person travelling eastwards with a speed of 4
miles per hour the wind appears to be north; on doubling
his speed, it appears to be N.E.; find the velocity of the
wind.
6. A body of 6 lbs. hanging vertically is connected by
an inextensible string with a body of 4 lbs. which is drawn
up a plane inclined to the horizontal at \n radians; find
the motion of the c. of m., k= \.
7. The hole in a boiler for the safety valve is a circle of
g in. diameter. The centre of the valve is If in, from the
fulcrum of a lever which keeps it closed; find where a
weight of 7 lbs. must be hung to the lever so that the valve
may not rise till the pressure of the steam is 3 atmospheres.
Take 7i = ^- and 1 atmosphere — 147 lbs.-wt. per sq. in.
185
8. In a Bramah press the diameters of the pistons are
2 and f inch. The smaller piston is also the piston of a
pump which supplies the liquid to the press. The arms of
a lever which moves this piston are 2j and 11^ inches: find
the mechanical advantage of the machine.
9. A ship is sailing eastwards, and it is known that the
wind is N.W.; the apparent direction of the wind, as
shown by a vane on the mast head, is N. N.E.; shew that
the speed of the ship is the same as that of the wind.
10. A ship sailing eastwards with a speed of 15 miles
per hour passes a light-house at noon; a second ship sail-
ing northwards with the same speed passes the light-house
at 1.30 p.m. When were the ships nearest to one another,
and what was their distance apart then?
11. ABCD is a parallelogram; forces represented in
magnitude and line of action by AB, BC, and CD act
upon a body; find the resultant.
12. A man's mass is 140 lbs., and he supports an Eng-
lish ton- weight (2,240 lbs.) by means of 4 moveable pulleys
fixed as in the first system; find his pressure on the
floor on which he stands, 1) when he pulls the free end of
the rope upwards, 2) when the rope passes over a fixed
pulley, and he pulls downwards.
13. The capacity of the receiver of an air-pump is 20
times that of the barrel; a piece of bladder is placed over
a hole in the top of it: the bladder is able to bear a pres-
sure of 3 lbs.-wt. per sq. in. ; how many strokes of the pump
will burst the bladder?
,14. A body floats in water with | of its volume above
the surface; the whole is put under the receiver of an air-
pump, and the air extracted; find the alteration in the
volume immersed.
15. How much cork (s. w. \) is required to float a man
of 152 lbs. in sea-water, his mean s.w. being l'l?
186
16. Find the lines of quickest descent between a point
and a line, and shew that they are at right angles to one
another.
17. An endless cord hangs over two smooth pegs in the
same horizontal line, and a heavy body is supported on
each festoon; if the one body be twice as heavy as the
other, shew that the angle between the lines of the upper
festoon must be greater than \n and less than it.
18. A body floats in a liquid two-thirds immersed, and
it requires a pressure equivalent to two lbs.-wt. just to im-
merse it totally; what is the mass of the body?
19. Oxygen at 0: and 76 cm. pressure has density 1"1056
with respect to air; find its density at 100 and 70 cm.,
1) with respect to air at 0 and 76 cm., 2) with respect to
air at 100 and 70 cm.
20. The mass of a specific gravity bottle is 20"5 when
empty, 70"5 when filled with water, 63 when filled with
turpentine; when 10 grams of salt are put into it, and it is
thereafter filled up with turpentine, the mass is 69-6; find
the s.w. of the turpentine, and of the salt to an approxi-
mation of the first degree.
21. A sunken vessel, whose bulk is half a megalitre and
mass 10° kilograms, is to be raised by attaching water-
tight barrels to it. If the mass of each barrel be 30 kilo-
grams, and the volume a kilolitre, find how many will be
required.
22. Find the height of the water barometer -under the
mean atmospheric pressure, when the temperature is 15 C.
the s.w. of water at 15c being 0999125 according to
Despretz.
23. One end of a string is fastened to a body of 10 kilo-
grams; the string passes over a fixed pulley, then under a
movable pulley, and has its other end attached to a fixed
hook; a body of 7| kilograms is attached to the movable
pulley, whose mass is 250 grams; if the three parts of the
187
string be parallel, and friction and the masses of the string
and pulleys may be neglected, find the accelerations of the
bodies and the tension of the string.
24. A body of 100 kilograms pulls by its weight 200
kilograms along a rough horizontal plane; if the coefficient
of friction be 02, find the speed after moving through a
hectometre, and the acceleration of the c. of m.
25. A stream is a feet broad, b feet deep, and flows at
the rate of c feet per hour; there is a fall of d feet; the
water turns a machine of which the efficiency is e; it re-
quires /foot-pounds per minute for 1 hour to grind a
bushel of corn; determine how much corn the machine
will grind in 1 hour.
26. Find what must be the area of a cake of ice 18
inches thick, sufficient to bear the aggregate weight of
three school boys whose aggregate mass is 280 lbs.; 1) in
fresh water, 2) in sea-water.
27. Three bodies P, Q, R, of masses 30, 15, 10 kilograms
respectively, are connected by strings AB and BC, whose
lengths are 5 m. and 70 cm. Q, R, BC, and half of AB lie
on the edge of a table vertically under a peg, over which
the other half of AB is placed holding P. If P be now
allowed to fall freely, find the motions of P, Q, and R. the
tensions of the strings after both become stretched, and
the measures of the impulsive tensions which set Q and R
in motion. Friction and the masses of the strings may be
neglected, and # = 980.
28. A string which passes over two pegs in a horizontal
line supports a heavy ring from falling; prove that the
string cannot be drawn so tight as to be horizontal.
2(J. A, B, C, and D are any four points whatsoever; find
a point O such that forces represented by OA, OB, OC,
and OD are in equilibrium. Hence shew from dynamical
considerations that the lines joining the middle points of
AB and CD. of AC and BD, and of AD and BC bisect
one another.
188
30. A cubical box is all but 1/wth part filled with water,
and is placed on a rough rectangular board so as to have
the edges of the base parallel to those of the rectangle; de-
termine in what order spilling, sliding, and toppling over
will take place, when the board is gradually inclined to
the horizon about an edge.
31. Find the least volume of a balloon filled with hy-
drogen that it may rise from the earth when the mass of
the solid parts of the balloon and the contents of the car
is altogether 250 kilograms.
32. The height of mount Fuji in Japan was found by
means of an omnimeter to be 12365 feet; the reduced
reading of the barometer on the summit was found to be
48 cm. when the temperature was 0°; shew that the height
of the atmosphere is at least 7 '3 miles high.
Answers.
1. (&-b)/(c-b). 2. 1939-2. 3. Increase by £/
very nearly. 4. 10 m., 102 sec. 5. 4 j/2
miles per hr., N.W. 6. (11 - 4 y3) v (13 - 6 t/3)#
-^100 in direction -tan-1 (3- yS) to the horizon.
7. 266 inches from the fulcrum. 8. 80.
10. 12.45 p.m.; 159 miles. 12. 1) 280 lbs.-wt., 2) 0.
13. 47. 14. An increase of 3-233/103 of the volume.
15. 325. 18. 4. 19. 07453, 11056.
20. 085, 2-5. 21. 516. 22. 10169.
23. 4 g, \ g, 5 kilogrs.-wt. 24. 1980"4; T\ g \ 5 at
- tan -1 1 to the horizontal. 25. 624 abode /60 f.
26. 37-4, 28. 27. Q starts with speed 1400/3, R with
420; 27T3T and Vd\\ kilograms-wt.; Q, AB 7 mega-
gramtachs; E, AB 28 and BC 42 megagramtachs.
29. The middle point of EF. E and F being the middle
points ot AB and CD. 30. It spills when
tan a — 2/n or n/(2n-2), slides when «=/, and
topples over when a — \7z. 31. 207,710 litres.
PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET
UNIVERSITY OF TORONTO LIBRARY
QA. Marshall, D
845 Introduction to the science
M35 of Dynamics
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