A MANTAL OF
PRACTICAL PHYSICS
I
A MANUAL OF PRACTICAL PHYSICS
For Students of Science and Engineering
VOLUME I. — Fundamental Measurements and Prop-
erties of Matter. — Heat.
By ERVIN S. FERRY and ARTHUR T. JONES
VOLUME II.— Wave Motion, Sound, and Light.
[In preparation
VOLUME III. — Electrical Measurements.
[In preparation
LONGMANS, GREEN, AND CO.
NEW YORK, LONDON, BOMBAY, AND CALCUTTA
A MANUAL
Ol
PRACTICAL PHYSICS
FOR STUDENTS OF SCIENCE AND KM.IMJ.i;
BY
KIIVIN SIDNEY I-TJJRY
PROFESSOR OP 1MM -:
AM.
ARTIiri! TAI'.KII JONES
: v ^ l 1 ROFE88OR OP Pin->I< -. ri KM I
\-M[.. I
FUNDAB1ENTAL Mi: \M !;IMI \ I s AND
I'l:- \ OF MATI
11KAT
ONOMAN8, GREEN, AND CO.
'.M in M ni MI ff
DON, BOM1 \^ VLCUTTA
1908
COPYRIGHT, 1908,
BY LONGMANS, GREEN, AND CO.
All rights reserved.
Nortoooft
J. S. Gushing Co. — Berwick & Smith Co.
Norwood, Mass., U.S.A.
PREFACE
Tm: aim of tin1 present work is to furnish the student of
science with a self-contained manual of the
theory and manipulation of those measurements in physics
which hear m<»t directly up«>n his Mil>M'<pient work in other
• •f study ami upon 1. ^uial career.
Only those experimental methods have heen included that
are strictly s< and that can he depended upon to give
good in the hands of the average student. Although
il pieces of apparatus, experimental methods, and deriva-
!ormul;e that possess some novelty appeal', our fixed
purpose has been to use the standard forms except in rases
e an extended trial in large classes has demonstrated the
;oposed imitation.
It has been assumed that the experiment is rare that should
1 before the .student ui iie theory in-
1 and the d -n <»f the formula required. Conse-
(pi.-mly the t .f each experiment is -.riven in detail and
the required formula developed at leiiLTlh. 'I'lie moiv important
ror are pointed out. and means are indicated by
which these :nay he miuimi/.ed or accounted for.
commenced during the second
'lie- I- presupposes a working knowledge of trigo-
1 college algebra, but does not require analytic
geoi dculus.
vi PREFACE
Most of the experiments here given were printed privately
some years ago and have since been in constant use, under our
direction, by classes of from one to two hundred students each
semester. They have all been carefully revised for the pur-
poses of this volume.
We are indebted to Mr. G. G. Becknell, Instructor in Physics
in Purdue University, for the method we have adopted for
solving the equation for the coefficient of expansion of a gas.
E. S. F.
A. T. J.
CONTKNTS
PART I
FUNDAMENTAL .u/;.isrA'/-;u/-;.v7'> AND PROPERTIES
OF U.I TTKR
< EAFTKB I
i MKABUREMI-N i
A«T. FAUB
1. Introductory 1
rore
• ;. M U of expressing Results s
1")
i i! M'lT.R II
<>D8 AND APPAR ATI'S FOR THE MEASICKMI NT OF
•\MI\l\I < * I ' \ N 1 I I I I -
i«;
\la»
of'J
< H kPTEB III
x . \ N . , I I
knew of a ite by Means of a Spberometer and an
1-j
;face . .
r»' and Sensitiveness of a \'>
. Ver I'.aro meter Scale 63
I
.ntintf of a Diviil.-il Circle 61
viii CONTENTS
CHAPTER IV
VELOCITY AND ACCELERATION
EXP.
7. The Change in Spee_d of a Flywheel during a Revolution . . 63
8. The Speed of a Projectile by the Ballistic Pendulum ... 64
9. The Acceleration due to Gravity by Means of a Simple Pendulum 66
10. The Acceleration due to Gravity by Means of a Compound Pen-
dulum 69
CHAPTER V
FRICTION
Introduction 75
11. The Coefficient of Friction between Two Plane Surfaces . . 75
12. The Friction of a Belt on a Pulley 77
13. The Coefficient of Friction between a Lubricated Journal and its
Bearings — Golden's Method 80
14. The Coefficient of Friction between a Lubricated Journal and its
Bearings — Thurston's Method 82
CHAPTER VI
MASS, DENSITY, SPECIFIC GRAVITY
15. The Calibration of a Set of Standard Masses . . . .86
Density and Specific Gravity ........ 88
16. The Density of a Solid by Measurement and Weighing ... 89
17. The Density and Specific Gravity of a Liquid with a Pyknometer 90
18. The Density and Specific Gravity of a Solid with a Pyknometer . 92
19. The Density and Specific Gravity of a Solid by Immersion . . 95
20. The Density and Specific Gravity of a Solid or Liquid with Jolly's
Spring Balance .......... 97
21. The Density and Specific Gravity of a Liquid with the Mohr-
Westphal Balance 99
22. The Calibration of an Hydrometer of Variable Immersion . . 102
23. The Relative Densities of Gases with Bunsen's Effusiometer . . 107
CHAPTER VII
MOMENT OF INERTIA
Introduction , .110
24. The Moment of Inertia of a Rigid Body 115
ix
UlAl'TKR VIII
ELASTICITY
»XP. PAGE
Introduction 1 1 -»
•J"). The Elastic Limit. Tenacity, and Brittleness , . .11!)
lY,. Tii«- Kiastifity. I '- Modulus — by
- rolling 1-JO
•J7. A - -angular III >d- li':1.
• •efficient of Elasticity. _;'s Modulus — l.y
I'-'-
- :n|.l.- lii^idity — Vibration Method l-'.l
H - Method l:'.7
:il. The Moduli. lie
CHAPTKB IX
\"i-. oirn
82. I >sity of a le's
. 11:;
ie Specific Viscosities oi I I . . 14€
PART II
///
< II \ri 1.1;
'I l MI-KI: \
1 ."..".
.
: [. Tl ' tM Tln'nnniii«-t«T , , I'l'i
'• Calil>ration of a Resistance Th'-itno! .... M'i
Oil . . . 17«>
. between the l; •! th*> c.
17:;
OHAPTBB XI
I. \I\N~
17-;
. 17ti
CONTENTS
PAGE
38. The Coefficient of Linear Expansion of a Solid . . . .179
39. The Absolute Coefficient of Expansion of a Liquid by the Method
of Balancing Columns 183
40. The Coefficient of Cubical Expansion of Glass . . . .186
41. The Coefficient of Expansion of a Gas by Means of an Air Ther-
mometer .... 188
CHAPTER XII
VAPORS
42. The Maximum Vapor Pressure of a Liquid at Temperatures below
100° C. — Static Method 194
43. The Maximum Vapor Pressure of a Liquid at Various Tempera-
tures— Dynamic Method 196
44. The Density of an Unsaturated Vapor by Victor Meyer's Method . 198
CHAPTER XIII
HYGROMETRY
Introduction . . 202
45. Relative Humidity with Daniell's Dew Point Hygrometer . . 203
46. Relative Humidity with the Wet and Dry Bulb Hygrometer . 205
CHAPTER XIV
CALORIMETRY
Introduction . 207
The Correction for Radiation —
1. Regnault's Method 209
2. Rowland's Method 212
3. Rumford's Method 214
47. The Emissivities and Absorbing Powers of Different Surfaces . 215
48. The Specific Heat of a Liquid — Method of Cooling . . .219
49. The Specific Heat of a Solid — Method of Mixtures . . .221
50. The Specific Heat of a Solid— Method of Stationary Temperature 226
51. The Specific Heat of a Solid — Joly's Method .... 229
52. The Heat Equivalent of Fusion of Ice 231
53. The Heat Equivalent of Vaporization of Water .... 235
54. The Heat Value of a Solid with the Combustion Bomb Calorimeter 237
55. The Heat Value of a Gas with Junker's Calorimeter 243
ENTS
XI
CHAPTER XV
THERMO i
ElP.
Introduction ........... L.M7
leal Equivalent of Heat by Rowland's Method . . 247
M M*»ehanu-:il E<iuivalent of Heat with Barnes's Constant Flow
Current Calorimeter ... , 250
TABLES
TABLB
Diversion Factors 254
•-a of Solids and Liquids 256
> of Water at Different T am . . . 257
•us Solutions of Alcohol . . . 857
>nsatl5°C -j:^
Scales ....
• ases and Vapors 259
8. Coefficients 260
Constants of S
-cosities of M<j i 1- 261
1. Cor on the Height of the
riwt.T . .... 262
i arometric Pressures . 263
1?'»1
ssure of >
. -ji;:.
»fficients< .... 266
<N 266
i II. M- \ : \ . . .266
.
!i . L'''7
•II . . . Mfl
rfaces
, }. The GriH-k \ 260
PRACTICAL 'PHYSICS
BENERAL ffOnONfl BBGARDING IMIVSICAL MF.AM'IIKMKNT
1. Introductory
KXIM -.1: IM F.XTAL work lias one of two objects; either to find out
;lt follows under given conditions, or to find out
itmerical 11 different quantities. The lirst
In th«- M y science qualitati\e experiments are
i unifi-ous ; w; s more ir ie majority of the
:ments are quantitatue. The determination of various
uair t of physical nicasun-nu-nt .
In making a ph\-iral nirasinvni.-nt. thr magnitude uf each
oantity oonoemed has to be exprened in terms of some unit.
nd the process of v.-nirnt . !v in tindiiiL,r
ow many tin, I in the ^ivi-n (juantity.
'he il points, for e\ may he expressed
us of the number of foot rules which omUl be laid end to
lid between tlloS.
Some quant ; thus be measured •/ others can he
ii-ed on] ; i be >> -Modulus of a
« rass wire cannot be expt-: mined l»y finding how
:ie unit of Vnun^'s niudiilus is c«»ntaincd in the
"oiin'/s modulus of the \\ii > niodnliisof the
^ »TO i mined l.y measuring a force and three
1 ugths, ;: : the Vuiiii^'s modulus.
^r»Mt uiajo: moasurements are indirect
1
2 PRACTICAL PHYSICS
2. Errors
Every measurement is subject to errors. In the simple case
of measuring the distance between two points by means of a
meter stick, a number of measurements usually give different
results, especially if the distance is several meters and the
measurements are made to small fractions of a millimeter. The
errors introduced are due in part to —
(1) Inaccuracy of setting at the starting point ;
(2) Inaccuracy of setting at intermediate points when the
distance exceeds one meter ;
(3) Inaccuracy in estimating the fraction of a division at the
end point;
(4) Parallax in reading, i.e. the line from the eye to the divi-
sion read not being perpendicular to the scale;
(5) The meter stick not being straight ;
(6) The temperature not being that for which the meter
stick was graduated ;
(7) Irregular spacing of divisions ;
(8) Errors in the standard from which the division of the
meter stick was copied.
Besides the above there are doubtless other sources of error.
It may be well here to note that blunders, such as mistakes
due to mental confusion in putting down a wrong reading, or
mistakes in making an addition, are not usually classed as
errors.
Of the above errors, (1), (2), and (3) can be very much
decreased by having fine divisions on the scale and reading with
microscopes ; (4) can be made small by bringing the scale on the
meter stick close to the object to be measured ; (5) can be made
very small by using a meter stick of special design, or, in rough
work, by holding the meter stick against a straight edge ; (6) can
be nearly eliminated by using the meter stick only at the proper
temperature, or, if its temperature and coefficient of expansion
are known, by calculating a correction to be applied ; (7) can be
diminished only by a careful comparison of the lengths of the
rriONS ni:<;ARi)i.\<; PHYSICAL MKASCKK.MKM 3
different divisions: and for (8) corrections can be applied only
when something is known about the accuracy of the standard
from which the meter stick was copied. llut even with the
refined methods and the most careful application of cor-
ns, different measurements -.mie distance usually
iitYereiit ivsults.
Er: . ^ :nay he d
which more or le>s am; - ,-an he calcu
lue to ( ! >init<
\ fur wliich cor: * amiot he calculated.
for whir ; ap]>lied,
>t- dut- to ( 1 ). ,
amount and will :
large and som. -tiim-s t-»o .hileotl. those due to
>t applied, will he
ccmtatit and will tend to ma! due ..htaim-d a
h rge iall.
Since the average \ N varialde err«'i'> \\lnch tend
t< m. ie niiiin
n easn - he al>.,ut ti ;is the average value of tlmse
\ ri.ti
(» a large in;1 from
\ riai 1 \\ith
t tistant errors, the >ame .juantity >houl' i>y M
n my
e t n. '. ill iiNiialh . all a
v lue can be e . alu<-
. . an an;.
i. -lined as the amount l»y
iicl» the vain 1 h.it !>>. if the
lue — wliieh i> n-.t usually known — is denoted 1»\ 7T, tin-
; by O, and • /.'.
/: <> T. ( i -
(»f the rnnvetioli wliiell oUUdlt to be applied
lied a.^ 'Unt which would ha added to
4 PRACTICAL PHYSICS
the value obtained in order to get the true value. That is, if
Q denotes the required correction,
0= T- 0. (2)
From (1) and (2) it will be seen that the error in a measure-
ment and the correction which ought to be applied to it are
equal in magnitude and opposite in sign. This does not mean
that the error is exactly equal in magnitude to a correction
which actually is applied, because for the correction itself only
an approximate value is usually known.
TRUSTWORTHY FIGURES. — Since all measurements are sub-
ject to errors, it is important to be able to determine how many
figures of a result can be trusted.
In direct measurements it is usually possible to make a fairly
accurate estimate of the extent to which a reading can be
trusted. Thus in reading by the unaided eye the position of
a fine line which crosses a meter stick, the reading will not be
in error by so much as a millimeter but pretty surely will be in
error by more than a thousandth of a millimeter. So the extent
to which the reading can be trusted will lie between these
limits. A person who is accustomed to estimating fractions of
a small division will be rather sure of not making an error so
great as the tenth of a millimeter, and he can often trust his
reading to a twentieth of a millimeter.
It is convenient always to put down all the figures that can
be trusted, even if some of them are ciphers. Thus the state-
ment that a distance is 50 cm. implies that there is reason for
supposing that the distance really lies between 45 cm. and
55 cm., whereas the statement that the distance is 50.00 cm.
implies that there is reason for supposing that the distance
really lies between 49.95 cm. and 50.05 cm. When the dis-
tance is said to be 50 cm. the second figure is the last in which
any confidence can be placed ; when the distance is said to be
50.00 cm., the fourth figure is the last in which any confidence
can be placed. If a distance is about 50,000 Km. and the
third figure is the last in which any confidence can be placed,
NOTION- REGARDING I'HYsir.M. MK.vsr KKMIA r 5
this fart may IK* indicated by saying that the dUtam -e is
50.0- 1' km'.
nent* the result is usually calculated by
;ie formula. To tind out how many figures should be kept
in the : osider the following t \\ •••
1. If the result is the al-vbraic sum of several quantities, sueh
:l 1. 128, 82.6, and 7.068, it is seen that in the sum. :',."> t.n'.U.
IK. tiinire heviid that in the first decimal place ean be trusted,
in the quantity which has the fewest t ru>t \\ort hv deci-
mal places, \i/. :>-.('. no figure b«-\ond the •', can be trusted —
;ild have been <-\ ^- the sum will not he
LI. Ti • be following rule : —
i; I. -In sums and differences HO mon decimal places
.iued than can be trusted in the »piaiitity haviiiLT
frweM trust wnrtliy decimal pla>
•1. If tlir ! product of two (jiiantitics. su.'h as
314. i_'^ and B2.6, then the product <'ann.»t l»c tru>tc«i t,. m.»rc
figures than apjM-ar in the quantit;, :nist worthy
, irn-sp«-ctive of the decimal pla- I • make this clear
the I'M 11
:;i L428 • B2. i 10187.4672
814.428 • 82.6 i
§14, •
I is seen that if tin- quantity \v Inch :
fignrefl the true
\ due «,f tl.. ae olitained.
'1 h- i and foiuth «.f the ab..\e prndiirts sh«>w that if more
1 an three \\^\ :n.t I.e • E tWO quantities
M iiich « multiplied, it is not wprih while to use n
t an three -Of a: th«- other. The>e
> suggest the following rule :
:.i; II. hi products and quotients no more figures should
b • kept th.i isted in the quantity havi tttrust-
>\ orthv ti-rures.
6 PRACTICAL PHYSICS
Until the final result is reached, it is often worth while to
keep one more figure than the above rules indicate.
For logarithms a safe rule is the following : —
RULE III. — When any of the quantities which are to be
multiplied or divided can be trusted no closer than 0.01 % use
a five-place table, when any of them can be trusted no closer
than 0.1% use a four-place table, and when any of them can
be trusted no closer than 1 % use a slide rule.
REQUIRED ACCURACY OF MEASUREMENT. — From Rule I.
it will be seen that if a small quantity is to be added to a large
one, the percentage accuracy of the measurement of the small
quantity need not be so great as that of the large one. Thus
if ff= a + b, and if a is about 100 cm. and b about 1 cm., a 1 %
error in a will produce in If no greater effect than a 100 % error
in b. When quantities are to be added or subtracted, they
should be measured to the same number of decimal places.
From Rule II. it will be seen that if a small quantity and a
large one are to be multiplied the percentage accuracy of the meas-
urement of the small quantity should be at least as great as that
of the large one. Thus if ff= ab, a 1 % error in a will produce
in H the same effect as a 1 % error in b. So that if a is about
100 cm. and b about 1 cm., arid if b cannot be trusted closer than
0.01 cm., there is no gain in accuracy by measuring a much closer
than to within 1 cm. When quantities are to be multiplied or
divided, they should be measured to within the same fraction of
themselves, e.g. all of them within 1 % and none of them much
closer, or all of them within 0.01 % and none of them much closer.
The last statement needs modification in the case of a power.
If the value found for a quantity a is 1 % too large, i.e. is 1.01 a,
then the value that will be obtained for a2 is 1.0201 a, which is
about 2 % too large, and the value obtained for a3 is 1.030301 a,
which is about 3 % too large. In general, if the value found
for a is Jc% too large, the value that will be obtained for an will
be nk % too large. So that a quantity which is to be squared,
cubed, or raised to some higher power should be measured with
more care than if it entered the formula to the first power.
[IONS I;I:<;ARI>I.\<; PHYSICAL MKASCKF.MKNT 7
INTRnlHVKl) BY m.MMnN A1TR< >XIM A TI-
Xl M
HER
TECE VALVE
AiM'i:..x.
VALIE
WHEN-
AP-
PLICABLE
II.. \v OKTAINKI>
•1, KD
i:Y THE
Ai-ri
1
1 -f a 4- a3
1 4 ./
<1 SIlUlll
ct a2
-a8
(a error
0.1 -
0.01
2
(l+a)(l + 6)
l+a+6
«« ami /.
small
s
-aft
3
(1 + a)-
1 + ma
,/ >mall
.'1 by binomial
•••ni. Neglect SIM ..ml
and higher powers
m(m-l) 9
2
4
(1 + a)«
1 + 2a
a small
Apply (3)
-a*
^ — (\ 1 n\~v
5
1
1 + a
1 - .1
• i >niall
l+«~
Apply (3)
6
vTTa
1 + ia
-/ Mil. ill
vTT^=(i ^
Apply (3)
-1- I
7
V56
*(«•«•»)
f.jil.il
to a
Let 6 = a + e. Then
Va6=Va*+a* \
Apply (6)
+ (b_^a£
8a
8
«•
'1 -Ml.lll
——!+!-•
Neglect third and 1.
powers
+ .'
9
OOti
1
»i small
--1-S+E-
N.--II-.-T leoond ;m.i
+ ia«
•
UBi
a«
« small
.-£+...
tana = *HL2 = [1
cosa ^1^..
Apply (5)
-ia*
ii
Una
sin »r
• t Mliall
Like (8) and (10)
-ja«
EzprMMd, of coarse. In ndl*n».
8 PRACTICAL PHYSICS
APPROXIMATE FORMULAE. — Beside the errors of observa-
tion, errors may be introduced into indirectly measured quanti-
ties by the use of formulae which are only approximate. Thus,
the sine and tangent of small angles are used as equal to the
angles, the reciprocal of (1 + a) is written equal to (1 — a)
when a is small, 3.14 is used for TT, a number of figures are
dropped from the end of a product, etc. Whenever such an
approximation suggests itself, the error introduced by using it
should be investigated and the approximation not made unless
the error thereby introduced is so small as not to affect any
figure that could otherwise be trusted in the result.
The preceding table of a few common approximations may
prove useful.
3. Methods of expressing Results
The object of a quantitative experiment is sometimes the
measurement of some quantity, and sometimes the determina-
tion of the relation between various quantities. When the
relation between several quantities is sought, the usual method
is, keeping all but two of the quantities constant, to vary
by known amounts one of these two and then determine the
changes produced in the other. Another pair of the quantities
is then varied while the rest are kept constant, and so on until
a sufficient number of pairs of quantities have been investi-
gated. The various relations found to exist between the
various pairs of quantities can then be combined to give the
relation sought.
When one quantity has been given various known values
and the corresponding values of a second quantity have been
determined, the relation between them can always be expressed
graphically; it can also be expressed more or less accurately
by means of an empirical formula; and when this formula is
sufficiently simple, the relation can without difficulty be ex-
pressed in words.
To illustrate these methods, suppose that it is desired to
determine the relation between the distance a body has fallen
NOTIONS REGARDING PHYSICAL MKAsri:i-:.MK.\T
9
from ivst and the time it lias heen falling. Suppose that a
Dumber of determinations are made, in each of which a hull is
allowed to fall a known distance, ami the time required is ol>-
!ues ohtaineil heinur those in the following table : —
IKBD
0064
0.101
0.1 I:',
<>.•_'«>_'
«»._
MM
<U«»I
PLOTTING 01 i; KMLI DIVIDED COOBDI-
PAPBB*- -These values may be plotted in the same \\.t\
Orvefl are drawn in A G iould
is to mal :id near! t In- sheet
ibothd, . unless by so doing a unit in the last place that
in 1. :epresei;' i than one of
"ii the paper. I f. Pot instance, time
etroc >ec., then t be scale f' <-n in
'ig. 1 times \\hat to be. If, ho\ve\er.
or closer, then the
ibl not always I..- drau n t hroii^b
,11 tbe points, but should I- >tli eur\e \\hieli tits the
Mints as ther scah" has been so ol
unit in the List plaee that can he tnist.-«l i . ntcd
• .smallest d '»u the paper, a deviation of
urve usually indicates errors ,,f ob.s.-rvat ion.
i !ice that the distance fallen
i i creases as the time increases; but >inc«- tl not a
nut proportional to the time,
ard tin- tiiii' . ' follows that
ie dii ases at a oonUnoally increasing i that
10
PRACTICAL PHYSICS
as the body falls it goes continually faster and faster. The
curve also serves to find the distance fallen in any time not
much exceeding 0.4 sec., or to find the time required to fall
any distance not much greater than 80 cm.
The next step is to find the equation which represents this
curve. Let the time which has elapsed be represented by £,
0.4
and the distance fallen by I. If I decreased when t increased,
the equation might be of a form
or
or
etc.
NOTIONS KKiiAKDINil PHYSICAL MKASl Kl-MENT 11
Iii the case in hand, however, since / increases when t inert
the relation cannot be one of the above forms; it may, perhaps,
be of a form
/ = a + It. (3)
or / = a + It -f (4)
or l = a + bt + ct* + tfr3, (5)
If the relation were of the form in (•''>), two points would sutliee
to determine <i and I. For if tin- coordinates of the two points
were (tv /j) and (/.,. /._, ». we should have
/j = a + f,tl
and /a = a -f btv
and from these two equations we eould determine the fcwp
quantities <i and b. Similarly, if the ivlatinn w.-re of the form
iii (4). three points would sutVice to determine a, 6, and <•.
'1 bus, if only two points are determined, there can always be
f -Mild a n-lation of the form in (:)) that \\ill he satisfied 1>\
1 itli tli-.se puints: if thive iK)int8 are determinc«l, a relation
c .n always be found containing three constants whieli will he
b tistitMl by all three points; if n points are known, a relation
( n a. unin^ n iiieh will be satis-
1 -d by all n p-ints. Hut an equation containing many con-
s , Hits is eiimbt-rsoim', and [1 uilly po»ibh- to find an
e nation with only three or four constants whieli i> vrry nearly
K tisli.d by nnber of points.
A r- -d of finding how many < -onMants should
I) • used in an equation like (8), (4), or (">) will he illustrated
b ' considering the curve plotted in Fig. 1. The maximum
a scissa is divided into some half dozen or more convenient
e ual parts, the ordinate at each division point is read, and the
c« nrespondin: of abscissas and ordinates are recordi <1 in
tl Srst two columns of a table : —
12
PRACTICAL PHYSICS
t
i
V
A,J
0.00 sec.
0.0 cm.
1.3 cm.
0.05
1.3
3.7
2.4 cm.
0.10
5.0
6.0
. 2.3
0.15
11.0
8.4
2.4
0.20
19.4
11.3 ,
2.9
0.25
30.7
13.4
2.1
0.30
44.1
15.8
2.4
0.35
59.9
18.4
2.6
0.40
78.3
In the third column are the differences of the first order, i.e.
the differences between the successive values of I ; in the fourth
column are the differences of the second order, i.e. the differ-
ence between the successive differences of the first order; in
a fifth column would be the differences of the third order, etc.
In the present case it is seen that in going down the columns
the differences of the first order continually increase, whereas
the differences of the second order, although varying somewhat,
on the whole neither increase nor decrease to any great extent.
By a simple application of the Differential Calculus it can be
shown that if the differences of the nth order neither increase
nor decrease, then (n+1) constants are enough to retain in a
formula of the general form
y = a + ox -\- cy? + dofi •+•
For the case in hand, then, three constants should be retained,
and the formula may be written
I = a + It + ct2.
The three constants are to be determined by choosing three
points on the curve as far apart as convenient, and so obtaining
three equations from which to solve for a, b, and c. If the
points selected are (0, 0), (0.20, 19.5), and (0.40, 78.2), the values
found for a, I, and c are respectively 0, —0.5, and 490. Con-
sequently the empirical equation is
Z «• 0-0.5* +490 A
NOTIONS REGARDING PHYSICAL MEASUREMENT 13
From theoretical considerations the formula for a hody fall-
St should
To compare the two equations compute by each of them the
ice fallen in a Lriven time. Thus if </ is '.K^Orm. per sec.
in a B6C., the distance fallen in O.o minted from the
::•--
Hi •':
.;
/
f
/
/
/I
/
/
/
/
f
f
f
f
/
~-
i
f
/
^
/
s
f
:
I
\
i 9 4 6 9 T 8 9 10 » » 40 80 90 T080 HO 100
,i.. wliile according to thr rmpiri-
equation t i }:'..'.»."> cm.
HMh'Ai.i.v DIVIDED
'K)RMV\i! I'M-l.i:. --In the ft] ivpivs.-ntatioii in
' j, 1. <>.•_> MO. 'f t'rnni the origin as 0.1 sec.,
14 PRACTICAL PHYSICS
0.3 sec. three times as far as 0.1 sec., etc., and similarly for the
distance fallen. Another method of plotting results is often
adopted, viz. to plot along each axis a distance which, instead
of being proportional to the value itself, is proportional to its
logarithm. In order to save looking up logarithms coordinate
paper having the rulings spaced logarithmically can be used.
Fig. 2 represents a sheet of logarithmic coordinate paper with
the values for times and distances fallen plotted upon it. The
curve connecting these points is seen to be a straight line.
If a straight line had been obtained when plotting on uni-
formly divided coordinate paper, it would be known at once that
the equation of the curve was I = a + bt, where a would denote
the intercept on the Z-axis, and b the tangent of the angle between
the curve and the £-axis. Since, instead of t and £, the quanti-
ties which have been plotted in Fig. 2 are log t and log I, the
equation of the straight line which is obtained is
log I = a + b log t.
Bat a is, of course, the logarithm of some number, and so the
equation may be written
log I = log A + b log t.
Whence I = At\
Since b is the tangent of the angle made by the curve with the
£-axis, its value can be found by dividing <?, in Fig. 2, by d.
On measuring these and dividing, the value found for b is
2.002. Since A is the number whose logarithm is the inter-
cept on the Z-axis, the value of A may be read off directly.
It will be noticed that the values of the times have been
multiplied by one hundred before plotting. This does not
alter either the shape of the curve or its slope, but merely
throws it far enough to the right to get it on the paper. If it
were moved to the left to its proper place, it is seen that it
would cut the Z-axis some place between 100 and 1000, and
since the triangle efg is equal to the triangle that would be
formed by the Z-axis, the curve, and the 100 cm. line, it follows
NOTKMi \i. Mi:Asrm-:.Mi-:xT 15
that the value of .-1 is the ^;de from decimal point, as the
interest /</ on the lU-sec. line. The point where the curve
crosso the LO-060. line > M I.1.' am. Moving the decimal point
so as to make the value lie l>et \\veii 1<>0 and 1000, the value
obtained by this method is about 4l»0. The empirical equation
i l»y this method is, then.
/ = 490 *i00a,
while the theoretical equation is
4. Notation
In ;"iit chapters frequent use will l>e made of the
The attention of the student is called to the
following symhnli.Mii and i.y means
of it : —
The symbol i1"* is an abbreviated way of writing
fc=i— »
, and is read: » The
m of tl.' 1 integral values fn.m 1 to ft*'1
Exp \\l»i( h are to be sumi be expanded as
own 1)\ tli ing example: —
2 (?--••• i' = v^-iv,-4.w.
»=| ... m tf.i ... n
By some one of the algebraic methods of summing sei
n he shown that : —
CHAPTER II
METHODS AND APPARATUS FOR THE MEASUREMENT OF
FUNDAMENTAL QUANTITIES
1. Measurement of Distance
THE vast majority of the measurements made in a physical
laboratory are ultimately measurements of distance. Two
temperatures, for instance, may be compared by the difference
in the lengths of a thread of mercury; a pressure may be
determined from the height of a barometric column, or from
the distance that the pointer of a pressure gauge moves ; a dif-
ference in time may be measured by the distance that the hand
of a clock has moved; etc.
The Meter Stick is the instrument most often used in the
laboratory for the measurement of moderate distances. Usu-
ally the smallest divisions marked on it are millimeters. Since
the last division at each end is liable in time to become worn
a trifle short, the ends are seldom employed. In use, the
meter stick is turned up on its side so as to bring its scale as
close as possible to the object to be measured, some line on the
meter stick is brought as nearly as possible into coincidence
with one end of the distance to be measured, and the reading
of each end of the distance is noted, the tenths of a millimeter
being estimated. The difference between the two readings
gives the distance sought. Division lines which are as close
together as a fifth of a millimeter are usually more confusing
than helpful. A very little practice, however, will make possi-
ble the rather accurate estimation of a tenth of a division, pro-
vided the division is not much smaller than a millimeter.
For the more accurate measurements of small distances, the
principle of the micrometer screw has many applications. A
16
KEMENT OF II M'AMKNTAL QUANTITIES 17
earefullv made screw with a divided head turns in an accurately
titting nut. An index mark close to the divisions on the head
shows through how many divisions the screw has turned.
The distance between the threads of the screw divided by the
number of divisions on the head gives the distance the end of
TCW advances when the head is turned through one of its
OUS, The principle of the micrometer screw is employed
in the micrometer ealiper, the spherometer, the dividing
engine, the tilar micrometer microscope.
Thr M • i ,8) consists of an accurately
made screw which can be advanced t'>\\ard or away from the
.1. whole
number of millimeters . »•!* p •rn-J£Ji""^i»l»»gBg>p*
b' ' 1 .1I1<1 ibflPteMMMMb^-. -' *":£&&£&&
indicated by the
millimeter divisions on
11 covered
l y the sleeve 1>* while
t ic fi,. i a milli-
!i by the
/- pitch of the screw is half a inilli-
; leter and if the head i^ divided into lifts e<|iial spaces, one
..ink will be uncovered by the .slee\e for ever)
*0 complete turns oi 8W, and . toe on the divided
of 0.01 mm.
hus if tenths of a division on the .sleeve ,
j iate to 0.000'» mm.
iiNtrunicnt. l.«, the reading when
• just touches A 1 always be recorded. In making a
: -ading, the sleeve is Qevei turned up tight, but only until a
ery slight |in->- It.
In • i rometer (Fig. 4) a inic; 101611 which has
a »ery large divided head passes vertically through a nut
i oiinted.it i ,111 equilateral tripod. The pitch of
tie S' .iieiitlv .1 mm. and the li«-ad divided into 500
18
PRACTICAL PHYSICS
equal spaces, so that by estimating tenths of a division, a read-
ing can be made within 0.0001 mm. However, with the type of
spherometer illustrated in the figure
several successive settings usually
show that they cannot be trusted
much closer than 0.001 mm., so that
it is useless to read the fractions of a
division. The spherometer is espe-
cially useful in measuring the radius
of curvature of spherical surfaces —
whence its name.
In the Dividing Engine (Fig. 5)
a long micrometer screw with a large
divided head A is mounted horizon-
tally in a massive base between a pair of tracks in such a way
that it has no longitudinal movement, but when rotated causes
a nut to advance parallel to the tracks. Attached to the nut B
is a carriage C which slides along the tracks with the advance of
the nut. Fastened to the base are one or two microscopes M,
with cross hairs in the eyepieces, which can be focused upon
FIG. 4.
FIG. 5.
an object resting upon the sliding carriage. In making the
measurement of the distance between two points, the carriage
is slid along until one point is under the cross hairs of a micro-
MEASUREMENT OF ITNDA.MKMAL ^I/AYI'ITIES 19
and then the mi. :vw is turned until the
other point under the cross hair. The difference
en the reading of the micrometer screw when one point
.uder the cross hair and the reading when the other point
• rnler the cross hair gives the distance between the two
points. If the pitch of the - 1 mm., the head divided
200 divisions, and t i to one tiftlt of a division, a
leading will be made within o.nol mm. The microscope should
lit magnifying power to show elearlv a move-
ment of n.ool mm. - ,ird paragraph bdov
The dividing engine receives its name from the fact that it
is most Of! i to rule divided BCal itened to the base
18 a s\ >v which a tracing point .V can be drawn
B the sliding in a direction normal to the motion of
'NT. l'>\ t: an be drawn upon an object
ted to the I . d\ aneed by a
• l-fmi- nt, anoti. ira\\ n parallel to the first, and so
• i until a scale is Co: 1. The mechanism earning the
t -acing point is often arranged with notched wheels /> which
i -i-mit li: ;th, so that in ruling
f scale e\ :i and tenth lint- may be dra\\n longer than
t ie o
«-ope is a microscope that has in
t ie fo .-1 cross hairs, <i and /•
( Kig. •) ), \\hidi caii be moved across t
pe by means of a mi-
I
/, the t«-,
\ hid; : he \\ hole
i imber of turns made by the n,
s rew. oscope
s age c<- FI<;
i «Qroine:- jied by focali/ing the micro-
s - pe rd sea! •mmoiily us<-d is a
s ale having ten divii :nillim«-; taken to
li we tlie lines of tin- standard scale parallel to the movable
20 PRACTICAL PHYSICS
cross hairs. Readings are made on, say, five consecutive lines
of the standard scale near the left side of the field of view,
and then on the same number near the right side of the field.
From the difference between the readings for the left-most
lines of the two sets is obtained one determination of the dis-
tance corresponding to one turn of the screw ; from the differ-
ence between the readings for the second lines in the two sets
is obtained a second determination ; and so on.
If the pitch of the screw is such that one turn corresponds to
a distance of 0.1 mm. on the microscope stage, and if the head
is divided into 50 parts, one division on the head corresponds
to 0.002 mm. With the best microscope it is impossible to
distinguish lines closer together than about 0.001 mm., but the
mean of a number of careful settings on a very fine line can
be trusted to about 0.0005 mm. In making a setting, the
screw should always be turned up from the same direction in
order to avoid errors due to backlash.
In the Eyepiece Micrometer a finely divided scale ruled on
thin glass is placed in the focal plane of a microscope. The
eyepiece micrometer is standardized in the same manner as the
filar micrometer.
Vernier 's Scale is a device employed for the estimation of
fractions of the smallest divisions of a scale. It consists of a
short auxiliary scale capable of sliding along the edge of the
principal scale. The precision attainable with the vernier is
about three times that attainable with the unaided eye. The
theory of the vernier may be made clear
. I I,, i i i i I i i by the following example : Suppose
i i
' that along a meter stick there slides a
— vernier 9 mm. long divided into ten
equal parts. Each division on the ver-
nier is then 0.9 mm. long, and if the 0-mark and the 10-mark
on the vernier coincide with lines on the meter stick, then the
1-line on the vernier lacks 0.1 mm. of coinciding with a line on
the meter stick, the 2-line lacks 0.2 mm. of coinciding with a
line, the 3-line lacks 0.3 mm., and so on. If, then, the vernier
Ml IIENT «>F FfNDAMKNTAL QUANTITIES 21
to be iin i VIM! aloii'_r 0.:] mm., its 0-line would be 0.3 mm.
beyond some mark on the meter stick, and the 3-line would
coincide with some mark; if the 7-line coincided with some
mark, the 0-line would be 0.7 mm. beyond some mark, etc.
The position of the 0-line is what is desired. In Ki*r. 7 the
:n.
In rising any vernier, we first lind how many divisions on
the vernier correspond with how many on the main scale, and
from this calculate the length of a vernier di\ision. The dif-
• •II the length of a >cale division and the length
of a vernier division is called tin- •• -unt " of the vernier.
i nit multiplied by the number of the vernier line
which coincides with a line on the scale <jives the distance he-
i the 0-. the vernier and the preceding line on the
I In the case of a circular scale divided into thirds of a
. the v« •: >ften made li t'ty-nine thirds of
ng and is di\ided ii . Its least count is
en one third of a minu: . s .shows such a vernier, and
;t -o illust: manner in which verniers are often num-
i : I can be made directly without computa-
t »n. In this particular case, since each vernier di\isioii corre-
s onds to one third of a min; natural to number the
ti teeiith division ~>, [}n- thirtieth division 1 In
• l -. s the : i ;:, lg< -
Tlf /' •'' a finely divided steel
'• (' \\\\ d jaw at 0110 end. and a u \v /! j,r..\ i.led \\ ith
a . l> ' ' i.- aloii'_r th,. 1,-n^th of the scale,
this instrument the jaw 11 is nearly closed upon the
22
PRACTICAL PHYSICS
object to be measured, the screw E is tightened, and the final
adjustment carefully made with the screw F. The zero read-
ing should always be noted, and care should be taken that F
is turned only until a slight pressure is felt.
The Cathetometer is an instrument for measuring vertical
distances in cases where a scale cannot be placed very close to
the points whose distance apart is desired. It consists essen-
tially of an accurately graduated scale, together with a hori-
zontal telescope capable of being moved up and down a rigid
vertical column.
In one pattern of the instrument (Fig. 10) the scale is en-
graved on the supporting column, while in another pattern the
scale is independently supported parallel to the object being
measured and close beside it. In the case of an instrument of
the first type, the carriage can be clamped at any point along
the length of the vertical column and its position read lay
means of the scale on the column and a vernier ( F", Fig. 10),
attached to the carriage. In the case of an instrument of the
second type, the position of the carriage is obtained by observing
through the telescope the point on the distant scale that appears
to coincide with the cross hair of the telescope.
Before taking a reading with a cathetometer three adjust-
ments are necessary. The first adjustment is to make the axis
AB vertical. To effect this, the telescope is set approximately
parallel to the line connecting two of the three leveling screws
in the base, and one or both of these two screws is turned until
the bubble in L is near the middle of the vial. The telescope
3UREMENT OF FUNDAMENTAL glAXTITIES 23
then rotated about Alt until it points in the opposite direc-
tion. If the bubble is n.it still in the middle, it is brought
bark to tin* middle by turning
or bntb i >f ti :v\vs. the
number of turns heiiiLj counted.
II Jf <>f that number of turns is
thru made in tin- opposite direc-
tion and the bubble brought back
be middle of the vial by DO
of tin- - ^cope
hen tinned so as to be 90°
from its original position and the
tlm in the base adj
until the bubble is in the middle.
iie bubble dors not n<»\v re-
n ain in the middle of the
h .\\ever tl. >pe may be
timed about . l / ntire ad-
j >tin. -nt ie .1.
The Se ijllstinrlit is tO
r ake the a
1 iriznntul. In lining thi-
t lescop
t trneil rnd for end. aced.
I tin- l»ul)i
r 8t at the middle .
i bro i,\ the
> I- -A />. the numbers of tin H^^
I (pliivd b.-ili^ eoimtrd. Half
t is numb<T of turns is made in l°-
t e opposi1 !.ubl)le tlien brought to tin- mid-
d e by m.-ans of the se: «,t tl,t- vial. The tele-
|'">pe ..-r>ed in tin- \\yes, and if the bubble do,
.iconic1 tin-mid. . .ire repeateil.
Ijiistm.-: lev,, pc. The front
'iitainii: it until the •
24
PRACTICAL PHYSICS
hairs appear as distinct as possible. Then, while sighting along
the outside of the telescope, the latter is brought to about the
right height and turned so as to point approximately at the
object to be viewed. The eye is then placed at the eyepiece
and the focalizing screw F turned until the image of the object
does not move with reference to the cross hairs when the
observer's head is moved slightly from side to side.
If the scale is engraved on the column which carries the tele-
scope, the latter is focalized first on one of the points and then
on the other, the final setting being made in each case by the
screw E. After each setting the height of a mark on the
carriage is read by the vernier F". The difference between
the two readings gives the desired distance.
If the scale is independently supported, it is placed vertical,
close to the object being measured, and so that the scale and
object are at about the same distance from the telescope.
The telescope may be focalized on one of the points and then
rotated about a vertical axis until the scale is in the field of
view, the height of the cross hair being then read directly ; or
a small mirror capable of rotation about a vertical axis may be
attached to the telescope just beyond the objective, so that by
rotating this mirror, an image of either object or
scale can be seen without rotating the telescope.
The height of the second point is then observed
in the same manner.
When the scale is independently supported, the
error introduced by lack of vertically of the scale
may be easily found as follows: Let AB (Fig. 11)
be a vertical line drawn through the point B of
the scale CB. Then in place of the real height
AB, we read CB, and the error is CB — AB =
CB- CB cos 0 = CB (1 - cos 0).
For a given inclination of the scale, this error
will evidently be greatest when CB is greatest.
If, then, the scale is 100 cm. long and readings are to be trusted
within 0.01 cm., the scale should be so placed that its departure
FIG. 11.
MKASI Ki:\ii:.\r OF FFNDAMFNTAL QUANTITIES 25
from vertieality is not greater than that given by 0.01 = 100
(1 — eostf). From this equation 6 is found to be somewhat
than 0.01 radian, whieh means that for the given degree
•urary the seale is nearly enough vertical when a plumb
irupped from its top would fall within 1 em. of its bottom.
2. Measurement of Mass
One of the m,.st eommou as well as the most accurate
methods for the eomparisoii uf masses is atYonh-d l.y the beam
lance (!•';•_:. 1- >. The beam BB' e;m rotate ab«.ut a knife edge
K. wbiei. :j.oii an agate j)!.!1--. ^uspeuded fr«.m knife
I\\ and /\., aiv the >eale pans p{ and /-,. A handle
// op.-rat.-s an a: :iiiL,r of a hoii/.ontal n»d and
tb *ee ' ' i ('.,, by means uf whieli the kn.
26 PRACTICAL PHYSICS
may be relieved of the weight of the beam and pans when the
balance is not in use and when the masses in the pans are being
changed. Fastened to the beam is a long pointer / which
swings in front of a graduated scale S. Whether the divisions
on this scale are numbered or not, it is convenient to assume
that the middle division is numbered 10, and that the divisions
are numbered from left to right. Projecting from the side of
the case is a rod R by means of which a. bent aluminium wire
called a rider can be placed at any point along the beam. This
rider is used in place of standard masses smaller than 10 mg.
The top of the beam is often divided into twenty equal
parts, the 0-line being over the central knife edge, and the
10-lines over the. other knife edges. If the mass of the rider
is 10 mg., and it is placed on one of the 10-lines, it produces
the same effect as if a 10 mg. mass were in the corresponding
pan ; but if it is placed at division 3, it has a turning moment
only three tenths as great, and so produces the same effect
as would a 3 mg. mass placed in the pan. Occasionally a
rider of some other mass is used and the beam divided accord-
ingly.
The Method of Vibrations is usually employed in making ac-
curate weighings. When using this method, the case is at first
left closed and the arrestment released. If the pointer does not
begin to swing, the case is opened, the hand waved lightly
over one pan, and the case again closed. With the pointer
swinging in front of the scale, but not beyond it, the zero point
of the balance is determined; i.e. the point at which the pointer
would finally come to rest, either with no load on the pans, or
with equal loads on the two pans.
This is done by observing an odd number of successive turn-
ing points of the pointer. As the pointer swings, the distance
between any two successive turning points on the same side of
the scale gradually decreases, but in a few swings the decrease
is slight. The zero point is about halfway from b (Fig. 13)
to a point midway between a and <?. It is also about halfway
from c to a point midway between b and c?, about halfway from
MKASlkKMKNT OF Fr.NDA.MKNTAl. Ql'ANTITIES 27
«7 to a point midway bet ween 0 and 0, etc. Sim-e the distance
from <f to <' is about tin- same as that from c to t% the average
, and t is nearly tin- The x.ero point, then, is
: tli.- point found by taking the
. <•. and 0, ami avera^im,' with
it the average of b and </. Suppose, for
in>tanee, that five turning
point- : —
- 11.8
[Tien 1 of the turning points at the left is S. .">:', and
of the turning points at the ri'_rht is 11.85. Consequently the
in tin- neighborhood of [J(8.53+ 11.8f>) = ]10.2.
Five ve turning points are usually enough to observe,
bet any odd number of successive turning points may be us,-d
in the same v. by averaging the left turning points and
a\ -ra^in-j' '.t turning points and tln-n finding t he a\ -
Of thf ; Imiild he noted that this method of
fii lin • accurate when the pointer s\\in^s
SJT :h a small amplitud S he zero p"int \ariesfrnmday
to day, and • . it should he determined
fo each expei-ini* •. I dmuhl be
d( -ennined b..th at the he^inning and at the end ..fa weigh-
in ', and the a \rraije value used.
Vfter thf /.«•!•.. p.,int has 1 ^rminrd and while the
ar estm-': 1 so as to lift the ln-am utT tin- knife edge,
tli obj«M-t is pi. i pan and standard masses on the
Ot er. IIi'_'ht-handrd I find it most coiiveni«Mit to
tin obj'M't nn thr h-ft pan so that the mass pan is in front of the
ha id t ha" th«- adjustment «•!' tin- .standards. Karh time
th; ta nrw mass : .m tin- pan tin- arn-M nn-nt is lowt-rrd
ju • iioii^ji t,, >(.r j,, \\hirh dirrt-tion the pninti-r would swiiiL^
bu no masses are ever put <>n or taken off while the pointer is
fp • to 8Wing, \VIn-n tin- masses are so nearly adjusted that
- 28 PRACTICAL PHYSICS
when the arrestment is entirely released the pointer swings
back and forth near the zero point, the position at which the
pointer would finally come to rest is determined from several
successive turning points in the -same way that the zero point
had been. . The rider is then moved so as to alter the effective
mass on the mass pan by one or two milligrams, and the new
position of rest determined. From these observations the mass
which would be required to make the point of rest coincide with
the zero point can be calculated without taking the time to
effect the balance experimentally.
Suppose, for example, that the zero point of the balance is
10.2 scale divisions, and that with the object on the left pan
and a mass of 24.166 g. on the right pan, the point of rest is
found to be 11.6 scale divisions. Since this point of rest is to
the right of the zero point, the mass on the right pan is too
small. Suppose that by means of the rider the effective mass
on the right pan is increased by 2 mg., and that the new point
of rest, determined as before, is found to be 7.4 scale divisions.
Then the addition of 2 mg. has moved the point of rest through
[11.6 — 7.4 =] 4.2 scale divisions, and 1 mg. would have moved
it 2.1 scale divisions. It follows that the mass which would
have to be added in order to move the point of rest through the
[11.6 — 10.2 = ] 1.4 scale divisions to the zero point of the
balance is [1.4-f-2.1=] 0.7 mg. Consequently the apparent*
mass of the object is [24.166 + 0.0007 =] 24.1667 g.
The sensibility of a balance is denned as the nilmber of scale
divisions through which the point of rest is moved by the ad-
dition of one milligram to the load on one of the pans. In the
above example the sensibility was 2.1 scale divisions per milli-
gram. The sensibility, however, depends upon the load and
should therefore be determined for each weighing. The fact
that it depends upon the load may be shown as follows : —
Let K^ _ZT2, JT3 (Fig. 14), denote the three knife edges
of the balance, and M the center of mass of the beam. Let
* See below, Errors in Weighing.
MEASUREMENT OF 1 INDA.MKN i Al. QUANTITIES 29
y>j and /'., be the respective masses of the left and right
pans, and M, the mass of the beam. Suppose that with a
.V1 <>n the left pan and a mass J/, mi the right pan
the beam enines
in the
i i inn indi-
i. Then,
-m
\{9ffl
the bal
is in iMjuilibriiini,
^\\\\\ of the
of
' }i - (M*
-}-//._,)//, and M.y
D al'Min I\
e.jnal
P
*3g
/, sin (0, - ft) - (Afa + pi)g x l^ sin (03 4- /9)
.— 37 = n,
; .-08/8-n.s^^ BUI
s /^ -h ens ^ sill /3)
- ^f,r sin 0 -- (10)
actual OU6 'nail, \v«- may n-place
lift ^/, = /, if ^.2= 0j = ^. and
Pi ~ V\ — I ' ve
(3/, + ;i)/(sin#- ^rnstf ) ( .V., -f /')/(Ml.^-h/^«-(,S^)
i/ ,;
"In-:
If.V,- ' , then/9 i if J/i-3f2= 1 m-.. then the
1< ft member of (11) d -In- movement <»f th.- |»..ii;ter for
1 mg. in the In.id. Thai i^. rach mrinlf.T i.f tiiis e<|iia-
t ' ) IB proportional in tin- sensihility of the balance.
If ^ --I'M. and uhalrvrr tin- value of the load,
J \ -f •/ .1 iin-ml. unaltered : that is. when
30 PRACTICAL PHYSICS
6 is 90°, the sensibility is independent of the load. If 0 is less
than 90°, cos 0 is positive, and as the load, Ml + M%, increases,
the right member of (11) decreases; that is, when 6 is less
than 90°, the sensibility decreases as the load increases. If 6 is
larger than 90°, cos 6 is negative. It follows that as Ml+ M2
increases, the denominator of the right member of (11) de-
creases, and the sensibility therefore increases ; that is, when 6
is larger than 90°, the sensibility increases as the load increases.
Since different loads necessarily bend the beam different amounts,
it follows that the sensibility is different for different loads.
The maker usually arranges to have the three knife edges in
line when the balance has about half its maximum load.
Errors in Weighing. — The errors to which a weighing is espe-
cially liable are due to (1) the buoyant effect of the air, (2) errors
in the standard masses, (3) difference in the lengths of the bal-
ance arms, and (4) difference in the masses of the scale pans.
(1) The buoyant effect of the air will be different upon the
bodies on the two scale pans unless their volumes are equal.
The true mass may be found as follows : Let M, D, and V de-
note respectively the mass, density, and volume of the body
the mass of which is desired, and m, d, and v, the mass, density,
and volume of the standard masses which just balance it in air
of density p. Then the difference between the weight of the
body in vacuum and its weight in air is equal to the weight
of the air displaced pVg, and the weight of the body in air is
consequently Mg—pVg. In the same manner, the standard
masses when in air weigh mg — pvg. Since the weight of the
body in air equals the weight of the standard masses in air,
or Mg-pj^g^mg-p^g.
tl-£
Whence M=m
(12)
MEASI i;i.Mi:.\l OF FUNDAMENTAL t>l AXT1T1ES 31
For ordinary temperatures and pressures p is about 0.0012 g.
per cc., so that if any solid or liquid is being weighed, p is very
small compared with />, and we may apply approximation (5),
p. 7. iibtaining from (12)
or. employing approximation <
or, since for the brass standards ordinarily used in weighing,
about 8.4 g. pei
It will ; rable error in /> can produce
i i the value t'.-r .!/ niily a small error, SO that a fairly rough
\ Aim ' /' i in (14). i '!' introdii«-e«l by the
proxiinationft employed in <.l)taining (I;1.) will almost al\\a\s
1 • :»!»•, bill nefl of p and </
^ lOuld l»e (h-t.-i iniiird and not assumed.
in the stan ssrs may be corrected as ex-
j lained nndt-i- K\p.-rimi-nt !•'>.
illd ( 1 i Brron due to difference in the lengths of the
1 il. us and to difference in the masses of the scale pans
< in he nt-arly eliminat- iurhin'_: the body first in one pan
; id then in tin- othri. |., : /( and /., denote the respecti\c
1 ngths of the h-ft and right anus nf th«- balance, and //, and />.,
t ic re*: masses of the left and ri-^ht pans. If an object
« ' ma>^ M is b :.ind,ii'l masses ?//, \\heli the object is
i the right pan. and by M.indard masses W2 when the object is in
I t i left pan, then in Pig. 14, /9 <». ltn.l if ^2 = ^,
. i/ (15)
mm c/'i- ;/ ,-f
32 PRACTICAL PHYSICS
If the pointer swings near the middle of the scale with no load
on the pans, we have also p\l\=p^ so that (15) and (16)
become 7 1/r7
m1l1 = Ml2
and Ml^ — m^.
Whence ^=VJ: (17)
In case of a balance in ordinary adjustment ra2 will so nearly
equal m^ that we may use approximation (7), p. 7, and in place
of (17) write ,,.-.
Precautions in the use of a balance.
1. Do not place on the pans anything wet, any mercury, nor
anything that might injure the pans.
2. Never change the masses on the pans nor move the rider
when the beam is free to swing.
3. Never touch any standard masses with the fingers — use
forceps.
4. Keep all standard masses in the proper compartments in
the box when not actually in use upon the balance pan.
5. Never raise nor lower the arrestment so quickly as to
cause any jerk.
6. When not actually altering masses keep the case closed.
7. Before leaving the balance bring the arrestment into play
so that the beam is not free to swing, set the rider at the zero
mark, dust off the pans and the floor of the case with a camel's-
hair brush, and close the case.
3. Measurement of Time
INSTRUMENTS. — In nearly all apparatus for measuring time
use is made of the principle that the period of vibration of a
body oscillating with harmonic motion is constant. The most
commonly used vibrating bodies are the pendulum, the balance
wheel, and the tuning fork. Any one of them may be kept
going indefinitely by a slight impulse given it each time it
passes through its position of equilibrium.
3UREMENT «>1 ITXDAMKMAL QUANTITIES
33
Iii order to ^ive this impulse to a pendulum or a balance
wheel, and also to count its vibrations, there is usually attached
to it a mechanism called dock work. A pendulum of such a
h as to make in cadi second one beat, /.<•. half a complete
.died a 10001 H. Such a pendulum is
'.n standard clocks. Where accuracy must he sacrificed to
dlity, the balance wheel is empl< - in the watch and
brODOmeter. A I watch provided with a
ii^ and stopping device M th.it the interval between two
11 be easily determin
The ini; keep the tuning fork v,r»»inij i* usually t^iven
bv an elect n •-magnet which is periodically act uated bv a cur-
rent i: 1 hn»k«-n by the motion of the tuning fork itself.
Alt, idled tonne pn .ULT < »f t he t nn ing fork is a sharp point which
lightly of smoked paper. The paper is
w apped riMind a metal drum which rotates and at the same
tine n >wly in tlie d . so that the trace
m ide by the vibrating Mini:, belix, Tin-
ill slants at which tw" occur may be marked by minute
h< !es made in the blackened surface of the paper by electric
sj irks which are
c.llscd '
fl 111
p< nt to the n
d urn. If the
jx -io<l of the tun-
in \ fork is kn-
tl • number of
W; vesand fraeti«»ii
of a wave in the
line
be ween the i . l,,,les shows the interval of time between
lh- *v \ luniii'_r fork and drum '1 in this
ID; .ill* : k chronograph.
The A*f differs from tin-
tin in^-fork chronograph principally in that the drum runs
A B
34 PRACTICAL PHYSICS
more slowly, the paper is usually not smoked, and in place of
a tuning fork there are one or more pens, A and B, which, by
means of electro-magnets, can be slightly displaced parallel to
the axis of the drum. One electro-magnet is included in a
circuit which is so connected to a clock pendulum that every
second a notch is made in the line its pen is drawing. In the
circuit containing the other electro-magnet a telegraph key can
be so placed that an observer can produce a series of notches
corresponding to a series of observed events, or the circuit may
contain some device whereby the successive events may auto-
matically close the circuit for an instant.
A clock is seldom read closer than to seconds ; a stop watch
is usually graduated in fifths of a second ; an ordinary watch,
due to eccentricity in the mounting of the second hand, can
usually not be trusted within three or four tenths of a second ;
an astronomical chronograph can often be trusted to a hun-
dredth of a second ; a tuning-fork chronograph may without
difficulty be made trustworthy within a thousandth of a second.
METHODS OF MEASURING TIME. — The measurement of a
short interval of time between two separate events is usually
made with a stop watch or chronograph. But for determining
the period of a regularly recurring event, like the swing of a
pendulum, there are several methods of procedure, the choice
between which depends upon the magnitude of the period and the
accuracy required. The movement of a vibrating point from one
end of its path to the other is called an oscillation. The complete
to-aiid-fro movement from the instant when the vibrating point
leaves any given position to the instant when it next passes through
the same position in the same direction is called a vibration. The
interval of time between two successive passages of the vibrat-
ing point in the same direction through a given position is called
the period of the vibration. The period of an oscillation is half the
period of a vibration. The most useful methods of determining
the period of a regularly recurring event will now be considered.
1. The Direct Method consists in noting by means of a clock
or stop watch the interval of time between two recurrences of
MKA>1 KKMKXr OF Fl X DA.MKXTAL QUANTITIES 35
vent and dividing this interval by the number of recur-
The accuracy of this method depends upon the accu-
racy of determining the times of beginning and ending the
count, and upon the time that is allowed to elapse.
"2. 7" .!/•'•' I "f Omitt — The preceding method
may be slightly moditied so as to increase somewhat the accu-
racy without materially increasing the time or labor required.
Suppose, for example, that a heavy horizontal disk is suspended
by a vertical wire, about the axis of whieh it can vibrate, and
that the instant at whieh a mark on the edge of the disk \.
through the middle point of its path is noted for sixty-one con-
secutive swings. Then the ditYerence between the tifty-tirst
time of passing and the first time of passing gives the time of
:i the lift v-second time of pass-
ed the seeond time of passing gives an independent deter-
mination of the time of : :igs ; and so on. Thus after
eo luting i independent determinations of the
tine of til- ^rs are obt average is IIIMIV trust -
w -rthy tlian a single drtermination. There is no need of noting
tl • times of tli.- tenth and the liftieth.
rieneed observer can
re .dily estimate times of t: to a tenth of a second. For a
cc icrete case consider again the vibrating disk that was used as
ai example fur the method of omitt.-d transits.
\fter focalizing a telescope on the mark on the edge of the
di k, the latter is set into vibration and the time of transit of the
ni rk past the cross hair of the telescope i> obtained as follows:
looking at the elock, the time in hours, minutes, and seconds
ed : then counting seconds, and while continuing the
CO int, the hour ami minute are recorded. Without interrupt-
in) the count, the eye is placed at the telescope and the time of
a ran time ean, with practice, if the mark
pa ses rapidly, be estimated to within a tenth of a second.
W out interrupting the count ti > recorded. Cou-
th, uing the count, the eye is again placed at the telescope ready
foi the next transit, and the tun transit observed and
36
PRACTICAL PHYSICS
recorded as before. After a little practice this method can be
used with ease and confidence for the observation of the times of
any number of transits. During the count one should occasion-
ally glance at the clock to confirm the correctness of the count.
4. The Flash and Stop-watch Method. — On a stand directly
in front of the disk D (Fig. 16) is placed a stop watch W,
and a few inches above the watch a mirror M is adjusted to
reflect an image of the watch into a telescope T. On the disk
FIG. 16.
a small bit of mirror, m, is so arranged that just at the/equi-
librium position of the disk the field of the telescope is brightly
illuminated by light reflected from the lamp L. On placing the
eye at the telescope, there is seen an image of the stop watch
which is illuminated by a flash of light every time the disk
passes through its equilibrium position. With the disk vibrat-
ing, the motion of the second hand of the stop watch is atten-
tively followed through the telescope, and when the flash occurs
the watch is read to tenths of a second.
5. In the Method of Passages the value which would be
found for the period if a large number of vibrations were
counted is obtained from the actual observation of a much
smaller number. To fix the ideas, consider the case of the
MKASI IM:.MI:M OF 1 1 NDAMKNTAI. QUANTITIES
87
vibrating disk referred to in tin* preceding paragraph. Sup-
:hat the instant at which a mark on the edge
nf the disk s\\in^s through its position of rest, once at the be-
l^innin^ and <>n«-e at the end of twenty complete vil>rations.
From th- ,in approximate value for the period
ean be calcul. ted. 8 that after a time the ohser\ ations
are resumed and the instant iiuted at which the mark is a^ain
•ion of rest — in the same direction as
tw«> pi- BagM were noted. Then the time
tin- lir.M and last ol>ser\ at ions divided by
:>eriod already found gives appmxiniatelv the
numi .i»rati..ns that neenrn-d b«-t\\eni tln.se two ,
vatic nple, th number of vibra-
1 if this \alue can be trusted within
•.umber of vibrations that
really oemm-d \\as a \\li.-le number, the actual number of
\ bratii.ns \vas II. >in.-e the lime thai n the
fi "St and last ol>- r than that between the
f\ -St and >•-, -.,nd. the time il.; ,M be tni.sted farther,
a ;d it is, therefore, j ier approximation
t the period. Alter tllll \aiue has been found.
a COli d be alloued 1 and
a iotl, time of passage \\oiild L(ive a si ill
ii ore act-lira; the p. I'his method may be
D ad. ihe followin-_r cxami
TIMK or PAMAOB
t>TRRV«l.
Arn
VIM.
1
2
1 17
10 [rountiMl]
Id
3 1 1 18 50.6
4 1 1
1 4
'HI
II',
77 '
5
1
'•;:r «
38 PRACTICAL PHYSICS
In using this method it is essential that the time between
observations of passages be so chosen that the approximate
number of vibrations can be trusted within three or four tenths
of a vibration. It will be seen that the successive values ob-
tained for the period are more and more trustworthy, so that,
instead of finding the mean of these values, the last one of them
is to be used.
6. The Method of Middle Elongations is another method by
which it is possible, without counting the number of swings
that occur, to obtain for a period of vibration a value of con-
siderable accuracy. The accuracy attainable is somewhat
greater than by the method of passages, but the method of
middle elongations is applicable only when the period is long
enough to allow of recording two readings during each vibration.
The point of its path where a vibrating body changes the
direction of its motion is called its position of maximum
elongation. The mean of the times at which any two succes-
sive passages through the position of equilibrium take place
gives the time at which the elongation between them occurred.
If ten successive passages are observed, the mean of the times
of the fifth and sixth passages, or of the fourth and seventh,
or of the third and eighth, or of the second and ninth, or of
the first and tenth, gives the time at which the middle elonga-
tion of the series occurred.
For a concrete case consider a disk suspended at the end of a
wire about the axis of which it rotates. Suppose that a mark
on the edge of the disk is observed to pass through its position
of equilibrium at the times indicated in the table on the follow-
ing page.
Suppose that the first transits in the two series occurred
when the mark on the disk was moving in the same direction.
Then the elongations considered were on the same side of the
position of equilibrium, and during the 790.31 sec. between
these elongations a whole number of vibrations occurred. This
O
number of vibrations is not counted, but by subtracting the
time of the first transit from that of the ninth and dividing the
• UK. MM NT OF FUNDAMENTAL <>l ANTITIKS 39
-
Tinu-
Mi.l.llo Klunpition
1
Ik20- :$.7'
1
lh:W U.o-
2
ll'.o
2
22X5
3
•_>o. I
3
4
4
38.6
5
5
17.n
6
(5.6) I'-Jn
6
(5.6)lh:j:J!:-:.l.-Jo-
7
4 7 IM.XM
7
1 :;i
1 4. 7) r.i.o:,
8
1 L'l 1.8
(3.8) 41.H>
8
12.0
(3.8) .-.!.•_>:,
9
(2.9)
9
20.1
(2.9) :,l.:;o
10
Lao
1.10) 40.85
10
(i,io :.i.:;o
M.-;in T :'>:'>'" :>1.±>'
ence l»y four, t: found to In- approximately
Bee., and th«- iiumh«-r «»f \ihraiions tliat nri'imvil IH-I
tie two given elongmtioos is, therefore, abooi [T'.'o.:1.! 4- !«'>. .")") = ]
4". 75. If this numlMT can l»c trustc<l \\itliin <».:', vihratimi,
tl e number oi \\lii.-h art ually ncctirn-d must Ix- -is.
'I h<> period ia, ] I1'-. MH sec.
If a still in<- . i iliini series
0 readin. :.il>lc tinn- hail fl.i
a id fr<»ni this thi: tfld either of thfl .'-iM-dinirlv
c 08e t«> tin- tru«- {M-ri-.d miild he nhtainrd.
'1 he «• • as above «-\,-«-].t that in
t idii. MillidM-; \\iuilil he
1 adr ;• tin- |H-ri«»d ti ..l.iainctl as the
r suit <>f tin* observations in tin- lir>t t\\o sc:
niiiot IM- tru .T than <>.:;
t .e time which elapses bet \\ mi tin- middle elongations of the
t fO St .iild Mot he ;han ahoiit -\ 7"-'. \\heiv 7'«le-
ii >tes the aj»pro\ima; i of the motion: if the times of
t unsit ran IM- trusted to 0.2 sec., the time l.ctwecn the middle
e »ngat ions may be allowed to be about \ 7"-': and if the times
10 0.1 SCO., the tilll." hct \Vcell the Mlid-
d e « -us may IM- allo\\i-.l to hrconn- x 7"- or '.» T2.
~. 'I'}-- MMod :' accurate method
40
PRACTICAL PHYSICS
for the comparison of two nearly equal periods of vibration.
Suppose the period of oscillation of a simple pendulum is to be
compared with that of a clock pendulum beating seconds. If
the simple pendulum swings slightly faster than the clock
pendulum, a moment will occur when both are at their lowest
points at the same time. But since the simple pendulum is all
the time gaining on the clock pendulum, after a certain inter-
val it will have gained a whole oscillation, and then both pen-
dulums will again be at their lowest points. If between two
such coincidences the clock pendulum has made n swings, then
the simple pendulum has made n + \ swings, and its time of
sec. Similarly, if the simple pendulum
oscillation is
n
were going slower than the clock pendulum, the time of oscil-
lation of the simple pendulum would be - — sec.
n — 1
One method of determining the instant of coincidence
employs an electric circuit containing the two pendulums,
a battery, and a telegraph sounder, all in
series as shown in Fig. 17. When the two
pendulums are in coincidence, they pass
through the mercury contacts A and B at
the same instant, and at this instant the
sounder clicks. It is to be kept in mind
that the n in the above expressions denotes
the number of swings made by the clock
pendulum — not by the simple pendulum.
Since one pendulum gains only slightly
on the other, and since the passage of the
pendulums through the mercury cups at A
and B is not instantaneous, there are often
clicks for several successive swings. The
mean time of the first and last of these suc-
cessive clicks is used as the instant of coincidence.
The actual instant of coincidence, i.e. the instant when each
pendulum is distant from its position of rest by the same frac-
FIG. 17.
Ml-: AS I KK.MKNT < >F IT N DAM KNTAL ^tlAXTITIES 41
tion of a vibration that the other is, may oeeur when l>oth pen-
dulums are i: -sit ion other than at their lowest points,
but it can never l>e more than half a s\vin^ from the lowest
point. If there are only a few sueeessive elieks. it will be safe
-;ime that in taking the mean of several sueeessive elieks,
not in error by so mueh as one swin^.
If the simple pendulum is swin^i; : than the eloek pen-
dulum, the error introdueed into the value for the period by
getting for ft One Swing ton tV\v is the dirt'erenee between the
I found, '' :id the true period, — , viz.
n t- 1
- 1
n rl " • 1 i
•Mpaivd with unity. t!i«- ermr is almost -- ^.
Thus if // = 7(l. :• inti-M.lu.-cd into the period by an
ei ror of 1 in the number of >.•«-, ,n,U b,-i \\.-.-u enineidence is
a .nut — <U)MOli see. If n IS Small, the aeeiiraev ina\ lie in-
( «-a>ed by . ..iintin'_r the numl>er od Ifl t<» some later eoin-
c lei)' '1 of to the se. ..lid. In this case one pendulum
v 11 1 the other more than one s\\in^, and the
a -ove formulas must be m idingly.
CHAPTER III
LENGTH, AREA, ANGLE
Exp. 1. Determination of the Thickness of a Thin Plate by
Means of a Spherometer and an Optical Lever
OBJECT AND THEORY OF EXPERIMENT. — The object of this
experiment is to measure the thickness of a microscope cover
glass by two methods and to compare the precision of measure-
ment obtained by the two methods.
The spherometer has already been described. The theory
of the optical lever will now be developed. The optical lever
to be used in this experiment consists of a piece of sheet brass
about 3 cm. long and 1 cm. wide mounted upon four pointed
legs, one at each end and the other two midway between the end
legs and in a line normal to the line joining the latter. Fas-
tened on the upper side of the optical lever, with its reflect-
ing surface in the plane of the two middle legs, is a small
mirror. The length of the four legs may be such that when
the optical lever rests upon a piece of plate glass all four legs
are in contact with the glass, or the end legs may be slightly
shortened so that the optical lever can be tilted forward and
backward about the ends of the middle legs. From the differ-
ence in the angle through which the optical lever can be tilted
when the middle legs rest directly upon a large plane surface,
and the angle through which it can be tilted when a thin plate
is interposed between the middle legs and the plane surface,
the thickness of the thin plate can be determined.
Let mna (Fig. 18) be the optical lever with its mirror
approximately normal to the base mn, T 'a telescope, and 0' 0"
42
LENGTH, AREA, ANGLE
43
a vertical scale about a meter from the optical lever. First
assume that the ends of the feet of the lever are all in one
plane, Imagine the thin plate j- placed under the middle feet
of the optical lever. When
the lever is tilted forward
an observer at the telescope
:ie point of the
'I in the inir-
.irror
is tilted backward the re-
'
t«» the crOSS
hair of tin- teleSC
;ime tin
has been tilted through the
argle 0. Consequently the ft] :i the normals to the
in rror in its two positions is also 6. And sim •»• the angle of
reflection equals ; of incidence, the angle between O'a
(V (y1
a: d 0"a' equals '1 0. When 0 is small, - 0 radians, and
o
Let
'
2mm'
thickness of the thin plate be denoted hy //, the dis-
ti ice ///// hetwccn t: • Ity '1 /. the distaii.
b« tween • ,ind mirror liy L. and the ditVcivncc between
tl J sc : 1? ami O" hy X. Since /• is midway be-
t\ een m ami is twice the thickness h of
tl -plate. ( )n substitnti: • rs in the above eqnat ion.
h=^L
(19)
I uis i«> for the case of an optical lever having the lower ends
all four : :ie plain-. Hut if the end feet are shortened
that ted enough to produce
44 PRACTICAL PHYSICS
a deflection Sf when placed upon a plane surface, the thickness
of the thin plate is given by
h = OS-f >*. (20)
4 jL
The development of this equation is left as an exercise for the
student.
MANIPULATION AND COMPUTATION. — In using the optical
lever, the telescope and scale must first be adjusted ; that is,
the telescope, the scale, and the mirror of the optical lever must
be placed in such relative positions that on looking through the
telescope a reflected image of the scale is seen. To make this
adjustment, place the scale vertical, facing the mirror, and
about a meter from it; standing behind the scale and looking
at the mirror, move the eye about until a reflected image of the
scale is seen; keeping the image of the scale in view, move
the scale and the eye toward each other until the telescope,
the eye, and the mirror are in the same vertical plane ; and
then, still keeping the image of the scale in view, move the eye
and telescope up or down toward each other until they come
to the same level. By sighting along the outside of the tele-
scope see that it is pointing at the mirror and then focalize —
first on the cross hairs and then on the scale — as is described
on p. 23. If the focalizing screw is turned so that the mirror
is clearly seen, the scale will not be visible. In order to bring
the scale into view the focalizing screw must be turned so as
to shorten the telescope tube somewhat.
With the optical lever on a piece of plate glass, adjust the
telescope and scale as above directed, and observe the scale
reading in the telescope when the optical lever is tilted forward
and when it is tilted back. The difference between these two
readings is S' . Now place under the middle legs of the optical
lever the plate the thickness of which is to be measured and
take two similar readings. The difference between these read-
ings is S. L may be measured with a meter stick, and I is best
obtained from the measurement of the distance between prick-
LENGTH, MM \ \v;i.K 45
marks made by pressing the feet of the optical lever on a sheet
of {i;t
mine first the y.ero point bv
phieing tlie instrument on a piece of plate 14; hiss and noting the
readings on the two .hen the point of the screw just
touci M the screw, place under it the thin
to be measured, lower the screw until
• touche> the thin plate, and note the readings on the two
& Th.' ditYeiviiee between the readings with and without
the plate gives the thickness of the latter.
Make live determinations by each method and compare the
Exp. 2. Determination of the Radius of Curvature of a
Spherical Surface
OBJ n> THBOBI 01 EXPERIMENT. There are three
I rincipal determining the radius of curvature of a
> .hcrical . I ire by means of (<i) the spherometer.
( >) the optical Li • llection ..f light. The last
i .-thud is ap; -ii«'d lUlfaoeS, and its
< 'iiMdcration wiL until th-- t of light is tak.-n
' .. i iineiit is to determine the radius
( ' curvature t.f a spherical surface by m.-ans «,f the spheroiue-
t r and by means of the optical le\er. and to cnmpare the two
i cthods as t- the t\\o
i ethods will
'it9 of tht \ '. 'I'hc ciir .
t .re « i the dimcii-
8 >ns ;h rough
\\ »ich the point <,f the screw must be moved from * Plo ,,,
ti 6 p li of the three legs in order
tl U all : may be brought into contact with the
8] l .Tit al
Let A" i lions of the three fixed feet,
ai d /' tl, :i of the point «>!' th- . when all four are
46
PRACTICAL PHYSICS
in one plane. Let the distances XY, YZ, and ZX be denoted
by a, 6, and c. A proposition in Trigonometry states that the
radius of the circle circumscribing the triangle XYZ is
abe
r =•
- a) (« — b)(s — c)
(21)
where
FIG. 20.
Now consider a plane passing through one
of the feet of the spherometer Y (Fig. 20),
the point of the screw E, and the center of
curvature 0, of the surface whose radius is
required. Then if R is the required radius,
and A, when all four points are in contact with
the spherical surface, the height of the point
of the screw above the plane of the ends of
the three feet, we have, from Fig. 20,
Whence
+ r2
On substituting in this equation the value for r obtained from
(21), we have
Tt sv27>2s>2
(22)
It should be noticed that in deriving this equation it has been
assumed that the axis of the screw is perpendicular to the plane
of the three feet of the spherometer, and also that when the
point of the screw and the three feet are in the same plane the
point of the screw is at the center of the circle circumscribing
the triangle formed by the three feet. Errors due to these
causes are, however, almost always so small as to be negligible.
If the distances between adjacent feet of the spherometer are
equal, that is, if we set I = a = b = c, (22) becomes
7? -u
'"5+6A
(23)
LENGTH, AREA, ANGLE
47
In practice a. b, and <• will not be exactly equal, but often
they will be so nearly so that instead of using (±2) it will be
permissible to substitute for / in (23) the mean of a, 6, and c.
(b) E;i - Figs.
l!l and 'I'l* which arc views at right angles
Me another, show the optical lever rest-
ing on the curved surface. The end points
of the lever touch the spherical surface at F
and />, and the middle points at H and //.
Let /,' ivpn-seiit the required radius of cur-
vature of the spherieal surface, -J / the dis-
tance between the end points, and '2 1 the
•nee. between the two middle points.
-1.
-= i if - A n\* a. ( /^m»
vhence 2
ilarly from i
_' R( I// • .I//,.
) f (A / Mall compared with (AD), and (AE ) small com-
i ued with ( Kll ), then (.!/>) and (A/i)\\ ill he approximately
« jual respectively to /and /-. and the above equations maybe
' 'rittt'n AB + B(7) = P (24)
; id 8 to I/.' - '.'P. (26)
MI the t : the ..pt ieal lever (p. 4-\ ) it has been seen
t tat x- V^NI
D/Tr^_ \ ^ ~ • /On^
~TT^
1 ut AH -1 equals A/-: ii :. On setting AB in
]. ace • : .I/.' , 26) and then eliminating AHi\nd BC from
( 24), (25), and (20'), we obtain
21
, ^ (26)
v here *9 and *8" denote the rwpectiYe deflections observed on
ti ie scale of a telescope distant L from the optical lever (1) when
48 PRACTICAL PHYSICS
the lever is rocked back and forth on the object being measured,
and (2) when it is rocked on a plane.
MANIPULATION AND COMPUTATION. — (a) When the sphe-
rometer is used, run the center point up out of the plane of the
other three, press the three outer points on a piece of bristol
board, and either by means of a glass scale laid face down on the
bristol board, or by means of a pair of sharp-pointed dividers
and a millimeter scale, measure the three sides of the triangle.
If the three sides are nearly equal, use the average for /.
Determine the zero point of the spherometer by placing the
instrument on a sheet of plate glass and noting the readings on
the two scales when the point of the screw just touches the
glass. Place the spherometer on the spherical surface whose
curvature is to be determined, and turn the screw down
until its point just touches the surface. Again read the two
scales of the instrument. The difference between this read-
ing and the zero reading is h. Substitute these values of I
and h in (23) and solve for R. Obtain the mean of five
values of R determined in this way from five sets of obser-
vations.
(7>) When the optical lever is used, press the end points of the
lever on a piece of bristol board and measure 2 I either by means
of a glass scale laid face down on the bristol board, or by means
of a pair of sharp-pointed dividers and a millimeter scale. In the
same way measure 2 6. Place the lever on the curved surface,
and, exactly as described in the preceding experiment, adjust
a telescope and scale, measure L, and take readings to deter-
mine S and S' '. Make five different sets of observations,
each time having the telescope and scale a few centimeters
farther from the optical lever. In each case find R by (26).
From the results obtained, compare the two methods as to
accuracy.
The spherometer is especially useful for finding the radius of
curvature of a surface of considerable extent, while the optical
lever is available for surfaces of limited extent and small
curvature.
l.KNClil. AREA, ANGLE 49
Exp. 3. Radius of Curvature and Sensitiveness of a Spirit Level
OB,JI:< T ANI> THI:«>I;Y < >F KXI-F.KIMKNT. — In many measure-
in. -nts in which a spirit level is used in connection with other phys-
ical apparatus it [fl try that the sensitiveness of the level
heat that of the other apparatus. An example
is tli- -i t lie telescope ami level of an em_rineer's transit.
When used in leveling or in measuring vertical angles, the
:cal motion of the telescope which can he detected hv
means of the -hoiild also make itself evident by a
displaeemelit of tiie level hnhhle. A text of the silitahilitv of
particular use includes the determination of the
I the run of the hnhl.le in the vial and the sensi-
•1. Th.- M of a spirit level
may he detin- • the hnhhle moves for an inclina-
<>f tl. >t" one minute. Since the sensitiveness can he
] -oven to be directly proportional to the radius of curvature of
1 ie vial, it :.-d h\ the radius of curvature. The
< jj. !o mak< A level.
In the lulu.: ;t level is usual: hv means of
Level 1 M-_r of a hase plate upon which
lhaped • -upport«-il l.\ two proj«-etin^r steel poini- /.'
; i the end oi the ai be T and a mi-
. Omet 'V AT at I of the T. The pitch of the
i ; 1 and also the perpendicular
the line coniieetin^ the
' /.' :. •; /'. I B ll /..::•'{ oil the
j .rid the jiMsitiun i.uhhle in the vial is noted hy means
a s< 1 upon the glass or hy a scale S attached to
t ie h-vel trier. In ifl i neon \ ci li.-n t to separate the level
50
PRACTICAL PHYSICS
G'
II
from a piece of apparatus of which it forms a part, the entire
apparatus, e.g. a telescope or theodolite, may be mounted in
the grooves ABC or DEF.
After the level tube is in place, the micrometer reading
is noted. The T is now tilted through a small angle
by turning the micrometer screw, and readings are again
taken of the micrometer screw and the position of the
bubble.
Suppose that by means of the micrometer screw the T of the
level trier is moved from the position FJ (Fig. 24) to the
position FJ' , the middle of the bubble moving meantime from
G- to H. If a vertical line „ J'
G-KWQTQ drawn through Cr
before the micrometer screw
was turned,
and if this
line were to
move with
the level, it would after the movement
be in a position 6r'P, such that the
angle through which it moved would
equal the angle JFJ' through which the
level moved. A vertical line through
the middle of the bubble's position of
rest has the direction of a radius of the
bubble vial. If, then,. HP is drawn
vertically through H, both HP and Gr'P are radii of the vial.
But since HP and CrK are parallel, the angle Gr'P IT equals
the angle between G-K and G-'P, which latter has just been
shown to equal JFJ'. It follows that
pIG> 24.
Let GPP be denoted by R, a' If by d, FJ' by a?, and JJ1
by y. Then
-—=0 radians,
R
(27)
LENGTH, ARKA, AXGLE 51
an- 1, since the screw is always perpendicular to the T,
y- = tan 0.
x
Since 6 is always very small, tan 0 = 0, and we have almost
exactly ±=y.
M ./•
Whence R=~^ (29)
y
Kliminating R from ell) anil (29),
<*. *<*
Sinc«' one minute differs from the :>438th part of a radian by
le,«« than <>."! D be reduced to minutes by multiplying it
bj 3438. The sennit iveness is, therefore, from its definition
ir 'veil by ,
a
T 3438
MANIIM I.AII..N AM. C/OMPl i \ri<>\. — Place tin- ^-shaped
<•; -ainij upon a pi»-«-e of bristol board, and by m.-ans of slight
p 088U r>- obtain an impivvM.in ««f the three supporting points.
^ ensure the perpendieular <li>tan.-e fr..m the impression made
b the «-nd ..f tin- micmiueter screw to the line eonneetiiiL; the
ii pressions <.f the other two supporting points.
The pitch of the micrometer sen-u may be obtained in the
!• ,lo\\inur DUUwer: After plaeiu-^ the spirit level on the trier,
a- just the mici-Min,-!. until on-- end of the bubble is
d ?ectly under a scale division near the middle of the vial;
tl -n insert under the mien. meter screw a small piece of plate
g 188 whose thiel, • - been alivady measured with a sphe-
re Tieter or micrometer caliper. and a'_fain adjust the micrometer
8( •« w until the bubble rests at the same point as before. The
tl ickness of the glass plate divided by the necessary number
oi turns nf the micrometer >< the pitch of the latter.
52
PRACTICAL PHYSICS
Again adjust the micrometer screw until one end of the
bubble is directly under a scale division near one end of the
vial. Observe the micrometer screw reading and the scale
readings at both ends of the bubble; rotate the micrometer
screw through a convenient number of spaces and take read-
ings as before. Continue this operation until the bubble has
been removed in some half dozen steps to the other end of its
run, and then return step by step in the same manner. Repeat
this series of readings three times. A series of such readings
may be conveniently tabulated in the following form : —
NUMBER OF
OBSERVATION
MICROMETER
HEADING
EEADINGS OF BUBBLE
DISPLACEMENTS
LENGTH OF
BUBBLE
Left End
Eight End
Left End
Right End
1
3.700 mm.
1.3 mm.
10.2 mm.
8.9 mm.
2
3.800
6.1
14.9
4.8 mm.
4.7 mm.
8.8
3
3.900
11.1
19.8
5.0
4.9
8.7
4
4.000
16.2
25.1
5.1
5.3
8.9
5
4,100
21.1
30.2
4.9
5.1
9.1
6
4.000
16.1
25.1
5.0
5.1
9.0
7
3.900
11.1
19.9
5.0
5.2
8.8
8
3.800
6.2
15.0
4.9
4.9
8.8
9
3.700
1.3
10.2
4.9
4.8
8.9
Mean 4.95
5.00
8.88
The values in columns 2, 3, and 4 are read, and those in 5,
6, and 7 are calculated from these readings. The values in
columns 5 and 6 show the uniformity of the run of the bubble,
or the variation in sensitiveness when the bubble is at different
positions in the vial. The average radius of curvature and sen-
sitiveness of the vial are obtained by substituting for d in (29)
and (30) the mean displacement obtained from columns 5 and 6.
Care must be taken to keep the entire vial at the same tem-
perature. It must not be touched by the fingers nor breathed
upon, as when unequally heated the bubble tends to move
toward the point of highest temperature.
LENGTH, Al;i:\. ANGLE
53
Exp. 4. Verification of a Barometer Scale
OBJK«T AND THI:<>KY <»F K.\I'I:I:IMKN r. — In the ordinary
form of Fortin's barometer, the lower end of the tube dips into
a men-ill >ir which can be raised or lowered by a screw,
liy this HUM >re taking an observation, the surface of
the mercury in li. is always brought to the level of
• •ini of an ivory pin extending (h»wnward from the cover
of th- dr. The barometric height is the length of the
in.-rcury column from the hottmn of this pin to the top of the
ni.-niscus at the upper end of the col-
umn. A brass scale attached to the
metal : tllhe
;.p«»>ed t«» he adjii.stt-d BO that its
divisions indicate di>tance8 measured
f -om the point of tli. in. The
object of thi> it to m,
1 lebaromet i ic height by acatl
; nd to compare with tin i the
i jading by the I>ra88 scale.
M \M! [>COMP1
- -Set tl. -tand
; bout a me: tfll from the ba-
i Hut-; ictoraeter has
1 een adjusted as described on ]
i use ' ' il the 1
( roes hair in the eyepiece is tangent to
; ie mmiscii- upper end of the
1 arometer column, and t i .ithe-
: .m.-1 -, of tin-
. i<l vernier. : the telescope until
i M- h ! u ith
level of the mer. in . reser-
ke th.- eathetom, ing, Tin- dilTcreiice
1 i3tween ' _TS is the barom.-ti-i.- height.
by means of the ion '/' ; the bottom of
nd
54
PRACTICAL PHYSICS
the barometer, bring the surface of the mercury in the reser-
voir to the level of the ivory point P, and read the barometric
height by means of the scale and vernier F(Fig. 26) attached to
the case. Attached to the sliding vernier there is a similar
piece of metal directly back of the barometer tube. These two
slides move together. In order to avoid parallax, the vernier
is moved up and down until the position is found where the
lower edge of the vernier, the upper surface of the meniscus,
and the lower edge of the rear slide are in line.
Find the error of adjustment by taking the difference be-
tween the mean of five determinations of the barometric height
by means of the barometer scale and the mean of five with the
cathetometer.
B
Exp. 5. Determination of the Correction Factor of a Planimeter
OBJECT AND THEORY OF EXPERIMENT. — A planimeter is a
direct-reading instrument which is used to determine the areas
of irregular figures on drawings. The correction factor of a
planimeter is that number — usually near unity — by which
the area read from the instru-
ment must be multiplied in
order to get the true area.
The object of this experiment
is to determine the correction
factor of a given planimeter.
Amsler's polar planimeter
(Fig. 27) consists of two
arms, a tracer arm AC, and a
pole arm EC, jointed at C.
The point E is fixed and the
point A is carried around the
boundary of the figure in
the direction of the hands of
a watch. Attached to the tracer arm is a small roller D, the
axis of which is parallel to the line AC. This roller and the
LENGTH, ARKA. ANGLE
55
points .1 and E are the only parts of the planimeter that touch
the paper. As the point .1 passes over the houndarv of the
figure, the roller rotates unless the mot id ntirely in the
direction of .1C, — in which ease the roller slides. It will now
he proved that when the tracing point circumscribes any (dosed
plane figure which does not contain the pole point, the cir-
cumference of the roller n • distance proportional to
the area circumscribed l»y the tracing point. This proof will
IM- in four p,:
-'.consider two concentric circular arcs AAn and .1 .1
. 28) cut l»y r.i.i i .1 /•'. Let the pole point of
planimetrr he i the
these arcs, and let the
tracing point he m«.\ed alonir the
radi >A'. Then
the roller will mo\ t- fn.m D to
y while a point in itx ciivum-
erence \\'\\\ rotate tlil-Mii^h the
DOC />//. A^ain, let
raei: : he mo\ed a!
he radio
• lie roll,-]' ti, : ,,m
ireiimtViviH «• j
he d /' II . Bin •
the figure
.'/> ///' '• i nne whetlier th- point has n
i / 'ongsoni*' i /'.it follows
hat //'//' e.piaU />//. T1 ntf i"'int
i asses «'ver t: iv radii interceptrd J»ei \\eni the
' Ulle • M-^r I 1,,. j / . : ,,llin«r
its of the in. 6 eijual.
Secoit'l. let i:<'.\ i I ,l tin- planimeter in one
1 '.Nition, and I be planimeter in another position, l
]B and .//> normal to A( :l l> \. . D «lra\\
.:.!. /•:/'. and /•;// . POJ brevity let t, denote the
56 PRACTICAL PHYSICS
through which a point in the circumference of the roller moves
with reference to AC wlien A describes any line x.
Let the instrument start from the
position ECA, and, keeping the
angle EGA constant, rotate about E
through a small angle A© into the
new position EC' A' . A A is, then,
the arc of a circle described about E
as center. The roller, meantime,
F 29 moves through a small distance DD1 ',
sliding through a distance HD ', and
rolling through a distance DR. Whence
BAA. = DH= DD' - cos HDD' = ED A0 • cos HDD1. (31)
Since HD is by construction normal to AC, and the very small
arc DD' is normal to the radius ED, the angle HDD' equals
the angle BDE. And since BE is by construction normal
to BD,
cos HDD'[ = cos BDE] =
Equation (31) becomes, therefore,
BAA. = ED . A® • — = A0 • BD. (32)
ED
Now BD = BC-DC=EC-GQ$ACE-DC. (33)
And since in the triangle ACE,
(AE)2 = (A C)2 + (J^)2 - 2 J. C - EC • cos
(33) may be written
Equation (32) becomes, therefore,
(34)
For this particular case, then, where the tracing point moves
over the very small arc of a circle described about the pole
LENGTH, AREA, ANGLE
57
point, SAA> is r I in terms of the radius of this circle, the
dimensions of the instrument, and the very small angle sub-
tended by the given arc.
lei any figure A'A.V.V i Fig. 30), not inclosing the
cut into a large number of
narrow strips by a series of ,-:
having A' for center. Let these strips be
cut into very small areas by radii drawn
from A'. Thus the entire figure is divided
into a great number of areas, each as small
as \\ If the tracing point
cum- :i the clockwise direction one
of these small area... <//-'. \\ from
-
»-;on of
I*.
And
lion
B,lb is described in the opposite
ill the second di\i this
the cllt
of
ls equal to
— B
' '
«_ „,,-!
Im-m.
1 ' L
i;m
/ -
* ,,,n
J
'
AH, ) = J (o^)(-
nid this last exjiression m- be circular sector
In the mne way .!A<-> - the area of the
ular sector- /.'. - that
area
- area(a'6';?) = area (jab1
AC AC
(35)
the angle ///'/ drawn Mute. If this
:i that the rolb-r then r«>-
in the oj.j,o>;' ( :otati«»n in this
58 PRACTICAL PHYSICS
site direction negative, and making the changes in sign involved
in the new figure, we find that (35) holds whether BDE is
acute or obtuse. That is, when the elementary area ab' is cir-
cumscribed by the tracing point, that area is given by the prod-
uct of the length of the tracer arm and the small distance
through which a point in the circumference of the roller has
rotated.
Fourth, let the tracing point move over the whole figure
KLMN (Fig. 30) in such a way as to traverse the boundary
once in a clockwise direction, and each of the radial lines and
circular arcs twice, once in each direction. By taking lines in
the proper order, this can be done without lifting the tracing
point from the paper. Describing these lines in the manner
indicated amounts to the same thing as going once in the clock-
wise direction around the whole figure ; it also amounts to the
same thing as going once in the clockwise direction around
each of the small areas into which the figure is divided. The
total value of Sx will then be
(KLMN) ,Q<»N
This equation shows that when an area which does not contain
the pole point is circumscribed by the tracing point, the area is
measured ly the product of the length of the tracer arm and the
distance through which a point in the circumference of the roller
has rotated with reference to the tracer arm.
The dimensions of the planimeter are usually so selected that
the product of the length of the tracer arm by the circumfer-
ence of the roller is equal to ten square inches or a hundred
square centimeters. That is, they are so selected that if the
tracing point circumscribes an area of ten square inches or a
hundred square centimeters, as the case may be, the roller
rotates once. The circumference of the roller is then divided
into a hundred equal parts, and these by means of a vernier
(F", Fig. 27) can be read to tenths. The counting wheel B
indicates the whole number of revolutions of the roller.
LENGTH, AREA, ANGLE 59
In the practical use of a planimeter, the figure the area uf
which is desired ma\ he so large that it cannot he c ire u in scribed
without placing the pole point inside it. In this case the area
may he determined as follows: —
If the angle A hi-! ( F I B a right angle, then BD is
zero, and, therefor,-, from (-\'2). 3, _,. t-«|iials /.cro. That is, as .1
moves ahont A' in the circular ar- .1.1 . the
roller slides, without rolling at all. The
generated hy the tracing point A
ahout tin- pole point A' as center when the
roller LOl rotate, and so makes no
record, is called the uzero" or "datum"
In Fig. -".I 1- t //Mlr he this datum CUT-
11 if the tracing point \\eiv to circi: the area
•hat
* _ area (7
OTUSRT— ^n — '
I
nid if now th- j point wer. amioribe the r<
•haded area, then
. shaded area (t/rp
AC
f th.->e two patlis were to be dt- rely, then, hy
dding (87) Mil] :id that
?, haded area
Ii tracing this whole path, the lines US and HT i
• •a.-h direet K.n, so that the r.
nt motion of the i-Mllei- prodiice.l l,y tracing t! ro, and.
! ilice /,'Ml' ^ tlie (hltU! . the rollel' did imt rotate while
WM traced. It • thai if the tracing point liad simply
I the peril : \\ollhl in the end
:e turned l: lion just as much as it did while
;ie more complicated niitline was he ed< That is, if
icing point were to the entire perimeter of the
60 PRACTICAL PHYSICS
figure, the area indicated by the roller would be the area of
that part of the figure outside of the datum circle. If the trac-
ing point were ever to cross the boundary of the datum circle,
the roller would move in opposite directions before and after
crossing. From this it may be shown, if proper attention be
paid to the sign of the roller reading, that whenever the pole
point is inside of the figure circumscribed by the tracing point, the
area actually circumscribed is greater than that indicated by the
roller, the difference being the area of the datum circle.
To sum up, if the tracing point circumscribe in the clockwise
direction any area, the difference between the final and initial
readings of the roller gives the area when the pole point lies out-
side the figure ; when the pole point lies inside the figure, the area
is obtained by adding to this difference the area of the datum
circle.
Equation (36) suggests at once a method of determining
the correction factor of a planimeter. If d denotes the diame-
ter of the roller, and I the length of the tracer arm A C, then
the area which can just be circumscribed by the tracing point
while the roller rotates once is, by (36), equal to Trdl. If the
roller is so graduated that the area indicated for one rotation
is J, the correction factor K is given by
(39)
MANIPULATION AND COMPUTATION. — With a steel scale
and a sharp pencil lay off a rectangular area of not less than 150
sq. cm. ' Make five careful readings of the length and breadth
of the rectangle. If the tracer arm is adjustable in length,
note the reading on its scale. Place the pole point outside
the rectangle, bring the tracing point to one corner, and read
the planimeter. Using the steel scale as a straightedge to
guide the tracing point, circumscribe the rectangle in the
clockwise direction, and again read the planimeter. In this
manner measure the area at least ten times. The product of
the average length and average breadth of the figure divided
LENGTH, AIM \. INGLE til
by tl. ge difference between tin- tinal and initial readings
of the planimetei -rivet ion factor.
With a micrometer caliper determine the diameter of the
roller. Witli the steel scale make live readings of the length
of the tracer arm. From these- calculate the correction factor
' . < >mpaiv the i >tained hy the two methods.
Exp. 6. Correction for Eccentricity in the Mounting of a
Divided Circle
< >i. D THI-:«»I:Y OJ EXFERDOENT. are ut'tcn
measured l>y means of a di\ided circle and an indc nier
atla-'hrd to an arm cat itimi alx.ut an axis passing
tlirou^li the eenter of tin- iethod is suhject to
a source of error due to the mechanical ditliculty of mount-
ing the ;irm carrviiiLT tin- >•• that ;ion
8;i .ill the noun , ided ci:
: inielit If i . iir\e
i >r a divided circle havi: cutrically niou: ier.
B cent,:
< i\ M! /{ the y.ern point-
t »c
t ion ahoiit t: /'. If the
// pAM6i through l>. and It coin, ;
:h (^ tl • in the
i ount ngs
ier.
1 ut in the neither ,.f
t ese conditions is fulfill,, :i be
o itained only
. ;:, : /;.
ILet A° and />' he the observed readings. Diau .1 /;
I- .I//. If there were no eccentricit \ in
i /: illy npjii.xite. the
voiild 1.. i 1 Bf. In othci and Bf
62 PRACTICAL PHYSICS
are the true readings corresponding to the observed readings
A° and B°. Through 0 draw the lines BE arid AF.
Since A1B1 is parallel to AB, and A 0 equals BO,
UCAl = Z CBA =^BAO= ^AOAr
Therefore Z XCAl = \ (Z JTC^ + Z JTOA),
or ^1° = i(^fo + A0).
If the division lines on the circle are numbered as shown
in the figure, E° = B° - 180°. Consequently the corrected
reading of the vernier A is
Af = ±(A°+B°- 180°). (40)
This is the corrected reading for the vernier giving the smaller
reading.
In precisely the same manner, since B^ = ^ (5° + F°) and
since F° = 180° + A°, the corrected reading of the vernier
B is
Bf = % (A° + B° + 180°) . (41)
This is the corrected reading for the vernier giving the larger
reading.
In this manner, by means of two verniers, is obtained the
reading of either vernier corrected for eccentricity of mounting.
MANIPULATION AND COMPUTATION. — Starting with one
vernier near the zero point of the circle, read both verniers.
Then move the verniers about thirty degrees and again read
them both. Repeat at intervals of about thirty degrees until
the entire circumference is traversed. The corrections for the
observed vernier readings are found by subtracting the observed
readings from the corrected readings.
On cross-section paper lay off the observed readings of one
vernier on the axis of abcissas and the corresponding correc-
tions on the axis of ordinates. The curve drawn through the
points thus obtained is the correction curve for this vernier.
From the form of this curve decide whether C and D are coin-
cident, and whether AB passes through D.
CHAPTER IV
VKLOCITV AND A( ( KLKRATION
Exp. 7. Determination of the Change of Speed of a Flywheel
during a Revolution
< > i •..!!•:« T AM. THI:MI:Y Ol K\ I-I:I:IM i:\ : . For many purp
§S when Used to drive hi^h->pecd machinery, it is important that
throughout a ivvolutioii the angular sj-.-i-.l i.f a flywheel shall
.irly constant. The ohject of this experiment is to deter-
mine tin- angular speed and the accch-rat i,.n of a flywheel at
•rent points in its revolution.
i to the shaft of thr llywlu'fl is a brass disk in th«-
of \\hich thir !ot8 of equal width have IMM-II cut and
h«-n lilh-d to the edge with pieces of hard rnhher. If one
• •nniiial of an rlt-.-trie circuit including a »-hn»im^raph he piv»rd
urain>t tli»- rd-_r«' «'t the «lisk. and tin* other against the re\..l\ in^
haft, then during each revolution of the tl\\\heel tlie «-ireuit
hrou^h the .•liroiioLrr;lpil will be made an<l l»rokeu thin
That is. at e\ei-\ In rotation of the flywheel a hivak
•ill he made in ti 1 line <>u the chronograph ilruin. It
tie tl i hrmiu'li «'<11: I '" «'«|M:l' l i'urs' th'1
h.-twiM-n iiMtehrs in the ivconl line will he equal.
vny irregularity "f umtinn will thus h«- rendered apparent.
MAMIM i..\ . 0 COMF1 FATXOK. — IMot a cin-ve with in-
n any select«-d notch and the succeeding
eg as abscissas, and the oonesponding angles ..f rotation as
( rdinato. If the angular q the tlywheel is unitonn.
t i > curve will he | t line.
i line he drawn tangent to this curve at a point
-ponding to any particular time, the speed of the tlywheel
63
64 PRACTICAL PHYSICS
at that instant equals the tangent of the angle between this
tangent line and the axis of abscissas. In this manner compute
the speed of the flywheel at points 20° apart throughout an
entire revolution.
Construct a second curve by plotting times as abscissas and
speeds as ordinates. If a straight line be drawn tangent to this
second curve at a point corresponding to any particular time,
the acceleration of the flywheel at that instant equals the tangent
of the angle between this tangent line and the axis of abscissas.
Construct a third curve by plotting times as abscissas and
accelerations as ordinates. Carefully interpret each curve.
Exp. 8. Determination of the Speed of a Projectile by the
Ballistic Pendulum
OBJECT AND THEORY OF EXPERIMENT. — The object of this
experiment is to determine the speed of a bullet from a rifle.
Newton proved that if two bodies are moving along the same
straight line, the speed of the first with respect to the second
after a collision between the two is directly proportional to the
speed before the collision, the proportionality factor depending
upon the elasticity of the two bodies and being called the coef-
ficient of restitution of the given bodies. He also proved that
if no external forces act upon a system of bodies, the total
momentum of the system is constant.
Imagine that a projectile of mass m and speed u strikes a body
of mass M and speed £7, and that after the impact the speeds
are u' and U' respectively. Then before impact the speed of
the projectile with respect to the other body is (u — £7), and
after impact it is (u' — Ur). It follows, then, from the state-
ments in the preceding paragraph, that
u' - U' = e (u- U) (42)
and mu' + MU' = mu + MU, (43)
where e is the coefficient of restitution of the bodies. If the
bodies are perfectly elastic, e = 1, and if they are perfectly in-
VELOCITY AND AU'KI.KlIATloN
65
elastic, < =0. If the experiment is so arranged that the initial
speed of the lar^e mass is y.ero. and so that after the impaet the
two ma>ses move together, thus aetin;_r like inelastie bodies,
then L '=0 and «' = 0. On making these substitutions in ( l'_!)
and (^4:)) and then eliminating tt' between them, we get
m
The conditions necessary to fulfill the requirements of this
equation are met by the use of the ballistic pendulum. This
t block of wood so suspended that it can
s\vin_ it ( ' as an axis. When a bullet strikes the
pendulum boh. the whole impulse may he used in Lri\ 'iuir t«» the
boh a motion of translation in the direc-
i in which the bullet \\ as moving, or
part of the impulse may be used in p:
duciii'_T torques which tend to set up \\oh-
fchal are not taken into
account in the above equations. If the
bullet strikes at a point called the
of percuwion* these ; pro-
duced, i at a
D the axis of r< equal
to tl ii of tin- equi\
pendulum, and when the masses of the .supporting OOTCJ
small compared with that of the bob, the lower end of this
equivalent si 111) »le pej)( 1 111 UIII is Vcl'V Heal' thecentcrof 1IIU88 of
the 1
lie angle through which the pendulum is delieeted by the
the bullet is denotrd 1 »y r1. the height through \\hicll
mass of tbe pendulum ''-d by //. and the
M-oiu the axis of D t.. the center of pi
v /. then // - ' 080), l»y the ti the bullet has ceased
10. .-minium bob they both have a speed {/',
consequently kinetic energy equal to £ (m + M ) A''-. When
tched, this kiliet \ has all been
66 PRACTICAL PHYSICS
used in lifting them through the distance h ; i.e. in doing work
equal to (m+M^gh.
Consequently J (m + M ) U'2 = (m
Whence Z7' = V2#A = V2^(l-cos(9).
On substituting in (44) this value for Z7', we obtain
cos0). (45)
MANIPULATION AND COMPUTATION. — In setting up the
apparatus see that the line of flight of the bullet is horizontal,
that it is perpendicular to the axis of rotation of the pendulum,
and that it passes through the center of percussion of the
pendulum. Weigh the wooden plug in .the center of the pen-
dulum bob both before and after the bullet is fired into it.
Weigh the rest of the bob, measure Z, and observe 6.
Exp. 9. The Acceleration Due to Gravity by Means of a
Simple Pendulum
OBJECT AND THEORY OF EXPERIMENT. — The object of this
experiment is, from measurements of the length and time of
oscillation of a simple pendulum, to find the value of the ac-
celeration due to gravity.
In elementary text-books on General Physics it is shown
that the period of a complete to-and-fro vibration of a simple
pendulum of length I vibrating through a small arc at a place
where the acceleration due to gravity is <?, is very nearly
9
Whence ff = (^- (46)
If the length of the pendulum is about 100 cm. and the
amplitude of vibration about 3 cm., the value that (46) gives
for^is about 0.01% too small. This error is so slight that in
VKUH'ITV AND ACCELERATION
67
the above equations the approximation si^n is omitted. More-
) is deduced on the assumption that tlie peiululuiii lias
its mass concent rated at a point on the end ot' a perfectly
flexible suspension. An increase either in the size of the bob
or in the mass of the suspending wire increases the error intro-
duced by usiiiL,' the above equation.
If the length of the pendulum is
taken as the distance from the
Supporting knife-ed-^e to the cen-
ter of mass of the bob. and if this
distant- is about 1MM cm., ami the
diameter of the bob about 3 cm.,
the value found for // is about
"."1 >o small. With the same
•minium, if the mas*
of the supporting \\iiv is about 0.8
if. and the mass of the bob about
value found for// is about
fo too large.
M AMI'I I. \ I !"N \\I» Cn.MI'l
. - In tindi:
.itjoll of t!ie expel illlelltal p,-M-
dulum by tl. »d of coinci-
dences, the time of roinridelle,
be observed by the electric method
described on p. 1«>. ,,r by the opti-
eal method used by I
In the apparat B
method ( \-\-_r. ;}{ ). tJH. cxi.erimen-
tal pendulum is suspend. -d di:
n froiit of a clock pendulum.
iidulums is
i screen C containing . Attach- d to the bob of
.3 clock pendulum is a small mirror which produces an image
"ii of the lil.unent of an incandescent lamp pla>
of the two pendulums. This ima^e is viewed with a
68 PRACTICAL PHYSICS
telescope placed a meter or more from the clock. When the
axis of the telescope, the points of support of the two pendu-
lums, arid the slit in the screen C are all in a plane perpendicular
to that in which the clock pendulum swings, a flash will be
seen in the telescope — if the lamp and mirror are properly ad-
justed — every time the clock pendulum passes its lowest point
except when the two pendulums are in coincidence.
To make this adjustment, set the experimental pendulum
swinging, and while looking through the slit in the direction
perpendicular to the screen, move the incandescent lamp until
a bright line of light fills the slit every time the clock pendulum
passes. Then place the telescope a meter or more from the
slit and in such a position that when the experimental pendulum
is deflected a bright flash is seen every time the clock pendulum
passes the slit, but when the experimental pendulum hangs at
rest no flash is seen.
If the slit in the screen 0 is too narrow — especially when
the periods of the two pendulums are almost the same — the
eclipse will last for several seconds. In this event, the time of
coincidence is the mean of the time of the beginning and the
time of the end of the eclipse. If the slit is too wide, no
eclipse will be seen, but only a dimming of the flash. Unless
the alignment of the two pendulums is better than is usually
attained, two eclipses will be observed within a few seconds of
each other, one on even-numbered seconds and the other on odd-
numbered seconds. The average time of these two eclipses is
very close to the true time at which the coincidence occurred.
Since the coincidences which occur when the two pendulums
are moving in opposite directions are more sharply marked
than those which occur when the two pendulums are moving
in the same direction, and since these two types of coincidence
alternate with each other, it is usually better to observe only
the coincidences when the pendulums are moving in opposite
directions, and pay no attention to the others.
When the apparatus is in adjustment, note the times at which
a series of coincidences occur. When the pendulums are swing-
VELOCITY AND ACCELERATION 69
ing in opposite directions, and it is seen that a coincidence will
9OOO OOCOT, HI >t,- the hour and minute and begin counting
is. Keeping both eyes open, put one eye at the teleseope
and hy the Hashes of light that are seen keep on counting
86COI] "rd the hour, minute, and second every time that
;•. liepeat about live limes.
The calculation of the period is explained on p. 4<>. In mak-
ing this ealeulatinn it is necessary to know whieh pendulum
goes faster. T rmine this, watch Itoth peiuliilums for a
fe\v moments imined. _;• a COilieideliee. It will
sun n be evident that one of them reaches the end of its path
re the other, and is, therefore, the one \\ faster.
To determine the length of the pendulum, mount a eathe-
tomrter in front of : rimmta! pendulum, make the ad-
justn. 'ii p. --. vad the positions of the
p of the bob. and the hotloin of the bob. For
ii of the pendulum use th«- >iii the knife
t the bob. Make at least two determi-
•is — each time read :neter — and take
the n,
Exp. 10. Determination of the Acceleration Due to Gravity
with a Compound Pendulum
M» Til! . The method
.Led in the ,ng experiment for determining tin-
acceleration d lires a clock h person
ii to p.-rf-.nn the experiment. If >. id.-nts are to
•• Tiiiinatioii.x at the same time, this \\oiild insnlve CX-
ve cost of apparatus. In the following method only one
clock i-s needed, and, in-t.-ad of tl.i»hes of light aj.pearing at
;-t when the pendulums are nearly in coinci-
bab ftppean only when the pendulums are nearly in
coincidence. The obj.-.-t of t : inn-lit U t .. determine t lie
hie to gravi; .-ompound pendulum.
70
PRACTICAL PHYSICS
Let A be an axis about which a body B is free to swing, and
0 be the center of mass of B. If B is swinging back and forth
about A) at some instant the line AC makes
with its equilibrium position an angle 0, and
if p denotes the distance from A to (7, and M
denotes the mass of B, then the torque tend-
ing to restore B to its equilibrium position is
L = — Mgp sin 0,
the negative sign being used because the
torque and the displacement are in opposite
directions. If K denotes the moment of
inertia of B about the axis A, and a denotes
the angular acceleration with which B swings
through the indicated position, then we know that
It follows that
Ka. = — Mpg sin 0,
or, if 0 is small (see p. 7),
Ka = - Mpgd. (47)
Since a is proportional to 0, the motion is simple harmonic.
If T denotes the period of a complete to-and-fro vibration of a
body which is vibrating with simple harmonic motion, it is
shown in elementary dynamics that
FIG. 35.
From (47) it follows, then, that
(48)
Now the moment of inertia of B is the sum of the moments
of inertia K{ of the n different parts of B. That is,
(49)
And if the masses of the different parts are Mj, if the centers of
mass of these various parts are at distances PJ from the axis of
VELOCITY AND ACCELERATION
71
ion, and if the lines PJ are all parallel to p, then from the
definition of center of mass we know that
= 2-'
(50)
^L ^
I t = !•->,
*V<r
>=i-»
On substituting in (4H) the values of A' and Mp from ( 41M and
(50), and writing in place of T its value '1 t, where t denotes the
time taken by a single oscillation, we get
nee,
pendulum to be us. sts (Fig.
of a stout piece of steel shaft /'. \hieh car-
it its upper end an adjustable collar to
which are fixed th«- knife edges on which the
[M-ndidum swings. I1
ill'_r l»cl<)W the Ihittnin nf |
rod is a liglit vertical
plate C', in the middle of
which is a vertical slit. This
pendulum swing- it of
in incandescent lamp I>,
light from which can be seen
Dnly through ;< -al slit
n ;t >n jacket abniu
\\ith the clock pcndiiliiii
• telegraph sounder which
- mounted with its anna-
tOTC tl. Connected
with the armature is a light shutter B, in which
a \
72 PRACTICAL PHYSICS
slit. When the sounder is actuated, this shutter moves just
far enough to bring its slit in line with another slit in a screen
A, mounted close to it on the base of the sounder.
When the pendulum is at rest, and the slits arid the filament
of the lamp are all in line, a person looking through the screen
on the sounder sees a flash each second. When the pendulum
is swinging, no flash is seen unless the pendulum passes through
its position of equilibrium at the same instant that the current
from the clock passes through the sounder.
Since a seconds pendulum of the type described above would
be about a meter and a half long — that is, too long to deter-
mine its length conveniently with a cathetoineter — the pendu-
lum used is a half-seconds pendulum. During the interval
between coincidences, then, the experimental pendulum makes
one swing more or less than twice the number made by the
seconds pendulum. .That is, if the number of clicks between
coincidences is w, the number of oscillations made in the same
time by the experimental pendulum is 2 n ± 1, and its time of
oscillation is, therefore, — •- — sec. In general, if n clicks occur
during the interval from any coincidence to the ,/th following
coincidence, the experimental pendulum makes j swings more
or less than twice the number made by the seconds pendu-
lum, and the time of oscillation of the experimental pendulum is
n
:sec.
2n±>
When the clicks of the sounder are counted, it will often be
observed that there is a series of flashes on odd-numbered
clicks, and then very soon a series on even-numbered clicks,
after a time a series on odd clicks, and then very soon a series
on even clicks, and so on. If the apparatus were more accu-
rately adjusted, the odd arid even sets of clicks would come
together instead of one after the other. The interval between
coincidences is, therefore, the interval from the middle of one
series of odd clicks to the middle of the following series of odd
clicks, or from the middle of one series of even clicks to the
VELOCITY AND ACCELERATION 73
middle of the following series of even clicks, or from the middle
of the time hetween a - odd and a - even clicks
to the middle of the time hetween the following two series.
MANIPULATION AM> COMPUTATION. — Take the diameter of
the pendulum rod once with a micrometer caliper. Then see
that the pendulum is at rest with room enough to swing freelv,
and with the knife • rpendicular to the wall and the plate
parallel to the wall. \Vith a calhettuneter (p. -o ) determine
the heights of the top of the rod, the kn:: . the hottom
of the rod. and the bottom of the plate. Make two sets of
determinate
Adjust the position of the lamp and its jaeket until the glow-
: filament in the lamp through the pendulum slit
when I v in front of the pendulum. Place the
i. or :Jn em. in front of the pendulum, and in
•inch a po>ition that when the armatuiv is held in its po>iti«»u
nearest the magnet, tin- filament in the lamp and the slits in
the jacket, pendulum, and sounder diaphragms are all in line.
Connect the sounder with the clock, and with the eye close to
:he shutter watch to see if a flash occurs every time the sounder
jlicks. Thru >••! the pendulum > with an amplitude
lot much e 1 » in., and with the eye again close to the
;hutter watch fur the flashc- dicate coincidences.
curs, begin OOUnting the click* of the sounder
ind record the number of each click on which a flash is seen;
>r, if this is too di!Vi< . ,ut the alternate < the
vound- : . i the in; cadi counted click on which a
lash is seen, and pay no ;r which occur on
;licks that are not counted. In one of these wavs make at
ea> >ets of obsei inning through the
in >ur or eight series of flashes. If all clicks are
jounted, the interval hetween c.iineidf!. i be obtained l»y
t'ndini: the interval from the middle of the tirst seriefl "f Hashes
the middle of the third eerie*, 01 the interval from the
uiddle of the second series to the middle of the fourth, or,
interval*. If only alternate
74 PRACTICAL PHYSICS
clicks are counted, the second and fourth series of flashes are
not recorded, but the interval from the middle of the first
series to the middle of the third series that was observed gives
twice the time between coincidences, and the interval from the
middle of the second observed series to the middle of the fourth
observed series also gives twice the time between coincidences.
It is to be noted that, if only the alternate clicks are counted,
the value for the interval should be doubled to reduce it to
seconds.
Whether the pendulum is swinging faster or slower than a
half-seconds pendulum may be determined by watching it for
a few moments at about the time when the clicks of the sounder
occur when the pendulum is at one end of its path.
The apparatus can be so designed that the moment of inertia
and mass of the collar may be neglected in comparison with
those of the rod, and, further, so that the moment of inertia of
the plate about an axis through its center of mass is neglible in
comparison with the moment of inertia of the rod ; that is, so
that the radius of gyration of the plate may be assumed to be
the distance from its center of mass to the axis of rotation.
The masses and moments of inertia to be taken into account
are, then, those of the rod and the plate. The masses will be
given by an instructor, and the moments of inertia are to be
calculated by the use of (6) and (2) on p. 111. These values
will complete the data necessary for the calculation of g by
means of (51).
CHAPTER V
FRICTION
IF a body resting on a plane surface is acted upon by a force
parallel to tin- surface, the l»ody dors not start to move until
force has reached a certain definite value. Moreover, the
/' which is necessarv to start the hotly is directly propor-
! to the force Fn which presses the two surfaces together.
That is, Ft> = ft /'t, in which the constant /z is called the <WfrV/f nt
rf std -he l>ody d.»es start to move, the force
which is required to keep it moving uniformly is somewhat less
the fore.- that is needed to start it. And this fur.
vvhid. ssary to keep the body moving uniformly is also
lirectly proportional t<> the force Fn which presses the two sur-
• th«T. That . /' /'. M which the constant b is
''///. Since /';, is greater than
Fpi /x i> than b.
Exp. 11. Determination of the Coefficient of Friction between
Two Plane Surfaces
AM. Tm .- BXPBRIMI I : t of
:iiiiic the cortliciriit of friction
plane surf.i
appara; of u hnri/untal plate having a small
•ul Icy fastened at one end, and a hloek that ran lie- drawn along
plate by means of a cord passing over the
Mine the pulley possess n, the weights on the cord do
n-s.-nt • •• re.juired to overcome the
and the l.loek. On this account the
76
76
PRACTICAL PHYSICS
force Fl (Fig. 37) is greater than the force Fp. The differ-
ence between these two forces (J^ — Fp) is a force fp which
has to be applied along the circumference of the pulley in order
to start it. But this fp is proportional to the force /2, — the
resultant of the two forces, F-± and Fp, — which presses the pulley
against its bearings. That is, /p = At2/2, where ft2, although not
the coefficient of static friction between the pulley and its bear-
ings, is a quantity proportional to that coefficient.
From Fig. 37,
Whence,
(52)
Since Fp is less than F^ the numerator of the quantity in
square brackets is less than the denominator. Since /*2 cannot
FIG. 37.
be negative, this means that the negative sign before the radical
is to be chosen.
P2 may be determined by passing the cord over the pulley as
in Fig. 38, applying to one end a weight IF, and finding what
weight, Jf, is required at the other end to start the pulley.
Then
where A is written for X— IF and B for X+ W.
tuting this value for /x2 in (52), it becomes
.-4
On substi-
(53)
FRICTION 77
the quantity in square brackets is the same for all values
of Fp and Fr it can he denoted by the single letter k and (53)
written in the abbreviated form
The coefficient of static friction between the two surfact
then, = = V
Wlu-n tin- OOeffitient of kinetic friction is to be determined,
:nust be modified by usinir in place of /'. /-'r A. and 7A
.uantities /'; . /\'. .-I', and li\ \\here tlie primed symbols
the same meanings as tin- nnprimed, except that the
primed arc taken when the bodies are moving uniformly instead
I hen they ar-
M \\IIM LATIOB AM. COMPUTATION.— After cleaning the
block and the surface of tin- plate and making the plate hori-
i with the aid of a Spirit level, place the block near one
•nd and add K to tlje j>an until the block on beiiiLT started
keeps in uniform motion. Make not less than live dctermina-
:ions with dillereiit weights on the block. Carefully clean the
plate and block before ea< -h observation. For each case cah-u-
be coefficient of kinetic friction. Having thus determined
;he u< Fp necessary to keep the block in uniform
:i \\h.-n it is pressed against the plate with various I
f'4. plot a curve showing the relation n ih.- t\\o. This
should be very nearly a straight line. and. if the normal
/" . are plotted as abscissas, (54) shows that the tangent
• f the slope gives the coeffi( ; ni.tion. Determine the
i the slope and see how the result checks with the
nean of the previous results.
Exp. 12. The* Friction of a Belt on a Pulley
Orui:< T UTD THXOBI 01 I:\I-I:I:IMI The object of
. ini'-nt is to determine th«- c«.,-ilicicnts of static and
vin- It and a pulley.
78 PRACTICAL PHYSICS
Let EG-HJ represent the portion of the belt in contact with
the pulley whose center is O. On account of the friction be-
tween the two surfaces, the tension
of the belt will vary all along the
length in contact with the pulley.
When the belt is just on the point
of slipping, let the tensions at the
ends of the arc CrH subtending the
indefinitely small angle A® be de-
noted by / and /'. Let F and I"
represent the tensions of the belt
FIQ 39 where it leaves the pulley.
By compounding the forces/and
fr — which are approximately equal because A® is very small
— it is found that the normal force against the pulley due to
the element of the belt Grffis equal to
'—
Therefore when the belt and pulley are in equilibrium, and the
belt is just on the point of slipping, the coefficient of static
friction is f — f
u,=J - J-.
' /'A®
Whence, £- = 1 - AtA®,
or log,/- log,/' = loge (1 - M@).
Expanding into a series the right side of this equation,
loge/- loge/ = - M® - J (/.A®)2 - 1 OA®)3 - etc.
But when A® is chosen very small, its second and higher pow-
ers become negligible in comparison with its first power, and
we may write , ,,.
If we write expressions like the above ^or all the elementary
arcs that the belt touches, and then write the sum of the left
members equal to the sum of the right members, we get
FRICTION 79
Whence, M - lofr *" ~ lofr *. (55)
in which the angle B is measured in radians.
The reason for not using the approximation sign in the last
• •qnatiuns is this: Approximations have been made in
two places, and in each one the approximate value approaches
the true value when AB approaches zero. That is, in the
limit, when AB = last two equations hold exactly.
In precisely the same manner is obtained the coefficient of
kinetic friction
b= °"f , (56)
6 -Fund /' : -iic tensions of the belt where it leaves the
pulley, when the belt is slipping at a uniform rate.
M ANIIM i. \ D COMPUTATION. — Stretch the belt over
t pulley that can be rotated l»y means of a crank. To one end
»f the 1). It apply a 10 Ib. weight and to the other end a verti-
•ally hanging balance whose lower end is fastened to
the floor. Now turn the < rank so as to cany the belt away
the spring balance until the belt is just on the point of
The sj.riiiLT balance reading plus the uei^ht of the
10 Ibs. weight, and e = 180° = 7r ra-
Consequently a value of p can be computed. Repeat,
Making / -sively equal to 20, 30, etc., pounds weight,
mtil the lim;- nee is reached. The mean of
- of ^ thus obtained is to be taken as the coefficient
:ri.-tion between the belt and the pulley. Determine
! for both the flesh side and hair side of the belt.
When the pulley is rotated until the belt slips and then the
!>t constant, the spring balance reading is
", as before, F equals the Wright acting «>n the other
>nd of the belt, and H equals TT radians. From these \&\\n^ /
computed.
80
PRACTICAL PHYSICS
Exp. 13. Determination of the Coefficient of Friction between
a Lubricated Journal and its Bearings
(GOLDEN'S METHOD)
OBJECT AND THEORY OF EXPERIMENT. — The object of this
experiment is to determine the coefficient of kinetic friction
between a cylindrical jour-
nal and its bearings for dif-
ferent loads, speeds, and
temperatures. The Golden
Bearing and Oil Testing
Dynamometer consists of a
spindle passing through a
bearing B (Fig. 40), form-
ing part of a yoke (7. The
spindle can be rotated at
various speeds by means of
a motor, and the yoke can
be weighted by means of
adjustable masses M' and
M" . As the spindle is ro-
tated the friction between
it and the bearing tends to
rotate the yoke also. This tendency to turn is measured by
the spring dynamometer D, which is essentially a spring
balance. For the testing of oils at different temperatures, the
collar A is cored in such a way that a stream of water or steam
can be passed through it and the temperature of the bearing
determined by means of a thermometer T.
If r be the radius of the shaft and F the total force of fric-
tion tangential to the surface of the shaft, the turning moment
resulting from the friction of the shaft and bearing is Fr. If
/ represents the force, having a lever arm Z, required to keep
the yoke from turning (Fig. 41), the resisting torque is/7. If
the center of mass of the yoke with its appendages is vertically
FIG. 40.
FRICTION
below the axis of rotation of the shaft, then when the shaft is
rotating and the yoke is held steady,
/•> -/I,
If the total weight on the bearing snrfaee due to the yoke
and : with the masses M' and M" he
denoted by /\ then \. iellt of
kinetie friction
If the surface of mntai-t between
the journal and bearing be projr
upon a hori/.ontal ]>lane. and if the
ihis projection be d< -noted
then the pressure on tin- hearing is
A
\\I1TI.A TI"\ AM> ( 'o.MlM I \ I IM\. \ '. ,nd
with calipers and sea!--. Kind tin- aiva .1 of the projection,
• ii a hori/.oiit.il j ,\cni the
onrnal and hearing.
journals and bearing \\ith beii/.ine, Inbri-
ate with the assigned oil and apply small and nearly ojiial
ike. l be dii'tVi-enee be-
\voloads should be sutVn-i.-nt t«> develop a tnni-
momenf due to gravity slight 1\ t than that due
o friction. Start the motor and : EU of the .spring
lynanioineter D measure the \- of the yoke to turn.
Averse t: tion o! is and take another dvna-
^>me- iing. By this operation the pull developed by
irtion between the n^ is. first added to
hf pull on the dyiiai: i on one
82 PRACTICAL PHYSICS
end of the yoke, and then subtracted from it. The difference
between the two dynamometer readings is 2/. The data are
now at hand for computing the coefficient of kinetic friction
between the given surfaces lubricated by the assigned oil, for
the particular speed, temperature, and pressure per unit area
of bearing surface, used in this determination.
Proceeding as above described and keeping the temperature
constant, determine, for a fixed speed of rotation s, the values of
b corresponding to a series of values of p. With these values
plot a curve coordinating b and p for the given speed and tem-
perature. Keeping the temperature fixed, now change the
speed of rotation and determine a new series of values of b and
p. Plot a curve coordinating these values on the same sheet
with the other. Proceeding thus, plot on the same sheet about
five curves for the same temperature but different speeds.
In the same manner obtain data for and construct on an-
other sheet of coordinate paper a series of curves showing the
relation between b and p for a given constant speed when the
temperature t is changed.
From the first set of curves construct another set coordinat-
ing b and s for different fixed values of p when the temperature
is constant. And from the second set of curves construct
another set coordinating b and t for different fixed values of p
when the speed is constant.
Care should be exercised that the direction of rotation of the
journal is frequently reversed, especially when the bearing is
heavily loaded, so as to avoid error due to inequality of the
wearing of the bearing.
Exp. 14. Determination of the Coefficient of Friction between
a Lubricated Journal and its Bearings
(THURSTON'S METHOD)
OBJECT AND THEORY OF EXPERIMENT. — The object of this
experiment is to compare the lubricating properties of different
FRICTION
oils from their relative eft'ect in reducing the friction between a
journal ami its hearings. The Thurston Oil Testing Machine
to be used in this experiment consists of a heavy pendulum
having at one end a bearing through which passes a horizontal
shaft capable of rotation. The bearings can be caused to exert
jiven pressure on the journal by means of a heavy coiled
spring and adjust in_ . forming part of the pendulum.
When the shaft is rotated, the pendulum is deflected through an
angle determined 1>\ the moment of the tangential effort at the
imference i.f tin- journal and the moment of the weight of
he pendulum.
• •sent the weight of the pendulum ; »9, the expan-
t'orce of the sprii : •/. the mean normal foive bet
ournal and bearings; A', th- 06 from the axis of the
"imial to th- miss of the pendulum : /«', the tangen-
:Tort at • be journal numerically
(jiial t<> the ; friction; r. the radiu- of the journal; /,
1 > length of the journal; and A. the coellicient of kinetic fric-
;on 1 -he journal and its bearings.
If the pendulu; jiiilibrium when dellected through an
84 PRACTICAL PHYSICS
angle 0, the couple due to the forces FF at the circumference
of the journal equals the moment of the weight W. That is,
2 Fr = WR sin 0.
Since the upper bearing exerts on the journal a force equal
to the sum of the weight of the pendulum and the expansive
force of the spring, while the lower bearing exerts a force due
only to the spring, the mean force between journal and bearing
is
j_(W+S) + S_2S+ W
Consequently the coefficient of kinetic friction between the
journal and bearings is
WR rjsin*. (58)
or b — k sin 0, (59)
where k represents the constant coefficient of sin 0 in (58).
This constant can be determined from a single series of care-
fully made measurements and used in any computation of 5, so
long as the force exerted by the spring is unchanged.
MANIPULATION AND COMPUTATION. — Measure the diameter
2 r of the journal with a pair of calipers. Obtain the weight TFof
the pendulum. Observe the angle 6 on the divided arc attached
to the apparatus.
Place the coiled spring in a testing machine and measure the
forces required to produce given compressions. Plot a curve
coordinating forces and resulting compressions. From this
curve may be read off directly the force S corresponding to any
compression measured by a scale and vernier attached to the
side of the pendulum.
The distance R from the axis of the journal to the center of
mass of the pendulum can be determined as follows : While the
pendulum is still suspended from the shaft, support the free end
on a knife edge resting on the platform of a balance. (See
Fig. 43.) The product of the weight observed and the horizontal
FRICTION Sr>
• listunce L between the supporting knife edge and the center
of the shaft equals II' A*.
Compute the pressure un the journal. Since the projection
«>f the journal surface equals -(-/•)/. the pressure is
P-TTr
After Cleaning the journal and bearings with ben/.ine. lubrieate
with one of the oils t«. !>•• trstrd. apply a ^i\rii force S to the
spring, set the shaft into motion, and observe the deflection 6
of the pendulum.
With the speed of rotation kept constant observe the deflec-
tions produced fur several values of the force S. Repeat the
series of observations when the other oils are used. Care-
fully clean the journal and bearings after each sample is
examined.
i >pecimen a cur linatin«_r the coefficient of
kind. force per unit a rea of bearing .surface p.
rou^ht out by ; aid be carefully
CHAPTER VI
MASS, DENSITY, SPECIFIC GRAVITY
Exp. 15. Calibration of a Set of Standard Masses
OBJECT AND THEORY OF EXPERIMENT. — In making a set of
standard masses it is impossible to get the mass of each piece
exactly right. Moreover, the handling of a set of masses, even
when carefully done with forceps, necessarily wears them a trifle.
In addition, some dust settles upon them, there may be chemical
action with vapors in the air, etc. It is consequently necessary
in accurate work to compare the masses with each other ; and
where absolute weighings are to be made, the masses in the set
must be more or less indirectly compared with the ultimate stand-
ard. The object of this experiment is to compare the masses
of the various members of a set and to construct a table of
corrections.
The method employed is by means of a sensitive balance to
find the differences between masses or groups of masses supposed
to be equal, from these results to form as many separate equa-
tions as there have been weighings performed, and from these
equations to find the masses of the different pieces in terms of
some convenient unit. In this experiment the unit of compari-
son will be the mass of one of the standards in the set being
calibrated.
Consider a set consisting of a 10-mg. rider, eight aluminium
or platinum masses ranging from 10 mg. to 500 mg., and nine
brass masses ranging from 1 g. to 100 g. Call the mass of the
rider r, the masses of the fractional gram pieces respectively
lOj, 20X, 202, 50j, lOOp 200X, 2002, 500X, and the masses of the
brass pieces respectively lx, 2X, 22, 5r lOj, 20V 202, 50V ioor
86
MASS, DI NSHV, SPECIFIC GRAVITY 87
First the position <>f rest is determined (a) when the balance is
unloaded and (?>) when r is at the 1-inark on the beam. These
observations give with sullieient aceuraev the sensitiveness of
the balance for the first few small loads. 1'^ is nu\\ placed on
one pan and r either on the other pan or at the 10. mark on
the other side of the beam. From the position of rest deter-
mined under these eonditiuns, the position of rest of the unloaded
balanee. and the I of the balanee the value of \(\ in
terms of r can be calculated : —
1 = , = ra +
where the al is a small number which may be either positive or
."j is now placed on one pan. Wl on the other, and r either
on tin- pan with 10j or at the 10-mark on the beam. As in the
ne of 20j can be determined in terms of r :
>r, substituting for 10j its value from the preceding equation,
= r( '2 4- rtj 4- a,).
In the same way is obtained
20, = 10! 4- r 4-<V*
= ; + «s).
Throughout the re*t of the <-alil. ration the rider is kept on
he brain, and the sensit i\ eness is determined f.»r each load.
A'ith .md :!<•,. -Jo.,, and !<>j in the other, the po-
ition .ind the seiisitiven.-ss are deti-rmint-d. '1'liis gives
lie value of 50j : —
= 20j 4- 20, 4- 10j 4- a4r,
•r substituting for 10r 20j, and 20, their values from the above
^nations,
50j = r(5 4- 3 a! 4- «a 4- a8 +
In the same way
)j = -iOj 4- 20j 4- 202 4- 10t 4- abr
4- - <t.2 4- 2a8 4- «4 4- 06)»
88 PRACTICAL PHYSICS
50j + 20X + 202 + 10X + a6r
= r(20 -f- terms involving ax . . . «6).
2002 = 100X 4- b01 4- 20X 4- 202 + 10X 4 a7r
= r (20 4- terms involving a^ . . . #7).
= 200X + 2002 4 100X 4 a8r
= r(50 + terms involving ax
+ 2002 + lOOj + agr
= r(100 + terms involving a^ . .
21 = 1 1 4- 500X + 200X + 2002 + lOOj +
= r(200 + terms involving a^ . . . 1Q
22 =5 it 4- 500X 4- 200j + 2002 4 100X 4 anr
= r(200 H- terms involving ax . . . «n).
etc.
In the above equation the a's are all experimentally observed,
so that if the mass of any one of the pieces in the set is known
in terms of the ultimate standard, then from the equation in-
volving the mass of that piece can be calculated the mass of the
rider. When the mass of the rider is known, the masses of all the
pieces in the set can be calculated from the respective equations.
MANIPULATION AND COMPUTATION. — Perform the opera-
tions indicated above. Make all weighings by the method of
vibrations, and with the brass pieces use the method of double
eighing. Assuming that the 100 g. mass is correct, determine
he masses of all the other pieces in terms of it. Record
fie results in a three-column table, putting in the first column
the symbol used to denote the particular mass considered, in the
second the value obtained for this mass, and in the third the
correction for the mass as obtained from (2).
DENSITY AND SPECIFIC GRAVITY
If a body of mass m occupies a volume v, then the average
density of the body is given by
»-" (61)
MASS, ni'.NSITY, SPECIFIC GRAVITY 89
From this expression it is seen that the number which ex-
presses a density depends upon the units in terms of which the
and volume are measured. For example, at 4°C\ the
density of lead is about 708 pounds per cubic foot, or 2868
grains per cubic inch, or 11. -'U grains per cubic centimeter.
Sinee deli quantity, the units in terms of \vhieh
the mass and volume of the body are measured must always
be sta: . Since must budies change their volume somewhat
with ehanges of temperature, the density of a substance depends
upon its temperature : ami so in accurate work the tempera-
ture at which a determination is made should alwa\s he stated.
'1 lie specific gravity or relative density of a substance is the
ratio of its density to the d.-nMty of some standard substance.
In other words, the specific gravity of a body is the ra:
ass to that of an equal volume of a standard substance.
gravity is thus a nuiin-rie.il ratio, an ah.Mraet number
which is indepeinh-nt of the units employed. For solids and
li<|ui<: at the temperature of its maximum densitx
i I C. Of ;''-r F. ) i> arl.it r .en as the standard sul»>iam ••.
Sine.- in the C.G.8. system of units the unit of mass is the
mass of a unit volume of water at the temperature of its maxi-
mum .it fulluws that t he density of a b rrams per
centimeter is n dly equal to its speci:
Exp 16. Determination of the Density of a Solid by Measure-
ment and Weighing
OBJECT AND THEORY <> 81) it will
be seen that tin- <h-n>ity of any >"lid c.,nld readily be determined
if a speci it cuiihl be obtained in a shape such that its
Volume could easily be compi;
M\NIITI.\ • PUTATIO -The specimen to be
• •y Under. Measure its diameter with a micrometer cali-
per and it> length with a \ • •••• pp. 1 T. L'l ) and cal-
culate the vuliime. I), i. rmine the mass by wei^hin^. u>in^ the
method nf vibrations (pp. ii»J '2*). In order to gut a very
90
PRACTICAL PHYSICS
accurate value for the density it will be necessary to correct
the weighing by allowing for the buoyancy of the air. First,
without making this correction, divide the apparent mass by
the volume and so get an approximate value for the density.
Use this value in (14) to get the true mass of the cylinder.
Calculate the density by (61).
Exp. 17. Determination of the Density and Specific Gravity of
a Liquid with a Pyknometer
OBJECT AND THEORY OF EXPERIMENT. — The pyknometer
is essentially a small glass vessel of definite volume. Various
FIG. 44.
FIG. 45.
FIG. 46.
FIG. 47.
forms suitable for determining the densities of liquid are given
in Figs. 44-48.
The pyknometers in Figs. 45 and 48 can be used only for
liquids, while the others can be used for either liquids or solids.
The most common form, that shown in Fig. 46, consists of a
MASS DENSITY, SPKCIFK' GRAVITY
91
small bottle tilted with a perforated glass stopper that always
comes accurately to a seat at the same point, so that the volume
of the bottle is definite when the stopper is in place. This form
is often call- ity hot tie.
The yolume of the pyknometer is obtained
i two weiijhin^s, first when empty, and
second when filled with a liquid of known
density, »-.//. water. If the mass of y\
contained in the filled pyknometer is denoted
by M,r and its density by p... then the vol-
Fit; 4S
let the water be replaced by the sp-
If the :: bhli second liquid tilling the pyknometer
meii.
l>e d- ]/ ;:d its density by p^ then
XT. M.p.
'• = TS .»/,
.mum density of water by £, we have for
ific gravity of the sp.
(68)
_r equations no account has b. on of the
Jt it HP-sphere on the liquids heili'_r Weighed
nd on the standard masses used in tl In pi
- this .soiirc. d. The
: MIC weight of an object eqt .'-rbl plus the
'eight of air . 1 when the balance is in
• luilihrium, th«- aj»p.: -ly equals theapj»ar-
i.ird masses. So that when the .specimen
; in air, its true weight minus it.s loss of wi-i-^lit due to
buoyancy of the air equals the true \\ei^ht of the standard
.a&8e> :ht. If the den>ity of air is de-
i Dted by pt and I I the standard masses by pb, this
92 PRACTICAL PHYSICS
last statement says that when the pyknometer was filled with
the first liquid,
and when the pyknometer was filled with the second liquid,
On eliminating v from these equations, we obtain
ft = -+,.. (64)
JXL
W
MANIPULATION AND COMPUTATION. — Weighing by the
method of vibrations, determine first the mass Mp of the empty
pyknometer; second, the mass (MP + MW') of the pyknometer
filled with recently distilled water ; and, third, the mass
(Mp + Mi) of the pyknometer filled with the liquid in question.
Take the values of pw and pa from tables.
Each time before filling the pyknometer, clean it by rinsing
successively with nitric acid, distilled water, and alcohol, and,
then dry it by putting into it the end of a tube connected to an
exhaust pump. Be sure that there are no air bubbles in the
pyknometer, that the outside is dry, that the stopper is in place,
and that the liquid fills the capillary tube in the stopper. In
order to avoid changes in volume due to changes in tempera-
ture, avoid touching the filled bottle with the bare hand.
Exp. 18. Determination of the Density and Specific Gravity
of a Solid with a Pyknometer
OBJECT AND THEORY OF EXPERIMENT. — The object of this
experiment is to determine the density and the specific gravity
of a solid in small pieces.
Three suitable forms of pyknometer have already been illus-
trated (Figs. 44, 46, 47). To determine the volume of a solid
by means of a pyknometer, four weighings are made : first,
when the pyknometer is empty ; second, after the specimen
M \- 3ITY, SPECIFIC UHAV1TV
93
introduced: third, after the rest of the space in the
py kilometer has been tilled with water or other liquid; and,
fourth, after the pyknoineter has Keen emptied and then filled
with the same liquid used in the third wei^hin^.
the mass c.f the pyknnmeter he denoted 1>\ Mf>. that of the
specimen liy J/«. that of the water which tills the pyknometer
1>\ M , and that <>f the water which was usrd with the speci-
men l»y m,r. Then if the : ind in the /ah weighing1
is denoted 1)\ M .
M{ - .V .
MI = Vp + M*
«
MI = Mp 4- Mw. ( " s i
e ma.xs ..f tli.- \\ater which displaces the specimen is
,. and from .'MI and <•
values of J/;, and < .)/ • M »
T i TUf \ f TUf i TLf \
'a i ^«4>) — v-"*i i -"*g/«
denotes the den- u>ed. the \..lnmc nf the
which displaces tlie specimen, and therefore the volume
if t;
the density of t: .men is
p>_JS_it_*i.
*\ eliminating r fmm tlie last two equations, we get
^ - • '' " -ft^-
(69)
94 PRACTICAL PHYSICS
If 8 denotes the maximum density of water, it follows that the
specific gravity of the specimen is
Sp. Gr. = &
B
This method is capable of very accurate results. In precise
measurements account must be taken of the buoyant effect of
the air on the specimen and on the standard masses used in
the weighing. The true weight of an object equals its appar-
ent weight plus the weight of the air displaced by it. When
the balance is in equilibrium, the apparent weight of the body
equals the apparent weight of the standard masses. Conse-
quently, when the specimen is weighed in air, its true weight
diminished by its loss in weight due to the buoyancy of the air,
equals the true weight of the standard masses diminished by
their loss in weight. If the density of air is denoted by pa and
the density of the standard masses by /o6, this last statement
says that
In the same way considering the water which displaced the
specimen,
vpw - vpa = (Mw - mj - (M™ ~ m^ .
Pb
On eliminating v from the last two equations, we get
| 0
""" P at
or, substituting for Ms and (Mw — m^ their values in terms of
the masses actually observed,
*
This gives the density at £°, the temperature at which the
experiment was performed. If 7 denotes the coefficient of
MASS, DENSITY, SPECIFIC <.RA\ITY 95
cubical expansion <>f the specimen, then its density at 0° is
given by
f- -
_ 2 - 4 - , 3
: men! of <7J> from (71) is left as an exercise for
student.
MANIITL. \TIMN AND COMPUTATION. — Make all weighings
by the method of vibrations. Observe the precautions that
are suggested in the last paragraph under Experiment 17.
Exp. 19. Determination of the Density and Specific Gravity
of a Solid by Immersion
OBJBOT AND THBOBI 01 BXPXBIMXNT.- The object of this
:iment is to determine the density and speeilic gravity of
a solid ,,f irregular form.
Since a >olid body immersed in a liquid is acted npon by an up-
thrust equal ' the li.jiiid displaced by the body,
it follows that it' this npthrnst is measured, the weight of the
displaced liquid is known, and if the weight of a unit volume
of the liquid is also known, then the volume of the liquid dis-
1 — and. then-fore, the volume of the body — can be cal-
eulat. the body in air is denoted by Ba,
and its weight when immersed in the liquid by /^, then the up-
: of the liquid —and. consequently, the weight of the
liquid displaced — is lia — Bt. So that, if //• denotes the weight
of a unit volume of the liquid at the temperature of the experi-
ment, the volume of the liquid displaced- and, consequently,
the volume of thebo<i
v =B*~B'. (78)
It follows, if m denotes the mass of the specimen, that the den-
sity of the specimen
i nation in ( 74) being true because m = /y,,///and u* = p,<j.
96 PRACTICAL PHYSICS
Since specific gravity is defined as the ratio of the density of
the substance in question to the maximum density, S, of water,
(75>
When the body is lighter than the liquid in which it is to be
immersed, a sinker is attached. Weighings are made to deter-
mine : first, the weight of the body in air, Ba ; second, the
weight of the sinker immersed in the liquid, Sl ; and third,
the weight of the two together when immersed, {B -f £),• The
weight of the body alone when immersed in the liquid is nega-
tive, but its value, sign included, is
B^as+sy.-s,,
and this value can be substituted in (74) and (75), giving
aad Sp.
MANIPULATION AND COMPUTATION. — The liquid in which
the body is immersed must be one which will not dissolve the
body, act upon it chemically, nor cause it to change its volume.
Whenever possible, use is made of water which has been freed
of dissolved gases by boiling. If the liquid contains dissolved
gases, bubbles will collect on the immersed body, causing an
increased upward" thrust, and therefore an error in the result.
Water should be boiled for about half an hour and then cooled
to the temperature at which the experiment is performed. As
water slowly dissolves air, it must be boiled on the day it is
used.
The motion of the balance beam is so much damped by the
immersion of the load in a liquid that it is useless to weigh by
the method of vibrations. The values of pl and S are to be
taken from tables.
M \s>. DENSITY, si'i UAVITV
97
Exp. 20. Determination of the Density of a Solid or Liquid
with Jolly's Spring Balance
<>I-,.IK<T AND Tm:"KY <>F K x i'i:i: i M KNT. — The Jolly spring
bahu. ••cially suited to tin- determination of the densities
of liquids and of solid bodies of small ma>>. The essential part
of the instrumen: :al spring which lianas vertically and
!es ut its Lowei did a weight pan. If the
limit of elasticity is not passed, any increase in
the length of this spring is proportional to i
Applied to it. In Line!
form of tin- instrument (FL . .-• sprin.:
d by i :n;_^ tubes.
inner tnl>.- can he adjusted up or down by
turning the milled head D. To tin- li»\ver end
)f tl. i an ind-
)f a doiiM-' OTCMI of aluminium, half of \\hic!
•ainted
ength of glass tubing whit h is wliitnud on the
•aek. and \\ bioh -e a
lorizontal hlaek hair line. This line se;
i zero, to \vhieh the line se|
• in- index may
lebrou^} .\ver end of tin- Li
ached a thin wire siipportini; t
9. ! iias been hroti^hi
Ixxly be placed on one
.•tl^ht to the
6TO marl. tin- inner
npporting tuln? A. On this tnhe is engraved
mil! ieli, by means of a vernier
t V, can !•«• read millimeters. The
laerence between the reading at I' when one of the scale pans
ii not In... M ion of the spring
to the \\, ; I,,- |,ndy NVlieii the h.idy does u«»t |
Itf
98 PRACTICAL PHYSICS
more than about 5 g., accurate results are possible with this
method.
In the case of a solid that will sink in a given liquid of
known density, the lower pan is submerged, and, after the in-
dex has been brought to the zero line and the reading at V
noted, the specimen is placed in the upper pan and the elonga-
tion of the spring, 6a, necessary to bring the index back to the
zero line, is determined. The specimen is then placed on the
submerged pan and the new elongation, bt, is found.
Since ba is proportional to the weight of the specimen in air,
and bt is proportional to the weight of the specimen when sub-
merged in the liquid, (74) and (75) can be put in the forms
BgPl . t>api ^7ox
si^s,* *^*
••[-:
and Sp.Gr.==, (79)
where Pt is the density of the given liquid and 8 is the maximum
density of water.
In the case of a solid that floats in the given liquid,
a sinker must be attached. With the apparatus arranged
as before, let the elongation of the spring when the speci-
men is in the upper pan be represented by 6a, the elonga-
tion when the sinker alone is in the submerged pan be
represented by s,, and the elongation when the specimen
and sinker are tied together and are in the submerged pan by
(£ + «);. Since these elongations are proportional to the forces
which produce them, (76) and (77) can be put in the forms
~L _ *£L _ (80)
&fl-(ft + 0i+«i
and Sp. Gr. = (81)
[*•—
In determining the specific gravity of a liquid by this method
a sinker that is unaffected by water and by the given liquid is
weighed in air, in water, and in the liquid whose specific gravity
MASS, DENSITY, >PKi IFIC GRAVITY <»!)
Miircd. If tli.' elongations of the spring when the sinker
is in the air, in water at 4° ('.. and in the given liquid, are de-
fectively by *a, 8^ and *,, it is easily shown that the
specific gravity uf the liquid is given by
^p. Gr. = ?fi=i. (82)
*a *tr
If tlie temperature of the water is t~ instead of 4°, the right
member of (82) is to be multiplied hy the specific gravity of
water a:
MAMITLATION AND ('..MIT TATK.N. — In determining each
make a reading <»f the scale at
V before the spring :ided. as well as afterward. The
•nded system must 1. without touching either the
glass tnl»f around the index or the beaker containing the liquid,
and the submerged part uf the suspended system must lie kept
From air bubbles and always submerge* I to the same depth.
The upper pan and its contents must be kept dry. After im-
•'ii in any liquid, the .sinker, specimen, and scale pan should
1 9 carefully dried with tilter paper.
Exp. 21. Determination of the Specific Gravity of a Liquid
with the Mohr Westphal Balance
ORH< r \\i> Tm:«)i:y MI K\|'I:I:IMI:\T. - The object of this
perimeiit is to determine the specific gravity of an aqueous
.- >lution by means of a Mohr- \Vestphal balance.
:n Archimedes' principle it follows that if a body of
• instant volume be immersed in various liquids, the conv-
- .unding losses of weight sustained by ih.- ill represent
t .e weights of equal volumes of the various liquids. Whence,
i a body ,ine t% when immersed in succession in two
1 juid>, ..f densities ^ ,uid py sustain the iv>peetive losses of
tt», and WT then
i £l. (83)
Pi
100
PRACTICAL PHYSICS
If the second liquid be water at the temperature of its maximum
density, then the ratio of w^ to w2 gives the specific gravity of
the first liquid.
If, therefore, a means be devised for measuring the loss of
weight of a given body when immersed in any liquid, and also
for determining what loss the same body would suffer if it Avere
immersed in water at 4° C., the specific gravity of the liquid
could be computed by means of the above equation.
A convenient instru-
ment designed for the
purpose is the Mohr-
Westphal balance.
This device (Fig. 50)
consists of a decimally
divided balance beam
at one end of which
is suspended a glass
sinker for immersion.
The other end of the
beam is so counterbal-
anced that the beam is
held in equilibrium
when the sinker is
surrounded by air.
The instrument is also
provided with five
riders which are ordi-
Fia go. narily equal in mass to
1.0, i.O, 0.1, 0.01, and
0.001 of the mass of water displaced by the sinker. Thus,
if the sinker be immersed in water, one unit rider placed
at the end of the beam would be required to compen-
sate for the loss sustained by the sinker and to bring the
beam back to a horizontal position. Again, if with the sinker
immersed in a certain liquid the beam is brought into a hori-
zontal position when a unit rider is hung on the hook J., the
MASS. DENSITY, SPKCIFIC GRAVITY 101
tenths rider on 'the second notch (7, and the hundredth* rider
un the third notch B* the theory of moments of forces show
that the upthrust on the sinker is l.UiM times as givat as in the
preceding case. Consequently the specific gravity of the given
liquid is 1.02
If, as the temperat . .ker were to expand at
- tint- rate that water dors, the temperature at which the
Mohr-Westphal balance is u>ed \\'<>uld make no difference, for
the sinker would always displace the same mass of water.
But, M ft matter of fart, at ordinary room temperatures water
expands more rapidly than ijlass, so that when the temperature
is a little above 20° C. the Mohr-\Vestphal balance reads 0, 1 ,
lower than it would at 1-", . Moreover, the temperature in a
laboratory is usually not BO low a- I < '.. and BO the riders are
usually adjusted to read specific gravities with reference to
at I")3 — about the temperature at which Kuropean
laboratoi : usually kept. In unh-r t«> use the balance
in a laboratory at a! and to get B] • with
• to water at 4° it will then be necosary to apply a
•orreetion.
find what this correction is, let bl6 and l>t denote the re-
spective r- ;ice when the sinker is immersed,
1 ) in water of dcnsil \ f>i:> at 1 "• .and cl) in the lit pi id whose
: v pt at t U . and let r^ and rt denote ihe i-i-
of the sinker. Then the \\ei-hts of liquid dis-
placed by the sinker in the two cases are respectively r.
lll(l Ptvt&' Since the readings of the balance are proportional
o these weights.
/\ "lsg = plbv$ (1 -h 7 ' 15) (84)
k . 1 + 7/ ,. (85)
s here v0 denotes the volume of the sinker at 0° and 7 its coeili-
•nt of expansion. On dividing (85) by (84), we obtain
102 PRACTICAL PHYSICS
Whence, since the balance is so .adjusted that b15 = 1,
. 15
or, employing approximation (5), p. 7,
*=Pi*M1 + 7-15) (1
or, employing approximation (2), p. 7,
(86)
If the specific gravity of the liquid is desired, we have at once,
if & denotes the maximum density of water,
Sp. Gr. [= |]= £i^[i_ y(t - 15)] . (87)
Since 7 is small and pl5 differs only slightly from S, it will
be seen that if only fairly accurate values are desired, (86) and
(87) give very nearly
Pt = Sbt (88)
and Sp. Gr.=bt. (89)
MANIPULATION AND COMPUTATION. — With the sinker in
air and no rider on the beam, the instrument is first leveled
until the pointer attached to the beam indicates zero. The
sinker is then immersed in the liquid whose specific gravity is
to be determined, and riders are placed in the notches on the
beam until the pointer again indicates zero.
Exp. 22. Calibration of an Hydrometer of Variable Immersion
OBJECT AND THEORY OF EXPERIMENT. — In the measure-
ment of the specific gravity of liquids for technical purposes
where great accuracy is unnecessary, some form of hydrometer
of variable immersion is usually employed. The hydrometer
(Fig. 51) consists of a closed graduated glass tube of uniform
cross section with a weighted bulb on the lower end. The
mass and volume of the instrument are so chosen that when
it is placed in the liquid whose specific gravity is to be deter-
mined it will float upright. The specific gravity of the liquid
MASS, DENSITY, SPECIFIC GRAVITY
103
is shown by the depth to which the hydrometer sinks. If the
graduations on the stem are so spaced and numbered as to give
directly the density of the liquid, the instrument
is called a densimeter. Often, however, the gradu-
ations are equidistant and are referred to some
arbitrary scale. Thus we have the scales of
Bauine. I'- : tier, and Twuddell. The specific
gravities corresponding to readings on these vari-
ous s- given in Table 6. Not infrequently
the stein of the hydrometer contains two or more
scales. When graduated with especial reference
to use with some particular class of liquids, the hy-
drometer is called the alcohol imeter, sal in imeter, etc.
A calibration curve for any instrument isacurve
in which the actual readings of the in>tniment are
plotted against the : that the instrument
ought to -/ive. The ol. exercise is to
Calibrate an hydrometer.
(a) Scale n$ of equal length. If an hydrom-
of mass m sinks to scale di\i>ion </t when placed in
i liquid of density pv and to di\iM.»n • ?., when placed
n a liquid of <lcn>ity p.,, then by Archimedes' principle the
volume of the first liquid displaced is — and of the second is
— . If 11 d.-notes the volume of that : the stem which is
J-j
ncluded between two consecutive scale divisions, then
Fio. 51.
Pi Pi
vVhence
(00)
>r
\om (90), if pr py and m are knoun, the value of u can be
onnd, and from (''1), if M, pr and m are known, p^ can be
ound.
104 PRACTICAL PHYSICS
If the maximum density of water is denoted by 8, the spe-
cific gravity of the second liquid is
Sp. Gr. [=^1= ^ .
L £J [m, — /0iM(^i — ^2)]^ \V^)
(5) Scale in which the successive divisions express equal dif-
ferences in density. Consider a wooden rod of mass m, of
uniform cross section <?, and so loaded at one end that it will
float upright. When the rod floats, the weight of liquid dis-
placed is by Archimedes' principle equal to the weight of the
rod. That is, if the rod sinks a distance Zj in a liquid
of density p^
--?- Whence ^ = — . (93)
Similarly, if the rod sinks a distance ?2 in a liquid
of density pv
I*=M' (94)
* Dividing (93) by (94),
* i = &. (95)
12 Pi
—*— That is, * the distances to which this hydrometer
FIG. 52. . ... ,. . . . J .. . ,
or uniform cross section sinks in various liquids
are inversely proportional to the densities of those liquids.
Consider now an hydrometer of the usual form, which is not
of uniform cross section throughout, but which is of uniform
cross section above some point IT (Fig. 52). For this hydrom-
eter there is at some unknown distance x below K a point
to which the hydrometer would extend if it had still the same
mass and volume which it really has, but if, instead of the
varying cross section which it really has, it continued through-
out with the same cross section which it has above jfiT. Sup-
pose that in one liquid this hydrometer sinks to a point distant
MASS. DENSITY, SI'KCIFIC GRAVITY
105
hl above K, and in another liquid to a point distant 7/2 above
K. Then from (95),
r~ i ~i 7
(96)
(97)
or
If the subscript 2 is dropped, (96) gives
h -fa:
(98)
If the maximum density of water is denoted by & the specific
gravity of the liquid is, then.
" h£l (99)
Thus if we determine to what distance above A' the hydrom-
eter sinks in each of t\\o liquids of known densities, we can
by (97 i determine x. And if we know to what distai
A' the hydnnne-te: liquid of known density, and
know also ./. then if we determine to what « list a nee above K the
>meter sinks in any other liquid, we can by (99) determine
the specific gravity of that liquid.
rniformity of cross section of the hydrometer may be tested
at various points with a micrometer cali-
It' the cross section is not uniform above A', the abo\e
method of calibration U n.,t applicable. In this case BOHM
having 'i
varying .somewhat uniformly \\ ithii.
the hydrometer should be made
up. the density of cadi determined, and
the readinur hydrometer in each
:i. This method of calibrat i<>n is, of
course, ac< -unite, but is more tedious than
the other.
M AMIM I.ATK'N A\D < '< >M IM T A I I < >N . — The surface <»f the
liquid about an h\drometer is usualU of a shape similar to that
in Fig. 53. AH is the stem of the hydrometer and < '/> is a tall
106 PRACTICAL PHYSICS
narrow jar in which the liquid is placed. First be sure that
the hydrometer is floating freely, and then place the eye below
the level of the liquid surface and raise it until it is sighting
the hydrometer along the dotted line. The point of the scale
crossed by this line is the required reading. The temperature
of the liquid should be noted at the time of each observation.
When changing from one liquid to another, the jar, hydrome-
ter, and thermometer are to be thoroughly washed and dried.
Determine the densities of two liquids either with a pyknometer
or with a Mohr-Westphal balance. Observe the scale readings
on the hydrometer when it is floated in turn in the two liquids.
(a) If the hydrometer has a scale with equal divisions, weigh
the instrument, place it in succession in two liquids of known
densities, and then by means of (90) calculate the value of u. By
means of (92) calculate the specific gravity corresponding to
each of the numbered scale divisions on the stem of the hydrome-
ter. Plot a curve with these calculated specific gravities as
abscissas and the corresponding scale readings as ordinates.
This is the calibration curve of the instrument. The calibra-
tion curve should be checked by comparing two or three values
obtained by means of the hydrometer in connection with the
curve, with values obtained by means of a pyknometer or a
Mohr-Westphal balance.
(5) In the case of the densimeter or direct-reading hydrome-
ter, lay a steel scale along the stem of the hydrometer and read
the steel scale at each numbered division on the hydrometer.
In addition, read the steel scale at the points to which the
hydrometer sank when floated in the two liquids whose densi-
ties were previously determined. From these last two readings
and the densities already determined, and taking K as any con-
venient point, calculate x by (97). Knowing x and the distances
from K to the various hydrometer divisions, use (99) to deter-
mine what the hydrometer readings ought to be at the various
points along its scale.
The quantity which has to be added to a reading in order to
obtain the corrected reading is called the correction for that
MASS. DKNS1TV. SPECIFIC GRAVITY
107
.:ILJ. Plot both a correction curve, coordinating the read-
"f the hydrometer and the corrections to be applied, and
.(•ration curve, coordinating the actual readings with the
corrected readii:.
Th- aions and results should be arranged in a table,
somewhat as follows : —
11 , l-KoMBTKB
RCA:
-//
STEEL SCALB
ABOVE A'
-A
A + a
e»Al<*l + «)
mg-M
*A + a.)
tp. 23. Determination of the Relative Densities of Gases
with Bunsen's Effusiometer
; IKNT. —-The object of this
cxpei : mine the ratio nf the dnisity of a gas to
i.-nsity of air or hydrogen. The density of a gas mi^ht !»«•
iiint-il l.y wri^l,ii|._r a large hull* <>f known voliinu-, lirst
• juite empty, and thru tilled with tlie L,ras nnd.-r investiga-
tinn. Hut on account of the ditheulty in ennij.lrt.-ly «-\ aeuatin^
10 first weigh 1 in olitainin^ an accurate
value of t !.-• mass of gas contained in the l.ul'o at t i;.- time of the
. this method recpiirrs nnnstuil care and many
r a gas of density p{ inclosed in a vessel at a pressure
of p dynes persq. cm. abo\ rounding at mosphrri-.
If tin-re l»e ;i >m;dl opening of area a in the vessel, then the g«s
'vill esca; ..-re at some speed *, om, per sec.
:i one set-mid there will issue fmrn the openinLT a column
of gas of length sl cm. and cross section a sq. cm. Conse-
«|iiently the mass of gas that escapes per second through the
108 PRACTICAL PHYSICS
opening is plas1 grams, and the kinetic energy of this mass is
Again, since the gas in the vessel is under a pressure exceeding
that of the surrounding atmosphere by p dynes per sq. cm., it
follows that the force producing the flow is pa. Consequently
the work done on the escaping gas in one second is pasv This
is the loss of potential energy of the gas in the vessel. Since
the loss of potential energy equals the gain in kinetic, it follows
that
(100)
Therefore the speed of efflux of the escaping gas is
*i=\/-£.
Pi
Similarly, if a second gas of density p2 is allowed to escape
through the same opening under the same difference of pressure,
its speed of efflux is
(101)
Dividing (101) by (100),
P2
(102)
FIG. 54.
where ^ and t2 are the times required for equal
volumes of the two gases to effuse through the
same opening.
That is, when under the same conditions as to
pressure, the densities of two gases are inversely
* proportional to the squares of their speeds of
effusion, and are directly proportional to the
squares of the times required for equal volumes to effuse through
the same orifice.
This is the principle of Bunsen's Eff usiometer. The apparatus
(Fig. 54) consists of a glass tube open at the bottom and sur-
mounted by an enlargement containing a diaphragm D pierced
with a small opening about 0.01 mm. in diameter. This tube
MASS, DENSITY, SPECIFIC (.KAVITY 109
is inserted in a larger vessel containing mercury. The gas
under investigation is inclosed in the tube and the time noted
that is required fur a certain volume of gas to effuse through
the diaphragm. C is a three-way cock by means of which the
.older can be put into direct communication with the
atmosphere, or with the orifice in the diaphragm, or can be
I entirely. F is a float for indicating the change in vol-
ume of the gas, and X is a stopper.
MAMIM i.Aii"\ .\M» ( '. .MIM TATIMN. — If the ratio of the
•f a gas to the density of air is to IK- determined, put
the gas holder into direct connection with the atmosphere hv
means of the three- way cock (7, and then, by raising the gas
•r, till it with air. Close the stopcock, depress the gas
:-, and clamp it into position. Next remove the stopper
and turn the three-way cock SO as to connect the gas holder
.vith the diaphragm D. As the gas effuses through the dia-
)hragm, observe the interval of time between the instant when
li.- upper point /' of the Huat arrives in t: ..f the upper
:' the mercury in the well, and the instant when the
nark at M on the float reaches the same level.
v empty the gas holder and till it with the other gas. I >• •-
•ress the gas holder as far as possible while it is in direct
ommunicatioii with the atmosphere. This will expel most of
with the gas being examined and elevate the
This opera '.1 till the gas holder. By
v refilling and emptying tin- gas holder, it will become
of air and tilled with a specimen of the gas
/hose dc: sought.
Proceeding as in the case of air, find the interval of time
en the infant when the apex of tin- tl-.at appears above
;rface ,,f the mercury in the \\ell and the instant when the
i lark ;/ The ti; equal volume of
t ie two gases under the >ame DI6lfOT6 to effuse through the.
t »ie opening have now l)een obtained. Their relative density
c in, therefore, be calculated by (10>
CHAPTER VII
MOMENT OF INERTIA
THAT property of matter in virtue of which a force from out-
side must act upon a body in order that either the speed of the
body or the direction in which it is moving may be changed is
called inertia. Similarly, that property of matter in virtue of
which a torque from outside must act upon a body in order
that either the angular speed of the body or the axis about
which it is rotating may be changed is called moment of inertia.
The inertia of a body is numerically equal to the sum of the
masses of its component particles. The moment of inertia of a
body can be shown to be numerically equal to the sum of the
products of the masses of the particles composing the body and
the squares of their respective distances from the axis of
rotation, i.e.,
K=^mi*. (103)
When a resultant torque is applied to a body there is produced
an angular acceleration a numerically equal to the ratio of
the applied torque L to the moment of inertia K of the body,
i.e.,
a=J. (104)
The moment of inertia of a body of simple geometric form
can be computed, but the moment of inertia of an irregularly
shaped body may often be determined most easily by experi-
ment. The experimental determination is usually made by
comparison with a body whose moment of inertia can be com-
puted. The computations for a few simple cases are effected
as indicated in the following table : —
110
MO MK NT OF 1NKHT1A
111
SHAPB
A \1S ABolT WI1K It
i:..i \
INF.KTIA
MEANING OK s \\ti.oiv
Kf. Monifiit of iiu-rt ia
Plane lamina
Any axis normal
tl'out u\\\ linr A A '
of any shape
to it
A-X+ A;
56) in tlu-
^lane of tlu- lamina
ami ini«>r>rrting the
2
Any shape
Any axis
m«Mit of inrrtia
il-oiit anotlu-r linr
ii the plan.- of
3
Solid circu-
1 -1 '
tin- lamina. ]MM-ptMi-
liciilar to A.V . ami
linder
4
Solid circu-
:iinl»T
Any axis parallel
to geometric axis
M/
A . M. 'in. 'lit of in.Ttia
al'ont tin- paral!'
•
DUM,
5
• ircu-
!• t»T of one
end
M | j£ + £]
/'. Distance bt't
th. two axes.
Solid
Through cmter,
r^ n
I/. M,l- of thr rylill-
lar cylinder
mal to length
M 116^ 1L>J
• r «if tln»
I'T.
7
HoQoi
Geometric axis
'-He/,1)
ill of th.- ryliii-
der,
fCOiitenl
'/,. In
expressions in tin- f«»urih
be obtaiiu-'l in the manner imli-
paragra]
< .t-le of mass
at P (1 ::. 66 - I b D by
the ;il»uv«- laltlc ma
-
112
PRACTICAL PHYSICS
2. Let the diagram (Fig. 56) be a section normal to the
axis, A the section of the axis about which rotation occurs, 0
the section of a parallel axis, and KA the
moment of inertia about the required axis.
Consider a particle of mass m at P. Then
or, since p is independent of the particle
considered,
By a proposition in elementary dynamics, ml = 0 when the axis
through <7 passes through the center of mass. Therefore,
KA = Kc + Mp\ (106)
3. Imagine the cylinder to be made up of n thin hollow cylin-
ders one inside of another. Denote the density of the material
by /?, the length of the cylinder by Z, the thickness of each hollow
cylinder by £, and the respective mass and moment of inertia of
the ith hollow cylinder, beginning at the center, by mt and ^ .
Then
mi =
i - 1),
where A is used in place of Trftlp. The moment of inertia of
this ith hollow cylinder is greater than the product of its mass
by the square of its inner radius and is less than the product of
its mass by the square of its outer radius. That is,
The moment of inertia of the whole cylinder K is the sum of the
moments of inertia of the elementary hollow cylinders. That is,
1'- 1)0' - !)¥>} < K<
t=i
or,
2 «» - 5 *a + 4 i - 1) <K< 4
MOMENT 01 IM-KTIA 113
On summing the series indicated in this last pair of inequalities
and substituting for ^4 its value Trt-lp* we get
or, since nt = r,
TT/K 1 r* - § & + i rt») < K< TrlpQ r* + } r*t - J rf *) .
The difference between tlif tliird and first members in this last
pair of inequalities is J?r/pr(4 r2 — r2)^, a quantity which, by
choosing f small enough, can be made less than any assigned
quantity. It follows that the value of I\ is the common limit
approached by th- and last meml)crs when t approaches
zero. That is. if •/ denotes the diameter of the cylinder.
D= \-rrlpr* = \Mi* = \Mffi. ( 107 »
4. Apply (:{) and c2> mi p. 111.
Imagine the cylinder cut into n thin lamina- l»y planes
normal U) - of the cylind.-r. If m is the mass of one of
lamina-, then by <1<>7, the moment of inertia <.f that
lamina about its ^eimu-tric axis is J md*. If tl»e thieknrss t of
the lamina wnv indelinit.-ly .small, tben, from the symmetry ..f
the figure, tlie iimin. ,'iertia of the lamina about any
diameter would e«jual its moment «f inertia ab«.iit any other
diameter, and therefore, by (105), Iti momeiii of inert ia about
any diameter \\<.uld be -j^-tnd*.
Consider n<.\v the moment of in.-r: A if the /th lamina
from on-- end of | inder \\hen I tiOD is a
diameter of that end of the cylinder. One side of this lamina
18 at a e(/— \ )t from the end of the cylinder and the
other at a distain--- i tin- end. The moment of inertia of
: ban it \\oul -ll the material in the
lamina were compressed into a thinner lamina at a distance
'/— 1 V from the end. and i> l«-ss than it \\oidd be if all the
i -vterial in the lamina were compressed into a thinner lamina
it a distance it from the end. From < L06) .' follow! that
m -<Ki<h™<&
114 PRACTICAL PHYSICS
The moment of inertia of the whole cylinder K is the sum
of the moments of inertia of the elementary laminae. That is,
i=l-n 1 16
or
Whence, on summing the series,
m
or remembering that nm = M, the mass of the cylinder, and that
nt = Z, the length of the cylinder,
The difference between the third and first members of this
last pair of inequalities is Mlt, a quantity which, by choosing
t small enough, can be made less than any assigned quantity.
It follows that the value of Kis the common limit approached
by these first and last members when t approaches zero. That
is, if d denotes the diameter of the cylinder,
6. Imagine the given cylinder to consist of two equal cylin-
ders set end to end. Then the length, diameter, mass, and mo-
ment of inertia of the given cylinder are respectively I' = 2 I,
d' = d,M' = 2 M, and K = 2 K. Substituting in (108),
7. Let p denote the density of the material composing the
cylinder, d0 and d4 its outer and inner diameters, I its length,
M0 and K0 the mass and moment of inertia which the cylinder
would have if it were solid, Mi and Kt the mass and moment
of inertia of the inner part of the solid cylinder that has been
MUMLNT OF INERTIA
115
removed in order to leave the hollow cylinder of mass J/ and
moment of inertia K. Then, by (107),
' -<7.2>
Exp. 24. Determination of the Moment of Inertia of a
Rigid Body
()i;.n:< T \M) TIII:<>I:Y Ol K\ i'i:i:i MF.NT. — The object of
this experiment is to determine a moment of inertia.
In any case where a body can be set into torsional vibration
about the axis about whieh the moment of inertia is required,
Dimple matter to determine
:<>raent of
of tlie body. I-'n.m < 1 t 1 » it follows
that if a body is suspended so that
: vibrate torsionally. its moment
of in- :ial to tlu* square
of its period of vibrat . <::. I : t;
the proportional:' ailed k\
If a mass of known moment of
>e added to the body
above 01 ive ^^^^
A K^kTJ (11 -J) F,o.B7.
wh. the new pe: ;brati«»n of the system.
Kliminatin^ k between (111) and ( 1 1 _' i.
A" '/'
I-TJ^T? (113)
MAMIM I.\UM\ .\M> ( .-MI .. .\ convenient form
"f a: this experiment consists (Fig. 57) of two
116 PRACTICAL PHYSICS
horizontal disks connected by three thin vertical rods.
From the center of the upper disk rises a short spindle for
attachment to the supporting torsion wire. The body whose
moment of inertia is required can be placed on the lower disk
in such a position that the line about which its moment of
inertia is to be determined coincides with the axis of the sup-
porting wire. The positions of the masses MM are then ad-
justed until the axis of vibration of the system passes through
the center of the two disks. Below the vibrating system is a
device by means of which the apparatus can be set into tor-
sional vibration with very little swinging motion.
Find the period of vibration, Tv of the apparatus ; then add
a body of known moment of inertia, 7f2, and find the new
period of vibration, T12. From (113) the moment of inertia of
the apparatus is
Now substitute for the body of known moment of inertia the
body whose moment of inertia is required, and find the period
of vibration as before. If this period be denoted by ^13, then
the moment of inertia Kz of the body under investigation is by
(113)
or, substituting the value of K^ from (114),
In finding the various periods of vibration, first with the
apparatus at rest set the pointer P directly in front of one of
the three vertical rods. Then set the apparatus into torsional
vibration with an amplitude of perhaps 90°. At some instant
when the vertical rod passes the pointer start a stop watch.
Count some ten or fifteen complete vibrations and stop the
watch. After recording the time that has elapsed, again at the
instant of a passage start the watch. After some ten minutes,
MOMKN 1 OF 1NKHT1A
117
during which time no attention has been paid to the vibrating
11, stop the watch at an instant of passage. Calculate the
period by the Method of Passages, given oil pp. 30-38.
Take all the required linear dimensions with a vernier ealiper
and make all weighings with a balance of moderate sensibility.
Calculate A', as indicated <>n p. 111. Determine A', both bv
( 11-", , ;lud a> indicated on p. Ill, and see how the values cheek.
CHAPTER VIII
ELASTICITY
WHEN a body is perfectly elastic, a given deforming force
keeps it distorted to the same extent no matter for how long a
time the force is applied. This means that the distortion calls
into play a restoring force which, so long as the body is at rest,
is exactly equal and opposite to the deforming force. It fol-
lows that, when the deforming force is removed, this restoring
force causes the body completely to recover its original shape
and size. When a body is imperfectly elastic, a given deform-
ing force produces a gradual yielding so that the restoring
force which the distortion calls into play is in this case not
quite equal to the deforming force. It follows that when the
deforming force is removed from a body which is imperfectly
elastic, the body does not completely recover its original shape
and size. It is said to have received a permanent set, or to
have been deformed beyond its elastic limit. So long as any
body is not deformed beyond its elastic limit it is perfectly
elastic.
The ratio of a force to the area on which it acts is called a
stress. The ratio of a deformation to the original value of the
length, volume, or whatever has been deformed, is called a
strain. When a body has not passed its elastic limit, the ratio
of the restoring stress to the strain which produced it is constant
and is called a coefficient of elasticity. Since forces applied to
a body in different ways produce different types of deformation,
there are various coefficients of elasticity.
If a wire is stretched or a pillar shortened by a load applied
118
J.I.ASTUTrY 119
to it, the strain is the change of length divided by the original
:i. In this case the ratio of the stress to the strain is
ealled the tensile coefficient of elazti'ltii <>r _)'«>///</',<< modulu*.
If a toy balloon were fastened under water and then pressure
applied to the water, the balloon would deerease in volume with-
out ehanging its shape. In this ease the strain is the ehange in
volume divided by the original volume, and the corresponding
'•ient of elasticity is called t:
If a rectangular parallelepiped of rubber a<> ( Fig. 5S) has two
opposite faces glued t" two boards, and if one of these hoards
is \n\- \ s in its n\vn plane, there is no change in the
volume of the block but its shape is changed to f<i<\L In this
case the strain is tin' nr ^
:id is called a shear or a then
be force applied, and
A the area face ab, the
livi«
ided 1. I - called a shear — FYo~58~
I, If ili«- l»'..M-k of rubber is very
thin in a . to the paper, and if it is bent around
intil ii<l 0 with /»•. it is seen that .1 - the kind of
arain involved in the twisting of a wire about its geometric
tioofa.v ; atnte to the shearing strain which
t produces is called the simple r c the slide modulus of
he material shea:
Exp. 25. Determination of the Elastic Limit, Tenacity, and
Brittleness of a Wire
Or..n-:iT AM» THEORY 01 -The elastic limit
if a material is the stress beyond which the material cannot go
vitlmnt be. •(>:niiiur permanently set. Since it is found that the
oirve _r the relation of a stress to the strain which it.
'•••(luces .t line until the < lied,
heel. it is the stress corresponding to the point on the
: im where the curve departs from being a
The tenacity or tei h of a material is
120 PRACTICAL PHYSICS
the greatest longitudinal stress it can bear without rupture.
The brittleness of a material is the ratio of its elastic limit to
its tenacity ; in other words, it is the ratio of the force just
sufficient to produce permanent set to the force just sufficient
to produce rupture. The object of this experiment is to plot a
curve showing the relation between the longitudinal stress and
strain of a wire, to determine from this curve the elastic limit
of the material composing the wire, and also to determine its
tenacity and brittleness.
MANIPULATION AND COMPUTATION. — Arrange a wire verti-
cally so that it cannot twist, with one end fastened to a rigid
bracket and the other end attached to a scale pan. Place on
the supporting bracket, directly above the wire, a number of
iron masses whose aggregate weight exceeds the breaking
strength of the wire. Focus the cross hairs of the telescope of
a cathetometer, or the cross hairs of a microscope containing an
eyepiece micrometer, on a well-defined mark on the lower end
of the wire, and take the reading. Take a weight off the
supporting bracket, place it on the scale pan, and take a new
reading of the position of the fiducial mark. Continue chang-
ing weights from the supporting bracket to the scale pan and
taking the corresponding readings until the wire breaks. On
coordinate paper plot the stresses applied to the wire as
abscissas, and the strains produced as ordinates. The stress
corresponding to the point where the curve departs from being
a right line and bends toward the vertical axis is the elastic
limit. The tenacity is the breaking weight divided by the
area of cross section of the wire, and the brittleness is the
elastic limit divided by the tenacity.
Exp. 26. Determination of the Tensile Coefficient of Elasticity,
or Young's Modulus
(FIRST METHOD, BY STRETCHING)
OBJECT AND THEORY OF EXPERIMENT. — From the definition
of Young's modulus (p. 119), it follows that if L denotes the
ELASTICITY
121
:i of a \viiv. 7 its diameter, and e the elongation produced
by a force /T. then the Young's modulus of the material compos-
ing the \viiv
If the force is measured in dynes and the
r quantities in centimeters, the value of
K will he in dynes per sq. em. The ohject
of this experiment is to determine the value
:iodulus for a metal in the form
of a v
the quantities which have to he meas-
ured, the only one that it is difficult to get
with mod, -rate accura< \ is the value of the
Cation ,•. On. -means of finding this is
The Upper end of the
lamped to a ri^id support
.d to the lo\\.-r end of the wire
r piece of metal S
terniinalini; in a h<» < hment
weight pan //. This rectangular
of metal is kept from i «,r
swiiiLMiiLf 1»\ IM-HILT let through a loos-
littii : pillar hole in a second '
;ied to the wall. One leg of the
optical lever is si. in the axis of the
hy the rectangular hook, while the other Fio. 59.
iHirted hy the hrack.
In I li the optical level- with its mirmr ••/• vertical,
ntal teh nd oo' is a vertical scale divided
• •ntimeters and millimeters. If the wire he .stretched hy
ill amount, the oj.tical lever will assume the position ///////
making an an^h- ^ \\ith its j ition. When li^ht is
reflected fidn a mirror, the an^le of reflection equals the alible
of in ., .. ,,' <i' i = v<i' i = 0. Conse(|iicntly tm'v=ml 6.
122
PRACTICAL PHYSICS
And since the small distance aaf is negligible in comparison
with ao,
tan 20 = 21.
ao
If 0 is small, approximation (10), p. 7, may be employed,
giving
FIG. 60.
The elongation is the vertical distance through which the point
m moves in passing to the position m' . So that
e = m'n sin 0 = mn sin 0,
or, employing approximation (8),
.__ Q ._ mn • oof
On putting this value of e in (116) it becomes
„ . 8 FL ao_
mn oo'
(117)
MANIPULATION AND COMPUTATION. — See that the wire is
straight and carefully suspended. Place three or four kilo-
grams on the supporting bracket directly over the clamp hold-
ing the upper end of the wire, and one kilogram on the pan
below. Put the optical lever in place and the telescope and
scale a meter or so from it, clamp the scale vertical, and adjust
the height of the telescope until it is at about the same level as
the optical lever. Move the head to such a position that the
image of the telescope is seen in the middle of the mirror of the
optical lever. If the eyes are not now at the level of the tele-
ELASTICITY 123
scope, turn the thumb screw beneath the front legs of the optical
until the image is seen when the eyes are at the same
level as \. "pc. This makes the mirror vertical. 1
ize the telescope as directed on p. 23.
I the telescope, move the masses from the supporting
bracket d<»\vn t" I Jit pan. read the telescope, move the
-•k to the supporting bracket, and read the telescope
again. If the elastic limit has not been exceeded, the last
reading should be about the same as the first. Repeat two or
three tin. \] shout live determinations, each one after
moving l: 'pc and scale a fe\v centimeters farther from
•ptiral If.
IfeMIlN tin- diameter i.f the wire in some half do/.en places
with a micrometf ; • :he length inn of tlie
optical lever by piv>- three feet upon a piece of card-
board, connecting the prick points made by the two front feet
!ine, and then m ( tin- normal diMance hetucni
.jiing prick point and this line by means of a milli-
meter scale. I). •:• ;he length of the wire with a meter
.stick, and the loads added to the weight pan \\ith a platform
balance weighing to grams.
each distance <t>> tind the average deflect ion oo' and
culate ""- Find the average of all the values for ~ ' and by
oo1
(117. Of the result in dynes per sq. cm., in
wt. p«-r s<j. mm., and in lit. u t . p-T ><j. in.
Exp. 27. Study of the Flexure of Rectangular Rods*
AM. Tm:«'i:v 01 r. — Fven before a
given phenomenon i> >ullici«-ntly understood to permit thederi-
vatinn by purely ana'. 'hods of a formula that will slmw
ors entering into the phe-
non. it il possible to construct from pun-ly experi-
ia taken with slight modification from Reed and ( .
Physical Measurement.**
124 PRACTICAL PHYSICS
mental data an equation that will give the law connecting the
various related quantities. An equation obtained from experi-
mental data is called an empirical equation. One of the
methods used in the construction of empirical equations is illus-
trated in the present exercise.
If a number of rods of any material, differing in length,
breadth, and thickness be supported on a pair of knife-edges
and loaded in the middle, it would be expected that the flexure
produced, that is, the displacement I of the middle point of any
rod, would be a function of the load F, the distance L between
the supports, the breadth of the rod B, and its depth D. The
law of flexure of rectangular rods of a given material might,
perhaps, be expressed by an equation of the form
I = kF-L^D', (118)
where k, «, j3, % and e are constants to be determined by
experiment. The object of this experiment is to ascertain
whether the facts warrant the acceptance of the above tenta-
tively assumed equation; and, if they do, to determine the
values of the five constants. The constants a, yS, 7, e, can be
most easily obtained by varying the independent variables one
at a time and noting the change of the dependent variable L
When, in this way, these four constants have been determined,
the value of k is obtained by solution.
First, let the load F be varied while the other independent
variables remain constant. This will give a separate equation
for each value of F used. Thus
F4t = .
Dividing the first of the above equations by the third and the
second by the fourth,
If-, JJ I , Ifn Jin
ELASTICITY r_>r>
Putting these equations into the logarithmic form,
log ln - log /,.3 = «13(log F1 - log FB) (lli» )
and 1. g /,-, - log /,.4 = «,4(log Fz - log JF4), (120)
in which «13 denotes the value of « derived from the equation
expressing the ratio of /^ to //-g. If the values of «13 and «.,4
obtained by solving (IT.*) ami (120) are nearly the same, their
be taken as the value for «. in ( 1 1 ^ i.
Second, the other independent variables remaining constant,
let the length L be varied, the flexure / bring observed when
- applied. Uy the pTO06M deseribed above we get
logfA1-log/a = 0Iar. L -logZ8) (121)
and log 1L^ - log l^ = /^(log Z2 - log L (1 22 )
values of /918 and #24 obtained by solving < 1-1 ) ;""'
, the same, their averag- l>e taken AS the value
for ft in
Third, let the breadth It be varied and the other independent
1 ariables remain constant. i 11 give
log 1RI - log lhz = 7,/log B, - log Bz) ( \ !>:'> )
nd log I,* - log //(4 = 7a4(log BI - log />' (124)
'roni the>e e^uati.'iis a value f«»r 7 \\ill be found.
the depth I> l»e i
log //a - 1. « /^ = «i/ log Dt - log Ds) ( 1 25)
• «d log //>, - log //I4 = < M(log Da - log J> (\
\ rom these equations a value for < will be found.
If the valUM found for «. ate nearly the same, for tf nearly
t ie same, for -/ the same, an<l for «. net] ime, this
M the form assume. 1 f,,r the deMi.'d ivlation. If the
uentally obtained for any one of the quantities «,
. . 7. € ar«- not nearlv the .same, this .at the form as-
- med is not the furni of the relation wliieh actually exi>ts, and
:"nn must lie trie<l.
equation obtained by substituting i',,r „. fo y, € their
\ ilues th ntally determined is called an empirical
126 PRACTICAL PHYSICS
formula. The statement of the facts expressed by this for-
mula constitutes the law of bending. The values obtained for
these four constants should be very nearly a = 1, ft = 3, 7 = — 1,
e = — 3. If exactly these values are obtained, the law of bend-
ing will be expressed analytically by the equation
i=*fg- cm)
Whatever the form of the empirical equation that is actually
found, the value of k is determined by substituting in this
equation a set of corresponding values for Z, .F, L, B, and D.
Several such sets of values should be substituted and the
average value for k used. If similar series of measurements
are made upon rods of different materials, the values of a, /3,
7, e are found to be very nearly the same for the different
materials, but the values of k are different. This means that
k depends upon the material of the bar and not upon its dimen-
sions, whereas the other constants depend only upon the
dimensions of the bar.
MANIPULATION AND COMPUTATION. — The rods to be ex-
perimented upon should be 70 or 80 cm. long and their trans-
verse dimensions so selected that the same bars can be formed
into two series, one in which the bars have constant depth and
variable width, and another in which they have constant width
and variable depth. The variable length is secured by adjust-
ing the distance between the supporting knife edges. After
a bar is placed on the knife-edges a weight pan is suspended
from the bar about halfway between the knife edges and
sufficient weight applied to insure good contact between the
bar and its supports. On the addition of a known load the
flexure of the bar, that is, the depression of the middle point,
is measured. This measurement may conveniently be made by
means of a microscope furnished with a micrometer eyepiece,
or by means of a micrometer screw fastened to an adjacent sup-
port directly above the middle of the bar. In the latter case
the instant when the micrometer screw comes into contact with
KI.ASTICITY 127
the bar can be determined either by means of a telephone
receiver in a battery circuit including the bar and micrometer
r, or by observing the image of some fixed object in a small
mirror, one end of which rests upon the rod ami the other
end upon some adjacent fixed support. In order to be certain
not to load the i <»ml their elastic limits, the student
should ask an instructor what loads may safely be applied.
'.owing the division of the experiment as outlined above,
make a series of -ions on a single rod by noting the
flexures produced by different loads on the pan. Add, say,
500 g. and observe the flexure, add 500 g. more and obfi
the flexure, and so on until six equal increments of load
D added. Then reverse the process, removing 500 g.
at a time and taking an observation for the tlrxiire after
each change of load. Combine the six values of load and cor-
•nding flexure as in < L19) ind < !-<>) so as to get three
i/., «14, a^, and o^.
- . -ml. by moving t jlt. knife edges a few centimeters, obtain MX
bhsof a single rod, and f«>r ca«-h Length determine the H.-MII-C
.roduced by the same load of, say ,2 Kg. Combine the values of
1 22) so as to obtain three values t
Third, with the distance between the knife edges constant.
ind the flexure pr.idue.-d by a constant h>ad of. say. -2 Kg. act-
ng on each of four or six bars of the same material and depth
• lit d i'i - Measure the l.readth of tlie hars \\ith
iniei-nn. .;j»er. Proceed 88 din-rted in the preceding
paragraph. • (128) and ( 1JI) t<> tind 7.
rth, \\ith th- .-e between the knife edges constant.
nd the ile\iii-e produced in each of four or six bars of the
ime material and breadth but different depth. Measure the
i epth of the :h a micrometer Proceed as di-
1 in the preceding paragraphs, using equations like
( 12;")) and t L26) U) find «.
'nsert the final I .^. 7, € in ( IIS), and formulate in
A ords a statement of the facts e\pre>sed by the resulting em
I irical equal.
128 PRACTICAL PHYSICS
Exp. 28. Determination of the Tensile Coefficient of Elas-
ticity, or Young's Modulus
(SECOND METHOD, BY BENDING)
OBJECT AND THEORY OF EXPERIMENT. — Consider a rec-
tangular rod of length L, breadth B, and depth J>, fixed at one
end and weighted at the
other. The rod will become
bent as in the figure. The
upper portion of the rod is
extended and the lower por-
tion compressed. Since the
rod is strained by a longi-
tudinal stress, and since
Young's modulus is defined
as the ratio of the longitudi-
nal stress to the longitudinal
strain, Young's modulus may be determined from an observa-
tion of the amount of bending which a given force produces in
the rod. The object of this experiment is, by the method of
bending, to determine the Young's modulus of the material
composing a rectangular rod.
Imagine the unstrained rod to be cut up into m laminae, each
of width w, by a series of planes normal to its length. Then
let the rod be bent slightly by the force F applied downward
at the end of the rod, and let the yth lamina from the free end
be thereby so distorted that its sides ac and Id make with each
other a small angle 0j. The restoring stress in this lamina
produces a couple which tends to bring the rod back to its
undistorted position, and is prevented from doing so only by
the distorting force F.
The first step in the development of the formula for deter-
mining the Young's modulus of the rod is to find an expression
for the restoring couple due to the stress in this yth lamina.
Halfway between the upper and the lower surfaces of the rod
ELASTICITY 129
is a neutral surface gh which is neither extended nor com-
pressed. Through the point e, where ac cuts gh, draw a'c'
parallel to bJ. Then the original length of any line vz in the
jtli lamina is that part of it included between n'c' and b>?. and
the increase in its length is the part of it between ac and a'c'.
ine the upper half of the jth lamina to l»e made up of
n layers, each of breadth B equal to that of the rod and of
depth t. Then, (-minting upward from </, the top of the /ih
layer is stretched itO} and the bottom «,f it is stretched
(7— l)*0y. The effective elongation, e& of this Ah layer lies,
therefor M limits. That is
1 ^j<e,<it^. (128)
If E denotes th«- Young's modulus of the material composing
><l,an«l /' tin- : intion developed in this tth
r, then from the definition of Young's modulus (j>. 11'.'),
layer, tin
/." w
Vhence /' /;/'"' '
w
M) that, from (I'JS),
1C W
;ince tin- a of th«? material in tin- Ah layer from the
• •utral >uiface is less than if and greater than (/ — 1 )f, it
')llo\\^. L :>-storing tonpic dn<- to tin- strain of
his layer, that
W W
"he restoring torque L developed by the straining of all the
icutral surface is the sum of the torques devel-
•d in the separate layers. Thai
w
130 PRACTICAL PHYSICS
or, on summing the series indicated in the above expressions,
BtWj 2n*-3n* + n L E£t
w 6 w
Whence, since nt = |- D,
EBB, JW-Zm + ZDt^T ^EBOj IP + Z D*t + 1 DP
' ^~j <L< * • —
w 24 w 24
The difference between the third and first members of (129) is
i t. Since — is the radius of curvature of the neutral
4w 0j
surface in the/th lamina, and since this radius is never very
small, it follows that, by choosing t sufficiently small, the dif-
ference between the third and first members of (129) can be
made less than any assigned quantity. It follows that the
value of L is the common limit approached by the first and
third members of (129) when t approaches zero. That is,
L = EB0;1P
24 w
Now the resultant moment of the restoring forces below the
neutral surface equals the moment of those above. It follows
that the whole torque due to the strain in the jih lamina is 2 L.
Since the bar is in equilibrium, this restoring couple equals the
distorting moment of F about e. If the rod is bent only
slightly, the moment of F about e is so little smaller than Fjw
that we may write
EBusD . -j-f* /'-t OA\
' = Fjw. (130)
12 w
The next step is to find the depression of the end of the rod.
The entire depression I may be regarded as made up of parts,
Zj, ?2, . ... , Zn, due to the bending in the different laminae.
At e and / draw es and fk tangent to the neutral surface and
equal in length respectively to the arcs eh and fh. Then the
angle between these lines equals the angle 03- between ac and bd,
and this angle is so small that sJc is practically the arc of a
circle two of whose radii extend in the directions es and fk.
ELASTICITY 131
It follows, since fk, if prolonged, would cut es between e and/,
that
Moreover, if the depression of the end of the rod is not more
than one hundredth as great as the length of the rod, it ean be
>hu\vn that *k diiiers fr«»ni /,, the depression due to the bending
in thet/th lamina, by not more than some »HH',' ,,r <>.<>•'>% of
racy so great as this is seldom required in a de-
termination of Young's modulus. and the bending is usually
than that indieat.-d. h ; >re. j.ermissilile to use sk
ual to /, . Since in addition. fk=fh = {j—\)W) and
,x = eh =jw, ( 1-11 ) may be rowrii
< )n snbstitntincr in tliese inequalities the value of Oj from
he\
-
EB&
i.-pivssion / of the end of the rod .hie to the straining of
.11 the Ian. m depressions due to the separate
uninae. Th.
</<
>r, on summing the series indicated,
, 2m8-f
"
611, i
/'./•/'" o
mtc = L. the leii'_rtli of the rod,
" A'/y//1
I
the tliird and the first members of
*«*«*•
quantity whielj, by choosing w small enough, ran be made
d quantity. \Vheii the rod is not bent
132 PRACTICAL PHYSICS
too much, it follows that the value of I is the common limit
approached by the first and third members of (132) when w
approaches zero. That is,
If the rod, instead of being fastened at one end and loaded
at the other, is supported on two knife edges and loaded in the
middle, the bending is practically the same as if it were fastened
at its middle point and had acting upward upon it at each end
a force half as great as the load actually applied. Let the dis-
tance between the knife edges be U = 2L, and the force ap-
plied be F1 = 21*. Then on substituting for L and F in (133),
we get
or, dropping the primes,
The preceding development of the above formula may be
summarized as follows :
The first step, after supposing the rod cut into laminae by a
series of nearly vertical planes, is to find the restoring torque
in one of these laminse. The upper half of the lamina is
imagined to be cut into a series of nearly horizontal layers.
From the general formula for Young's modulus, the restoring
torque due to the strain in this layer is found. These torques
are then summed, and, since an equal torque in the same direc-
tion is exerted by the lower half of the lamina, the result is
doubled, the total restoring torque in the lamina being thus
obtained.
The second step is to equate this restoring torque to the
distorting torque due to the force at the end of the rod, thus
getting an equation by which the angle 0 between the two
sides of the strained lamina can be found.
The third step is to find the depression of the end of the rod.
This is done by first getting an equation connecting the angle
ELASTICITY 133
0 with the depression due to the strain in one lamina, eliminat-
ing the aiujk' 0 from this equation and the last equation ob-
tained in tin1 second step, and then summing the depressions due
to the strains in all the lamina-. This ^ives the formula for
a rod fixed at one end and loaded at the other.
The fourth step is to modify this formula to fit the case of
a rod supported at hoth ends and loaded in the middle. This
is done by ima^inin^ the rod to be made up of two rods placed
end to end. their inner ends bein^r fixed ami the outer ends
pushed upward.
MAMITI. \ri"\ AND COMPUTATION. — Measure B and />
Lumber «>f points aloiiic the rod by means of a micrometer
ealiper. M • M IW I.- U106 between the two ki,
with a in- •• the rod on the knife edges and sus-
pend from the middle point a pan containing sutlicient load to
\ riiiLT the rod into good contact with the knife ed^es. The
• f the rod produced by an additional load /•' may he
1 leasured by >f a microscope lilted \\ith an eyepiece
i lierometer, or by mean- of a mi kbOY6
: "d and mo\ in;^ in a nut fastened to a
: Ipport.
A mi< roscope is focal i/rd by tirM brin^in.ur it too near to the
d then, \\ ith ' at the e 1 he whole
e slnwl;. 1'rom the object until the latter is in
M •«!>. In the present case it • r to move the rod than
;icroscope. ; the length of the microscope tulie
fvinpr power of the instrument, and if this
. -n^tli i> the r\p«-riment the eye-
] iece must IKJ re see p. 20).
If the mir. .meter > '.. the instant \\ln-n tlie screw
c >mes into contact with the nxl can be detei-mined either by
i cans of a telephone in a battery circuit including the rod and
i icrometer screw, or by observing the fixed
C ^octin a small mirror one end of which rots upon the rod
\ hile the other end iv>ts upon an adjae.-nt fixed support.
d the po.sition < • mark or pointer near the
134
PRACTICAL PHYSICS
middle of the rod, add, say 3 Kg. and read again, remove the
3 Kg. and read again. Repeat several times, both to be sure
that the elastic limit has not been exceeded and to get a num-
ber of determinations of the flexure. Then alter by a few centi-
meters the distance between the knife edges, and repeat. Take
about five different lengths, and for each length, using the
average flexure for that length, calculate the ratio
I
Find
the average of the five values of — , and by (134) calculate
(/
the Young's modulus of the rod. Express the result in dynes
per sq. cm., Kg. wt. per sq. mm., and Ib. wt. per sq. in.
Exp. 29. Determination of Simple Rigidity
(VIBRATION METHOD)
OBJECT AND THEORY OF EXPERIMENT. — The object of
this experiment is to determine the simple rigidity
of a thin wire.
Consider a cylindrical rod or wire of length I and
radius r with one end fixed and the other end
twisted through an angle <£. This will cause an ele-
ment of the surface as AB to be displaced to AB' .
From the diagram the shearing strain in the outside
7? 7?'
layer of the cylinder is — — • And since BB' = $r,
it will be seen that at every point of the wire dis-
tant T-J from the axis and I from the fixed end, there
is a shearing strain equal to ?U. If S denotes the
shearing stress developed at a point distant r^ from
the axis and I from the fixed end, and JJL the simple
rigidity of the wire, it follows from the definition of simple
rigidity that
FIG. 62.
ELASTICITY 135
Whence S=?lp. (135)
This is the value of the stress at any distance rx from the axis
of the wire.
Tlu* next step is to find what torque would he needed to
the wiiv twi- : is in the figure. Imagine any cross
n divided into // concentric rinirs, each of width Ar. The
ii of the outer boundary of the Ah of these rin^s. be^innin^
at the center, ia - - Ar. and the length of its inner boundary is
— 1 )Ar. If, then, t .>f the /th rin- :iuted
by .1 .
-(e-l)Ar-Ar<^<-J7r/A/- Ar. (1
If the average stress on this ring is denoted by ,v, then < 1 •'.."• )
shows that
Eft*- l>Ar < ^ <gjpr> (137)
On multiplying (186) bj .id denoting by Ft the force
.Inch acts on the ring,
<F < 2
Ft<
t follows that if ili.- toppie which acts ou tin- ring is denoted
J
» - 1 )8(Ar)« .
)n .siiininini: the torques which act on all the rings and letting
, denote the resultant torque, we obtain
Ary g ._ 8<L<2^(Ar)« V
t *-l | ^4-
r, on summing the series indicated by the summation signs,
see p. I
_ '
\ Inch may be rewritten
- 2 n« + n«] < L < [»« + 2 n» + ,/- 1
136 PRACTICAL PHYSICS
- 2(nAr)3Ar + (wAr)2(Ar)2] < L <
— '
+ 2(nAr)3Ar + (nAr)2(Ar)2] .
Since % is the number of rings of width Ar between the center
and the circumference of the wire wAr = r, and the above ex-
pression may be rewritten
[r4 - 2 rSAr + r2( Ar)2] < L < [r4 + 2 r3Ar +
'
The difference between the two outer expressions in the above
inequality is ^ • 4 r3Ar, a quantity which, by choosing Ar
— /
sufficiently small, may be made less than any assigned quantity.
It follows that the value of L is the common limit which both
of the above expressions approach when Ar approaches zero.
That is, if d denotes the diameter of the wire,
TT/ 7T
21 321
This is the torque that must be applied at the end of the wire
to keep it twisted.
If a massive body B is suspended from the lower end of the
wire and twisted about the axis of the wire through an angle
<£, the wire and B react upon each other — B exerting upon the
wire this torque L that keeps the wire twisted, and the wire exert-
ing upon B an equal and opposite torque L' that tends to swing
B back to a position such that the wire is not twisted. That is,
(139)
The second negative sign in (139) means that the torque I/ and
displacement <j> are in opposite directions. If B is twisted
about the axis of the wire and then released, the torque L' will
swing it back towards its original position with an acceleration
which by (104) and (139) is
where K denotes the moment of inertia of B.
i-:i.. \STICITY 137
Since the quantities within the parenthesis in (140) are all
nits, it is seen that the angular acceleration is proportional
to the angular displacement, and that the acceleration and the
displacement are in opposite directions. From this it follows
that the motion of B is simple harmonic. Its period of com-
vihration is, therefore,
(141)
a
Whence, the simple rigidity of the material composing the
wire is
M AMlTi. ATION AND ('< »M IT T A 11 • »\ ;. — Suspend from the
«-nd nf the wire a massive body of such a shape that its
mom, -nt nf in- :i easily be computed, a solid iron cylinder,
or in with it- ident with that of the wire.
•'ind tin- period of viliratinii nf tin- suspended system hy one of
he methods mitlined in Chaptrr II.
Tal. witli a micrometer ealiper.
Mlier tlie dia: eqUAtfoll t«» the fnlirth p..\\
mst be determined with con>. care. Measure it in imt
988 than iistrilmted about equally along the length
f the wire and lake tin- DM UL x! ' li of the \\ ire
• ith a m- >teel tape, and take the necessary dimen-
'ided body. iss of the snsprnded hody
honld he determined within 0.1%. Express the value for the
imple rigidity in dynes per N -xj. mm., and
». \\ ' . . in.
Exp. 30. Determination of Simple Rigidity
(STATK Mi: I IK- 1)
• M:.II:« r A\I> 'rm:«»i:v or K\ i-i:i:i \i r.\r. — The method
:ven in tin- ju-rerdiiig »-.\ pe ri m* • 1 1 1 is a]'plicahle only to wires
mis so small that if a 1 i>pnided 1»\ the
138
PRACTICAL PHYSICS
wire, the period of torsion al vibration will be large enough to
admit of accurate determination. The present method is ap-
plicable to heavier rods.
In the method here employed there is fastened to the lower
end of the rod a massive disk which has its upper face grad-
uated in degrees, and has around its
edge a series of pins placed 20° apart.
In front of the disk and in back of it
are two horizontal scales. The twisting
couple is applied to the disk by hori-
zontal forces acting tangentially at its
circumference. Masses mL and w2 are
suspended by cords which pass in front
of the two horizontal scales. Tied to
each supporting cord at about the level
of the pins in the disk is another short
cord which has at its other end a loop
that can be slipped over one of the
pins, thus twisting the graduated disk
through an angle which can be read by
means of a pair of pointers fixed above it.
Let the forces in the horizontal cords
be denoted by F1 and F2. Then from
the diagram (Fig. 64)
= -, (143)
and since F1 and m^g are perpendicular
to each other, and Fv m^g, and the
FIG. 63. tension in the supporting cord are a
system of concurrent forces in equilibrium,
nt-,0 s-\ 4 ,IN
-32 = tan w. (144)
~E1 >« s
F\
From (143) and (144) it follows that
,, _
l~
ELASTICITY
139
This is the force with which the horizontal cord pulls on the
point where the three cords join. The pull JFy which the cord
exerts on the disk is equal tu this, but in the
direction. Thai
(it.-,,
The second negative sign in (14-~> > denotes
that /',' and jr have opposite dii If
:idwi2are equal ami their
supporting tlireads looped ever diametrically
opposite pins, and if the points from which
the upright cords han«j '*"'*-' equidistant from
the plane of the wire and support in.i:
t\ - /'.. I: w.« «lrop the subscripts and
by It the diameter of the disk i IK leased
>y txvi.-c th«- MC hui-i/Miital cords,
lie nioiii.-nt of the couple that tend-
the disl. :n its equilibrium po.sition is
(146)
pp. l:;i-1:',»; it has been shown that it a « y Under of length
I made of a material of simple rigidity ^ is
d tliroiiLrh an angle of <f> radians, the tor.pit- which this
wist exerts on the b< twisted
L'=-
(117,
inber of ( 1 !''• » is the tonpie that tends to
i-om its position of equilibrium, and the ri^ht
IT > is the tor<pie which t.'iids to restniv it to that
on. When the one of these is equal and
, p<>- . That is,
irD _ 7r
"~
CU8)
140 PRACTICAL PHYSICS
The negative sign in this equation occurs because x and </> are
measured in opposite directions. Using simply the numerical
values, and writing in place of <£ radians its value -£—. 2 TT
360
radians, where y8 is the number of degrees in </> radians, (148)
gives
mx (149)
MANIPULATION AND COMPUTATION. — Carefully measure
the diameter of the rod or wire in at least ten places with a
micrometer caliper. Take the diameter of the disk with a
vernier caliper. Measure h and I with a meter stick or steel
tape. Use such loads and loop the cords over such pins as to
get a series of some half dozen values for /8, each somewhat
larger than the one before it, but the largest not much more
than 90°. The loads in the two pans must be equal, and the
cords should be looped over pins far enough around to give
fairly large values for x. In getting each end of the distance
#, record the reading on each side of the cord and use the
mean as being the position of the middle of the cord. Find
the average value of ^, and by (149) find p. Express the
result in dynes per sq. cm., Kg. wt. per sq. mm., and Ib. wt.
per sq. in.
Exp. 31. Determination of the Modulus of Elastic Resilience
of a Rod
OBJECT AND THEORY OF EXPERIMENT. — Resilience of a
body is the energy it possesses du.e to a strain developed in it.
The ultimate resilience or modulus of resilience is the strain
energy of the body when strained up to the. elastic limit. Cor-
responding to the different types of strain are different types
of resilience : tensile resilience, flexural resilience, torsional
resilience, etc. The resilience of a material is usually given
either in terms of work per unit mass or work per unit volume.
KLASTICITY 141
'1 In- object of this experiment is to determine the flexural
iei ice of a rod.
~is mi two knife filers and is distorted by a force
applied at the middle point. Let the length nl' rod between
;., the area of cross section be A sq.
cm., the density he p grains per cu. em., the mass of that part
nf the md between the kn /// grains, the load li-
sa ry tn strain the n-d to its clastic limit he /'dynes, and the
displacement • >int nf the rod by ti. /-'he
. until the elasti- limit is reached, the distortion
is proportional to the force applied, the averai:«' force acting
while the di>i i increasing from zero to / is \ /•'. There-
fore the .strain energy stored up in the specimen, that is, the
ulns of tlexural : 6 of the rod.
// | /V • rgs.
The modiilu >• per unit of volume is
R Fl
ergs per cc.,
us o! uce per unit of mass is
i? '•'
Jtm = ergs per gnim,
« • . if f«r d in Arams' weight, /•". instead of dynes,
/•' ntimeters per gram.
MAMI \ LOT COMPUTATIO --The aj.|.aratns con-
Bi ts of the rod to be examined \\ ith i: ng upon knife
e« gee, and a microscope fitted with an e; micromel
n iasii All of the apparatus must
IH placed upon astij.p«»rt n \ibratioii. NVei^h the b,u.
111 ;asure its le: :<>ss section, and calculate the mass of
ti t : ,-n the knife edges. < the mi«i<>-
upon a tine cross engraved upon the center of one of the
v< rtical faces of the bar, or upon the point of a needle fastened
ri -i<lly to the middle ,,f t: ( ly add weights to the
142 PRACTICAL PHYSICS
pan suspended from the middle of the bar, taking a reading of
the deflection after each addition. During the progress of the
experiment carefully plot weights and deflections on cross-sec-
tion paper — the weights as abscissas and deflections as ordi-
nates. As would be expected from Hooke's law, the line
connecting these points is straight from the point of zero load
up to the point representing the elastic limit, and from there
it bends toward the axis of ordinates. Thus, from the curve
can be obtained both the value of the load necessary to strain
the bar to its elastic limit, and the deflection produced by this
load. All the data are now at hand for determining the value
of the modulus of flexural resilience per unit volume, or per
unit mass.
CIIAl'TKK IX
VIM -usiTY
IN an elastic solid a shearing stress produces a shearing
strain, and this strain, in turn, produces a restoring stress. If
tin- body is subject to a given stress that is not beyond its
limit, the strain does not change with the lapse of time,
;he ratio of tin- > tress to the strain is a coeilicient «»f
In a liquid a shearing stress produces a shearing strain, and
with >ped a stress that opposes tin dis-
tortion but does not tend to restore tin- liquid to any former
shape. In fact, any shearing stress, however slight, pic < luces a
<• 'iitinuously increasing strain, and the ratio of the shearing
s ress to the shearing Miain thereby developed in one second is
< lied the <-.,. t° viscority of the liquid.
lid layers of the liquid // em. apart, the
1 \V.T layer at rest, and the upper moving x em. J-.T see. It .1
i thean-aof the upper layer and /-'tlie f..r<-e \\liieh is in
F
\ forward, then the shearing stress applied is -, and the strain
• I
9
per second is -. Consequently the coefficient <>;
-sity of the liqu!
" = - =
Eip. 32. Determination of the Absolute Coefficient of
Viscosity of a Liquid
POBBDILUra MKI HMD
OB.FI:< r \M» Tn -Consider a column
liquid ilnwini^ through a tuhe of length /, and witli a radius,
144 PRACTICAL PHYSICS
r, so small that there will be no eddies in the liquid column.
Imagine this column to be made up of a large number n of
concentric hollow cylinders of very small thickness Ar. Sup-
pose that all of these hollow cylinders but one could be made
solid, so that there would be a solid rod surrounded by a thin
layer of the fluid, and this again surrounded by a solid tube.
While the rod was moving, two forces would be acting on it
— one due to the viscous resistance in the tube that was still
liquid, tending to retard the motion of the rod, and the other
due to the difference between the pressures at the two ends
of the rod, tending to accelerate it. If the radius of the rod
were #Ar, the viscous resistance in the liquid tube surround-
ing it would, by (150), be
Ar
and \ip denotes the difference between the pressures at the two
ends of the rod, the force to which this difference in pressure
would give rise would be
If the rod were moving uniformly, Fv would equal Fp^ i.e.
77 • 2 TTX&rl - S
— - -- = p •
Whence . = . (151)
"2 rjl
This equation shows that the difference in speed between the
outside of an axial cylinder of the liquid of any radius and the
outside of the adjacent layer is small near the axis where x&r
is small, and increases in direct proportion with the radius of
the cylinder.
If it is imagined that these concentric layers of liquid are
congealed without interfering with their ability to slip past one
another, then on account of their difference in speed, at any
instant after the flow has begun, the end of the inmost cylinder
will protrude beyond the end of the adjacent layer, this second
VISCOSITY 145
\vill protrude beyond the end of tlu'thinl layer, and so on.
in the speed of the inmost cylinder relative to
; x.r the speed of the seeond layer relative to
the third. BtO. Also 1,-t r{ represent the volume of the portion
of the inmost cylinder that protrudes heyond the end of the
the volume of the seeond solid eylinder that
• ides he\ «»nd the third layer, ete. Then the entire volume
_;vd hv the capillary in time / is
r-r1 + r,-hr;J-f- ... +Vm.
Row
9tm . , rri -JA/-I-V. r3 = 7r(3Ar>
I'uttinir these \alues in ( 1 .VJ ». \\
and o i tilting for *p 9V 8V etc., their values from ( 1.">1 ),
v/e obtain
v
|.2»+*+... + i
i
Ar) +
r. Therefore in the limit, when A/-
F
pre0fore is dm; to a column of li<iui<l of heiglit // and
= p<jh. On putting this value in (1.V1), and
I- /;. \\r i
i
| (
i that in dn ; ; has IMM-II tacitly
umed («) that the viscous resistance to th,- tl..\v of the liquid
uniform t hmu^lmiit tin- «-ntin- length <>f the tnln-, ( I ) that
i ie lines of tl"\\ ..f li<piid in the tube are parallel to the axis
' ihe tul.e throughout its length, f^) that no part of the energy
• plied to the liquid in the tul»e appears as energy of motion,
( d) that then- i> no .-rt'ret at the nutlet due to surface tension.
XI
146 PRACTICAL PHYSICS
The conditions demanded by (V), (6), and (<?) can be realized
to a sufficient degree of approximation by using a tube that is
both long and of narrow bore and having the liquid flow through
at a uniform rate. Condition (d) is met by immersing the dis-
charge orifice in a portion of the liquid having a
considerable free surface.
MANIPULATION AND COMPUTATION. — A vis-
cometer that fulfills the above conditions is illus-
trated in Fig. 65. The vertical tubes AB and CD
are of uniform bore and are graduated in millimeters
throughout their length. The capillary tube BE
is straight and of uniform circular bore. In order
that the temperature of the liquid being investigated
shall be constant and definite, the viscometer is
supported in a suitable water jacket supplied with
a thermometer.
The length I of the capillary tube is measured
with a meter stick. The mean radius of the bore is
determined by measuring the length of a known mass
of mercury at different positions along the length of
the tube. An amount of mercury sufficient to make a
thread about four centimeters long is drawn into
the tube by suction applied at the opposite end, and
this thread is measured in length at different
FIG. 65. equally spaced positions along the length of the
tube by means of a dividing engine. Knowing
the mass of the mercury thread and the average length,
the average radius of the bore of the tube is determined.
A tube with a bore departing very much from uniformity
must be rejected in determining the absolute coefficient of
viscosity.
In order to determine the V in (154) it is necessary to cali-
brate the lower part of the tube CD. This may be done by
putting a solid stopper at E, removing the one just above (7,
and dropping into CD known volumes of water from a burette.
VISCOSITY 1 17
After each small volume of water is dropped in, a reading is
made of the t<»p of eaeh water column — the one in CD and the
one in the burette. From these readings a curve is to be
plotted coordinating the volume of water in CD with the read-
ing of its surface on the (.'/> scale.
After thoroughly cleaning and drying the parts of the vis-
nbled, a quantity of the liquid under investiga-
tion is introduced, and this liquid column run back and forth
until it is free of air bubbles and the tubes are coated with
a thin film of the liquid. The quantity of liquid introduced
should be such that it will form a column extending from a
point n. -ar the upper end of the tui a point near the
end of CD.
With all the rubber stoppers tight, and the stopcock S open,
run the liquid into the position mentioned above, close the stop-
,uid place the viscometer in the water bath. After the
temperature has become constant and of the desired value, with
\\atch in hand open the cock S, and when the meniscus in
. 1 // lies some previously selected scale divi>i"ii X. start the
the meniseus reaches some second selected scale
diviv topthewatol be valuator tin (154).
When the upper meniscus was at X the lower nit-nix -us wmfl
at some point //. and when the upper meniscus had fallen
riaen to some point y'. The p«»si-
i //' can be obtained by opening .s1 and again run-
ning the liquid into A /> to the points jr and ./•'. The mean of
the v.-rtical distances bet\\ -the value
for h in (l.r>4). I • es can be obtained by the scales
.!// and CD. p can be obtained by means of a
balaner and a 5oC« pi|>ette. Vis obtained by finding from the
i the volume of water that would be held
:he marks ij and
At ve sets of observations should be taken and tin;
i
tvera. . in ( 1-~>1) to get rj at the temperature
>f the experiment.
148 PRACTICAL PHYSICS
Exp. 33. Determination of the Specific Viscosities of Liquids
(COULOMB'S METHOD)
OBJECT AND THEORY OF EXPERIMENT. — On account of
such experimental difficulties as that of obtaining a capillary
tube of uniform bore and circular cross section, of accurately
measuring its diameter, and of keeping the capillary free from
minute air bubbles and particles of foreign substances, the
determination of a coefficient of viscosity by the method given
in the preceding experiment is very troublesome. Coulomb's
Method is especially suited to the determination of the relative
viscosities of those liquids used in engineering and the arts
which are liable to contain particles of solid substances in
suspension, e.g. lubricating oils. By specific or relative vis-
cosity is meant the ratio of the viscosity of the liquid to the
viscosity of water. The object of this experiment is to deter-
mine the specific viscosities of a series of liquids.
If a massive disk suspended axially by a thin vertical wire
be immersed in a liquid and set into torsional vibration, it will
be shown in the chapter on Damped Angular Vibration, Vol.
II, that the ratio of the lengths of any two successive swings
from one end of the path to the other is a known function of
the " damping constant," a, which is proportional to the viscos-
ity of the liquid surrounding the moving body.
Let o-j represent the ratio of the lengths of any two successive
oscillations of the disk when immersed in the first liquid ; T^
the period of vibration of the disk when immersed in the first
liquid ; and av the damping constant for the first liquid. Let
<r2, 7y, and a2 represent the corresponding quantities for the
second liquid. Then, from (31), Vol. II,
and ^ = e ^ (156)
where e is the base of the natural logarithms and K is the
VISCOSITY
149
moment of inertia of the suspended system. Dividing (155)
bv ( l-~'h) and putting the resulting equation
into the logarithmic form, we obtain
Whence the relati\ -ity of the two
liquids, f,
a, = TV log <r,
a, 7,' log «r,'
(167)
If the second liquid is water. z is the specific
; he first liquid.
MANIIM LATION AND < \TioN.-In
.ipparatus here emj . one
of a thin piano \\ .-d to a
r.gid support while the othe; ittached
t) a '. ro<l carr\ : •! eii-eh-
;i -id the mas>i\c di>k wlii<-li is to l.c imm.
i i the vari.ms liquid*. Ti..- di.sk has a thin
j- em by whi<-h it tad to the rod car Fl<; ^
i ig the divid- : --ssel containing
i ie liquid l)'-iir_r .studied is surrounded by an oil hath heated
1 v in.
\sthe :uaii\ liquids U very different at ditlr rent
i mpcr.ituivs. it !• always oeoeMft] Jte the determination
; the temperature at which the liquid is t.. he used. For in-
s an. ..f cylinder oil should be made at about 1"»(> to
1 *5° ('.. whil- Jieoils should he tested at ahoiit .",M ( .
^ iH-e the n-lative viscusities of many pairs .,f ipeoimei
(< 1 art- even reversed with a change of temperature ,,f less than
1 >0° < !e to jud_L,'e tlie relative lubricating values
o oils from t: :.-s determined at a tempera-
t ?*e much ditTerent from the temperature at \\hich they are to
At'; ;n^ and as^.-ml»lin^ the appai.ilus and allowing
tl e temperature of the s[- to attain the re«piired value,
150 PRACTICAL PHYSICS
twist the disk through about 180° by rotating the rod above
the divided circle. With a stop watch observe the time of ten
complete vibrations. One tenth of this time in seconds is the
period T± . By means of the pointers P and P' make a series
of readings of the turning points of successive swings to the
right and to the left. The number of scale divisions through
which the disk turns in rotating from one end of its path to
the other is the magnitude of that oscillation. Calling the
magnitudes of these successive oscillations fj, f2, f3, etc., we
have
Whence
= 0"
4!'
and, in general, fn = f^™-71.
If, say, twenty oscillations were observed, we have then
etc.,
and by finding the average of ^-, |*-, !*-»••• f^i and taking
Sll ?12 ?13 ?20
the tenth root of this average, o~l is found.
In the same manner find T% and <r2. These values of T^
T^, (TV and o-2 substituted in (157) will give the relative vis-
cosity of the two liquids. With liquids having viscosities not
very different, the value of T± will be so nearly equal to T2f
that their ratio may approximate unity ; but it is never allow-
able to assume their ratio to be unity without experimental
verification.
Instead of reckoning viscosity in absolute units or with ref-
erence to water at some standard temperature, the viscosity of
VISCOSITY
151
a liquid is sometimes rated in comparison with the viscosity
of an aqueous solution of sugar having a definite concentration
and temperature. For example, the viscosity of a certain oil
at •~>(>: ('. may In- specified as being equal to the viscosity
of an 18% aqueous solution of pure sugar at _(l C, Values
for tl Mtifs of aqueous sugar solutions of various con-
centrations, referred to water, are given in Table 10.
1'AKT II. IIKAT
TART II. HEAT
CIIAl'TKl! X
TEMPERATURE
THE comparison of temperatures involves several arbitrary
conventi* Miperatuivs cannot he directly measured, —
• •an be compared only in terms of some other phenomenon
which depends upon temperature. Of the various phenomena
which are used for the comparison of temperatures, the follow-
;ie most important 8 («) change of the volume of a gas
or liquid kept at constant piv>sure, (£) change of the pressure
<>f a gas kept -ant volume. ( ,- ) change of the electric re-
ice of a metal wire, (</) production of an electromotive
at the junction of two dissimilar metals, (.•) quantity of
nergy radiated hy the hot body, (/) luminous intensity of the
liated by the hot body. In
ases (<i), (6), (c?), and (-/ fcessary to select a particular
• substance. In all opaei .-cessary to adopt
AO ; ires as standard or lixed points of a
scale, and to divide the interval Let ween these
oints int.. a d. tinite number of spaces or degrees.
The scale of temperatures that has been adopted as standard
i . based on the change of pressure \\hi.-h a change of tempera-
• ire : iii a fixed mass of hxdm^,.,, kept at constant vol-
T me. By means of a gas thermnmcter temperatures as hi^-h as
1 700° C. can be compared. However, as the standard ^as ther-
i <ime .th hulky and fragile, it is seldom used except in
8 ijititic work and fur the purpose of standard i/in^ other ther-
n ometei-s. All other th. . -T8 are calihrated in terms of
t ie gas tl.
HI
156 PRACTICAL PHYSICS
On account of the comparatively simple technique necessary
in its use, the mercury-in-glass thermometer is employed when-
ever the conditions of the measurement permit. By using a
very hard glass, and filling the space above the mercury with
an inert gas at a pressure sufficient to prevent boiling of the
mercury, a mercury-in-glass thermometer can be used up to
about 550° C. (1000° F.). Mercury -in-quartz thermometers
having the space above the mercury filled with gas at 60 atmos-
pheres pressure can be used up to 700° C.
Thermometers available for temperatures above 500° C. are
often called pyrometers. The electric resistance and the ther-
moelectric instruments are available for temperatures up to
about 1500° C. For temperatures above this, radiation pyrome-
ters are available.
Measurements of temperature, even by means of the mercury-
in-glass thermometer, are subject to so many sources of error
that an accurate determination of temperature is a task of some
difficulty. Nevertheless, the thermometric methods have be-
come so highly developed that if proper precautions are taken
and proper corrections made, a measurement of temperature
made with a mercury-in-glass thermometer between 0° C. and
100° C. can be trusted to 0°.005. The methods described in
the following pages correspond to an accuracy of about 0°.05.
The principal sources of error in the use of a mercury-in-
glass thermometer are : —
1. Errors in reading the thermometer due to parallax. Usu-
ally the scale of a thermometer is at some distance in front of
the capillary, so that, iinless the line of sight is normal to the
length of the tube, the reading is too high or too low. The
two principal methods employed for keeping the line of sight
normal to the length of the thermometer tube are, (#) to hold
a small mirror against the back of the thermometer, and to place
the eye in such a position that the top of the mercury thread is
in line with the image of the eye seen in the mirror; and (b) to
observe the thermometer at a distance by means of a telescope
containing a cross hair in the eyepiece, the telescope being fas-
TEMPERATURE 157
iciied normal t«> a rod placed parallel to the tliermometer tuhe.
Tin- i iiiu.Ni he arranged ><> that when it is moved alon^
!pportin<_r rod t-> thf end of the moving column at
different heights, it will always remain normal to the support-
ing rod. A cathometer is usually most convenient for this pur-
pose, l>ut a short open tube without lenses, having crosshairs at
WO ends, and sliding cither on the thermometer itself or on
a parallel rod, serves the purpose very well.
Krrors due to the changes in the volume of the hull) la^iiiL;'
liehind tl. . temperature. A ri>in^ thermometer in-
0 low. and a falling thermometer too hi^h, a tempcra-
Tliis hiLT is du- of the ijhiss of which the
t hcrniom. ! be kept for some davs
iniform temperature of. sa\. L!" <'., and then plunged int»>
,i hath of melting 1 the temperature observed ; and if it
•rat lire of 100° (\, and a^ain plunged
lito the hath of melt;: ; re n«»\v obeeTVed will
be lower than tlie one piwioii>ly ol.tain«-«l. The increase in the
volume of the hull) du .iture docs n,,t at once
pear, and the zero point may he depressed as much as half
i degree for some kit. tss. This dcpiv«vxi«.n of th.
hen t lie tempera' which the thermoine-
ar has beeni , and when the time u rthai
at the higher temperature. The dr-
don {Hjrsists t I and even months hefore the normal
i-.-^ained. It follov. hih- the ther-
is hein^ )tts temperatures the zero point is
This i. !].erature determinate
the /ero point at this particular time is
;im\vn. , uc of the /.ero point can he ol, t;iincd hy cool-
ng the i ,i) to the temperature of nicltin
miii- !• the d.-Mivd temp. r.-adin^ has been
'iade. Then, if no other errors aflfed th- :tion. the true
: iajM iirtwccn the nhserved temperature
nd the \ the d.-pi.->.-d /ern. T: '.led the '•<!«•-
temperature, and is the
158 PRACTICAL PHYSICS
only method capable of yielding the most accurate results
attainable.
3. Errors due to the exposed column of the thermometer
being at a temperature different from that of the bulb.
Let T denote the true temperature of the bulb ;
£, the temperature indicated by the thermometer ;
s, the temperature of the exposed part of the stem ; and
e, the reading where the stem emerges from the bath.
Then the length of the exposed column is (t — e) degrees, and
the difference between its temperature and that of the bulb is
(T — s). Since the coefficient of apparent expansion of mercury
in glass is about 0.000156 per degree C., this exposed part of
the column, if it were to be raised in temperature (2T— s)
degrees, would increase in length 0.000156 (t — e) (T — s)
degrees. That is,
T= t + 0.000156 (t - e)(T- «).
m t — 0.000156 8 (t — e)
Whence T= ______J.
or, employing approximation (5), p. 7,
T=[t- 0.000156 s (t - *)] • [1 + 0.000156 (t - *)],
or, neglecting the term which involves the square of 0.000156,
T = t + 0.000156 (t - «)(« - e). (158)
4. Errors due to inequalities in the bore of the tube. These
errors are corrected by calibrating the tube as described in
Experiment 34.
5. Error in the graduation of the stem ; that is, although
the divisions are of equal length, their length is not such as to
make just a hundred divisions between the boiling point and the
freezing point of water. Let Tv denote the true temperature
of the vapor above boiling water as determined by reading the
barometer (see pp. 176—178, and Table 12), tv the temperature
indicated by the thermometer when it is immersed in the vapor
above boiling water, and t0 the depressed zero reading taken
ll.MPKKAH 159
immediately after t,. was observed. Then the number of de-
- that oujrht to be between the point where the thermome-
ter r- ud that where it iva«ls / ifl '/'.. and the number
..I decrees that really are between those points is (ti—t0). It
follows* that any temperature difference read from the ther-
mometer is to be multiplied by a faetor
*--• (159)
6. Errors due t a tin- pressure to which the bulb
is subjected. \ liaise of pressure will cause a change ot
; of the mercury column independent of any change
Miperatii!''. I -ually the experimental nietliod can be
• eliminate this source of err-
7. Error due to capillarity. IM a thermometer of very small
- not move smoothly but m
in little jumps. Tiii- Lfl much greater when the
temp. :iin^ than when rising. In fact, the
capillary action D impossible to measure necur-
:!liu^ temperature by means of a mercury -in-
glass thermometer.
Tn i:. — A thermometer
penitures to thousandth-
i a long space for each degree of
. that. ; '1 on the ordinary plan, the
668. This would require the use of a numb
f ordinary laboratory
\Vhen it mine definite
trm: nail temperature differences,
bermometei I .Miami can be used at
temperature for which n mercury-in-^Iass tlier-
mom< idiarity of this ther-
7 ) at the Upper end >'"
of the tube, by in. vhid, the .|iiantitv of mercury in the
bulb can he inereas.ed or dimini.shed. i usually about
160 PRACTICAL PHYSICS
five centigrade degrees in length and is divided into hundredths
of a degree.
In setting the instrument, a sufficient amount of mercury
must be left in the bulb and stern to give readings between the
required temperatures. First invert the thermometer and tap
the tube so that the mercury in the reservoir will lodge in the
bend B at the end of the stem. Now heat the bulb until the
mercury in the stem joins the mercury in the reservoir. (See
Fig. 67.) Place in a bath one or two degrees above the upper
limit of temperatures to be measured. If now the upper end
of the tube be flipped with the finger, the mercury suspended
in the upper part of the reservoir will be jarred down, thus
separating it from the thread at the bend B. The thermometer
is now set for readings between the required temperatures.
Exp. 34. Calibration of a Mercury-in-Glass Thermometer
OBJECT AND THEORY OF EXPERIMENT. — If the bore of a
thermometer is not uniform in cross section, the length of the
tube corresponding to a degree difference in temperature will
not be the same at different parts of the tube. And as it is
impossible to get a perfectly uniform capillary, it is necessary to
determine the correction to be applied to any particular reading
to take account of the irregularity in the bore of a thermometer.
Again, if the fixed points are incorrectly placed on the stem,
this will introduce an error throughout the scale. The object
of this experiment is to construct for a given thermometer a
curve by which to correct errors due either to the irregularity
of the bore or to the location of the fixed points.
The experiment consists of two parts. First, the length of a
short thread of mercury is measured at different parts of the
tube, and from these lengths points are found throughout the
whole length of the tube that separate equal volumes. Second,
the position of the fixed points is determined by placing the ther-
mometer in the vapor of boiling water and also in melting ice.
RE 161
MAN i IT LATH >\ AND COMPUTATION. — The length of tin*
I t.» be broken off depends upon the thermometer. If the
thread is too long, local irregularities of bore are not evident;
if the thread is too short, its changes in length are minute. If
a dividing engine is available, a thread not more than a centi-
:• long is advisahle. If no magnification is to be used and
.ermometer is an ordinary Centigrade thermometer grad-
1 from 0° to 100° in degrees, a thread some lifteen degrees
long is perha: <ry.
.tration i,f the calibrating thread requires some dex-
terity. In blowing the bull) on a thermometer tube, a slight
notion is usually left where the bulb and tube join. If
such a thermome; >-d and then given a sudden jar,
1 i> likely to separate at this point. If there be no
s idi constriction, tin- thread may be separated by laying the
•le and striking the upper end of the tube
v itli a small block of wood. If this is not carefully done, how-
be produced in- stem near the bulb.
1 ' the bore has an enlargement at the upper end, the column of
i ercury that has been broken ntV is allowed to run into this
t ilar- .uid to remain there while the tube is b.-in-jr eali-
1 -ated. The be ightly warmed until a thread of
i ercu ito the tube, and this, in
1 irn. is separated from ti. : ry in the bulb. This is the
t iread that is used in the fcioiL If the capillary has no
t • ihir. .e upper end in which to store, part of the
ii ercury, it may be necessary to use two mercury threads to
c librate the t \\ o ;lir tube. When this is the case, the
I lib is <•" ' !i a mixture of ice and salt if necessary, until
;i ' the mercury lias run into the bull) except the length that is
t( be br< , thread is separated and run to the
f rth- f the tube. In order to make measurements in
ft! \lowerend of the tube, this part of the thermometer must be
fi cd of n ie r. in y and another thread separated as before.
When a tli; off. it is to be brought nearly
t on,- end of the t u be and the position of both ends carefully
162
PRACTICAL PHYSICS
read, then moved along through a quarter or a third of its
length and the positions of both ends again read, this process
being repeated until the thread has been moved to the other
end of the tube. Suppose that when this is done — a mirror
being used as suggested on p. 156, and readings being made to
twentieths of a degree — the readings are those in the first,
second, fourth, and fifth columns of the following table : —
LOWER END OF
THREAD AT
UPPER END OF
THREAD AT
THREAD
LENGTH IN
DEGREES
LOWER END OF
THREAD AT
UPPER END OF
THREAD AT
THREAD
LENGTH IN
DEGREES
- 12°.30
+ 3°.60
15°.90
39°.95
55°.65
15°.70
- 7 .25
8 .65
15 .90
45 .30
60 .95
15 .65
- 2 .05
13 .85
15 .90
50 .00
65 .65
15 .65
+ 3 .00
18 .85
15 .85
55 .20
70 .80
15 .60
8 .15
24 .00
15 .85
60 .15
75 .75
15 .60
14 .30
30 .15
15 .85
64 .90
80 .45
15 .55
20 .00
35 .80
15 .80
70 .10
85 .60
15 .50
24 .55
40 .35
15 .80
74 .85
90 .35
15 .50
30 .05
45 .80
15 .75
80 .05
95 .50
15 .45
34 .90
50 .65
15 .75
83 .30
98 .75
15 .45
The quantities in the third and. sixth columns are calculated
from the observed quantities. The quantities in the first and
third columns give the following curve, which shows the length
of the thread when at different points of the capillary. From
this curve the lengths of equal volume portions of the tube can
be determined as follows : —
— 10
90 100
The curve shows that if the bottom of the mercury thread
were at 0° its length would be 15°. 87, so that the top of the
1 EMPERAT! UK
163
•1 would be at 1")°.8T; it also shows that if the bottom of
the thivad wen- at 15°. 87 its length would be lo.°81, so that the
top uf it would be at 31°.»'»S: if the bottom of the thivad were
at 31°. 08 its length would be 15°.73; etc. These values are
recorded in the f<>llo\vin_ : —
VLE BE-
TWEEN Wli!
I'MES OF BORB ARE
Ey<
LKKOTHS or THREAD
BETWEEN Kyi'AL VoL-
t ME POINTS
POSITIONS or K y i \ i
K POINTS ir
BORE HAD BEEN
Rnvoa
CORRECTIONS roR
r..iM- is FIK-I
0.00
±0.00
1.-..M
15.69
-0.18
fJ8
o.:;i
17.41
15.66
17M
- o.:;-j
yaw
1&48
7-.ll
0.90
ua
±0.00
v .
1
t
t
The quantities in tin* third column arc fmiud by inultipl\ iu^
1, 'In- average length of thread U-t \\vrn iMjual
Ininr ] i tin- fourth rolumn arc found
- subtracting the quantites in the first column from those in
c third. The quantities in the first and fourth columns ^ive
e upper cnr\«i in Fi^. •'>'.). This curve gives the corrections
tl at must be up] Mings at different puints along the
B< lie ' unt uf irregularities in the bore.
i'V displacement <>f the fixed points will
n< w b B l.-tinitiun, the lower fixed point (0° C.
01 3l; i iie temp (rf melting ice. The upper fixed
164
PRACTICAL PHYSICS
point (100° C. or 212° F.) is defined as the temperature of the
steam produced by water boiling at sea level and latitude 45°
under a barometric pressure of 76 cm. of mercury when the
barometer is at the temperature 0° C.
Observe the barometric height, noting the temperature of the
barometer by means of the thermometer attached to the instru-
ment. Ascertain from the laboratory instructor the latitude
and altitude of the laboratory. From these data compute,
in the manner explained on pp. 176-178, the corrected baromet-
ric pressure H reduced to standard conditions. From a con-
sideration of the number of figures that can be
trusted in the uncorrected readings determine
which of the corrections on pp. 177, 178 it is
worth while to make.
Suspend the thermometer in the vapor of
boiling water. It must not be immersed in the
water itself nor be so near the surface that the
bulb will be spattered by drops of water, because
the temperature of boiling water is influenced by
the nature of the surface composing the vessel
and by the presence of slight quantities of dis-
solved impurities. But the temperature of the
vapor depends only upon the pressure. Re-
gnault's hypsometer consists of a reservoir R
(Fig. 70) in which the water is boiled, sur-
mounted by a tube in which the thermometer is
suspended. After passing through this tube the steam passes
through the jacket J and escapes into the air at E. M is a
water manometer which serves to measure any difference of
pressure between the steam inside and the air outside. If the
manometer indicates a pressure of d mm. of water, i.e. TTT~^ nim.
lo.b
of mercury, then the total pressure on the surface of the boiling
water is H -f- 75-^- mm. Call the observed boiling point T0.
lo.b
Draw the thermometer up until the upper twenty degrees or so
FIG. 70.
TKMPKRA'lTKi: 165
of the stem is exposed. After about live minutes note the read-
ing and also the reading at the top of the stopper, draw the
thermometer up some twenty degrees farther, and, after about
live minutes more, ivad again at both points. Repeat until the
zero point is at the stopper. The ditVereiiee between the read-
ing at the top of the mereiiry when the thermometer was wholly
immersed and the reading in any of the other eases is the
in exposure eorreetion for that partieular ease. Note ap-
proximately the temperature of the air in the neighborhood of
the hyp.soineter. From (1">S) and (-) calculate the stem ex-
posure correction for each OM6. Plot on the same sheet two
en; rdinating the thermometer reading at the stopper
and the observed stem exposure eorreetion, and the other coor-
dinating the thermometer : a the stopper and the ealeu-
d >t« -in exposure cornet i MM.
ttOVe the thermometer from the hypsometer, allow it to
1 in the air ; : 40° C., and then immerse it in a vessel
illed with snow or shaved ice which contains enough water to
ill the i: -. Tliis gives the depressed /.TO point.
•> Table 12, obtain the temperature of the
-apor of water boiling at a pressure of -^+7o~j»' Call tnis
rue temperatui-e T.. Then (T0— Tt) is the error «,f the upper
ixed point, and ( 7*, — T ) is the correction to be applied to the
eadr - !i the above example the error of the
•niling point is found to be +0°.»J, and the error of the free/ing
•oint -f n°.,S. Then the oorreotion for the boiling point is
— 0°.8. If now on the game
:dinate ; n which the corivtion curve for
iTeg!. of bore was plotted, the free/ing point correction
• •iitered alung the axis of ordinates opposite the, zero of ab-
issas, and tlie boiling point correction be entered opposite
|;e observed boiling; ;d these points be connected by a
aiight line, as sho\\n by the dotted line in Fig. «'»!', this line
;he con odiftte points of the scale due
• the displac«-meii-
ives
. th«
166 PRACTICAL PHYSICS
By adding the ordinates of the correction curve for the irreg-
ularities of bore — the upper curve — to the corresponding
ordinates of this correction curve for displacement of the
fixed points — the dotted line — the lower curve in Fig. 69
is obtained. This is called the Calibration Curve of the
thermometer.
If this calibration is done with two mercury threads instead
of one, the calibration should extend from each end to a dis-
tance past the middle of the tube. The curve analogous to
that in Fig. 68 will be a continuous line, but along the region
where data were taken with both mercury threads, one branch
of the curve will be above the other. In this region find the
ratio of the ordinates of the two curves for three or four posi-
tions on the thermometer scale. This ratio must be really the
same for all points on the thermometer scale. By multiplying
any ordinate of one curve by the averages of the values found
for the ratio, the corresponding ordinate of the other curve will
be obtained. Proceeding in this manner, a continuous curve is
obtained, just as though all of the calibration had been per-
formed with a single mercury thread.
Exp. 35. Calibration of a Resistance Thermometer
OBJECT AND THEORY OF EXPERIMENT. — The mercury-in-
glass thermometer is unavailable for the measurement of ~tem-
peratures much below — 30° C., or above + 300° C. Although
the gas thermometer can be used for any temperature for which
a suitable material to construct the bulb can be found, it is such
a large awkward instrument, and the difficulties of the manipu-
lation are so considerable, that it is suitable only for stand-
ardizing more convenient types of thermometer. Since the
electrical resistance of metals varies continuously with the tem-
perature according to definite laws, and since the accurate
measurement of resistance is attended with no considerable
difficulty, thermometers depending upon this change of resist-
ance are in common use for measuring high and low tempera-
TEMPER ATI' RK 167
tures. Platinum is the material usually employed, both because
its resistance at any given temperature does not change with
time, and because the law connecting the temperature and
nice of a wire made of it is expressible by a simple for-
mula throughout a very wide range of temperatures. It has
been shown by experiment that if /»'., represents the resistance
of a piece of metal a < .. then throughout a more or less
definite range of temperatures the resistan< B /«'. at t° C. is
Me by the equal;
/;. // [1 + at + W], (160)
where R& a, and b are constants. In order that a resistance
thermometer may be u> N three constants must be
known. The object of this experiment is to determine the
I of these OOOOtantfl for a given 106 thermometer.
If the : .-(»f the wire at three different temperatures
ie known, three equations of the form oi are obtained,
ind 1: ons tlie values of the three constant-
>e calculated. '1 .t and boiling point of water
il temperature! for the e\j.i-riment. The re-
naming temp.-rat .•_; point of any convenient
ubst. /. sulphur, which boils at -lit Hut from
at very low tempera-
\arand I ; have shown that at t he absolute zero
re it is highly probable that the resistance <
TO. Assuming t ktion, the resistance of
B need be m i at but two tempcra-
-implitied process gives the values of the constants
ii <l'i(>) with iracy for most purposes. L'
•ntii. of the thermometer wire at the tempera-
; d the absolute zero by the symbols Rt^ Rtj and
•'
(161)
168
PRACTICAL PHYSICS
From these three equations the values of the three constants a, 5,
and RQ can be obtained. These constants once being known,
the resistance of the wire at any unknown temperature can be
determined experimentally and the temperature calculated
from (160). A convenient way of finding the values of the
three constants is to set R^= — , and then solve (161) by
determinants. °
The Wheatstone bridge is to be used to determine the re-
sistances. In texts on General
Physics it is shown that, if the
symbols have the meanings indi-
cated in the figure,
i = _3. (162)
Battery
Gal
Bl
FIG. 71. R l'
If the resistance R^ and the lengths 13 and Z4 are known, the
other resistance R± can at once be calculated.
MANIPULATION AND COMPUTATION. — The resistance ther-
mometer consists of fine platinum wire wound on a mica frame
enclosed in a wrought-iron capsule. In order to diminish
errors due to a change in the temperature of the leads which
run down into the capsule, a second pair of leads precisely like
the first is placed side by side with them but short circuited at
FIG. 72.
the bottom. By measuring at each temperature the resistance
between the terminals of the coil and also the resistance between
the terminals of the dummy leads the change in the resistance
of the coil alone can be obtained, so that any change in the
resistance of the leads does not need to be taken into account.
The particular form of Wheatstone bridge called the " slide
wire " or " meter " bridge will be used in this experiment. This
apparatus is illustrated in Fig. 71. A uniform wire AC is
stretched over a divided scale. The ends of this wire are con-
ITMPERATnu: 169
I to a parallel copper rod in which are two gaps. In one
of these gaps is inserted the resistance to be measured 7?p and
in the other gap a resistance box /{.,. The current enters at
one end of the bridge and leaves at the other. One side of the
galvanometer is connected to the binding post B and the other
to a key A', which can slide back and forth and make contact
at any point on the wire AC.
WL :he keys are oioaed, /\\ is closed first and then A',.
Until the bridge is nearly balanced /v, is dosed for as short a
time as possible, and as soon as AT, is open h\ is opened. If a
mirror galvanometer with its telescope and scale are used, the
to be adjusted as described on p. \ I.
With the resistance thermometer packed in a bat h of melting
the connections indicated in Fig. 71. With no re-
toe in the r. -• box and A'._, about halfway from A
co C, close the circuits just long enough to see in which direction
:he pointer of the galvanometer swings. Put a la: malice
n the box, and again see in \\hich direction the pointer >\\ ings.
If tin direction of swing is the same as before, something us
•vrong with the connections or else a larger resistance is needed
n the box. If the pointer swings out in the opposite direction,
•led in the box lies betw 0 and the re-
HOW in the b v half the resistance now in the
the din ' -ceeding iii this way, a
/able of the resistance can soon be found for which there is not
niich moveii. the pointer. The remaining adjustment
l»e made by ; A',, finding two positions of l\'., such
hat the d. lie, t ions for the two are iii opposite directions,
ind then closing down upon a point where there is no deflec-
tion. When no drtleeti..n is obtained, note the resistance in
he box and the reading on the bridge scale. Note also how
'ar the key can he moved in each direction without producing
ny o •• deflection. Then, from ( 1»JJ), the value of the
> Distance being measured is
//,-/d8. M';:IO
~'l
170 PRACTICAL PHYSICS
In the same manner find the resistance of the dummy leads.
If the temperature of the resistance thermometer when in
the bath of melting ice is represented by £1? then the difference
between the two resistances just found is the value of Rti in
(161).
Proceeding in the same manner, find the resistance of the
platinum coil when immersed in a steam bath. If this tem-
perature be denoted by t%, the resistance will be the value of
R(2 in (161).
The values of the three constants can now be determined
from (161). On substituting their values in (160), an equa-
tion is obtained which gives the relation between the tempera-
ture of the coil and its resistance. Such an equation, containing
experimentally determined constants, is called an empirical
formula.
Substitute for t in this empirical formula the values — 200°,
- 100°,. 0°, 100°, 200°, and compute the corresponding values
of Ht. With these values plot a curve coordinating R and t.
The accuracy of the preceding work should be tested by meas-
uring the resistance of the thermometer coil at two or three
known temperatures and comparing these observed values with
the corresponding values given by the curve.
Exp. 36. The Flash Test, Fire Test, and Cold Test of an Oil
OBJECT AND THEORY OF EXPERIMENT. — If an inflam-
mable gas is mixed with air in proper proportion, the mixture
will explode on ignition. The air above a volatile oil is satu-
rated with the oil vapor. If the temperature of the oil is
slowly raised, the proportion of oil vapor in the air will in-
crease until, at a certain temperature, the saturated air will
become an explosive mixture. This temperature is called the
flashpoint of the oil, If the temperature of the oil is still
farther increased, a point will be reached at which the oil will
evolve vapor so rapidly that, when ignited, it will burn con-
tinuously. This is called the fire test of the oil. The cold
TEMPERATURE
171
f an oil is the lowest temperature at which the oil will
tl.»\v. The object of this experiment is to make a flash test,
tire test, ami cold test of a sample of oil.
The general method of determining the llash point is to heat
•eeimen gradually in a covered cup and at frequent inter-
nail flame near the surface of the oil. In making
a fire teet, the specimen is heated in an open cup and the tem-
ire is noted at which the vapor will hum continuously
when ignited. The llash point depends upon ( </) the rate of
h.-at in-. ( /. ) the depth and diameter of the cup, ( f) whether
the cup [fl (.r open, (•/) the quantity of oil used. ( »•) the
size < ._C flame and its distance from the surface
of the oil. Consequently, the size and design of the testing
apparatus and the method of carr\in^ out a determination are
expli "d in the legislative enact-
ments nf tin- various star
M AMI'I I. \T!"\ .\M> ( lOMPI •
a of apparatus most commonly used
in this country for the tla.sli point is the *4New
VMI; 1 of Health Tester.'
by a glass plate perforated with two
holes ni of the ther-
mometer and another for the testing llame.
in a water or air bath //
by n .in alcohol lamp or small I'.n
l)iin]' :. The whole apparatus should 1>.
plared in a >heet-iron pan Tilled with sand.
In u>iii^ this apparatus to test illuminating
v Vm-k State Board .,f Health
publish* the following regulations: —
•• I; ' he oil cup and till the water
•iath with col.l \\at.-r U{. to the mark on the inside. Replace
tae oil cup and pour in enough oil to fill it to within one eighth
• Report of N. Y. State Board of Health, 1882.
172 PRACTICAL PHYSICS
of an inch of the flange joining the cup and the vapor chamber
above. Care must be taken that the oil does riot flow over the
flange. Remove all air bubbles with a piece of dry paper.
Place the glass cover on the oil cup, and so adjust the ther-
mometer that its bulb shall be just covered with oil.
" If an alcohol lamp be employed for heating the water bath,
the wick should be carefully trimmed and adjusted to a small
flame. A small Bunsen burner may be used in place of the
lamp. The rate of heating should be about two degrees per
minute, and in no case exceed three degrees.*
"As a flash torch, a small gas jet one quarter of an inch in
length should be employed. When gas is not at hand, employ
a piece of waxed linen twine. The flame in this case, however,
should be small.
" When the temperature of the oil has reached 85° F., the
testing should commence. To this end insert the torch into
the opening in the cover, passing it in at such an angle as to
well clear the cover, and to a distance about halfway between
the oil and the cover. The motion should be steady and uni-
form, rapid and without a pause. This should be repeated at
every two degrees' rise of the thermometer until the thermom-
eter has reached 95°, when the lamp should be removed and the
testings should be made for each degree of temperature until
100° is reached. After this the lamp may be replaced if neces-
sary and the testings continued for each two degrees.
"The appearance of a slight bluish flame shows that the
flashing point has been reached.
" In every case note the temperature of the oil before intro-
ducing the torch. The flame of the torch must not come in
contact with the oil.
"The water-bath should be filled with cold water for each
separate test, and the oil from a previous test carefully wiped
from the oil cup."
Make five determinations of the flash point and take the mean.
* This refers to degrees Fahrenheit.
Tli.Ml'KKATl UK 173
After each determination, remove the cover from the oil cup and
blow the burnt ibises out of the cup.
After the flash [mint has been determined, remove the cover
from the oil cup and continue to heat the oil at the rate of t\\o
degrees per minute. About every half minute test the oil with
nail tlame as above deseribed. The lowest temperature at
which the vapor of oil will burn continuously is the lire test.
Remove tin* thermometer and smother the tlame by plaeing on
top of the oil eup a piece of asbestos board. Sueh a damper
should always be at hand for eine:
In the ease of lubricating oils the method of lindin^ the Hash
point and the iiiv • ly as ah ribed except that
•f heating should he 1 •"• I'. per minute and the testing
should be applied first \\hdi the oil is about lMi)° F.
In making tin- cold test, a glass vial or boiling tube of about
100 c :ty is one fourth tilled with the oil under in.
i, and then placed in a : mixture of ice and
When all «•!' the oil has 00 i. it is removed from the I
iir_r mixture and i • -d \\ ith a t hcrmometer until it
iici.-ntly 1 to tlo\v from one end of the tube to the
other. The temperature at \\hieh this occurs i> the cold t-
'il.
Exp. 37. Relation between Boiling Point and Concentration
of a Solution.
Mi Till -The object of
: i m. -nt is to Hi id the relation bet \sc.-n t he boil ing point
and the emu -ent rat ion of a solution of common salt.
The boiling point of a solution of a non- volatile siilotaiic.- i>
T than the boiling point of the pure solvent. If a current
bfl passed into an aqueous solution below its boiling
•oint. steam will be condensed in the solution until the h«-at
.itjreby liberated :ie temj.er.it uiv Q "lution to its
point. Consequently >team that passrs through a solu-
ion will : .ni of the solution and not at
174
PRACTICAL PHYSICS
that of pure water. However, as the steam escapes into the
space above the solution it cools somewhat by expansion, and
wherever it comes into contact with the walls of the vessel,
with the thermometer, or with any body that can gradually
conduct away the heat given up by the condensation of the
steam, this cooling continues until the steam becomes saturated,
that is, until its temperature falls to the boiling point of pure
water. Consequently, in determining the boiling point of the
pure solvent the thermometer is suspended in the space above
the liquid, while in determining the boiling point of a solution
the thermometer bulb must be immersed in the solution.
Buchanan has recently utilized the principle stated in the
preceding paragraph for finding the boiling point of a saturated
solution. A quantity of the pure solute is placed in the bottom
of a tall test tube containing a thermometer. A current of
steam is sent through a glass tube extending to the bottom of
the test tube until a saturated aqueous solution of the given
solute is obtained. As long as any of the solute remains un-
dissolved and the current of steam is uninterrupted, the tem-
perature of this saturated solution remains at
its boiling point.
MANIPULATION AND COMPUTATION. — The
apparatus used in determining the boiling point
of a dilute solution consists of a flask provided
with a cork fitted with a thermometer and con-
denser. Without the condenser the solution
would gradually increase in concentration
through the loss of steam. To prevent "bump-
ing," a handful of clean dry pebbles or pieces
of broken glass is placed in the flask. With
the flask about one third filled with a solution
of some 50 g. of sodium chlorid to each liter
of water, insert the thermometer until its bulb
is 5 cm. or more above the surface of the liquid.
With the condenser in place, heat the solution until it boils
fairly rapidly. Read the thermometer and also the laboratory
FIG. 74.
TKMPERATfKi: 175
barometer. Push the thermometer down until its bulb is in
the boiling solution and when it has become steady read it
tin. From the corrected barometer reading (see pp. 17»'>-
17s and Table 1-) the true temperature of the vapor of boiling
water can be found. This value minus the reading of the ther-
mometer \vheii in the vapor is the eorreetion to lie applied to
the given thermometer in the neighborhood of 100°. This cor-
rection added to the reading of the thermometer when in the
boiling SolutlOl ;•_: point of this solution. In the
ue manner find the boiling points of solutions which contain
•ively three and live times as large a proportion of salt.
To find the boiling point of a saturated solution, I'aMen a
large boiling tube ;i.-al position in a retort stand, and
fill the tube to a depth of one or two centimeters \\itli salt.
Suspend in the tube the same thermometer used before so that
the bull) just touches the 1 salt and then push half\\;iv
through tli. .f salt tlie end of a glass tube in \\hieh ll
acunvnt' When the bulb of the thenuo: -.ub-
mer^ed in • by the salt and eon densed steam,
observe the temperature. When the , :i determined in
the tirst p \perimeiit is applied to this reading, it
6 required boiling point.
1'lot a curve showing the relation between concentration and
COi: • ••iling p. .int. The e. .neent rat ions may lie expressed
in grams nf salt p.-r liter .and th- itrationofa
sat dium . -hlorid may be taken as 396.5 g.
p.-r liter. The temperature of the \ ov« the boiling
solutions i> iin-^ point of a solute »noeotration«
on pp. 1«» 1-J ftnd .111 empineal e.pia-
tion e«,iuieetinur th- ng point of a sodium
chlorid §olll1
CHAPTER XI
EXPANSION
IF a body has at 0° a length Z0 and at t° a length lt it is found
that the relation between length and temperature is usually ex-
pressed very nearly by the equation
lt = 1Q (1 + at), (164)
where a is a constant for any one substance and is called the
coefficient of linear expansion of that substance.
Similarly, if a body has at 0° a volume VQ and at t° a volume
vt, it is found that the relation between volume and tempera-
ture is usually expressed very nearly by the equation
vt = v0 (1 + 7*), (165)
where 7 is a constant for any one substance and is called the
coefficient of cubical expansion of that substance.
In the case of gases it is shown in texts on General Physics
that if jt?, v, w, and T denote respectively the pressure, volume,
mass, and absolute temperature of the gas,
pv = RmT, (166)
where R is a constant which depends only upon the units
chosen and not at all upon the nature of the gas nor any
other condition. (166) is obtained by combining Boyle's and
Charles's laws and is known as the Fundamental Law of Gases.
REDUCTION OF BAROMETRIC READINGS
If the Torricellian vacuum above a barometric column were
devoid of matter and if there were no capillary force between
the mercury and the tube, the weight per unit cross section of
EXPANSION 177
a barometric column would equal the pressure of the atmos-
phere at the place where the barometer is situated. The space
above the mercury eolumn is, however, filled with mercurv
vapor which exerts a small pressure depressing the mercury,
and the capillary action between the mercury and the glass tube
iiminishes the height to which the mercury i
Again, even if tin- prosure of the atmosphere does not change,
the actual height <>f tlie mercury column may be altered in two
: first, by a change of temperature which not only alters
the density <>f the mercury in the barometer and conse.jiientlv
•ight, but which also alters the length of the Male used to
IN the height: second, by a change in the force of irravitv
acting on the mercury as it is moved t<> different parts of the
earth's surface. ('oiisequently, in order that barometric read-
ik.-M .it diiTeivnt temperatures and at ditTerent parts of the
earth's surface mav be compared with one another, thev must
luced to the heights that would have been oh>er\ed if the
barometer had been at some standard temperature and at some
stan. lard position mi : .'s surface. The standard comli-
arbitrarily • \n the temperai lire of melting iee and
the altitmle of the sea level at latitud • I
In pre< ; iroinetrie reading must be adjusted in the
four particulars, of \\lii.-h | doiW and two
are red lie- i conditions. The method of making
i ivdiic; lard cMiiditiuns \\ ill
now be oonsidered,
1. '/' 1 p represent the observecl height
and the density of themercm and l.-i r represent th.-
volume of mass in of mercury at this temperature. i
and r,, represent the . :iding «jiiantities at 0° C. Then
p*/h = p0f/h9 and m = rp = r^f. « 1
If £ denotes the coetVicient of cubical expansion of meron
t; = r0(l -h &).
NVhttnce --
178 PRACTICAL PHYSICS
And since, from (167), £ = and £ = ^,
/>o ^ Po v
it follows that ^ = — L— . (169)
On using approximation (5), p. 7, the above equation reduces to
But the brass scale used to measure the height h is ruled so
as to be correct at 0° C. That is, a space on the scale having
at 0° unit length has at t° a length (1 -f ctf), where a is the co-
efficient of linear expansion of the brass scale. Whence, a dis-
tance which at t° is presumably h units long is really h (1 + af)
units long. Consequently, the barometric height that would be
observed if the barometer were cooled to 0° C. is
h, = h (1 + O (1 - #),
or, employing approximation (2), p. 7,
A0 = h [! + (<*- £)*]. (171)
Since £=0.000182 and a = 0.000018 per degree centigrade,
(171) becomes
h0= h(l-Q. 00016Q
if the temperatures are taken in centigrade degrees. If, how-
ever, the temperatures are taken in Fahrenheit degrees
hS2 = h[l- 0.00009 (t - 32)]. (173)
2. Depression due to Capillarity. This depends upon the
diameter of the bore of the glass tube. Its magnitude may be
taken from the following table : —
Bore of tube in mm. 2 4 6 8 10
Depression in mm. 2.18 0.70 0.25 0.10 0.04
3. Reduction to Sea Level at Latitude 45°. This is most easily
effected by means of Table 11.
4. Depression due to Pressure of Mercury Vapor. This may
be neglected except in the most refined work. The values of
I:\PA.\SIO.\ 179
apor pressure of mercury at different temperatures are
:i in Table 1 t.
Exp. 38. Determination of the Coefficient of Linear Expansion
of a Solid
:> THI:<U:Y <>F KXI-KKIMENT. — The object of this
•i nit-lit is to determine the coefficient of linear expansion
of a m.-tal. If /t d«Miot,-s the length of the body at tempera-
'i and 1.2 its length at temperature £2, wo have, from (bit »,
/^/oO-f, (171,
and /2 = /0( 1 + «t%). C175)
On dividing (175) by ( 171 » uv obtain
• /, 1+*,
is very small, we m.i\ « mploy approximation (5) on
p. 7. obtaining ,
or, employing approximation
i-
*'
nee «= JjLZ_[l_. (17(11
'•i ~ fi>
[f the specimen being studied u in tin- f.-nn ,,f along \\n-<-
!, it may be suspended vertically, surrounded by a st.-am
uige of length obtained by means of th.- t<>rm
of optiral h-verdescribeil in K \periment 26. If the
is in rod or tube, either of tin
•<ls may be used.
In the first method OQfl end of th,- sj.rcimni is supported by
i Fig. 75), while the other end /? is sup-
1 by a form of ..ptiral h-ver devised by Dr. Miiller.
This uptiral h \ci is composed of a stirrup-shaped piece of
180
PRACTICAL PHYSICS
steel CEFD(¥ig. 76) on which are ground three parallel knife
edges, (7, EF, and D The stirrup is supported on the edges
FIG. 75.
C and D, and the specimen rests on the edge EF. Attached
to the stirrup are a mirror M and a pair of counterpoising
masses HH'. When the specimen changes
its length the optical lever is tilted through
a small angle.
The length ?x of the rod at f1° is obtained
directly by measuring the distance between
the knife edges A and B. The change of
length is obtained by measuring the angle
of rotation of the optical lever and the
distance from the line of the knife edges
C and D to the knife edge EF. The angle
of rotation is obtained by means of a tele-
scope and vertical scale. If the telescope is
at the level of the miror and the latter is vertical, the angle of
incidence of the light ray that comes to the telescope is 0°.
When the temperature of the specimen rises, the optical lever
is tilted through an angle 6. Thus the angle of incidence be-
comes 0 (Fig. 77), and, since the angle of reflection equals the
angle of incidence, the angle OpO' = 20. Denoting the dis-
tance of the scale from the mirror by L and the deflection 00'
by «, we have
tan 20= 4.
FIG. 76.
EXPANSION
181
It' tin- distance from the line of the knife ed^vs C and D to
the knife edge A'/' < Fig. 7''. > he dt-noted by ///. \\ e have
• •fore A, — /j = in sin f ^ taiT^y-J.
And, from (170) and ( 17
wi sin [ -^ tan •— i
(177)
(178)
, - <j
Another method of d.-l.-nninin^ ( /._, - /, ) is to 086 a small
roller to indicate the fliaiiLje ••!' h-n^th uf tlie rod and at the
>;iiiif tim- 'v it l»y ;i known amount. Tin- speciim-n is
iiori/ontally in two wyrs, mic of which, M ,"S), is
. while tin- othrr. A', is fastmrd to a horizontal pl;r
On t\\o rollers made of harden. -d stcrl rods. Tins
i ovahl «• support with its t\\.» n»ll, :I<T on a glass hctlplate
- a rarriai;«' whi«-!i mo\«-s when the length of the
specimen chaii'_Tcv. \ jht pointer li\«-d \» one of the rollers
182 PRACTICAL PHYSICS
moves over the face of a divided circle. If the roller carrying
the pointer is situated directly below the wye supporting the
movable end of the specimen, the indication of the pointer will
be unaffected by any change in the temperature of the carriage.
When the rod is heated, the carriage is pushed forward a
distance (Z2 — ^) and the pointer is turned through an angle 0.
During this motion the carriage has advanced a certain distance
with respect to the roller, and the bedplate has moved backward
an equal distance with respect to the roller. That is, with
respect to the bedplate the roller has moved forward half as
far as has the carriage, i.e. the roller has moved a distance
FIG. 78.
If the diameter of the roller is called d, then the
distance that the roller has moved on the bedplate is also
0
-- ird. Whence
360
and, therefore, from (176),
« = __ 07rd (119)
•
MANIPULATION AND COMPUTATION. — In the case of Miil-
ler's form of optical lever, find the distance from the line of the
knife edges C and D to the knife edge EF with a dividing
engine. After assembling the apparatus set up the telescope
and scale about a meter from the optical lever and see that the
telescope is about at the level of the mirror. After adjusting
EXPANSION 183
the t< Ifl as directed on p. 44 set the optical lever
in such a position that when the eye is at the level of the
telescope and close to it, the image of the telescope in the
mirror can be seen. This makes the optical lever vertical,
ire /r the distance between the wyes on which the specimen
. and L. the distance from the mirror to the vertical scale.
the readings of both thermometers, take the scale reading
in the telescope, send a current of steam for some minutes
;_rh the jacket surrounding the specimen, ami then take the
new s ling and agftin read both thermometers. I'»\
correct the thermometer readings for steam exposure. Calcn-
• bj < 178).
In the case of the roller method, measure the diameter of the
roller with a micrometer caliper. Assemble the apparatus,
being careful that the roll the pointer is normal to
the length of the bedplat '.so thai it is at the center Of
Lmded Circle. It U well to .start with the pointer about
as far to one side of t u it \\ill DOOM i«> be on the
other side of the m be set in this way after a
preliminary experiment in \\hich the angle through which it
will turn IN determined muchly. I iag6 >hoiild be so
ie wye i> y above the roller that carries the
pointer. Measure /,. I : the edges of th<
wyes on which the .spi-eim.-n rests, and note the readings of b.>th
IOmeter.^ .ter. Send a current of steam for
some minutes through the jacket surrounding the speeimeii ami
observe the new position of the pnint.-r and a^ain read both
uometers. By (158) correct the tin-nun meter readings for
steam exposure. Calculate .- T'J).
Exp 39. Determination of the Absolute Coefficient of Expan-
sion of a Liquid by the Method of Balancing Columns
Hi. |, Tin;, , . The object of this
experiment is to determine t he coefficient of expansion of mci-
a method which is independent of the change in
184
PRACTICAL PHYSICS
volume of the containing vessel. The method employed in
this experiment is to determine the coefficient of expansion of
the liquid from the ratio of its densities at different tempera-
tures.
The apparatus used by Regnault is illustrated in Figs. 79
and 80. Consider a W-shaped tube ABCDEF containing
mercury, having the branch A kept at a high temperature by
means of a steam jacket, and the remainder of the apparatus
FIG. 79.
at the temperature of the room by means of water jackets.
The mercury columns A and F are connected at the top by the
tube 6r, so that the pressures of the mercury in both columns
are the same at this level. At the bottom the two columns A
and F are kept separated by means of compressed air in the
tube CD. Let ffv Hv hv and A2 represent the differences in
level indicated in the figure. Let the temperature and density
of the mercury in the hot part of the apparatus be denoted by
KXI'ANSION IS")
f., and />., rely, and ih*.1 temperature and density of the
remainder of tin- mercury by tl and pr Let /' denote the
atmospheric pressure and P' the pressure of the air in If.
Then the pressure at tin- bottom of the /^-eolumn is ( P + Pi'jl^ ).
and the j : it the bottom of the 7>-eolumn is (/*' 4- p^/J^ ).
Nu\v these are pressures at the same level in u thud at rest and
are therefore e<|Ual. That is,
(180)
In the same way for the ^-column and the C-column,
I' + PjIL-r'+wlr (181)
On subtracting (181) from (180) and >nlving for ^»
b = ** • (182).
From the figure //j 4- /• = //., -f >t
//j - //2 = <i — b.
611 ) - //j - (<!-&) = #, -a.
comes
» f
ft «§ — •
the dfiisity of a given mass iry is inversely
proportional i denotes the eoe Hit -ient
of expan>
fi> = ri = l +^/ (184)
Pi
ind fin = 'a=i + tf. (1
On diviiling (185) by M - I , a value is obtained fur^i. If this
ft
value 1^ i-.jual.-d to that in . d the r- .•.ju;itinii
Q obtain
/S-T -4r a*6)
186 PRACTICAL PHYSICS
MANIPULATION AND COMPUTATION. — An instructor will
pump air into the reservoir R so that the mercury rises in the
tubes A and F until its surface stands about at the axis of the
tube (r. Water should flow through the water jacket only
slowly and its temperature, t^ should be near that of the room.
This temperature is to be taken by a thermometer, and the tem-
perature of the hot jacket calculated from a reading of the
barometer. (See pp. 176-178, and Table 12.)
Measure ff2 with a meter stick and a with a cathetometer.
Make at least five independent determinations of a, readjusting
the cathetometer before each one, and use the mean.
Exp. 40. Determination of the Coefficient of Cubical Expansion
of Glass
OBJECT AND THEORY OF EXPERIMENT. — When a vessel
filled with liquid is raised in temperature, the apparent expan-
sion of the liquid depends upon the coefficients of cubical ex-
pansion of the liquid and the containing vessel. If the apparent
expansion is observed and the coefficient of expansion of either
the liquid or the containing vessel is known, then the coefficient
of expansion of the other can be determined. The object of
this experiment is to determine the coefficient of cubical expan-
sion of a glass bulb from an observation of the apparent expan-
sion of mercury contained in it. The absolute coefficient of
expansion of mercury is supposed to be known from the pre-
ceding experiment.
Let the bulb be filled with mercury at the temperature t^ and
then be heated to £2°. This will cause some of the mercury to
be expelled. A bulb used in this manner is called a weight
dilatometer.
Let Ml and pl represent the mass and the density of the
mercury contained in the bulb at ^°;
M2 and />2, the mass and the density of the mercury
contained in the bulb at .t2° ;
r.xi 187
'•j and >•.,. the respective volumes of the mercury in the
bulb at t^ and
v^, the volume at £2° of the mercury which at t^ tilled
the bulb;
7. the mean coefficient of cubical expansion of L,rlass be-
ns <
£, tlie mean coetlicient of cubical expansion of mercury
between t^ ami £2°.
AC have
M M, , M* . -
' ! = — 1, f.£ = — 3, V = —1. (187)
Pi ^2 ^-2
Since A3 and 7 an- Imth small, it ma\ In- shown from < 1 !'.."• > in
A;IS d«-\«-lop.-d from ( H'»{ ), that
3n subs
an 7
' •
sub>titutini; in ( 1S8) the values of vr r... and r.2r from (187),
iminatiiig £2 fr«»m the resultin-.: equations, we get
• r, writing
rwa for the ma.ss <.f the mercury expelled \\hen the
mlb is heated from tf to *,°,
M\\n i. COMF1 IATION. — A convenient fnrm
•_'ht dilaton. this experiment is a specific gravit\
;.ited glass stopper. Aft liinur the
M.ttle, nearly till it with mercury, and, without in>ertiii<j the
V Carefully Until all observable air bubbles
;\ve b .1. An iron wire will greatly facilitate the
i.ixving out of these bubbles. Have under the bottle a vessel
•••h the mercury in case the heating should be done too
i apidly and t: b.- biol
188 PRACTICAL PHYSICS
After the bottle has cooled enough to be held comfortably in
the bare hand, place it on a cork stool in a beaker and pack it to
a little below the opening with shaved ice. After five or ten
minutes insert the stopper, being sure that there is enough mer-
cury in the bottle to fill the capillary and leave a tiny globule
above it. If this globule does not decrease in size in a few
moments, brush it off, and then begin slowly heating the beaker.
As fast as mercury comes out of the capillary, brush it off into a
piece of paper bent into a cup without any hole in the bottom.
When the mercury has stopped coming out, remove the flame
and allow the beaker to cool.
Meantime read the barometer in order to determine the tem-
perature of the hot mercury (cf . pp. 176-178, and Table 12), and
then fold the paper containing the extruded mercury so that
the latter will not run out and by the method of vibrations
weigh paper and mercury. Then weigh the paper alone, and
after the specific gravity bottle has cooled to the temperature of
the room, weigh the bottle and the mercury which it still con-
tains. The difference between the weight of the paper with
the mercury in it and the weight of the paper alone is m2 ; the
difference between the weight of the bottle with the mercury
left in it and the weight of the bottle alone is M2 ; ^ is zero, and
£2 is the temperature of boiling water. The value of y8 may be
taken as 0.000182 per degree C.
Exp. 41. Determination of the Coefficient of Expansion of a
Gas by Means of an Air Thermometer
OBJECT AND THEORY OF EXPERIMENT. — If a given mass
of any substance is heated from 0° to 1°, and the pressure upon
it kept constant, the ratio of the increase of volume to the initial
volume is called the coefficient of expansion of the substance. If
it is heated from 0° to 1°, and its volume kept constant, the
ratio of the increase of pressure to the initial pressure is called
the temperature coefficient of pressure of the substance. In the
succeeding paragraph it will be shown that for a perfect gas
EXPANSION 189
two coellieients are equal. Sinee it is MUUer to measure
IUT6 of a Li'as under eonstant volume than to measure
the volume umler eonstant pressure, the eoetlicient of expansion
will be determined iu the present experiment from observations
of the ehaiiures produeed in the pressure of a ijas when its mass
and volume remain nearly eonstant and its temperature elian^es.
Mder a ^i\cn mass of <jas at temperature 0° C. or TQ
absolute. piv>sure /',,, and volume r,,. When it is heated to
. (7^-f t )° absolute, let the pressure be represented
and the volume by vt. From the fundamental law of
_ .-• -. • 1- ••'. >.
-. (190)
If the \.»lume lie k.-pt constant, which is denoted below by the
i ipt r outside of the parenthesis, the t», in (190 ) beeoim •>
• jual t<> the r,r and (190) solved for •—- gives
r, if the pressure be kept constant,
(192)
'.in the coefficient of expansion of a gas, A is, from it> definition,
t follows that
'hat oeilieient ot -.in of a gas is equal to the
'cipr> absolute tempera tun- and is also i-ipial to ii>
icssure coeflieient.
An appai atus well .siiit.-d for determining the pressure coeffi-
is some form of J.»ll\'s Air 'I'liermometer. This consists
190
PRACTICAL PHYSICS
(Fig. 81) of a glass bulb B filled with air or other gas, connected
to an open manometer tube M filled with mer-
cury. Immediately below the bulb is a tube
containing an index finger F made of colored
enamel. The volume of the gas is made defi-
nite by adjusting the plunger P until the mer-
cury surface is brought to the point F. The
pressure of the gas in the bulb is measured by
the difference between the levels of the mer-
cury surface at F and the mercury surface in
the manometer tube. The bulb is inclosed by
a vessel in which can be placed water or ice.
D is a drying tube used in filling the bulb.
On account of the temperature of the small
amount of gas in the exposed part of the bulb
being different from that in the jacketed part
of the bulb, and also on account of the change
in the volume of the bulb when its temperature
is changed, (193) cannot be used in its present
FIG. si. simple form. The corresponding equation in
which these facts are taken into account will now be derived.
Let PQ denote pressure in bulb when at 0° C. (T0° absolute) ;
pt, pressure in bulb when at t° C. ;
v0, volume of jacketed part of bulb at 0° C.;
vti volume of jacketed part of bulb at t° C. ;
77i0, mass of gas in jacketed part of bulb when at 0° C.;
mt, mass of gas in jacketed part of bulb when at t° C.;
vj, volume of exposed part of bulb when jacketed part
is at 0° C.;
v/, volume of exposed part of bulb when jacketed part
is at t° C.;
w0', mass of gas in exposed part of bulb when jacketed
part is at 0° C. ;
in/, mass of gas in exposed part of bulb when jacketed
part is at t° C.
KXPANSION" 191
Without great error, the temperature of the exposed part of
the bulb may be assumed to be constant and equal to that
of the room: Let this temperature be denoted by t'°C. It
follows that vt' = VQ.
Applying ( I'1.'1.) to («) the jaeketed part of tin- bulb when at
0°, (& / the jarketed part of the bulb when at r , ( • • ) the exposed
part of the bulb when the jaeketed part is at 0°, and (»/) the
exposed part of the bulb when the jacketed part is at £°, the
following four equations are obtained : —
the mass of ga constant, we
also
' = m +
• i :.
'eliminating from these five equations the four unknown
lasses, we get
7 tin- corlVicicnt of t-ubic.' Ljlass
tin- tnnprnit uivs O° an«i ,. have
utiiiij this value in ( l!M), reinei: that, from ( 1
ing the constant i obtain
P vo
(195)
Wl i is solved for 0 the resulting t..nnnla is \« i\
1 ag. 1 re, seldom adopted. One of
t ie in \\hidi may . !•<• niiployt-d to tind ft is th«i
. It \\ ill lx- seen that as long as th«- P-MMI t.-mpt -ratnn;
192 PRACTICAL PHYSICS
remains the same the left member of (195) does not change.
It follows that the right member is also constant. That is, if t
had some value t^ this right member would have the same
value as if t had some other value £2. That is,
(196)
If t2 is chosen as the room temperature t\ if the subscript 1 is
dropped, and if p' is used to denote the pressure in the bulb
when all the gas in it is at the temperature t1 ', (196) becomes
,10~
(197)
Whence ft- p'(l + 7** + *) -ftO + 7* + *) rl98N
^-' ' '
It will be seen that if k and 7 were both zero and if the tem-
perature t were 0°, (198) would reduce to (193), as it should.
MANIPULATION AND COMPUTATION. — The air in the bulb
has been dried once for all and the upper opening of the bulb
permanently sealed. Hang a thermometer in the air just out-
side of the bulb and adjust the plunger until the mercury
touches the index F. After a few minutes, when the tempera-
ture seems steady and the top of the mercury stays at F, set
the slider S at the top of the column in M and read both the
position of the slider and the temperature in the jacket. Then
fill the jacket with shaved ice. Notice the index frequently for
several minutes, readjusting whenever the mercury is not just
touching it. When no more adjustment seems necessary, set the
slider S at the top of the mercury in M and read its position.
Read also the laboratory barometer.
For the ice substitute water at a temperature of about 40° C.,
and after allowing a few moments for the bulb to aquire the tem-
perature of the water, begin slowly heating the water by pass-
ing steam into it. While the water is heating, read on the
manometer scale the height of F. This can be done best with
EXPANSION
193
a cathetoineter. but may be effected by a straightedge held
horizontal with the aid of a level. As the water warms adjust
the plunger occasionally, and when the steam is bubbling rapidly
through tin- water in tin- jacket and no further adjustment
seem- iry, set tlie slider and read it again. Read also
a thermometer which is pretty well immersed in the jacket.
When through witli the apparatus, draw oil' the water in the
jacket and leave the mercury at about the same height in both
tubes.
The barometric height plus or minus the difference bet
the heights of & and /'when the bulb was at the room tempera-
ture gives//. The cMi-responding quantity when ice \\
the ja«-k«-t Lrives one value for ;>,, the t in this case being 0°.
k will be given by an instructor, and 7 may be taken as (».nniniJ7
per degree C. Use (198) to get" one value for $. Find a
second value t ^' />, tin- pn-xsure found when the
>ulb was surrounded by the hot water and for t the tempera-
ure of that water.
CHAPTER XII
VAPORS
Exp. 42. Determination of the Maximum Vapor Pressure of
a Liquid at Temperatures below ioo°C.
(STATIC METHOD)
OBJECT AND THEORY OF EXPERIMENT. — When a liquid
evaporates in a closed space, the vapor formed produces on the
surface of the liquid and on the inclosing walls a pressure
which increases with the mass of vapor and with the tempera-
ture. For a given temperature the vapor pressure"* reaches a
maximum value when the space is saturated. The object of
this experiment is to determine the pressure of saturated
aqueous vapor at temperatures from about 50° C. to 100° C.
In the method to be used in this experiment, the vapor pres-
sure is determined from the decrease in the height of a barome-
ter column produced by a small quantity of the specimen which
is introduced into the vacuous space above the mercury. The
apparatus (Fig. 82) consists of a barometer tube, the upper
end of which is enlarged into a narrow bulb and its lower end
joined to an open manometer tube, M. Opening into the hori-
zontal tube joining the barometer and manometer is an iron
cylinder filled with mercury. The height of mercury in the
two tubes can be varied by means of a plunger, P, in this cylin-
der. A small enamel finger, F, in the bulb of the barometer
tube serves as a convenient fixed point from which heights can
* The expression vapor tension is sometimes used instead of vapor pressure
to denote elastic stress exerted by a vapor. .Careful writers, however, use the
word pressure to denote a push, and tension to denote a pull. Since vapors and
gases cannot exert a pull, the term vapor tension is a misnomer.
194
VAPORS 195
be measured. The vapor being studied can be brought to the.
-.1 temperature by means of a water bath surrounding
the bulb.
The pressure in the tube J/ at the level of
the mercury surface at S is the atmospheric
<>f the vapor in the
bull - than this by the pressure due to
the column of liquid 1- the lev-
and /".
MAMITLA i !<>\ AM. COMPUTATION, — The
bulb has hem fived from air, a specimen of air-
free mtrodu- ; the upper end of
the bulb permanently sea
Obs. from tlie
•ry .standard barometer. Fill the water
jacket witli water and pass steam int.. it until
t reaches .! .tturo of about -1.V ( '. ( )M
vccount of danger of cracking the glass, the
•ill-rent of >• uld not be < against
he bulb nor against its jacket. 1 1. .Id
emp.-rature as : y as possible for
ew minutes, and by means of the plunger
:;<• mercury in the barometer tube until it
s brought just into contact with the .tip of the iiid>
>tir the water in the juket, observe its temperature, readjust
.s1 until it> index lino
;he menixMis in the man • ube, and read the
>n of this index line. Determine to the nearest millime-
er t! : of the water column above the mereur\. l)i-
I gravity of mercury, and
dd ; the ditleivnce between the levels of the mer-
017 in the two tubes. ^ult bom the barometric
ure as given by the standard barnm.-i.-r. 'I'he result i> the
>SUre of wat< temperature of the experiment.
ngs every ten degrees up to about '.
n through theexperin •:'!' the water in the jacket,
196 PRACTICAL PHYSICS
and adjust the mercury to about the same level in both arms.
Plot a curve with vapor pressures as ordinates and corre-
sponding temperatures as abscissas. On the same coordinate
axes plot another curve from the values given in Table 13.
This method is liable to several errors. The surface tension
of the dry mercury in the manometer tube is different from that
of the wet mercury in the barometer tube. This will cause a
rise of the column having the wet surface of 0.10 to 0.15 mm.
The fact that the lower part of the barometer tube is at a
lower temperature than the upper causes the final result to
be too low. This error will be of the order of 0.15 mm. If
the position of the end of the index finger is read through the
water jacket, the refraction of the glass and water will intro-
duce an uncertainty that may amount to 0.5 mm. This error
is obviated by carefully measuring the distance from the end of
the index to a fine scratch on the tube below the water jacket
before the apparatus is assembled. In order to use this scratch
as the fiducial line from which heights are measured, the posi-
tion of the line is read on the meter stick by means of a cathe-
tometer (see p. 23). The greatest limitation to the use of
this method, however, is due to the large error introduced in
the depression of the barometer column by an impurity of the
specimen.
Exp. 43. Determination of the Maximum Vapor Pressure of
a Liquid at Various Temperatures
(DYNAMIC METHOD)
OBJECT AND THEORY OF EXPERIMENT. — The object of
this experiment is to determine the maximum vapor pressure
of water at various temperatures from about 50° C. to about
120° C. The dynamic method to be employed in this experi-
ment is based upon the following two laws of vapors: first, a
liquid boils when the pressure of its vapor equals the external
pressure ; second, if the pressure does not change, the tempera-
VAPORS
197
tu re of the boiling liquid remains constant as long as there
is any liquid to vaporize.
In Reguault's apparatus (Fig. 83) the water is inclosed in
a boiler B, from the top of which runs a lube through the
condenser C to a large metal reservoir
inclosed in a water bath kept at con-
stant temperature. The reservoir is
tilled with air the pressure of which is
i by means of a pump connected
to P. The air in the reservoir »
to equalize any sudden changes of pivs-
sure due !• .ariiies in boiling. It'
ire not for tin- condenser, most of
the steam formed in the boiler would
not be condensed, hut instead \\ould
increase tin j in the boiler and
.tnd tlius prevent much 1>»
'.'hat is, when the burner was lighted
•«.th temp.-rat urc and pi- .\ould
TadualK •• of tin-
apor is iii«-;ixur,.«l i.v nn-ans of the open
ia!in]i].-t«-r at tin- ri^ht. 'I'hr trmpera-
iif <.f tin- vapor in tin- l»niler is ob-
1 from th«- four tin-nun1
1 iu tubuluivs \\liirli ju-ojiTt into
i ic b<«. : l\vo of thcsr tuliulurea are
•ML:, projecting into tba-ntor, and two
; ne abort, prqjeoti -vapor only.
h<- bottoms of all of tin-in an- lilh-d with inrrcury, so that tin-
1 dbs of the th.Tinoiii.-tcrs qiiirkly acquire the tenipt-rat ures of
t ie water and vapor in the boiler.
MAMITI.A D CoMPUTATIOir. — The boiler aln-ady
c ntJii ,im of cold \\ater flowing
t lOUgh the QOnd M<1 then light the burner under the
b >iler. rump air out of th- .ir until tlie pressure is
I'' dlleed to about 1<I em. of ll.. /., . 11 lit \\ l , • 1 1 ( -i • I M •-
198 PRACTICAL PHYSICS
tween the heights of the mercury in the two arms of the manom-
eter is about 65 cm. Then close the stopcock in the tube con-
necting the pump and the reservoir. When the thermometers
have become steady, record the reading of each thermometer
and of each of the manometer tubes. Note also the temperature
of the manometer and the barometric height. The corrected
barometric height diminished by the corrected difference of level
between the manometer columns gives the pressure of the vapor
at the temperature indicated by the thermometers in the boiler.
Assuming that the manometer scale is correct at 20° C., reduce
the pressure to 0° C., i.e. so correct it as to make it the pres-
sure that would have been observed if barometer and manom-
eter had been at 0° C. This can be effected for the barometer
by (172) and for the manometer by (170).
Allow air to enter the reservoir until the difference in the
heights of the mercury columns is about 20 cm. less than before.
This increase of pressure requires that a higher temperature be
attained before the water will boil. When the temperature has
reached the new boiling point, a second set of observations is to
be made. In the same manner, the boiling points correspond-
ing to about eight different pressures are to be determined, the
difference in pressure in passing from each case to the next
being about 20 cm. of mercury.
Plot a curve with pressures as ordinates, and temperatures as
abscissas. On the same coordinate axes plot for the same range
another curve with values from Table 13. This curve, showing
the way in which the pressure of saturated water vapor changes
with temperature, is called the steam line.
Exp. 44. Determination of the Density of an Unsaturated
Vapor by Victor Meyer >s Method.
OBJECT AND THEORY OF EXPERIMENT. — Probably the
most accurate method for determining the density of an unsat-
urated vapor is to allow a known mass of the liquid whose vapor
density is to be determined to vaporize in the Torricellian vacuum
VAPORS
199
of a barometer, and then to observe the volume occupied by the
vapor. The ratio of the mass of the liquid vaporized to the
volume occupied by the vapor is the density of the vapor at the
temperature and pressure of the experiment. But if the ivsult
need not be trusted more closely
than to within some three to
five per cent, a method due to
Victor Meyer will be found
much more conveni
apparatus used in this
method is shown in Ki<*. S4. It
uprises a gas mea>urin^ tube
/.'. and a vapor chamber con-
:IILT (>f * l(>ng glass tube ter-
minating in a bulb //.surrou:
by a bath containing a liqu;
\ i'_rher boiling point than
s distance under examination.
' he sp- in a
g nail bull) \\-hich can be sup-
I >rted in tin- upper cooler part
« ' the vapor ehamher by means
« a n»d If capable of a back
ft id forth motion in a side tube.
^ rhen the bath has attained
a const, int temperature, lii^h
• |OU«_rll to \a|M>ri/r the s;
ilrawn :
as to allow tlie little bulb
C ntainini: the Q to fall
t< the bottom ..f the ehaml.er.
I eW it either breaks l»y .
Ci -Hion with the bottom, or
b ;>ts due to the ,n of the contained Irquid. \Vlien
tl e contained liquid : pushes out of the vapor
cl amber a vulume .ual to it> o\\n volume, and the vol-
200 PRACTICAL PHYSICS
unie of this expelled air is determined by means of the measur-
ing tube E.
In bubbling up through the water in the measuring tube the
air expelled from the vapor chamber becomes cooled and so con-
tracts. Since all gases have nearly the same coefficient of. ex-
pansion, the vapor, if it could be cooled to the temperature of
the air in the measuring tube without becoming saturated,
would after this cooling occupy the same volume that the air
does. Therefore the density of the vapor at the temperature
and pressure of the air in the measuring tube equals the mass of
substance vaporized divided by the volume of air thereby forced
into the measuring tube. The temperature of the bath sur-
rounding the vapor chamber must remain constant during the
vaporization of the specimen, but its value need not be known.
Since the densities of gases and vapors vary greatly with
changes of pressure and temperature, it is customary to reduce
the values to what they would be at some standard pressure and
temperature. The pressure usually adopted as standard is the
pressure of 760 mm. of mercury, and the temperature adopted as
standard is 0° C.
Let m be the mass of substance vaporized, and vt and v0 the
volumes of the vapor when at the respective pressures, tempera-
tures, and densities pt, p& t, 0, />„ and /o0. From the funda-
mental law of gases, (166),
pQvQ = EmTQ (199)
and ptvt = Rm ( To + 0 . (200)
Dividing (199) by (200),
Po «+t
Whence >0 = *= 76°™ [™ + *\ . (201)
MANIPULATION AND COMPUTATION. — The substance to be
selected for the bath will depend upon the temperature of vapori-
zation of the specimen being examined. The following sub-
VAPORS 1201
stances will be found convenient to use: water, whose boiling
point is 100° C. ; analin, Is2°.5 ('. ; bromonaphthalin, 280° C.
The specimen is inclosed in a thin <jluss bull) C, which must
tirst be weighed and may then be tilled as is illustrated in the
figure. A hot metal rod held below C will cause some of the
contained air to bubble out, and when the bulb cools it will be-
come partly filled with tlie specimen. l>y repeating this opera-
i ion the bulb can be entirely tilled. If the liquid is volatile, the
>tem of the bulb must be sealed in a tlame or plumed in some
manner. The mass of the specimen is determined by wei^hiiiLT-
The bulb is then supported in the cool part of the vapor cham-
ber by the rod /,'. When the temperature of the bath becomes
constant, no more air bubbles up through the water in the
trough r. When this state is attained, the measuring tube K.
tille(l with water, is placed over the outlet of the discharge tube
and the n>d It is withdrawn, allowing the bulb to fall and
break. The volume of air entering the measuring tube and its
temperature are «»b>.-rv«-d. The lur.>metic height is al><> noted.
The pre»mv of the moist air in the measuring tube is equal
to the barometric pressure diminished by the sum of the pres-
• he column of water within E above the surface of
I', and the pressure of aqueous vapor at the temperature E.
This latter may be taken 11.
CHAPTER XIII
HYGROMETRY
HYGROMETRY or Psychrometry is the theory and art of meas-
uring the amount of moisture in the atmosphere. The mass of
water contained in unit volume of air is called the absolute
humidity. Absolute humidity, then, is simply another name for
the density of the vapor which is present. The ratio of the
mass of moisture contained in unit volume to the mass which
would saturate the same space at the same temperature is called
the hygrometric state or relative humidity of the atmosphere.
Let p be the pressure and v the volume of a mass m of
aqueous vapor at the absolute temperature T. Let m' be the
mass of vapor at the pressure p' necessary to saturate the same
volume at the same temperature. Then since for ordinary
atmospheric temperatures and up to the point of saturation
aqueous vapor obeys approximately the fundamental law of
gases, we have from (166)
pv '—. RmT ',
and p'v=Rm'T.
That is ™ ~ 4
m' ' p (202)
Consequently relative humidity equals the ratio of the actual
pressure of the aqueous vapor in the air to the maximum
pressure possible at the same temperature.
It thus appears that there are two general methods of deter-
mining the relative humidity of the atmosphere. The first
requires the measurement of the actual mass of aqueous vapor
contained in a given volume of air. This can be done by
drawing a given volume of the air through a drying tube and
202
HY GEOMETRY 203
.iiiLj tlu' drviiiLT tube. The mass of aqueous vapor required
: unite the same space at the same temperature eau be
obtained from tables. The more common method, however, is
to determine the actual pressure of the vapor in the air, and
from tables lind what the pivssmv would be if at the same
temperature the vapor were saturated.
Exp. 45. Determination of Relative Humidity with Daniell's
Dew Point Hygrometer
OB.II:« r AM> THF.OI:V o» KXPKKIMKNT. — The temperature
to which the atmosphere must be cooled in order that the water
vapor present may he saturated is called the ilfic j»n'nf. The
object of this experiment is to determine the relative humidity
of the atmosphere from an observation of the dew point.
Mder a mixture of air and water vapor which has volume
ma>s // . and absolute temperature '/'. Lrt
ber vapor in this mixture have mass m and exert pre»ure/'.
to the temperature of .saturation, both \\ater vapor and
ir obey approximately the fundamental law of gases. T;
ore, fron;
fi T (203)
R / (204)
vs long as p" does not change, (203) shows that -~ does not
hange, and, theref..re. from • does not change. That
-, whatever change there may be in the i ure of a part
f the atn. . if the piv»ure «.f \\ -phere as a whole
i . not altered, then the pressure of . apor in it is not
1. Consequently, the pressure of the water vapor in any
[ 'rtioii of air can mined by cooling the air down to the
« -w point and looking up in the proper table the pi-e^ure of
- -turaled water \ a p. T corresptmdiii^ to this temperature. From
( !^2) and the dclinitioii of ; humidity it follows that
t ;e relative humidity of any portion of air is ^iveii by
204 PRACTICAL PHYSICS
where p and p' represent respectively the pressures that satu-
rated water vapor would exert at the dew point arid at the actual
temperature of the air.
MANIPULATION AND COMPUTATION. — Daniell's hygrometer
consists of two glass bulbs connected by a bent tube as shown
in Fig. 85. The lower bulb contains ether and a thermometer.
The upper bulb is wrapped with a piece of muslin.
In determining the dew point with this apparatus all of
the contained ether is passed into the lower bulb and then
the upper bulb is moistened with ether. The evaporation of the
ether poured on the upper bulb causes the
bulb to cool and a part of the vapor in the
apparatus to condense. This condensation in
the upper bulb decreases the pressure which
the ether vapor exerts on the surface of the
liquid ether in the lower bulb. Under the de-
creased pressure some of the ether in the
lower bulb evaporates and so cools the lower
bulb. Thus the temperature of the lower
bulb gradually falls until dew is deposited on
its surface. The beginning of a dew deposit
is usually detected by watching to see if any
effect is produced on drawing a fine brush lightly across a
gilded surface on the bulb. The temperature of the lower bulb
is then read. After this the apparatus is allowed to remain
until equilibrium is restored and the temperature begins to
rise. The temperature at which the deposit of dew disappears
is noted. The mean of the temperatures when the deposit
appears and when it disappears is taken as the dew point.
The temperature of the surrounding air is noted by the
thermometer attached to the wooden stand supporting the
hygrometer.
From Table 13 find the pressure of saturated aqueous vapor
at the dew point and at the temperature of the room. Make
at least five determinations and take the average.
HYGROMETRY 20f>
Exp. 46. Determination of Relative Humidity with the Wet
and Dry Bulb Hygrometer
OBJK«T AND Tm-:ni:v «>F KXI-KIMMKNT. — If two exactly
similar thermometers, the bulb of one naked and the bulb of
ther OOYered with a wet \\ick, are placed near each other
:invnt of air, the thermometer with the naked bulb will
indieate the temperat ure of the air while the other will indieate
a lower temperature. The difference between the indications
of the two thermometers is due to evaporation at the surface of
• ulb and depends upon the derive of saturation of the
air. The relation between the relative humidity of the air and
the thermometers has never been obtained in
an entir. ,ry manner from purely theoretical con-
siderations, lint liy comparing \\ it ions of this liy^nnn-
'. ith the indications of hy^i of other t \ pcs. tables
I by means of which the relative humidity
• f tlie air can rea< lily be determined from a single pair of si-
) uili .iiid dry l>ull> thermometers.
se\eral years simultan the I>,iniell and
• f tin .d dry inilli hy^n • meters Wei :. and from a
• mipari.NMii of these : | the niimh. i kUe 1 •"• \\i-re
• itained. A- . Q instruments nc,u-
i ic another, the ; multaneoofl readings \M re made :
iperature of th«- air, ill •
(rf the wet bulb, I1.' '
Dew p"int. Is
I i Tal.l- ! pressure of saturated aijueoiis vapor at 18° 18
U ven as 15.33 mm. It follows that corresponding to an atnms-
p icric temperature of L'! and a \\et bull) temperature -1 lower,
t. 6 pressure \\hich the aqneoiU mpoi in the almosph.-n- would
e ert at the dew point and. eon>e(pientl\, does exert at the
Lr • n trmprratmv i> 1 ."».:;:', mm. of meiviirv. In Table \.~>
tl is number in placed in the line numbered '21° C. and in the
f' luiiin numbered ~1 .
206 PRACTICAL PHYSICS
MANIPULATION AND COMPUTATION. — The wet and dry
bulb hygrometer, sometimes called August's psychrometer, con-
sists of two similar thermometers, one with a naked bulb and
one with the bulb covered by an envelope of wet muslin. A
current of air is caused to blow over the two bulbs with a fan
or some other means. In one convenient arrangement for this
purpose (Fig. 86) the two thermometers are supported by a
frame that can be rotated by hand.
With the muslin envelope dry
take simultaneous readings of
both thermometers for several
minutes. Then see that the
muslin envelope about the bulb
of the wet thermometer is kept
FIG. 86.
thoroughly moist, and with a fan
or the rotating device shown in the figure change rapidly the
air about the instrument. When the wet bulb has reached a
stationary temperature, read each thermometer again. From
the corrected difference in the readings of the two thermometers
find in Table 15 the pressure p of the aqueous vapor present
in the atmosphere. In the 0° difference column find the pressure
p which the aqueous vapor would exert if there were enough
of it present to be a saturated vapor. Calculate the relative
humidity by (205).
Make not fewer than five determinations and take the average.
Before each determination be certain that the muslin envelope
is thoroughly moist.
(IIAPTKR XIV
CALORIMETRY
C U.OKIMF.TKY is tin- theory and art of measuring quantities
of heat, rnfortunately th»-r. ogle quantity of heat that
is universally adopted as the unit. A emnmon unit in scientific
work iMiMiiut <>f heat required to raise the temperature
of MM*- L^ram of water ( '. t.. 1- This unit is called
dorie or simply ; or the ^raiu-de^ree-eenti-
grade thermal unit. In the liriti m the unit adopted is
tie amount of heat required to raise the temperature of one
I <>und «•! 1'. Thii i- i-alled the British
t lermal unit or the poi, ! ahrenheit thermal unit.
will he i: nsively.
numher of thermal units required to raise the teni]
• i iv of unit mass of a subs tan <• from . all«-d its
> >ecit The >j is slightly diiTer-
t it at different temperatures, hut the difference is HM minute
t iat e\< rpt in the most : neuts it need not be
considered. The average spe- n any
t fO t- lie numlM-r i.f heat units rr.jiiii-rd to
a unit mass of it : -e temixjratures to the other,
(i vided l»y tin- d a p.-rat ures. That is,
t e quant // :,-d to raise from tf t«» A/ the trm-
j rat ure of m grams of a substance of average spe< iti< heat « is
// = »,HV-O-
1 i^oii^lmut the ahovr paragraph it is as>umed that hei
tl • temp- :iot melt, solidii'y,
V! p< idrlise.
When a hody <loes melt, solidif \ . vapori/e, or condense, with-
208 PRACTICAL PHYSICS
out changing at all in temperature, the amount of heat required
or given out is proportional to the mass of the substance that
changes state and depends upon what that substance is. That is,
H=mf, (207)
where / is a constant called the heat equivalent* of fusion,
solidification, vaporization, or condensation, as the case may be.
The mass of water which requires the same amount of heat
as a given body in order to change its temperature by the same
amount is called the water equivalent of the body. Thus, if
e represents the water equivalent of a body, and c the mean
specific heat of water between t£ and £2°, the quantity of heat
required to raise from t± to £2° the temperature either of the
body or of e grams of water is
H=ec(tf-tf). (208)
Dividing (208) by (206),
e = — - (209)
c
Ordinarily, the specific heat of water may be taken as constant
and equal to unity. In this case
e = ms. (210)
That is, the water equivalent of a body equals the product of
its mass and its specific heat.
In determining the water equivalent of a thermometer, only
that part of it which changes in temperature is to be consid-
* From the fact that the heat absorbed by a body during fusion or vaporiza-
tion does not change the temperature of the body, it used to be supposed, when
heat was considered to be a form of matter, that the heat absorbed during fusion
and vaporization existed in the melted or vaporized body in a latent, i.e. a
hidden, form. This heat was then called the latent heat of fusion or vapori-
zation. Now that it is known that heat is a form of energy, viz. that form
which changes the temperature of bodies, we prefer to say that the heat absorbed
by a body during fusion or vaporization does not exist in the melted body as
heat but as some other form of energy. Consequently, the expression
latent heat is now obsolescent and is giving place to the term heat
equivalent.
I \\LoHIMr.TKY 209
This may be taken as somewhat more than the part
immersed. Fortunately, the product of the density of mercury
by its speeitic heat is nearly the same as the corresponding
product for glass. That is, the water equivalent of a given
volume «»f mercury is about the same as that of the same vol-
Since the value of this product is about 0.5 g.
. i he water equivalent of a thermometer in grams may
;en as somewhat more than half the volume of the im-
d part in cubic centimeters.
Although simple in theory, calorimetric experiments require
gr«-at care and many precautions. One of the most important
sources of error is radiation, i.e. there is a gain or loss of heat
beOftUl .boring bodies are at temperatures different from
Of the body being studied. The principal methods of
diminishing this error are (a) to compute the amount of heat
a-tually gained or lost by radiation; ( /, > to determine the
t -mperature \vhidi tin would have attained if there had
I sen no radiation; (V ) to employ a method in which the tern-
I jrature is kept the same as that of the surroundings.
Tin: ( ORRB rmv FOR RADIATION
1. kegnault's method is based on Newton's Law of Cooling.
'i his law may be stated as follows: The rate at which a body
C Ols is proportional to the d. betUcell its temperature
a -d the temperature of its surroundings. If. then, Af denotes
t e fall <>f temperature due to the radiation which occurs in the
8 ort tiiii.- A 7'. ; iperatures of the body and its sur-
i undings are denoted respect ml t» Newton*s law of
c oling may also be stated by the equation
is the pi-' ,,dity factor. If both members of this
e< nation are multiplied by the water i-quivaleiit. >•' * «»f tin- cool-
in j body, then since e'&t is, from (210) and (-<»)), th<- heat, A //,
210
PRACTICAL PHYSICS
lost while the temperature falls A£°, the equation becomes
AJ2WA2%-O, (211)
where r is written in place of the product e'k'. This r is a con-
stant which depends only on the nature and area of the radiating
surface, and is called the radiation constant of the body. New-
ton's law is now known to be only a rough approximation to the
true law of cooling; but it is simple, and, if the difference in
temperature between the body and the surroundings is not
greater than 15° or 20°, it holds fairly well.
Let CD and EF in Fig. 87 represent the respective changes
in temperature of the body and its surroundings while the body
cools by radiation. Since (tb—ts) is at any instant the vertical
distance from EF to CD and AT7 is the horizontal distance be-
tween two of these vertical lines which are near together, it fol-
lows that the product A I7
(tb — t*) is represented
very nearly by the shaded
area A A. That is, from
(211),
FIG. 87.
where k is a constant
which depends upon the
scales chosen in plotting.
If the temperature of the
body, instead of falling
the small amount A f, falls
from £2 to ^, the entire fall being due to radiation, the total
heat lost H involves the sum of the elementary areas A 4, that
is, the area CF. If this area is denoted by ACF, we have
H=rkACF. (212)
Now suppose that the body has heat given to it in such a
way that its rise of temperature can be represented by the
curve ABC in Fig. 88. The maximum temperature is reached
when the body ceases to receive heat from the source faster
CALORIMETRY
211
than it nuliatrs h< at t«> the cooler BUrroundingS. After this
point is ivarhrd. the hotly falls in temperature in a manner
that can he represented hy the line CD. While the body is be
lo\v the temperature of its surroundings it ahsorhs heat from
thrni, and while it is above thr temperature of its surroundings
> iu-at to them. The r<nli«tion correction, now tu he found,
Is Ul6 difference brt \\vrn thr amount of heat lost by the hodv
through radiation and thr amount ^aiiied hy ahsorption while
thr body \vas rising from ual to its maximum teinprra-
ture.
100 1 2 3 4 5 6 6 9 10 II 12
Curve showing rate of change of temperature of heated body.
I. • // t< thr heat lost by radiation during thr tim- 7"
t iat thr body is above thr trmp.-rat mv of iN sun-oiindin^s, and
! by ahsoi-ption during tin- tim.- T" that thr
1 >dy is h.'lou- thr t.-mp of its .siirrniiinlin^s. Tlirii from
( Ml.') and Rg. 88,
// ,- .1 . : i // r
v 1. ; lenoteat B / 1 .1" the area A-1l». Since,
uldition. thr radiati«»n ron-.-rtiou R is, from its drlinition,
ive
, .1").
(218)
212 PRACTICAL PHYSICS
If the radiation constant r were known, (213) could be used to
determine the radiation correction R. The purpose of allow-
ing the body to cool for a time by radiation after the other
heat changes have taken place is to make possible the determi-
nation of this r. From the part CD of the curve we have from
(212), if H denotes the heat lost by radiation while the body falls
in temperature from tf to £",
H= rkA,
and from (206) and (210) we have also
H=e'(t'-t"),
where e1 denotes the water equivalent of the body. It follows
that e'(t'-t") = rkA. (214)
On substituting in (213) the value of r from (214), we have
R = e'(t' - t") - A'~A". (215)
A
2. Instead of finding the number of heat units lost by the
body due to radiation while the temperature of the body is
rising to its maximum value, the effect of radiation can be
accounted for if the temperature is determined which the body
would have attained if there had been no radiation. In the
following modification of a method due to Rowland this tem-
perature can be obtained to a close approximation by a simple
graphical construction.
Suppose that a body at a temperature below that of its sur-
roundings is given a quantity of heat H such that its tempera-
ture rises to a value above that of the surroundings. While
the temperature of the body is lower than that of the surround-
ings the body absorbs heat, and while the temperature of the
body is above that of the surroundings, the body loses heat.
The way in which the temperature changes before the heat H
is added is represented by the line AB in Fig. 89. The line
ED shows how the temperature changes while the body is
absorbing the heat H. From B to C the body is, in addition,
CALORIMETRY
213
Fi.;. v.i.
receiving heat from the surround ings, and from 0 to D is
: heat to the surroundings. The line It I] shows how the
temperature of the hotly changes due to radiation alone.
Through li and C
draw vertical li i
Prolong DK hack ward
until it cuts the vertical
line through (.' i:
Through /'draw a li;
parallel to AH until it
•Ttical line
through li in //.
tempt-ratiin- indicated by
: cmpera-
To see tliat the above
method of finding
clesin n is reasonable, en: following. If
t 1C heat // had n ' • w«»uld hav«-
t nued to rise in temjKirature in the same way that it was n>ing
f om A /.'. ' uned tlie tempera-
t ire indicated by f it would have reached the temperature
i dicated l>y h. Tliat is, while the h..dy \ M in tempera-
t ire f: in temperature from // t.. // vrafl due
t heat from tin- surroundings and tlie rise from h to C was
« ie t«. a ; //. Again, if the hody had not been
p ven the : • //. hut if it had been at first at such a trmprni-
t re tliat as it coo. iched the temperature indicated l.y I>
a the same in>tant that it r«-ally readied that temperature
- -and then-after cooled as sh /'/.'— it would have been
a a temperature / at the instant when it really was at the
t. inperature (.'. Th.it is, while the hody really ro.se in tempera-
•• fall in tein dm- to radiation was
tl fall fi • D, so that if there had been no loss of heat
b radiation, the ;ure during this time would
h; ve been from C to f. If, then, there had been no gain nor
214 PRACTICAL PHYSICS
loss of heat by radiation, the body would have risen in tem-
perature the amount indicated by the distance from h to /.
But the temperature when it began to receive the heat H was
that indicated by B. So that the temperature which would
have been reached if there had been 110 radiation is a tempera-
ture as far above B as f is above h — that is, the temperature
indicated by g.
While the temperature of the body rose from C to D it was
really at a lower temperature than if it had been cooling from
/ to D, and so did not really lose as much heat by radiation as
has above been supposed. That is, the point / is higher than
it ought to be. For a similar reason h is also somewhat higher
than it ought to be. If the time from B to C is about the
same as that from 0 to D, these two errors will nearly balance
each other.
3. Another method should be referred to, although it is con-
siderably less accurate than the two already discussed. In this
method, first suggested by Rumford, the initial and final tem-
peratures of the body are so arranged that the difference be-
tween the temperature of the surroundings and the initial
temperature of the body equals the difference between the tem-
perature of the surroundings and the final temperature of the
body. The idea is that, by this arrangement, the heat absorbed
from the room while the body is colder than the surroundings
equals the heat lost to the room while the temperature of the
body is higher than that of the surroundings. That this, how-
ever, may be only a rough approximation can be shown as
follows : —
When a body is heated and then immersed in cold water, the
temperature of the water rises in a manner very like that repre-
sented by the curve HA in Fig. 90. During the first part of
the time, the temperature rises rapidly because the body is at
a temperature considerably higher than that of the water,
whereas when the temperatures become more nearly the same,
the temperature of the water rises more slowly. This means
that the first half of the temperature rise is accomplished in
CALORIMETRY 215
i me than the second. And this, in turn, if the tempera-
ture of the surroundings is half \\ ay from the lowest to the
highest temperature of the water, means that less heat is gained
• lion during the first half of the temperature rise than
is lost bv radiation during the seeond half. oruuu;,ui...u....ii..i,i..
— I""-
In fact, from cM:.') it follows that in L
the case represented in Fig. l»o the ratio
of the heat h»st by radiation to the heat
gained by absorption equals the ratio of
:••• tf /'.I '/ ftnd //AT. The radiation
in would compensate the radiation out if 20j
the temperature of the surroundings were
i to BD,&0 that the ftroM ''.!/> and
HBC were equal. That is, in the given
case, tli- temperature bet h- Fl° "'
ing the temperature of the surroundings should be about two
; nd a half times that after passing the temperature of the
: tirroundings.
Exp. 47. Determination of the Emissivities and Absorbing
Powers of Different Surfaces
. — The miJarivA
', idiating power of a surface is defined as the number of heat
io>t by : i at atmospheric pivssinv, per srrond, per
nit area, per deg: Me of temper* ton cooling body
B the temperature of the surrounding air. Similarly, the
( b%orbing poicer of a surface is denned as the number of hrat
i nits absorbi-d at atmo>ph«-i-i«- j.- p»-r M-mnd, p«-r unit
{ rea, per degree excess of temperature of the surroumling air
f ix>ve tin- t« mp. rature of the absorbing body. The object of
i iis experiment is to detern. emiariTitlef and the ab-
*• "bing powers of ditTerent surfaces for various temperature
( U'erences betw.-rn the surfarrs and the surrounding air, and
Jisotocompa: -ivity and ab>.)i-liing power of a given
h irface under similar condition*.
216
PRACTICAL PHYSICS
Consider a mass M of water filling a closed vessel, the water
equivalent of which is e and its external surface area A. If
during a short time AT the vessel and its contents cool A£°,
their mean temperature during the time being tb and the tem-
perature of the surrounding air being t# then from the above
definition the emissivity of the surface is
(216)
Similarly, if the vessel and its contents rise in temperature
A£°, due to absorption of heat from the surrounding air, the
absorbing power of the surface is
a —
(M+
ATA(ts-tb)
(217)
MANIPULATION AND COMPUTATION. — For this experiment
there are provided two or more metallic cylinders C (Fig. 91),
exactly alike except for the nature of their external surfaces.
One cylinder, for example, may be highly polished, one may
have a tarnished surface, and one may be
coated with lampblack. In finding the emis-
sivities of these different surfaces, the cylin-
ders are in succession filled with warm water
and suspended inside of an inclosure formed
by two concentric cans </, K, the space out-
side ^Tand inside J being filled with water
at the temperature of the room. The temper-
atures of the water in the central vessel and
of the water in the jacket are observed every
two minutes for at least half an hour. The
water in the central vessel and in the jacket
is kept thoroughly stirred throughout the
whole experiment. From these observations
are plotted two curves coordinating temper-
ature and time — one for the radiating body and one for the
water jacket.
FIG. 91.
CALORIMETRY
217
Suppose that in a particular experiment the curve shown in
'_ was obtained and the following data found: —
35 t
20
FIG. '.'_'.
Carve showing the rate of change of temperature of the hot body
and of the water jacket.
of copper radiating- vessel a ml stinvr, 78.1 g.
Mass of ••'•MtaiiM-d in vessel, 126.3 g.
Area of external surface, 1.~>:J. 1 sq. cm.
Si ice the sp> at of cuppor is known to be 0.093, it
.. *i,t.
218 PRACTICAL PHYSICS
follows from (210) that the water equivalent of the radiating
vessel and stirrer is
e = 78.1 -0.093 = 7.3 g.
In computing the emissivity by means of (216), ffe, t# and A£
may be taken from the curve. For a curve of this sort a con-
venient value for T is five minutes. For example, to find the
emissivity of the surface of the radiating body when at 34° C.,
while the inclosure was at 20°. 23 C., proceed as follows: To
the right and left of the point where the 34° line crosses the
cooling curve, lay off distances corresponding to 2.5 minutes.
At A and B, the ends of this line, erect perpendiculars until
they intersect the cooling curve, and at the points of intersec-
tion draw two lines parallel to the time axis. The distance be-
tween the last two lines represents the Atf° through which the
radiating body cooled during an interval of five minutes. In
this particular case A£=l°.34. Substituting in (216) the
values thus obtained, we have
_ (126.3 + 7.3)1.34 _
300x153.1(34.0-20.23)
In the same way are found the emissivities at other temperatures.
Proceeding as described above, with each of the surfaces being
studied, plot on a sheet of cross section paper an emissivity
curve for each surface.
Now fill with cold water the vessels heretofore used as radi-
ating bodies and by means of (217) and an experimental method
similar to that used to find emissivity, determine the absorbing
powers of the different surfaces for various temperature differ-
ences between the absorbing surface and the inclosure.
State in words the conclusion reached from a comparison of
the emissivity and the absorbing power of the same surface.
At the beginning of the radiation experiment the tempera-
ture of the water in the radiating body should be about 15° C.
above that in the water jacket, and in the absorption experiment
the temperature of the water in the absorbing body may be
about 15° C. lower than that in the water jacket. But in no
CALoKIM KTRY 219
should the temperature of the absorbing body be so low
that dew will be deposited on its surface.
Exp. 48. Determination of the Specific Heat of a Liquid
: . nini) <>F CM..LING)
OBJK< r AND Tm:<>i:y <>F KXI-F.KIMKNT. — Suppose that a
mt of a liquid of a specific heat st is contained in a \
which lias a \vaterequivalent *• and a radiation constant r. If
the temperature of the liquid is somewhat above that of
irn>undin<rs. and if the temperatures of both liquid and
surround; d for some little time, ami then the
tempi-ran. plotl.-d against times, two curves like those
_r. sT will be obtained. While the liquid and vessel fall
in tmiperatuiv through a range of A^°, the heat which they
is, from
S nee this heat is lost by niiliati..n. ( _ 1 _' i ihowi that it is also
g ven by
ff, = r/ .-'19)
\N iere r, / •• the same meanings as the r, k, and 1
ii (iMiii. I J 18) and (219) we have at M:
(m^-f-OA^/ = r J-'O)
I: the same way, if the liquid in question by warm
\v ter, and if t.s w mean that the symbols to which they
ai > appt-ndrd r.-h-r to tliis w.i
It' the scales chosen in plotting an- the same for both pai
CU WCS, the k in < '-!" > ami ( L'-Jl ) is the same; ami if the nature
of the sin essel remains the same, tin- /•
is '>e same. On dividii,. b\ <^_1) and idlTing for s^
wt get
e e
220
PRACTICAL PHYSICS
FIG. 93.
MANIPULATION AND COMPUTATION. — The apparatus used
in this experiment consists of a closed metal radiating vessel
suspended in an inclosure surrounded by an ice jacket. The
radiating vessel is provided with a stirrer
for agitating its contents and a thermometer
for reading temperatures.
Weigh the radiating vessel and stirrer
and, by multiplying their mass by the spe-
cific heat* of the material of which they
are composed, determine their water equiv-
alent. Fill the radiating vessel just to the
bottom of the neck with the liquid whose
specific heat is to be determined and set it
in water in a saucepan over a burner until
its temperature is about 40° C. Then wipe
the outside of the radiating vessel dry, sus-
pend it in the inclosure inside the ice jac-
ket, and while continually stirring, observe
the temperature every minute for quarter of an hour or longer.
Throughout this time keep the jacket full of ice. Then remove
the radiating vessel from the jacket and weigh it, thus finding
the mass of the liquid.
Clean the vessel, rinse it out with water, and fill to the
bottom of the neck with water. Then heat, dry, and suspend
in the ice jacket as before, and again observe the temperature
every minute for a quarter of an hour. Remove from the jacket
and weigh, thus finding the mass of the water.
Plot the cooling curves for both substances on the same sheet.
Since the jacket is packed with shaved ice, the temperature of
the surroundings in each case is zero. If, then, times are
plotted as abscissas, Aw is the area bounded on the top by the
water curve, on the bottom by the temperature axis, and on the
sides by any two convenient ordinates, — one near the beginning
and one near the end of the time employed, — and Al is the
* If the radiating vessel or stirrer is of unknown composition, the water
equivalent can be obtained experimentally by the method of mixtures, p. 224.
CAI.ORI.MKTHV 2m
same -.- -ept that its upper boundary is the other curve.
These IT6H may be obtained with a planimeter, determined by
counting the millimeter squares, or, perhaps most easily, found
by the method of average ordinates. A is the difference
between the temperatures at the points where the water curve
crosses the ordinates whieh hound the area Air. and Af, is the
eoiTespondin. nee for the other CUl
All of the data are now at hand for calculating the specific
heat <>f the specimen by means of ( Ji'i' ). Two or three cooling
curves for each sub>: taken, and two or three
values for the specific heat thus obtained.
Exp. 49. Determination of the Specific Heat of a Solid
I 1I«>I) MK MIX 11 H
OBJBOT \\i> TIIK-'KY Of i:\ri:i:iMi:\ r. -The Method of
> ixtures dep.-nds upon the principle that when a number of
I. .dies of dirtcreiit temperatures are brought toother, the
a noiint of heat lost by the bodies that fall in temperature
f [Uals the amount of heat gained by tl. s that l ;
t mperatuie.
Consider a body of mass 7/1, specific heat «, and temperature /,
t be plac.-d in a mass m} of liquid of sp* .1 *: and t.-m-
I ;raturc / OOnteined in a fenel «-f mass m made of a material
v hos«- il temperature of the mix-
t re be if. Tht-n. if / is higher than t ., the heat lost by the
•. \.-n body equals th«- su: n.-d by the vessel and
ii * contents and that gained by the surrounding air. That is,
I »m i -" .ation correction,
1 1 ™K'-'/)=^/+'M,X</--'/)+ R
'I he correction for radiation may be applied either by Re^nault's
II thod o: : //. . v: ;:<liically by the im.dilicat ion «,f
R >. viand OB pp. -l'J--ll. \Vhenthespi-eimen
i- in small pieces of temperature is rapid and Rowland's
n -thod i> pcrhap-> to !)••
222
PRACTICAL PHYSICS
If water is the liquid used, 8,= !. For purposes of ab-
breviation the water equivalent of the vessel, wcsc, will be
denoted by the single letter e. Then if tj denotes the tempera-
ture that the mixture would have reached if there had been no
gain nor loss by radiation, (223) gives
(224)
It should be noted that e represents the water equivalent of
the vessel in which the mixing occurs, together with any ac-
cessories it may contain, such as a stirrer or thermometer.
MANIPULATION AND COMPUTATION. — The special apparatus
used in this experiment consists of a calorimeter and a heater.
FIG. 94.
FIG. 95.
A calorimeter is any apparatus used to measure quantities of
heat. The ordinary "water calorimeter" used in this experi-
ment (Fig. 94) consists of a thin polished copper vessel held
centrally within a jacket by means of non-conducting supports.
The inner vessel contains a thermometer T' and stirrer $, while
a second thermometer T is suspended in the air space between
the two concentric vessels. A convenient form of heater, shown
in Fig. 95, consists of a closed copper can in which water can be
boiled. Extending through one side and projecting nearly
through the boiler at an angle of 45° with the bottom, is a tube
sealed at the lower end and having the upper end closed with a
cork through which extends a thermometer. The specimen to
CAI.OUIMKTKY
223
be heated is placed in this tube, and when the temperature
indicated by the thermometer T has become steady, the specimen
is dropped into the calorimeter. If the specimen is in small
. '. lead shot, it ran be poured into the calorimeter by
simply tilting the heater, if the specimen is in a single piece, it
i> drawn out of the heater with a thread and quickly lowered
into the calorimeter.
A ray Compact form of apparatus designed by Ke^nanlt is
In this apparatus the calorimeter is on a
little carriage that can easily b ', up to the heater and with-
drawn. The tube />'/.' \ • nds entirely through the heater.
At i: Quitter
.1 b\ vhich the
(pi ickly be opened
or closed. A thermom-
eter extending through the
s opp. /; permits the
< bservatioii «.f the tern-
he specimen.
specimen d in
t ie mid-: : he tube
1 Y a - :
t irough /•'. ci-
i en i ontained in a small wire basket.
1 /'hen '1 t-. drop the .specimen into the calorimeter.
t ie l;r / 1 ''lied, and the
s riiiL: leased so as to allow the i
• •n to I'all (luiekly into ; : imeter.
\\ater «Mjui\ | the inner vessel of the calorimeter
t be determined. If the mass and specific heat be
low:. . part oft that chan^o intempera-
1 easily and m<>M accurately
o fained liy ' the sum of the prnduets of th.-se masses
a n the < _f Specific he;it^. \\'hell this method WUI-
•t be applied, the ID | mixtures can l>c emploved. In
•lie inner \esseland >tinvr, half iill
224 PRACTICAL PHYSICS
the inner vessel with water at a temperature some 10° or 15°
below that of the room, and weigh again to determine the mass
mc of cold water. With one thermometer in the water in the
calorimeter and another in water at a temperature some 15° or
20° above the temperature of the room, watch both thermome-
ters for a few moments, and immediately after reading both te
and £A, the temperatures respectively of the cold water and of
the hot water, pour rapidly into the inner vessel enough of the
hot water nearly to fill it. Meantime stir briskly and watch
the thermometer in the calorimeter. After noting the tempera-
ture of the mixture tm, weigh again to determine the mass mh
of hot water added. Since the heat lost by the hot water
equals that gained by the calorimeter and contents plus that
lost to the surroundings,
m&* ~ O = C^c + 0 Om - *c) + R'I (225)
where e represents the water equivalent of the calorimeter
and R' the radiation correction. From the discussion on pp.
214-215, it follows that R' can be made very small by selecting
proper values for the temperatures. Since e is iisually small
compared with m^ and since in the only place where e is used
in (224) it is added to ra,, a smaU error introduced into e by
failure entirely to eliminate Rf would cause in *, a very small
error. If this very small error is neglected, (225) gives
tm}-mc. (226)
The satisfactory determination of a water equivalent by this
method requires deft and rapid manipulation and careful deter-
mination of temperature.
Be sure that the specimen is dry, and place it in the tube in
the heater until its temperature assumes a constant value t.
While the specimen is heating weigh the inner vessel of the
calorimeter, if this has not already been done. Then pour into
this inner vessel water at a temperature three or four degrees
below that of the room until the vessel is somewhat more than
225
half full and determine the mass of the water. Assemble the
I of the calorimeter, placing one thermometer in the water
contained in the inn. nid another thermometer against
the inner surface of the jacket. The thermometer in the water
should have its bulb entirely covered by ihe water, but should
not be low enough to be touched by the specimen. The tein-
• ure of the water should now be observed at quarter or
half minute intervals, and the temperature of the jacket every
minute or two. For each reading the hour, minute, and second
at which the reading is made should be recorded. The readings
;ken continuously but belong to three successive periods.
ore beginning the first period be sure that the thermometer
in the heater is steady in the neighborhood of 100° and record
its reading.
<t period. While stirring the water read times and eorre-
iing temperatures for some three to live minutes before
ran- .men to the , .-tor.
l>eriod. \ en instant transfer the specimen rap-
dly to the calorimete ;inue to .stir the water and to take
emperature reading* lart.-r <>r half minute. While the
ieated specimen isghing up its heat, the \\ater rises rajiidly to
maximum temperature t,. :>eriod is frequently over
n fifteen nr twenty see,- maximum temperature is
ttained when the rate at which i -.idiated by the water
i) the air equals the rate at which tin; water receives heat from
he BJ . The temperature may remain stationary at this
• for an app: length of time. Then-after, if the
: has risen to a temperature above that of the jacket, the
yss by radi.it ion exceeds the gain of heat from the specimen.
f the water does not rise above the temperature nf the jacket
of a minute after the .specimen is dropped into thecalo-
i imeter, the sprcimen is tobe :<1 the experiment begun
i l/4n. If the temperature rises rapidly and then almost at once
! ills again somewhat, the speeimm has come too close to the
t lernioineter and th- M-nt should be begun again.
/>eriod. Without interruption e..ntinue to stir the,
226 PRACTICAL PHYSICS
water and to take readings of temperature and time for at least
five minutes during the cooling of the water in the calorimeter.
After the third period weigh the inner vessel and contents,
and so determine the mass of the specimen.
With these readings, plot on the same sheet two curves — one
coordinating temperature and time for the water in the calo-
rimeter, and another coordinating temperature and time for the
surroundings. A pair of such curves is shown in Fig. 89.
From these curves the temperature which would have been
reached if there had been no radiation can be determined by
Rowland's method. The data are now at hand which when
substituted in (224) give a value for the specific heat of the
specimen.
One or two preliminary experiments may be necessary in
order to determine just how much water to use and at what
initial temperature to have it. After a satisfactory set of read-
ings is obtained, another set should be taken and two values
found for the specific heat.
Exp. 50. Determination of the Specific Heat of a Solid
(METHOD OF STATIONARY TEMPERATURE)
OBJECT AND THEORY OF EXPERIMENT. — The object of
this experiment is to determine the specific heat of a solid by a
modified form of the Method of Mixtures in which the water
equivalent of the calorimeter is avoided and the radiation cor-
rection is eliminated. This is accomplished by maintaining
the temperature of the calorimeter throughout the experiment
the same as that of the surroundings.
Consider a body of mass m, specific heat «, and temperature £,
dropped into a calorimeter containing m1 grams of water at the
temperature of the surroundings tr Let cold water be added
to the calorimeter at such a rate that the temperature of the
calorimeter remains constant. If the mass and temperature of
this cold water, be represented by mc and tc respectively, then
the heat emitted by the specimen is, by (206), ms(t — t^, and the
CALORIMETRY
227
.rained by the cold water added to the calorimeter equals
Since the water originally in the calorimeter has
not changed in temperature, the heat lost by the specimen
.rained by the cold water. That is.
<m9(t-tl) = ///,.(<!-
Whence
(227)
MANIIMI.A i i«\ \M» COMPUTATION. The apparatus use*!
iu this experiment includes a calurimeter of special design C
KI.J. '.w.
( <Hg. ;etl,er with a heater If and a water dropper />,
1- th . 1 1 axes. The water d r«» |>-
I>- r consists of a reservoir li ( Fig. 98), having a valve I", by
in »ns <»f which the t! through tlu-<M\ ui In-
n mla ^iirruun.:. reservoir is an ice jacket J. P.y
HI Ulis nf the thrnni-: temperature <»f the water at
tl • iiimiH-iit Li from the water dropper can be obser\c<l.
T ie calorimeter i > is essentially the metallic bulb <>! an
228 PRACTICAL PHYSICS
air thermometer into which projects a copper tube X for the
reception of the water and specimen. Any change in the tem-
perature of the calorimeter is indicated by an open manometer
tube M. To prevent any effect due to changes in the tempera-
ture of the surrounding air, the calorimeter is placed in a
water bath !Fat the temperature of the room.
After the apparatus has been assembled ready for use, the
specimen is weighed and placed in the heater. The mixing
tube of the calorimeter is unscrewed arid weighed, first
when empty, and then when filled with enough water at
the temperature of the room to cover the specimen. The
mixing tube is now replaced and the stopcock attached to
the manometer is opened for an instant. By this means
any difference of pressure between the inside of the air ther-
mometer bulb and the outside air is equalized. When the
thermometer in the heater indicates a steady temperature near
100°, the water dropper is made ready for use by allowing cold
water to escape until the thermometer in the
escaping steam indicates a stationary tempera-
ture. The temperatures of the specimen in
the heater, the water in the mixing tube of the
calorimeter, and the cold water in the water
dropper are now noted. The heater is rotated
f% into position over the calorimeter and the
specimen quickly lowered into the mixing
tube. The heater is immediately rotated out
of position and the water dropper rotated into place. By
operating the valve F", cold water is now allowed to fall into
the mixing tube at such a rate that the index in the manometer
tube of the air thermometer remains stationary. The proper
rate can be ascertained only by previous trials ; it depends
largely upon the conductivity and fineness of division of the
specimen. When no more cold water is needed, the mixing
tube with its contents is again weighed. All of the data
necessary for the computation of the specific heat of the speci-
men by means of (227) are now at hand.
FIG. 99.
CALORIMETRY
229
Exp. 51. Determination of the Specific Heat of a Solid
(JOLY'S MKTHOD)
OBJECT AND THI:MI;Y OF KXPHUIMKNT. — Acold body placed
in an atmosphere of steam absorbs heat until its temperature
i> the same as that of the steam. A certain amount of steam
is thereby condensed. It' the steam is at the boiling point of
water. the amount of 1 equals the product of the mass
I and thf jiiivalent of condensation of steam.
\'> "heat equivalent of condensation" of steam is meant the
number of heat units L^iven up
by the condensation of unit
mass "f sleam. '1'his is nu-
merically equal to the ul
equivalent of vapori/ation
vater, /.,•. the number of heat
units required to \apori/e unit
188 «>f water. The nhje.
t is experiment is to determine
t e specific ht-.it of a solid
f om a nieaxn «>f tin-
tss of s: ndfn>f<l on
t e body as it rises in tempera-
•iling point of
July's apparatus ( Fig, 1"" .
'* of a steam chan
i dosing one pan of a delicate balan I • ;
: >m thf l.al.uife b.-am by a tine \\ir<- passing tbroujrh a small
h( le in t). : the steam chandler. Steam is first passed
in O the steam chamber ami the mass of steam which condenses
Oi . he scale pan is weighed. The apparatus is n»\\ allo\\ed to
en . to the temperature of the room. The scale pan is dried
an 1 upon it is placed t : ;i.- n whose specific heat is required.
St -an; ii passed into the steam chamber and the mass of
230 PRACTICAL PHYSICS
steam which condenses on the specimen and on the scale pan
is weighed.
Let s denote the specific heat of the specimen ; e, the water
equivalent of the scale pan and suspending wire ; t1 and £2, the
respective temperatures of the room and of the steam ; A, the
heat equivalent of condensation of steam ; and mv mv w3, and
m^ the respective masses required to balance (1) the empty
scale pan, (2) the pan with the steam which condenses on it,
(3) the pan and the specimen, and (4) the pan and specimen
with the steam which condenses on them both.
The amount of heat absorbed by the scale pan and suspending
wire as they rise in temperature from tl to t2 is 0(£2 — ^i)-
This heat is supplied by the heat liberated in the condensation
of the mass (w2 — wij) of steam. Therefore,
e 02 - *i ) = <>2 -m^h. (228)
Similarly, the amount of heat absorbed by the specimen, the
mass of which is (w3 — w^), together with the balance pan and
suspending wire is (ra3 — m^ s (£2 — ^) 4- e (t2 — ^). This heat
is due to the condensation of the mass of steam (w4 — w3).
Consequently
O3- m^s(t2 - ^)+ e(t2 - ^) = <>4 - ™3)^'
Subtracting (228) from (229),
Whence .-- (230)
It will be noticed that (rw4 — w3 — -w2+ T^) is the mass of
steam condensed on the specimen, and that (w3 — m^) is the
mass of the specimen.
MANIPULATION AND COMPUTATION. — A common source of
error in this method is an uncertainty in weighing produced
by steam condensing on the suspending wire where it emerges
from the steam chamber. In the apparatus illustrated in the
231
figure, this trouble is diminished by having the suspending wire
Through a small tube surrounded by
•am jar!. £, 101). l»y passing
the steam through tliis jacket before it
elite: -learn chamber, the neighbor-
hood of the aper; itViciently heated
to prevent a large amount of condensation
on the suspending wire outside of the , 1()1-
chamber.
Take care to have tin- >u>peiiding win- hang free. Then
with standard masses nil balance tlie lower scale pan. When
: in a detached b •;-«»usly, note the teni-
p'-ratuiv tl of the iii>ide of the steam chamber and then eounect
• iler t<» tin- steam chamber with a go. rubber tube.
D will immediately be condensed on the object pan. After
one or two minutes diminish the flow of strain to sueh an
i that t: :it will not dist nib the object pan. With
s andard masses HI, aga i n bring tin- balance into equilibrium.
'in tin- steam chamber and allow the
< lamber to cool to the temperature of the room f,. Dry the
( >ject pan, place up..n it the specimen irhoae ^ecific heat is
, and dct' nass m^ imw required to bring the
into equilibri-iin. Again connect the boiler to the
s earn chamber, and after four or ti\e minuto. when the speci-
1 en and [.an have aeipiin-d the tempera: of the
s eam, diminish the flow of |1 :id with standard masses
;- 4 again l»ring the balance into equilibrium. Since //. th-
e nivalent of vaporization of water, is known, all of the data
,: e now at hand for computing the specific heat of t lie .specimen
b • means of (230).
Exp. 52. Determination of the Heat Equivalent of Fusion
of Ice.
I:IMENT. — The Heat Equiva-
le it of I-'MMMM of a substance is the number of heat units re-
qi.iivd to melt unit mass of it without changing its temperature.
232 PRACTICAL PHYSICS
Suppose that when m^ grams of ice at 0° C. are dropped into
mw grams of water at tw°, the ice melts and the temperature of
the mixture of the two becomes £2°. During this operation, the
ice has absorbed the heat required to melt it and also after
melting to raise its temperature from 0° to £2°, while the calo-
rimeter and its contents have lost heat. If there were no gain
of heat from the surroundings nor loss to them, the heat gained
by the ice in melting and then rising to the temperature t2 would
equal the heat lost by the calorimeter and contained water.
That is, if e denotes the water equivalent of the calorimeter,
and / the number of heat units required to melt unit mass of
ice, we should have, from (206) and (207),
™,/+ mt(tz - 0) = (mw + e) (tw - *a) .
That is, the heat equivalent of fusion would be
/=0"»+0ft.-«,)_fr (231)
In most cases, however, the error due to radiation is too great
to be neglected. This error may either be computed by Re-
gnault's method or determined graphically by the modification
of Rowland's method given on pp. 212-214. If the latter
method be selected, it is necessary to determine the tempera-
ture that the mixture would have attained if there had been no
radiation nor absorption. Denoting this corrected value by £2',
we obtain the corrected equation
(^ + e)fe-V)_
rrii
The simple theory given in this experiment applies only to
a solid whose temperature is at its melting point at the moment
it is introduced into the calorimeter. In the general case not
only will the temperature of the specimen be below its melting
point at the moment of its introduction to the hot water of the
calorimeter, but in addition its specific heat will be different
in the solid and the liquid states. Even though neither of
these specific heats is known, by means of three experiments,
CALORIMKTKV 233
similar to the above, in which the masses of the specimen and
the water, as well as the original temperature of the water, are
different, the heat equivalent of fusion of a substance can be
found. We have thus three simultaneous equations containing
but three unknown qantities. vi/. the required heat equivalent
of fusion and the specific heats of the specimen in the solid and
in the liquid states. l>y eliminating the specitic heats, the
heat equivalent of fusion can !•«• determined.
MANMM LATK'N ANI» ( '< 'M I'l T A TION. Wei^'h the inner
! of the calorimeter and the stinvr. Tlie product of their
and the spe.-ilic heat of the material of wliirh the vessel and
:iiposed gives the water cqnival.-: . Fill their
Teasel somewhat over half full of water at about 60° ('., weigh,
and then a^cinMe I
Cut a piece of ice having a ma>s somewhat over a fourth
that in the calorimeter. Keeping the water in the calorimeter
veil i the temperature of the water about •
li ilf minute ami of the surroundings every minute or kwt>«
1 ecord the hour, minute, and >••<•, ,nd at which each reading
i made. At a given inMaii:. temperatures for
f ur or five minutes, dn>p the carefully dried ice into the
c lorimeter and continue reading temperatures and stirring
f r seven or eight minutes longer. The ice must be Kept
s Emerged, jind under n» -tances must the temperature
<> the inner vessel fall so low t : ins on it. Now
v -igh the inner vessel witli its contents. The data for deter-
n ining ww and ///, are now at hand.
The ted temperature of the mixture can hr detern
i: aphieally. as follows. On a sin^l. t' em. rdinate axes
p >t two curves — one coordinating temperatui-e and time
i th. \\at« -r in the calorimeter, and the other for the air
b< twee n tlie two ves- 9 ;eli a pair 0 shown in
F r. I"-'- Through the puint of intersection of the two curves
di ,v a line parallel to tlie temperature a\ duce the
•lini,' OOrre -I// -intil it intersects this line PS at soiii,-
.MI j\ Pi 1>I rard until it intersects the line
234
PRACTICAL PHYSICS
PS at some point R. Then in the manner given on p. 213
it may be shown that the point corresponding to the tempera-
ture t2' is as far above R as w is above x. That is, to find t2r
add to the temperature indicated by R the temperature differ-
ence represented by wx. It may be necessary to make one or
45
40
35
15
100 24 6 8 10 12
FIG. 102.
Curve showing rate of change of temperature of calorimeter.
two preliminary experiments to determine just how warm to
have the water and just how much ice to use. After the ex-
periment has been performed successfully it should be repeated
once or twice, two or three values for the heat equivalent of
fusion being thus obtained.
( Al.uRIMKTRY 235
Exp. 53. Determination of the Heat Equivalent of Vaporization
of Water
OB.HJ r \\i> TIIKOUY Off Kxi'Kiii.MKN r. - - If heat be applied
to S liquid, the liquid rises in temperature until its niaximiiiu vapor
- a tritle j :!i;in tin.' external pressure on
\ap.>r pre>xiuv is then great enough to make
in the liquid expand in spite of the pressure of the
liquid olit>ioV of them. As the bubbles gro\v they rise to the
surface and burst and the liquid is said to lx.il. Further addi-
tion of 1; mperature. but simply makes
• n into ' Q on faster. i.> . produces
• rapid boiling. The number of heat units required to
v tpori/.e unit mass of a liquid is called the /
vaporization of the liquid. The object "f this experiment is to
determine t. ; vapori/ water.
i ins of steam be condensed in m Drains ,,f water
e )!itained in a ••alorini.-t.-r »»f w.it.-r .-quivah-n: tt dem.te
t ie temperature of the st« niperature of the ealoriiu-
e ,-r and OOUt the moment the steam began to enter; (.,.
t e tern j -era ture <»f the two aft.-r t he v are mixed ; and r, th<
e nival. -nt of itioii ot Th.-n the h.-at uri\en up
1 the steam in 0 i'_T and then cooling to th,- t.-mpei'at ure
t. equals the heat taken up by the calorimeter and content s plus
t e 1 md (^»-
= (wi.+ OOi-O-
\\ ier- // t'liation ,n.
\ hence,
. - (, ^ (233)
M \\IITI. \ i i"\ \\i» COMPUTE 'I'lie apparatus used
in thi nt e..mpri>.-> a b..il.-r in which the li(piid is
v. pori/.-d and .1 calorimetei' containing a copper worm in which
tli 3 vapor is coml.-ns.;d. Tin- liquid in the boiler A ( Fig. 103)
236
PRACTICAL PHYSICS
FIG. 103.
is heated by means of an electric current passing through a coil
of wire. The arm holding the boiler is attached to a vertical
rod supported by the tubular column B. Below
the clamp D there is a horizontal slit extending
through an arc of about 90°, and from one end
of this horizontal slit there is a vertical slit ex-
tending about halfway down the tubular column.
A pin in the vertical rod supporting the boiler
extends through this slit. By means of this
arrangement, the boiler can be rotated quickly
into a definite plane and dropped in a vertical
line so as to cause the outlet 0 of the boiler to
register with the end W of the copper worm
contained in the calorimeter 0.
Weigh separately the condensing worm and
the inner vessel of the calorimeter with the
stirrer, and determine their total water equiva-
lent e. Pour water into the inner vessel of the
calorimeter until all the convolutions of the condensing worm
are covered. The temperature of this water should be below
that of the room, but not so low as to cause dew to be deposited
on the calorimeter. Determine the mass mw of this water.
Assemble the apparatus and adjust the position of the calorim-
eter until the outlet of the boiler will register accurately with
the opening in the rubber stopper oh the end of the condensing
worm. Raise the boiler, thus disconnecting it from the calorim-
eter, rotate it to one side, and pour into it enough distilled
water to cover all the turns of the wire. Connect a 110-volt
circuit to the terminals of the wire spiral and adjust the rheo-
stat until the water boils rapidly but does not spatter over into
the outlet tube.
Now commence stirring the water in the calorimeter, every
half minute recording its temperature, and every minute or
two recording the temperature of the air in the jacket. Record
the hour, minute, and second at which each reading is made.
After reading for two or three minutes rotate the boiler into
CALORIMKTKV 237
position, drop it into place, and, without interrupting the stir-
ring and reading of temperatures, allow steam to tlow into the
condensing worm until the temperature of the water in the
calorimeter rises to 45° or 50°. Disconnect the boiler from
alorimeter, rotate it to one side, throw oiV tlie current, and
continue stirring and taking temperature readings at one min-
ute intervals for about ten minute^. K.-move the condensing
worm from the calorimeter, carefully dry the outside, and weigh.
The diiYeren< .-n this mass and the mass of the worm,
iy determine(l. is the m M of the condensed steam.
the barometer, Correct it as indicated on pp. IT'I-ITS. and
by Table ]-2 find the temperature of the steam.
Compute tli. • tlie radiation correction /{ by lleguault's
method in the manner ^i\cn nn pp. -Jn'.i-iMl?. In determining
.1 notice that t he , ' ; M ( _' I .", » j> t he water equivalent
of everything that cooled along the curve ('/> ( l-'ig. ss). In
tl e pr.->.-nt case *•' is the ••piivalent of the inner vessel
of calorimeter, contained water, >tirrer, thermometer, worm, and
c< -id- .im.
«xp. 54.
Determination of the Heat Value of a Solid with
the Combustion Bomb Calorimeter
)r..i i-:< r AM* TiiKniiV Off K\i'i-:i:iMKsr. — The object of
ifc s experiment !i •• amount of h.-.n developed
b\ the complete eombostioi] of a unit mass of coal. The heat
Vn uc ; liquid .<T in liritish thermal
Uii ts per pound • r ^ram.
'he method to be . 1 in tliis experiment is to burn
a nown mass oi' -tam ••• in a strong steel bomb
filial with oxygen under lii-^li pressure. During the combns-
tio i the bomb remains immersed in a water calorimeter and
tb. hi- :in.-'l by the ordinary method «,f mix-
Thn- • that by the combustion of m grams of
th- sub>tan'-e, ih»- bumb t- -. ith t he calorimeter, its acces-
; \\ater ri>e in temperature from tf to
238
PRACTICAL PHYSICS
£2° C. If the mass of water in the calorimeter is mw grams, the
total water equivalent of calorimeter, bomb, thermometer, and
stirrer is e grams, and the radiation correction is R calories,
then the heat value of the substance is
TT
H=
m
calories per gram. (234)
K
The superiority of this method is that since in it complete
combustion is attained and all the products of the combustion
remain in the apparatus, the quantity of heat developed is
readily computed.
MANIPULATION AND COMPUTATION. — The apparatus used
in this experiment consists of a water calorimeter, a combustion
bomb, a press for molding the specimen into a small coherent
pellet, and a retort for generating
oxygen.
Hempel's combustion bomb consists
of a soft steel or cast-iron capsule
D (Fig. 104), closed by a massive
plug (7. The inside surface of the
bomb is coated with enamel. The
plug is pierced by two passages —
one JH for filling the bomb with
oxygen, and the other for the in-
troduction of an insulated conductor
KF. The gas passage is controlled
by the compression valve A. The rod
KF is insulated from the metal plug
by the rubber packing M and asbes-
tos packing N. G- is a metal rod
screwed into the plug. A little basket
E, made of incombustible material,
is suspended by means of heavy plati-
num wires from the ends of the rods
G- and F. The ends of the rods Gr and F are connected by a
thin platinum wire.
FIG. 104.
CALORIMETRY
239
In preparin_ imen of coal for a determination, the coal
It pulverized in a mortar and then molded into a compact
coherent pellet by means of a screw press ( Fi^. 1 '.»">). The mold
of the pn f a block of steel s ( Vig. 1 <>*'.) bored out
to the required si/c. The
upper portion of this hole
vlindrical and is titled
with a cylindrical p!u_: .1.
Lower portion of the
:iued nut to a
al form and is lilted ^_ ^^
with a conical plu^ It. ( hi
of the
plu<_: narrow channels \\hi.-h extend from on
in oil
Loop a / s / .ei- thr conical phiLf and lav
it; ends in tl L.26 ^. of pulverized coal
about thi- . pin thr • -slimier .1 in place and the
p unger P on t< n tin- screw down until the
8] ecin .mpressed into a compact pellet. I
tl 6 s» p thr mold ml • the upprr : .1 guides, and
a ain depress the s.-r.-w until tli- t mil through
t: • b t' the iimid. \\'i;h a «.harp knitr parr down thr
p' llet until it wri^hs about one gram. Cut off one end of the
tl read cl nenu li.-:m>ve any. loose partii
C« ll by DO a .small brush, place the pellet mi a «
g MB. . hoiiMt touch the pellet with the fingers, but
h; tldlr by 11: ; hr thivad.
Dome* the plug' : th« b..mb. mount it in a
rt ort stand, con IM-.I ihe t«-r.v. trie circuit to the
bi id ing posts A' and L. and can-full \ d.-en-asr the resistance in
th) circuit until the current will just brin^ the platinum wire
ctin^ (1 ami /'to a r.-d glow. \Vithmit disturbing the
re ..->tancr in I . open the switch and disconnect the
te mi: MI the binding posts A' a ml A. Place the specimen
Of coal in the basl I /-' i of the thread to
240
PRACTICAL PHYSICS
the wire connecting Gr and F. Without disturbing the speci-
men, remove the plug from its support and screw it tightly
into the bomb. The bomb is now ready to be filled with
oxygen. Into the gas generating retort R (Fig. 107) put a
mixture of about two hundred grams of potassium chlorate and
some fifty grams of manganese dioxid. Put a tightly wound
roll of copper or brass wire gauze into the tube leading from
the retort, and connect the retort and a pressure gage Gr to the
combustion bomb in the manner shown in the figure. Before
beginning to heat, shake the retort so as to spread the mixed
potassium chlorate and manganese dioxid along its whole
FIG. 107.
length. The pressure gage and combustion bomb are immersed
in a vessel of water for the purpose of detecting any leak in the
bomb and also for the purpose of cooling the oxygen coming
from the hot retort.
Open the gas valve in the combustion bomb and apply a
Bunsen flame near the farther end of the gas generating retort
until the gage indicates a pressure of about 1 Kg. per sq. cm,
(14 Ib. per sq. in.). If the flame be now removed, the heat
already given to the retort will generate enough oxygen to
raise the pressure to about 5 Kg. per sq. cm. (70 Ib. per sq. in.).
Now loosen the flange coupling P so as to allow the mixture
of oxygen and air contained in the apparatus to escape. By
QALORDCETRY I'll
tightening the coupling P and repeating this operation the
entire apparatus can be freed of air. Now tighten the couplings
and slowly heat the retort until the gas pressure rises to about
1- Kj. pel Bq. em. (170 Ib. per sip in.). Close the gas valve
on the combustion bomb ami immediately afterwards discon-
:he boml) at the coupling // from the remainder of the
apparatus. Cool the boml) to about the temperature of the
room and carefully dry it with a towel.
Place the bomb in a water calorimeter (' ( Fig. 1<>7 ) containing
m* gram- r at abmit the room temperature. Connect
the terminals of the pre\iou>ly arrau trie circuit to the
binding pOfttfl A' ••'.-. i A. and see to it that A' and I, are not
-circuited by the cover of the calorimeter. liclWe closing
\itcli in the electric circuit take temperature ivadii:
the continuously stirred water at half minute intervals l'..r at
least live minutes. At a given instant olotM the switch so that
t ie electric current will ignite the specimen. The switch .should
1. 3 closed for a moment only or the heating r fleet of the current
v ill need to be taken into ;iccou nt . Continue the water
a id taking half minute tenij | for ;it least ten
i inutes ;i | bomb OUt of the water, open
t ie va -crew the head, wash out the inside, and oil the
8 rew ti
From a en: r temperature and time lind by
t 6 graphical method described <>n pp. _' \- -1 \ the highest t,-m-
p nature tliat would have been .;' by the calorimeter it
tl ere had been no Ion nation. Let /.. i,p
tl ia oorreeted temperature, rbenin \vrite
& — -i' calories per gram.
in this equation tl. t equivalent •' is still unknown.
: mined in any of three - iking
im of the products- of the masses ami the assumed specific
hilts the apparatus. In an apparatus
liletl many different materials of uncertain
242 PRACTICAL PHYSICS
composition, this method is unreliable. (£>) Experimentally,
by the method of mixtures. The large amount of water re-
quired in this experiment and the difficulty of obtaining
temperatures accurately make this method unsatisfactory for
inexperienced observers, (e) By means of a supplementary
experiment in which a definite amount of heat is developed in
the apparatus by the combustion of a known mass of a sub-
stance having a known heat value. There are a number of
substances the heat values of which are accurately known and
which can easily be obtained pure. The last method is the
one that will be employed in this experiment.
Suppose that when using the same apparatus as before, the
burning of m' grains of a substance of heat value H' raises the
temperature of the apparatus and of m'w grams of water from
fj° to t%°G. Let £3 be the temperature that the calorimeter
would have attained if there had been no loss of heat by radia-
tion. Then
H, = (X/+gXV-*i') calories per gram. (236)
Whence, on solving for e,
Wl' H I Xf)OTX
e = - — -,-wiJ. (237)
Naphthalin is a suitable substance to use in this supplemen-
tary experiment. Make a pellet of somewhat smaller mass
than that of the coal already used and proceed exactly as in
the experiment with the coal. Use (237) to find e, and then
(235) to find H.
Before putting away the apparatus dig the remaining solid
substance out of the gas retort, rinse out the combustion bomb
with water, and carefully oil the threads of the bomb and all
parts of the press. Be certain that no water or oil is left inside
of either the retort or the bomb. If 'oil or any other organic
substance is heated in the retort with the oxygen producing
mixture an explosion is liable to occur.
CALORIMETRY
243
Exp. 55. Determination of the Heat Value of a Gas with
Junker's Calorimeter
OB.II.' r AND THI:<>I:Y o* KXI-KKIMKNT. - - The object of
this experiment is to determine the numher of lieat units devel-
liy tin- eoinbustion of unit volume of a Driven sample of
In Junker's metho<l the heat developed hy a steady llame
•cniiiin-d l>y mt-asurinLT the heat ahsnrhed by a steady
:n <>f water incli»in_f the Itame.
lus.
a ratus consists of an accurate gas met. •;• V < 1 i_^. 108),
a eras piv^un- iv_Mila; '.»rimeter (7, of
244
PRACTICAL PHYSICS
design. The calorimeter consists of a combustion chamber A
(Fig. 109), inclosed by a water jacket B, traversed by a large
number of tubes for the pas-
sage of the products of com-
bustion. The water jacket is
surrounded by a closed space
L filled with air. After trav-
ersing the meter and pressure
regulator the gas is burned in
the burner Q. The products
of combustion after passing
through the tubes traversing
the water jacket escape
through the vent Y. The
temperature of the gas as it
enters the burner and the tem-
perature of the products of
combustion as they leave the
calorimeter are given by the
thermometers T" and T1". A
stream of water flows from the
supply pipe D into a small
reservoir kept at constant
level by means of the overflow
pipe 0. From this regulator
the water passes down the
tube E through the control
valve V, thence through the
water jacket jB, thence through
Gr and the discharge nozzle
H into the measuring vessel
U. The temperatures of the
water as it enters and as it
leaves the calorimeter are
given by the thermometers T'
and T. Water vapor formed by the combustion of the gas
FIG. 101).
( AI.oRl.MKTRY 245
condenses on the inside of the combustion chamber and escapes
through the outlet •/ into the measuring vessel W.
The tlow of water ami < -o adjusted that the tempera-
ture of the products of combustion escaping at Y is approxi-
mately the same as the temperature of the gas entering the
burner at /'.
Let r represent the volume (reduced to standard conditions")
of the gas burned during a certain time. Let the ma-- of
which passes through the calorimeter during this time be
denoted by IK,, and let its temperatures on entering and on
leaving be represented by £ and? respectively. Let the mass
am condensed during the combustion be represented by
.d let the temperature at which it condenses and the tem-
perature of the condensed steam as it leaves the calorimeter he
d-'iiotrd b\ ?. and t . respectively.
u the heat value of the gas // is given by the equation
*« ~ ( L,.;S (
is the heat equivalent of vapoi i/ati<>n of water. If m,r
a d in, are measured in grams, v in liters, and temperature- in
d grees centigrade, th.-n // m in gram calories per liter
o kilogram calor
MAMITI.A ri<>\ AND COMPUTATION. — After assembling the
a paratu-. : /> to the water supply so that any leak in the
c; or: .dent. The tlo\v of water into t he
8] parat us inn s be sufficiently great to overflow through
tl ) pipe 0. With gas valve at the burner Pclosed, conneet the
gj } regulator to the gas supply and notice whether the index of
,th i meter moves. If it does, seek out th- .d remedy it.
With the water still flowing through the apparatus, take the
burner out of tlie i-alorimeter, light the gas. and replace the
bu ner. If the gas is lighted while the burner is inside the com-
bo lion chain' danger of an explosion- Haveti
of ,he bu m u t., 1") cm. above the lower opening to the
en: iliustion chamber. The dani] mid t"- from one half
246 PRACTICAL PHYSICS
to completely open, depending upon the draught required for
the flame.
Arrange the flow of water by means of the valve V and the
flow of gas by means of the valve P so that the thermometers
T" and T'" indicate practically the same temperature. For
ordinary illuminating gas the proper rate of flow of water is
from 1.0 to 1.5 liters per minute.
After all of the thermometers indicate nearly stationary tem-
peratures, note simultaneously the gas meter reading and the
temperatures indicated by the thermometers T and T' . Then
immediately place suitable vessels U and W so as to catch the
warmed water escaping from ^Tand the condensed steam escap-
ing from J. Note the temperatures of the ingoing and the out-
going water every 15 seconds until two or more liters of water
have flowed into the vessel U. Then remove the vessels 27 and
TFand at the same time take the gas meter reading. Note the
temperature tc of the condensed steam in W. Determine mw
and ms by weighing.
From the difference between the two gas meter readings to-
gether with the temperature and pressure of the gas passing to
the burner, the value of v is found by means of the fundamental
law of gases. The temperature is given by the thermometer
T" . The pressure is the sum of the barometric reading and
the height of mercury corresponding to the difference in the
levels of water in the manometer V.
All of the data are now at hand for substitution in (238.)
By substituting a properly designed lamp for the gas burner,
Junker's calorimeter can be used for finding the heat value
of a liquid.
CIIAPTKR XV
THERMODYNAMICS
IT is found that whenever Tr units of mechanical energy are
entirely used in producing heat, the amount of heat produced is
always the same, being independent »>f tin- particular wa\ in
which th- 1 to produce the heat : that whenever
// units of i entirely used in producing mechanical
v the amount of mechanical energy prodm-ed is always the
s uiie. l»eing independent of tlie particular way in which heat
i? used to produce the energy; and that if FT unite of meohan-
i« al e: / . // mills of h.-at produce
\7 units of : : -e facts may all he
i dicated l»y the one equation
• TT= JH.
<7repre> ^ of mechanical energy
t at a I _riven
t e name mechanical equivalent of heat. Its value depends only
u >on the uiiii- i which • . an ical energy and the
li at • d.
I cp. 56. Determination of the Mechanical Equivalent of Heat
by Rowland's Method
OBJECT A \i» Tiir.i>- , --< >ne m<-tho<l of de-
[ning the me.-hanical equivalent <»f h,-at i-> to measure the
ar OUlit of heat developed when a ^ivm amount of meehanieal
en ;rgy is used to stir wate In the apparatus used
by Joule and improved l»y Etowlan is done in the
847
248
PRACTICAL PHYSICS
inner vessel 0 (Fig. 110) of a calorimeter. From the inner
walls of this vessel pro-
ject vanes W between
which the paddles PP
have just room to turn.
These paddles are fas-
tened to a piece of brass
tubing that carries at its
upper end a disk which
is driven by the belt
from the small motor
seen at the right. The
vessel 0 is supported
below on a point with
very little friction and
on top carries a disk
D. Around this disk
is lapped a cord which
passes over a pulley P
and carries at its end a
mass M. If there were
nothing to prevent it,
the weight of M would
cause O to turn until a projection on D came against one of two
stops between which it plays. But the motion of the paddles
PP throws water against the vanes FT' so rapidly that when
the adjustments have been properly made 0 remains nearly at
rest, the projection on D playing between the two stops.
Let M denote the mass of M and d the diameter of D. Then
Mg X J d is the torque that M exerts in keeping 0 from turning
with the paddles. When the paddles have turned n times they
have turned through an angle of 2 trn radians. From the propo-
sition in elementary dynamics which states that the work done
by a rotating body is measured by the product of its rota-
tion in radians and the torque which opposes that rotation,
we have then
FIG. 110.
THERMODYNAMICS
249
(240)
where W denotes the meehanieal energy used in stirring the
water.
m denote the mass of water in the calorimeter; e, the
water equivalent of the vessel (7, the paddles, and the immersed
part of the thermometer ; tr the initial temperature of tlie water
in tlie calorimeter: t.,, its temperature after the paddles have
// turns, and // the net amount of heat lost from C by
radiation. Then from (206),
where // denotes the amount of heat developed by the churning
of the water.
From ( L 10), and < -J U > w h..\e then
'
(in 4
(242)
i» CnMi-i TATIOK. — In order to deter-
: /.' 0 know how
ie readings of the two thermometers compare. Adjust tin-
inn tin -rmometer as directed on p. 160, and Mi>p, nd
>th tii'Tiiioinrt'T^ in a hath <•' nrar tin- tt-mprrat nre of
e room. Stir the water o.-rasioiialh , and after a time record
,e reading of each UHTII;' Meantime take the diameter
(i D with a caliper and met»-r stick, & lie pulley /'runs
c sily, and he sure that the vessel C and the paddles are dry
a d wei^h them together. Then fill C to within a few niilli-
i: -ters of the top \\ ith water at a temp.-iMt ni-e - i>elow
tie temperature of t ': :i. Assemhle the
a paratus. >et the motor running, and hy means of the screw A
n >ve the motor until the ten>ion of the belt is such as to keep
tl • p: i on It playing about halfway between it> itopt.
nit read the therm.. meter T and imme-
. tte! the speed eoll!lt«T X. |-'or some tell or fifteen
111 nutes after that iiotant read 7'and 7\ every minute — always
250
PRACTICAL PHYSICS
reading one of them half a minute after the other. At the end
of this time open the switch that supplies power to the motor,
note the reading of the speed counter, and continue reading the
thermometers for five or ten minutes. During this time the
water in the calorimeter ought to be kept stirred. This can be
done by turning the paddles steadily and very slowly by hand.
The paddles must not be turned faster than about one revolu-
tion in two minutes.
On the same sheet plot two curves coordinating temperature
and time' — one for the thermomter ^and the other for Tl — and
by Regnault's method determine R. In finding n note that
the speed counter reads 1 for every four turns of the paddles.
Determine M by weighing and e by (210).
Without throwing out the water or repeating the weighings
make three determinations and find the mean.
Exp. 57. Determination of the Mechanical Equivalent of Heat
with Barnes's Constant Flow Current Calorimeter
OBJECT AND THEORY OF EXPERIMENT. — In text-books on
General Physics it is
shown that when a
steady electric current
of /amperes flows from
one to the other of two
points between which
there is a potential dif-
ference of V volts, in t
seconds there is trans-
formed between those
two points from elec-
tric energy into heat
the amount of energy
FIG. ill.
W—IVt joules =
IVt • 10? ergs. (943)
THERMODYNAMICS
251
If the current, the potential difference, the time, and the heat
produced can be accurately measured, (-43) suggests a method
of determining tlie meehanical equivalent of heat.
hi this experiment the electric current flows through a wire
I inside of a glass tube B-^B^ (Fig. 111). Through this
same tul a«ly flow of water. The heat developed by
irrent warms the water during its passage through
the tube so that the thermometer 7^ indieates a higher tempera-
ture than T.2. If /// Drains of v . in t seconds, and
during the passage through the tube are raised from tempera-
.iperature Tr the amount of heat developed in the
wire during the same t seconds is, b\
// . IK 7\ - 7^) caloii (244)
ibstituti values of W and 11 from <
iind ( - H ) \\e obtain
L0>
ergs per cal
C246)
'\ ~
Sine.- the triii{» T{ and 7^ are practically steady dur-
i ig tl '-listTvations are l>eing taken there are no conv.--
t ons for t iiennometers,
i or anything else. With a good flow of water and the mean
« :" the tnuperatures of the inflowing and outflowing water
^ ithi: • room t.-mj.erat mv • losses by condne-
t on and :i are negligible.
M\\M M> COMPUTATIO -The rate at which
^ ater tl ept #, ir B-
C instant by a small
r servoir inside a
1; rg- >\\ 11
a th«- top
1 1. Ti. -up-
p is so arranged
tl it u i ys
Overflowing gently from the small reservoir, and the head of
W iter is th«-i- .nstant.
1 HJ ll'J.
252 PRACTICAL PHYSICS
After the apparatus is set up as shown in Fig. Ill and the
water is started, the electric connections are to be made as in-
dicated in Fig. 112. Bl arid B^ are the binding posts shown in
Fig. Ill, and W is the wire inside the glass tube. V is a volt-
meter, A an ammeter, R a rheostat, and EE' the terminals of
an electric circuit. In connecting the ammeter and voltmeter
care must be taken that the positive wire is connected to the
side marked + . If it is not known which terminal is positive,
one wire may be connected and then the other flicked quickly
across the other terminal.
After making the electric connections and adjusting the flow of
water and the electric current to suitable values, open the switch
in the electric circuit. After a few minutes, when the readings
of the thermometers have become steady, record their readings
every minute for four or five minutes. Make all thermometer
readings to hundredths of a degree. Close the switch, and when
the thermometers have again become steady, put under the out-
let a weighed vessel and at the same instant start a stop watch.
After fifteen seconds read the voltmeter, afcer fifteen more the
ammeter, after fifteen more one thermometer, and after fifteen
more the other thermometer. Continue taking readings in the
same order every fifteen seconds for five or ten minutes. At
the end of this time remove the vessel from under the outlet and
at the same instant stop the watch. Find the mass of the water
that flowed through. To get (2\ — ^2), subtract the difference
between the averages of the temperatures indicated by the two
thermometers before the electric current was turned on from
the difference between their average readings while the current
was flowing.
Take five sets of observations for different rates of flow of
water and different values of electric current.
TABLES
254
PRACTICAL PHYSICS
TABLE 1. — Conversion Factors
LENGTH
1 centimeter
1 meter
1 kilometer
1 micron
= 0.39371 inch
= 3.2809 feet
= 0.62138 mile
= 0.001 mm.
= 0.0000394 inch
1 inch
1 foot
1 mile
Imil
= 2.53995 cm.
= 0.30479 m.
= 1.60931 Km.
= 0.001 inch
= 0.00254 cm.
AREA
1 sq. cm.
1 sq. m.
= 0.15501 sq. in.
= 10.764 sq. ft.
1 sq. in.
1 sq. ft.
= 6.4514 sq. cm.
= 0.092900 sq. m.
VOLUME
1 cu. cm.
1 cu. m.
1 liter
= 0.061027 cu. in.
= 35.317 cu. ft.
= 1.76077 pints
1 cu. in.
1 cu. ft.
1 quart
= 16.386 cu. cm.
= 0.028315 cu. m.
= 1.13586 liters
1 gram = 15.43235 grains
1 kilogram = 2.20462 Ib.
" MASS
1 grain = 0.064799 gram
1 Ib. (7000 grs.) = 0.45359 Kg.
ANGLE
1 radian = 57.296 degrees | 1 degree = 0.017453 radian
DENSITY
1 g. per c. c.
= 62.425 Ib. percu. ft.
1 Ib. per cu. ft.
= 0.016019 g. per c.c.
FORCE
1 dyne = 0.000072331 poundal
1 g. wt. = 0.0022046 Ib. wt.
1 poundal = 13825 dynes
1 Ib. wt. = 453.59 g. wt.
1 cm. g. unit
MOMENT OF INERTIA
1 ft. Ib. unit
= 2.3731 x 10~6ft.lb. units
= 421390 cm. g. units
TABLKS
255
STRESS
1 dyne per sq. cm.
= i).n.;7i!i7 j.oundal per sq. ft.
1 g. \\ in.
= -J.ols-j n,. wt. per sq.ft.
1 <-in. of mercur
= 1 r sq. cm.
= 0.19338 Ib. wt. per sq. in.
1 poundal per sq. ft.
= 14.^1 r, .lynes per sq. cm.
1 Ih. wt. ]>.T >-i. t't.
\vt. | KM- sq. cm.
1 in. of iwMvury at 0° C.
= 34.f>o3 £. wt. per sq. cm.
= 0.49117 Ib. wt per sq. in.
\V..i:K 01 Ba
1 erg = L>.:J7:U x 10-« ft. poundals
1 joule = 107 ergs
= L' lals
, ; x 10-* ft. 11..
1 ft. poumlal = 1 _'!:;: MI ,.rgs
1 ft. Ib.
5486 joult-s
1 II. T. hour i.ml.'s
I'nWI K
10T
n.-j ; P.
force de cheval
= 0.9863 horse power
j.-r miu.
il per sec.
li»1390 ergs per sec.
1 ft. Ib. IHM- miu.
in. [«T miu.
1 horee power • 741
= 1.0139 force de cheval
1
Tm it MOM ETR ic SCALES
F =
iv «'i HEAT
calorie = 0.0039683 B. T. U. | 11.11 -j.vj.00 g. calories
Mi . it \M. M l'...i i\ \n N r ..i II
1 .calorie
i les
= 1HMMJ
1 B. T. 1T.
iav> joules
778.1 ft. Ib.
Loo vi:i HIM-
| log. JV= 2.3026 log,0
mpute<l with the value of g at Greenwich.
256
PRACTICAL PHYSICS
TABLE 2. — Densities of Solids and Liquids
Since density varies with the temperature and with the specimen, these numbers
are to be regarded as approximations only.
SUBSTANCE
GRAMS
PER C.C.
LBS. PER
CU. FT.
SUBSTANCE
GRAMS
PER O.C.
LBS. PER
CU. FT.
Aluminium ....
2.7
170
Lime
52.3
140
NH4C1
1 52
95
) 3 2
200
Antimony
671
419
125
150
Asbestos
52.0
$2.8
125
175
Limestone ....
Marble
)8.0
(2.6
190
160
Asphalt
U.O
62
]2.8
175
J1.8
110
Mica
<2.6
160
Beeswax
0.96
60
) 2 9
180
Benzene
070
44
Mercury at 0° C
13 596
8487
Bismuth ...
980
612
Nickel
8 90
556
Brass . . . .
(7.7
480
Oil, Linseed . . .
0.94
59
Brick
18.7
(1.6
540
100
Oil, Olive ....
Paraffin
0.91
J0.87
57
54
Bronze
|2.1
86
130
540
Phosph orus
") 0.93
1 83
58
114
CaCl2 . . .
2.2
140
Platinum
21 5
1340
CS2 at 20° C. . . .
Chalk
1.264
(1.8
78.9
110
Porcelain ....
K2Cr04
2.4
2.72
150
170
1 2.8
1.2
175
75
K2Cr2O7
Quartz
2.70
265
169
165
Coal
1 1.8
110
Resin
1.07
67
Copper .
892
557
(99
140
CuSO4 ....
2.27
142
Sandstone ....
} 25
150
Cork
0.24
15
( 1.1
70
Shellac
Diamond. .
3 52
220
1 1 2
75
Ether at 0° C. . .
German Silver . .
0.736
8.62
$25
45.9
538
150
sHEEf::
Slate
10.53
10.38
27
657
648
170
Glass
Glycerin
} 3.9
1.26
250
79
Soapstone ....
Solder (soft) . .
2.7
8.9
170
555
Gold pure
1932
1206
NaCl
2 15
134
Granite
J2.5
1 Q A
150
1QO
Sulphur, rhombic
Tin
2.07
7 9Q
129
4KK
Graphite
23
140
Turpentine
0 87
54
Ice at 0° C
09167
5722
Vulcanite . . .
1 22
76
cast ....
pure ....
Iron<
steel ....
wrought . .
Ivory
(7.0
J7.7
7.86
(7.6
J7.8
< 7.79
17.85
5 1.83
440
480
491
470
490
486
490
114
Water at 4° C. .
ash . . .
cherry .
YeH^ • • •
soned pine . . .
poplar .
walnut .
1.000013
0.75
0.67
(0.7
jl.O
0.5
0.4
0.7
62.4252
47
42
45
62
31
25
45
I 1.92
120
Zinc
7.15
446
Lead Ccast)
11 34
708
ZnSO
2 0
125
TABLES
257
TABLE 3. — Specific Gravity of Water at Different Temperatures
Referred to Water at 4° C.
•a
SP. GR. I
°c.
SP. GR.
"C.
SP. OR.
°c.
SP. GR.
°<\
Or •
-4
0.99945
17
090688
38
0.99303
59
0.98382
80
0.97191
-3
58
18
39
60
:i:il
81
129
-2
70
19
40
61
280
82
066
-1
20
41
195
62
83
004
0
21
42
157
63
17:.
84
I'll
1
22
788
43
117
64
12]
85
2
23
44
(•77
65
067
86
812
3
M
24
45
66
012
87
717
4
l.oOOOO
25
710
46
046998
67
7<)57
88
682
5
26
;M
47
68
89
616
6
07
27
i.>7
48
905
69
90
7
N
28
!L'!»
49
BOO
70
91
188
8
<ss
29
100
50
813
71
92
416
9
30
."•71
51
767
72
93
10
71
31
MO
52
7*1
73
Olfi
94
11
M
32
509
53
74
666
95
212
12
64
33
177
54
75
96
L48
13
34
III
55
57flT
76
97
<>7I
14
35
410
56
77
98
15
n
36
57
481
78
:;ll
99
:<:;!
16
049896
37
887
58
}::•_'
79
100
'ABLE 4. — Specific Gravities of Aqueous Solutions of Alcohol
%
8r>< •> AT
%
gpKcirtn GRAVITY AT
\1, .,!,..,
»T*
1"
•*>
80°
•T
UH..M1
100
200
800
0
0.99975
049681
OJ9679
55
0.91074
0.90275
0.89456
5
J6946
60
.89944
<!•_'!•
10
46400
46196
65
48790
,87126
15
.97
.97142
70
.87613
20
.97
.9<;
75
.86427
.8.v
.84719
25
46072
46166
46628
80
.86210
.81
.83488
30
46406
.94751
85
,!i:>
35
.9.r)171
90
41801
t>
.!»i2.v,
46611
42787
95
,8129]
40488
.79668
45
42498
.91710
100
,76946
50
41400
40977
258 PRACTICAL PHYSICS
TABLE 5. — Specific Gravities of Aqueous Solutions at 15° C.
Referred to Water at 4° C.
%
HC1
HNOg
H2S04
NaOH
NaCl
CuS04
ZnS04
SUGAR
AT 17°. 5
%
0
0.9991
0.999
0.9991
0.999
0.999
0.999
0.999
0.9987
0
5
1.0242
1.029
1.0334
1.056
1.035
1.050
1.052
1.0184
5
10
1.0490
1.058
1.0687
1.111
1.072
1.103
1.108
1.0388
10
15
1.0744
1.089
1.1048
1.166
1.110
1.161
1.168
1.0600
15
20
1.1001
1.121
1.1430
1.222
1.150
1.225
1.236
1.0819
20
25
1.1262
1.154
1.1816
1.277
1.191
1.307
1.1047
25
30
1.1524
1.187
1.223
1.133
1.382
1.1282
30
35
1.1775
1.220
1.264
1.387
1.1526
35
40
1.2007
1.253
1.307
1.442
1.1780
40
45
1.287
1.352
1.496
1.2041
45
50
1.320
1.399
1.548
1.2313
50
55
1.350
1.449
1.2593
55
60
1.377
1.503
1.2883
60
65
1.402
1.559
1.3183
65
70
1.424
1.616
1.3494
70
75
1.443
1.675
1.3813
75
80
1.461
1.733
80
85
1.479
1.785
85
90
1.497
1.819
90
95
1.514
1.839
95
100
1.530
1.838
100
TABLE 6. — Reduction of Arbitrary Hydrometer Scales
LIGHT LIQUIDS
SCALE
READING
HEAVY LIQUIDS
Hiiiune
Beck
Cartier
Baum6 *
Baumet
Beck
Twaddell
Sp. Gr.
Sp. Gr.
Sp. Gr.
Sp. Gr.
Sp. Gr.
Sp. Gr.
Sp. Gr.
1.000
0
1.000
1.000
1.000
1.000
0.971
5
1.035
1.036
1.030
1.025
1.000
0.944
10
1.073
1.074
1.062
1.050
0.967
0.919
0.970
15
1.114
1.116
1.097
1.075
0.936
0.895
0.936
20
1.158
1.161
1.133
1.100
0.907
0.872
0.905
25
1.205
1.210
1.172
1.125
0.880
0.850
0.876
30
1.257
1.262
1.214
1.150
0.854
0.829
0.849
35
1.313
1.320
1.259
1.175
0830
0.810
0.824
40
1.375
1.384
1.308
1.200
0.807
0.791
45
1.442
1.453
1.360
1.225
0.785
0.773
50
1.517
1.530
1.417
1.250
0.764
0.756
55
1.599
1.616
1.478
1.275
0.745
0.739
60
1.691
1.712
1.545
1.300
0.723
65
1.795
1.820
1.619
1.325
0.708
70
1.912
1.920
1.700
1.350
* Original scale for liquids denser than water. t Newer or so-called " rational " scale.
TABLES
259
TABLE 7. — Specific Gravities of Gases and Vapors
Referred to Water at 4 C. ; also to Air and Hydrogen at 0° C. and 760
mm. of mercury pressure.
All results art- given for ft pressure of 7t» mm. <>f mercury.
FOKV
TBMPKRATURB *C
IV KEKEKKRD !••
Air
HxilrnL'cii
Air
o
0.001!
1 .ii i
14.446
Ammonia ....
Ml ....
0
:.;!,;
0.6890
!•• .
0
0.001966
21.966
t'hlui: . Cl,
0
11074
2.4*
f\ __I __ J I
0
0.000421
0.8266
1.718
. <
0
0.6168
7.468
I I
0
1.000
1
o
1 l.()l:5
.
0
'in
L6.964
!«;.•_'
i ii i >
- -
UAxn
.>. |o
l.r.7
0.001 U
8.18
(6.0
MO
,1..
Vmmonium
Ml.ri . .
800
0.00188
0.986
!!.-_':;
rhl..ri.l.-« . .
800
1139
0.944
18.68
IIS
190
18.46
. . 1 I, . .
IIS
880
0.01064
118.8
868
IK;:.
1006
7.01
1014
84.0
•
1HW
6.06
78.0
'itrogen
l.-J
2.27
82.77
'!•>
L.99
27.72
90.0
J_"_>
i .:•_'
100.1
0.00917
1.66
•j i .-.'.-.
154.0
0.00204
L.68
22.81
Ammooiiun ehlorida rspor ftv«c ftbnormal vapor densities only * hen lu presence of moisture.
260
PRACTICAL PHYSICS
TABLE 8. — Coefficients of Friction
SUBSTANCE
STATIC COEFFICIENT ft.
KINETIC COEFFICIENT b
Metals on metals (dry)
from 0.2 to 0.4
from 0.18 to 0.35
Metals on metals (wet)
from 0.15 to 0.3
from 0.14 to 0 28
Metals on metals (oiled)
from 0.15 to 0.2
from 0.14 to 0.18
Wood on wood (dry) *
from 0.5 to 0.7
from 0.2 to 0.3
AVood on wood fdry) t . .
from 0.4 to 0.6
from 0.18 to 0 3
Leather belt on wood pulley ....
Leather belt on iron pulley
from 0.45 to 0.6
from 0.25 to 0.35
from 0.3 to 0.5
from 0.2 to 0.3
* Motion in direction of fiber. t Motion normal to fiber of sliding block.
TABLE 9. — Elastic Constants of Solids
. B. — Flexural Resilience per unit volume equals one ninth the Tensile Resilience per unit volume.
SUBSTANCE
YOUNG'S
MODULUS
ELASTIC
LIMIT
BREAKING
STRESS
SIMPLE
RIGIDITY
TENSILE
RESILIENCE
dynes
sq.cm.
Ibs.
dynes
Ibs.
dynes
Ibs.
dynes
sq.cm.
Ibs.
erps
ft. Ibs.
cu.ft.
sq.in.
sq.cui.
sq.in.
sq.cm.
sq.in.
sq.in.
cu.cm.
Multiply by •
1011
106
108
103
108
103
1011
10«
104
1
BRASS:
cast . . .
6.5
9
4.5
6
20
30
2.4
3.5
16
330
wire . . .
10
14
11
16
60
80
3.7
5.4
60
1300
COPPER :
annealed .
10
14
3
4
31
43
5
100
cast. . . .
12
17
4.5
6.3
18
25
4.0
6.0
8
170
wire . . .
12
17
7
10
40
55
4.5
6.5
20
420
GLASS ....
6.5
9
2.3
3.2
2-9
3-12
2.4
3.5
4
80
IRON :
annealed .
21
30
5
7
50
70
6
130
cast . . .
12
17
7
10
15
20
5.3
7.6
20
420
wire . . .
19
26
20
30
60
85
8.0
12.0
100
2000
wrought .
20
28
20
30
40
55
7.7
11.0
100
2000
STEEL :
Bessemer .
22
31
33
46
70
100
250
5200
cast . . .
20
28
40
60
8.0
12.0
{5600
{120000
hearth . .
21
30
70
100
wire . . .
19
26
*40
*60
110
150
*420
*8800
WOODS :
oak ....
1.0
1.4
2.3
3.2
W
f7
27
560
pine . . .
1.1
1.6
2.4
3.3
H
t5
' 26
540
poplar . .
0.5
0.7
1.5
2.2
t3
H
23
480
* Unannealed. t Parallel to grain. % Hardened.
TABLES
TABLE 10. — Viscosities of Liquids
i) denotes the coefficient of viscosity in «.«..>. units,
viscosity relative to water at 0° t
::._,,,. otc.. the
i>ou>ity,
(a) Water at Different Temperatures
TBMI-.
n
*0
Tor.
T
<r
1.000
30°
0.00812
0.449
5
OuOl
o.>
40
o.ooi ;•; i
10
n.n:
5O
O.oo:.7o
815
15
L160
60
'1-7
• !.•_>•;!»
20
o.i*'
70
•llM
0886
25
;i»9
(b) Aqueous Solutions of Sugar of Various Concentrations at 20 C.
%8C«A«
•»
% SIT.AB
*»
%s.
*»
2
•21
12
1.4110
22
2.05.r)2
4
1.1104
14
24
•J.LM.M
6
1.1H40
16
l.'J196
26
2.4540
8
18
1.7ISI
28
•J.7'
10
20
-95
30
8.0<J7 1
(c) Various Commercial Lubricating Oils
hi th<* following tal are taken at 20° C., and viscosities
TKADK NAMK
Sr. GK.
.- «
:-. .
"t°C.
100° C.
125° C.
150° C.
." UIIIIMT LuliriiM-
• >
O.W8
1 1-;
0.148
(Mis
0.066
0.045
0.024
0.023
007
009
ii":.
106
,000
.< >L'.-'
.024
006
080
Ills
026
ii"|
J06
Mi
,10
.06
.<>!
.026
.021
.021
} irius (
^ ix c
li nov«-
P l;ir !<•«• M
400
£06
.887
.11
.186
.!'!•
078
.115
.Hi*;
,081
.nl
.048
,026
.021
«;< ) En'
.885
.1 1
.042
B86
.H»
.n(i7
,047
1) tmond Paraffin . .
'in.l.T . .
'
.889
.880
.876
.07.:
1 1"
.11
.086
.012
.148
.(•7
.042
.046
.ii-M
.0:17
.042
.022
N . 1 Dvn i
G< Idfn
M6
.10
II-'"
ill!)
.ni'l
.018
.022
.017
262
PRACTICAL PHYSICS
TABLE 11. — Corrections for the Influence of Gravity on the
Height of the Barometer
(a) Reduction to Latitude 45°
From 0° to 45° the corrections are sub tractive ; from 45° to 90° the correc-
tions are additive.
LAT.
BAROMETRIC HEIGHT IN MM. REDUCED TO 0° C.
LAT.
670
680
690
700
710
720
730
740
750
760
770
780
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
0°
1.74
1.76
1.79
1.81
1.84
1.86
1.89
1.92
1.94
1.97
1.99
2.02
90°
5°
.71
.73
.76
.79
.81
.84
.86
.89
.91
.94
.96
1.99
85°
10°
.63
.65
.68
.70
.73
.75
.78
.80
.83
.85
.87
.90
80°
15°
.50
.53
.55
.57
.59
.61
.64
.66
.68
.70
.73
.75
75°
20°
.33
.35
.37
.39
.41
.43
.45
.47
.49
.51
.53
.55
70°
25°
.12
.13
.15
.17
.18
.20
.22
.23
.25
.27
.28
.30
65°
30°
0.87
0.88
0.89
0.91
0.92
0.93
(K95
0.96
0.97
0.98
.00
.01
60°
35°
.59
.60
.61
.62
.63
.64
.65
.66
.66
.67
0.68
0.69
55°
40°
.30
.31
.31
.31
.32
.32
.33
.33
.34
.34
.35
.35
50°
45°
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
45°
(b) Reduction to Sea Level
Corrections are subtractive.
BAROMETRIC HEIGHT IN MM. REDUCED TO 0° C.
ELEVATION
660
680
700
720
740
760
770
m.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
100
0.01
0.01
0.01
0.01
0.02
200
0.03
.03
.03
.03
.03
0.03
300
.04
.04
.04
.04
.04
400
0.05
.05
.05
.06
.06
.06
500
.06
.07
.07
.07
.07
.07
600
.08
.08
.08
.08
.09
700
.09
.09
.10
.10
.10
800
.10
.11
.11
.11
.12
900
.12
.12
.12
.13
1000
.13
.13
.14
.14
TABLES
263
TABLE 12.— Boiling Point of Water under Different
Barometric Pressures
(a) Temperatures in Degrees Centigrade and Pressures in Millimeters of
Mercury
°c.
.0
J.
.2
.3
.4
.5
.6
.7
.8
.9
90
.YJ1U
531.4
r>33.4
537.5
53f).l>
541.6
548.7
91
547.8
540.9
554.0
556 1
562.4
564.6
92
660.7
668.8
578.1
."'77 . 1
581.8
584.0
686J
93
588J
601.6
806.1
608.4
94
815.2
617.6
610.8
822.1
624.4
881.4
95
840.7
96
857.4
667.1
060.6
674.6
877.0
870.4
97
881.8
r.M.i
686.0
680.4
601.0
807.0
600.6
702.1
704.8
98
7074
7o!i.7
714.0
717.:.
7-J...1
722.7
727.8
780.5
99
741.8
740.2
751.0
754.6
LOO
760.0
770.2
7*2.0
784.8
b Temperatures in Degrees Fahrenheit and Pressures in Inches of Mercury
F.
.0
J.
.2
.3
4 3
.6
.7
.8
.9
: 94
90.68
90.89
20.00
21.04
21.08
: 95
21.18
91.17
L'l •_"_'
•jl/j'i
21.44
21.48
21.58
: 96
21.58
91.71
91.76
21.80
21.88
21.04
21.90
: 97
92.17
22.40
22.46
: 98
22.92
: 99
28.40
: X)
98.60
•J3.S!>
: U
28.80
24.00
94. 14
94.18
24.24
24.84
24.88
2 )2
2-1.41
24.50
•Jl.f.l
94.60
94.74
•Jl>0
24.85
24.80
2 )3
94.06
25.21
25, 1 1
2 )4
95.67
25.94
2 )5
95.00
26.(M
96.20
26.42
26.47
2 «
96.68
96.68
26.06
27.01
2 »7
L'7.07
•J7 1 -
27.40
27.45
27.51
2 8
27.81
28.01
28.07
28.12
2 >9
98.18
98JO
98J6
28.41
98.46
28.64
L'.S.IJ!)
2 0
2H.7^
98.81
98.87
98.09
98 J8
20.10
29.21
29.27
2 1
90 J8
90.51
20.74
29.80
2 I ; 29.92
90.08
30.01
80.10
60.16
80.40
80.46
264
PRACTICAL PHYSICS
TABLE 13. — Pressure of Saturated Aqueous Vapor
In millimeters of mercury
°c.
PRESSURE
°c.
PRESSURE
°c.
PRESSURE
°c.
PRESSURE
°c.
PRESSURE
1
4.91
31
33.37
61
155.95
91
545.77
121
1539.25
2
5.27
32
35.32
62
163.29
92
566.71
122
1588.47
3
5.66
33
37.37
63
170.92
93
588.33
123
1638.96
4
6.07
34
39.52
64
178.86
94
610.64
124
1690.76
5
6.51
35
41.78
65
187.10
95
633.66
125
1743.88
6
6.97
36
44.16
66
195.67
96
657.40
126
1798.35
7
7.47
37
46.65
67
204.56
97
681.88
127
1854.20
8
7.99
38
49.26
68
213.79
98
707.13
128
1911.47
9
8.55
39
52.00
69
223.37
99
733.16
129
1970.15
10
9.14
40
54.87
70
233.31
100
760.00
130
2030.28
11
9.77
41
57.87
71
243.62
101
787.59
131
2091.94
12
10.43
42
61.02
72
254.30
102
816.01
132
2155.03
13
11.14
43
64.31
73
265.38
103
845.28
133
2219.69
14
11.88
44
67.76
74
276.87
104
875.41
134
2285.92
15
12.67
45
71.36
75
288.76
105
906.41
135
2353.73
16
13.51
46
75.13
76
301.09
106
938.31
136
2423.16
17
14.40
47
79.07
77
313.85
107
971.14
137
2494.23
18
15.33
48
83.19
78
327.05
108
1004.91
138
2567.00
19
16.32
49
87.49
79
340.73
109
1039.65
139
2641.44
20
17..36
50
91.98
80
354.87
110
1075.37
140
2717.63
21
18.47
51
96.66
81
369.51
111
1112.09
141
2795.57
22
19.63
52
101.55
82
384.64
112
1149.83
142
2S75.30
23
20.86
53
106.65
83
400.29
113
1188.61
143
2956.86
24
22.15
54
111.97
84
416.47
114
1228.47
144
3040.26
25
23.52
55
117.52
85
433.19
115
1269.41
145
3125.511
26
24.96
56
123.29
86
450.47
116
1311.47
146
3212.71
27
26.47
57
129.31
87
468.32
117
1354.60
147
3:501.87
28
28.07
58
135.58
88
486.76
118
1399.02
148
3392.98
29
29.74
59
142.10
89
505.81
119
1444.55
149
3486.09
30
31.51
60
148.88
90
525.47
120
1491.28
150
3581.23
TABLE 14, — Pressure of Saturated Mercury Vapor
In millimeters of mercury
PC.
PRESSURE
°c.
PRESSURE
°c.
PRESSURE
°c.
PRESSURE
°c.
PRESSURE
0
0.00047
10
0.00080
20
0.00133
70
0.050
120
0.779
2
.00052
12
.00089
30
.0029
80
.093
130
1.24
4
.00058
14
.00099
40
.0063
90
.165
140
1.93
6
.00064
16
.00109
50
.013
100
.285
150
2.93
8
.00072
18
.00121
60
.026
110
.478
160
4.38
TABLES
265
TABLE 15. — The Wet and Dry Bulb Hygrometer
:n Smithsonian Tables
•rnperature of the atmosphere ijiven l»y a dry l>ull> thermometer
be den (.'., and let th" wet bulb thermometer be de-
noted by (/-A/). In the following taMe. oorreoponding to tlie various
-l ••!! in the t»p lii!.-. \ve i. i ill.- pressure (in nun. of
• aqueous vajH>r in the atmosphere at the temperature t° C.,
• pressure that would be exerted l>y tin- aqueous vapor in the atmos-
if the t«-in|>erature wei- to the de\\ point.
Ki:\« K HKTWREX TUB DRY AM> Wr.T I'. 1 1- 111 \
i ii
m
0
4.6
U.7
2.9
2.1
1.3
1
1.1
8.4
2
\A
1.1
3
\A
l.i
4
1.8
5
17
2J
l.-J
6
i.-j
•j.i
ir,
7
1.:.
1.1
8
1.1
1.4
9
:,.:,
{A
10
4.0
•_••_'
i.:;
11
1.7
12
7.1
5.0
I.o
2.1
l.-j
o.:;
13
11 J
.;.:.
•J..-.
l..;
0.7
14
2.0
1.1
15 i 1-J.7
11.1
lo.l
6.6
1.6
l',
rj.-j
I.o
1.9
17
11. 1
11.7
KM
!'.!
8.0
l.:.
2.4
18
1.-..I
ll.J
8.6
!.'>
19
11.!.
lo.7
L6
20
1 1 A
8.8
•; l
:,.-_'
J.I
21
r_M
11.0
-.\
7.1
1.7
22
ll.>
ll.:»
L0.fi
!'.!
ft.4
23
11.::
L£8
LO.O
8.6
8.1
24
!•_'.:;
104)
!'.!
8.1
8.8
25
11.-
HJ
lo.:;
9.0
7.6
26
21.1
L9.4
17. •;
1!.:;
12.8
11. 3
9.8
B.4
27
1&8
17.1
l.vi
L0.8
28
!!.:•
L1.8
10.2
*
•J7..;
21.fi
17.-
16.1
11.1
L2.8
\\:2
• 31.5
2141
19.1
i:-:-
L84)
!•_'.:;
266
PRACTICAL PHYSICS
TABLE 16. — Coefficients of Linear Expansion of Solids
SUBSTANCE
TEMP. °C.
a
SUBSTANCE
TEMP. ° C.
a
Aluminium . .
Brass
40
0 to 100
0.000023
0.000019
Iron (softj .
Iron (cast) .
40
40
0.000012
0.000011
CopD6F . . .
40
0 000017
Lead
40
0 000099
German silver
Glass (crown).
Glass (flint) .
0 to 100
0 to 100
0 to 100
0.000018
0.000008
0 000009
Nickel . . .
Silver ....
Zinc
40
40 •
40
0.000013
0.000019
0 000029
TABLE 17. — Coefficients of Cubical Expansion of Liquids
SUBSTANCE
TEMPERATURE
£
0,
03
Alcohol* .
Analin . . .
Glycerin. .
Mercury . .
Water . . .
-39 to 27° C.
27 to 46
7 to 154
24 to 299
0 to 25
25 to 50
50 to 75
75 to 100
0.001033
0.001012
0.000817
0.000485
0.0001818
-0.00006106
-0.00006542
-0.00005916
-0.00008645
0.00000145
0.00000220
0.00000092
0.00000049
0.00000000018
0.000007718
0.000007759
0.000003185
0.000003189
0.000000000628
0.00000000035
-0.00000003734
-0.00000003541
0.00000000728
0.00000000245
* 93.3% (by volume) pure.
TABLE 18. — Heat Values of Various Fuels
h indicates the number of gram calories of heat developed by the complete
oxidation of one gram of substance, k' indicates the number of gram calo-
ries developed by the burning of one liter of gas measured at 0° C. and 760
mm. pressure. If water is one of the products of combustion, the heat value
is given when the water is in the liquid form.
SUBSTANCE
h
SUBSTANCE
*'
Cane Sugar . . .
3866
Acetylene
14460
Carbon (charcoal) . .
Cellulose
8080
4140
Benzene vapor ....
Coal gas
33496
1000 1400
Coal ....
5500-9000
Daw son g«is
5500 7000
Naphthalin
9692
Hvdrofiren .
3090
Petroleum
10200- 11500
Natural jras (Ind )
9500
Peat
4000-4500
Waiter gas
2000 3500
Wood*
4000-5000
(carbureted)
3500-7000
* Containing 10-12 % of moisture.
TABLES
267
TABLE 19. — Specific Heats of Solids and Liquids
Unless otherwise stated, the following values express the mean specific
heats from 0° to 100° C.
-
SP. HEAT
Bvwi
SP. HEAT
\luininiuin
0219
Lead
0 03°
:i.jl. i-thvl
0.'
M' ivurv
0.088
Anti'
0.(K
bfe
0.218
Bismuth
0.080
0.113
.uii Milphate . . .
0.0
Paratlin (-..,i.l " [•
(liquid 50°-100°)
Platinum
100
0.710
ii 11:5:5
Copi* . .
'in silv»*r
110
0 c
Rock salt
Sand-t iiii'-
0418
ii •>•>!
i
Glass (cr<i\\ :i )
o..
0.161
Sugar
0.060
<i:;nl
0.117
it :
i-fiitiu*'
Tin
0.4(57
0.056
....
In.i
0.1
0.117
Xilir
<>.:;:;!
0.094
TABLE 20. — Melting Points and Heat Equivalents of Fusion
8CMTAXCB
MELT-
ma
HEAT
SCBOTAJtOE
MELT-
HEAT
PoIXT
or
Pom
or
Cd.
P
Cal.
perf.
i
perg.
leeswax
»)1 >
.!'V
80
2.8
'•enzol .
5.4
so
•iial.n
70.9
Ism nth
M18
1 ' ' ' i
L450
4.6
•rominf
Palladium .
:5i; :{
* adininin . .
Paraffin
B8. l
35.1
( lycerin . .
I'h.-nol
25.4
24.9
•e . . .
80
Platinum
177!»
-7:2
•din*- . . .
116
11.7
!»!>!>
21.1
on, cast i
ra^t ( \\ hit*?) .
1 1 < M 1
Sulphur
Tin .
L16
233
9.4
14.3
•SU) *f* 3V ^ Tf » »»*=/ «
:{-ji;
28.1
. . .
268
PRACTICAL PHYSICS
TABLE 21. — Boiling Points and Heat Equivalents of
Vaporization
SUBSTANCE
BOILING
POINT
HKAT
EQUIV.
OF VAP.
SUBSTANCE
BOII-ING
POINT
HEAT
EQUIV.
OF VAP.
Acstic acid
C.
188
Cal.
perg.
84.9
Benzol
C.
79.6
Cal.
perg.
93.5
Acetone ...
56.6
125.3
Chloroform ....
61
58.5
Analin
182.5
93.3
Ether, ethyl
35
90.5
Alcohol ethyl
78
205
IVIercury
350
62
Alcohol, methyl ....
64.5
267.5
Water
100
536
TABLE 22. — Thermal Emissivities of Different Surfaces
The following results were obtained by Bottomley for a cooling copper
globe surrounded by air at atmospheric pressure in an inclosure kept at a
constant temperature of 14°.5 C. The emissivities are expressed in gram
calories of heat lost per second per square centimeter of surface per degree
centigrade excess of temperature of the body above the temperature of the
surroundings.
TEMPERATUBE OF
GLOBE IN °C.
SURFACE POLISHED
BRIGHT
SURFACE POLISHED
BRIGHT AND
THINLY LACQUERED
SURFACE THINLY
COATED
WITH LAMPBLACK
21
165 x lO-6
246 x 10-6
278 x 10-«
22
170
250
281
23
174
254
284
24
178
257
287
25
181
260
290
26
184
263
293
27
187
265
296
28
190
268
299
29
192
271
301.5
30
194
273
304.5
31
196
276
307
32
198
278
310
33
199.5
280
313
34
201
282.5
316
35
202
285
320
36
203.5
287
323
37
205
290
326
38
206.5
292
329
39
207
294
332
40
208
297
335
41
299
338
42
301.5
341
TABLES
TABLE 23. — The Greek Alphabet
LETTER
NAMK
I.K.TrBR
NAME
Lnm
NAME
A, a
Alpha
I t
Iota
P, p
Rho
Beta
Gamma
K. K
A. X
Kappa
Lambda
T,T
Tau
A d
Delta
M./x
Mu
Y. u
Upsilon
E,e
Bprilon
Xu
4>. <^>
Phi
Zeta
— ^
Xi
X, v
Chi
H, rt
Eta
0,o
Oinicron
^, ^
Psi
0.0
Th.<ta
Pi 0,0,
Omega
INDEX
Mng power. _M">.
Aeeuracy re.jiiired. f>.
Air ti..
HJft,
r_'.
Approximations. •
\r.-:i. ."4.
August's peychromeu-r, 206.
.lolly-l.in.
M.-hr-Westphal. '.•'.i.
Balancing column-, method of. '
Ballistic iH'iiiluluiu. >>\
Barnes's current calori:
Bearings and journal, friction 1
Beekin:iiin t)i.Tiii..n..
;!nl pulley, frich
|H.Jnt tab!-
incidences.
Break
MM, I'.1".
119.
itiun of an r, 102
a : 1,.- MIL. in. -i. -i-
standard mass.
WT*i liViln.iiM.t.T s.-alf, 258.
< '.-Ilt.T 'if ;
riin.ii'
ticity, 11^.
Coertit-ii-nt of expansion-- Continued.
of air, 188.
of a rod. 17'.'.
of glass. l.s<i.
of lllf!CU!-\
of f rid ion. 7:..
of restitution, dl.
of viscosity. 11."..
Coinci.l.'iK-.-s, method of. :'.«.», (\8, 72.
Cold test of an nil, 170.
( 'oinliiistion bomb. -'.'>'.
ConilH.und pendulum, r>! i.
< 'onccntration ami boiling point . relation
brtwrcn. 17.;.
Coiixtant em
Dt*i numl.cr to kcop in empirical
formula. 11.
M..II factors. •_'.'.».
C..oli||.4. l;i\V of. 'Jll'.l.
•; km, i.
forr.M-tion factor of a planimctcr. M.
Coulomb's m.'tliod for viscosity, 148.
Cubi.-al .-xpaiisiou. 17«J. -jr.U.
Cui-r.-nt calorimeter. •_'."><).
Damping con.stant. 1 1-
.'s li\.;r..|li.-|,T.
Datum circle of plaiiiim-ter. .".'.».
hfiisim.-t.-r. 1":;.
Densi:
b\ imm.-rsion, i>5.
l)\ measurement ainl weiu'liin^, S!».
of an unsat united vapor. T.'S.
with .Jolly l.alance. !»7.
with Mohr-NVrstplial balance, (.i(.t.
with ]iyknometer, '.«>, '.L'.
-.•.I E6TO, 1">7.
Determinate emu
potet, •_'«>:;.
I UtTer.'iH'i-s of various order-, 12.
Dilatometer, 18(i.
Direct measurements. 1.
Distance measurements, If.. }•_'.
Divided circle, eccentricity in mounting. HI.
271
272
INDEX
Dividing engine, 18.
Dynamometer, oil testing, 80, 82.
Eccentricity in mounting of circle, 61.
Effusiometer, 107.
Elasticity, 118, 260.
Elongations, method of middle, 38.
Emissivity, 215, 268.
Empirical equations, 10, 124.
Errors, 2.
Expansion, 176, 266. See also Coefficien
of expansion.
Exposed stem correction, 158.
Eye and ear method, 35.
Eyepiece micrometer, 20.
Filar micrometer microscope, 19.
Fire test of an oil, 170.
Flash and stop watch method, 36.
Flash point of an oil, 170.
Fly wheel, change of speed, 63.
Friction, 75, 260.
Gases, fundamental law- of, 176.
Golden's oil-testing dynamometer, 80.
Gravity, acceleration due to, 66, 69.
specific. See Specific gravity.
Greek alphabet, 269.
Heat equivalent, 208.
of fusion, 231, 267.
of vaporization, 235, 268.
Heat value of coal, 237, 266.
of gas, 243, 266.
Hempel's combustion bomb, 238.
Humidity, 202.
Hydrometer, 102, 258.
Hygrometry, 202.
Hypsometer, 164.
Immersion, specific gravity by, 95.
Indeterminate errors, 3.
Indirect measurements, 1.
Inertia, moment of, 110.
Jolly's air thermometer, 188.
spring balance, 97.
Joly's steam calorimeter, 229.
Joule's method for mechanical equivalent
of heat, 247.
Journal and bearing, friction between,
80, 82.
Junker's calorimeter, 243.
Latent heat, 208.
Law of cooling, 209.
Least count of a vernier, 21.
Length, 16, 42.
Level trier, 49.
Lever, optical, 42,47.
Limit of elasticity, 118.
Linear expansion, 176, 179, 266.
Linebarger's spring balance, 97.
Logarithms, accuracy, 6.
Lubricated journal, friction at, 80, 82.
Mass, 25, 86.
Maximum elongation, 38.
Mechanical equivalent of heat, 247.
Melting point table, 267.
Mercury vapor pressure, 264.
Meter bridge, 168.
Meter stick, 16.
Meyer's method for vapor density, 198.
Micrometer, 16-20.
Middle elongations, method of, 38.
Mixture, method of, 221.
Modulus of elasticity, 119, 260.
of resilience, 140, 260.
Mohr-Westphal balance, 99.
Moment of inertia, 110.
Muller's optical lever, 179.
Newton's law of cooling, 209.
New York Board of Health tester, 171.
Notation, 15.
Oil, test of, 148, 170.
Oil testing machine, 80, 82.
Omitted transits, method of, 35.
Optical lever, 42, 47.
Oscillation, 34.
Parallax, 2, 156.
Passages, method of, 36.
Pendulum, ballistic, 64.
compound, 69.
seconds, 33.
simple, 66.
3ercussion, center of, 65.
Period of oscillation and vibration, 34.
~'ermanent set, 118.
'lanimeter, 54.
'lotting, 9, 13.
'oiseuille's method for viscosity, 143.
ressure, temperature coefficient of, 188.
rojectile, speed of, 64.
sychrometry, 202.
[NDEX
273
Pulley, correction for friction .•:
1'ulley and belt, friction between, 77.
Pyknometer. '.«>.
, 156.
Mtive experiment^. 1.
Quantita' DttltB, 1.
. •_'!."..
-lant. '-'ID.
correction for. ]
~j>irit level.
Regnault'-> MI. -tlio«l of eon : radi-
atio! .
Regnaul' . for \apor pressure,
Restitute
methods ol £, 8.
r. of :i !.;«'
7, 260.
Rov : r:uli-
itlM
•ii.nl f.n- m.-.-liaiiiral rijiiiva-
l.-nt Oi
ra.li-
8al
SecM.n.ls IH-II.II.
i.irit level,
Series, suiiiinii .
Set.
Shear. 11'.'.
J(W.
Solution, KoiliiiL: point of
wiii
ui-
\\ith pykr
1
with 8t:t?
with »t«-:i
Sj .1. •;:'.. hi.
Sphoroiueter. 17, 4.'>.
Spirit level, sensitiveness. l'.>.
Staiular.l masses, calibration of, 86.
Stem exposure correct ion, 158.
Stop watch. :;::.
Strain, 118.
us.
Summation notation, 15.
..IjnstnuMit. 41.
Temperature, 1 .".">.
Tmacity. '. :
Tensile \-,,enicieiil of elasticity, ll'.». ll'O,
l-js. -Ji«>.
Thermodynamics. -J17.
lliermometi'r, air, 188.
calibration, KK). HMJ.
errors, l.^i.
Miice. li;»',.
Thickness of a thin plate. »•_'.
Tlinrston's oil testing machine, 83.
Transits. metho.I ,.f oinitte.l.
Trust \v..rtli\
Twa.i.l.'ll'.s hy.lrometer scale, 258.
ritimate resilience . L40,
Yaria!'1'
at i..n of a baroineier scale, 53.
nietlio,! of. •_'<;.
. L'.il.
r.,uivalent.L>Os.
01 pn-ssiirr. I'.H. UN;, '2(\\.
• \ <h > imlb h\ u'rometer, 205, 265.
Voun-'smo.lulu,. 11" J«».
eh- of plaiiimeter. .V.i.
point of balanr.-. -Jii.
read in- of caliper, 17.
UNIVERSITY OF TORONTO
LIBRARY
Do not
remove
the card
from this
Pocket.
Acme Library Card Pocket
Made by LIBRARY BUREAU, Boston
•j=£
o
&$
g
*-3
P3
cr
o
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11