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PHYSICAL LABORATORY EXPERIMENTS MECHANICS, OPTICS and HEAT
H. M. GOODWIN, Ph.D.
PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY.
Printed for the use of students of the Massachusetts Institute of Technology, not published.
SIXTH EDITION.
NGRTHEASTERH UNWERS'iTY
BOSTON:
Geo. H. Ellis Co., Printers, 272 Congress Street.
1917
Copyright, 1904, L.Y H. M. Goodwin.
Part I -MECHANICS.
CONTENTS.
PAOB
Units of Length and Estimation of Tenths 3
Verniers 7
Area Measurements 10
Measurement of Thickness 16
The Micrometer Caliper 16
The Spherometer 17
The Optical Micrometer 19
The Spirit Level 21
Measurement of Mass. (General Discussion) 27
Theory of the Balance 29
The Use of a Balance 33
Methods of Weighing 34
Double Weighing 36
Reduction of Weighings to Vacuo 37
The Balance, Part I. (Methods of Weighing) 40
Tlie Balance, Part II. (Sensitiveness and Ratio of Arms) ... 42
The Balance, Part III. (Calibration of Set of Weights) .... 45
Specific Gravity. (General Discussion) 47
Specific Gravity of Solids, I. (Specific Gravity-bottle) .... 49
Discussion of Corrections 52
Specific Gravity of Solids, II. (Archimedes' Principle) .... 55
Specific Gravity of Liquids, I. (Specific Gravity-bottle) .... 56
Specific Gravity of Liquids, II 58
Sprengel-Ostwald Picnometer 58
Mohr-Westplial Balance 60
The Jolly Balance 62
Barometric Measurements. (General Discussion) 67
Types of Barometer 71
Comparison of Barometers 79
Meteorological Instruments 81
Boyle's Law, 1 86
Boyle's Law, II 90
Law of the Pendulum 94
The Method of Coincidences. (General Discussion) 97
The Physical Pendulum 99
The Metronome Pendulum 104
Law of Freely Falling Bodies 108
The Tuning-fork Chronograph 108
Moduh^, of Elasticity 115
Appendix , i-Lx
l^\G3
UNITS OF LENGTH AND ESTIMATION OF TENTHS.
Object. — The object of this experiment is twofold — first, to determine the value of the inch in terms of the millimeter, and second, to give the student practice in estimating by eye, tenths of a small division. The ability to make accurately such estimations is of great importance in all scientific work as the experimenter is continually obliged to estimate fractions, usually tenths, of the smallest division of his instrument e.g. tenths of a degree, tenths of a millimeter, etc. The computation also introduces the student to some fundamental matters in the precision discussion of direct measurements.
Apparatus. — ^The apparatus consists of a micrometer screw, M, mounted horizontally and carrying a movable nut on
|
M |
|||
|
S.|,i.i,iii,l, 1,1. I.I.I |
PI |
||
|
■n_^1» : |
|||
|
"^ W 80 30 1 |
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Fig. 1.
which is a fine reference mark, A^. The nut moves between two horizontal scales. Si and ^2; the graduations of which are flush with the reference mark on the nut, thus eliminat- ing errors in reading arising from parallax. The scale Si is graduated in twentieths of an inch. The scale S^ is gradu- ated in millimeters corresponding to the pitch of the micrometer screw. One complete revolution of the screw moves A'' through exactly one division of S2, and this scale may, therefore, be used for recording the total number
4 PHYSICAL LABORATORY EXPERIMENTS
of revolutions made by the screw. Fractions of a revolu- tion are read off on the graduated head of the micrometer (divided into 100 parts), with respect to a fixed reference mark not shown in the figure. The instrument is so adjusted that when the zero of the micrometer head is set on the refer- ence mark, A^^ coincides with the divisions on S^.
Procedure. — First. — To measure the inch in terms of the millimeter.
Set the reference mark A^ exactly on the zero division of the inch scale and record the position of N on the milli- meter scale to thousandths of a millimeter by estimating the position of the micrometer head with respect to the reference mark at the base of the instrument. For example, if N stands between divisions 2 and 3 on S2 and the reference division for reading M is half way between the 43 and 44 graduations of the micrometer head, the reading will be 2.435 mm. In setting N always bring it up to the mark from the same side, e.g., turning a right-handed screw clock-wise. This is necessary to eliminate backlash, i.e., the play of the screw in the nut, which is likely to be present, sometimes to a large extent, in instruments involv- ing the use of the screw. It is evident that if all settings be made from the same direction this source of error will be eliminated in taking the difference of two consecutive series of settings. This point should be borne in mind in all subse- quent work. Make nine independent settings of N on both the first and last division of the inch, recording the position of N each time, as described above. The difference of the means of these two series of observations will evidently be the best representative value of one inch expressed in millimeters.
Computation. — Find the mean, the average deviation of a single setting and the average deviation of the mean setting for each series. From these, compute the value of the inch in millimeters and the average deviation and percentage deviation of this result (see precision discussion).
Second. — Estimation of tenths of a small division (one- twentieth of an inch).
Compute (in the laboratory) from the value of the inch
UNITS OF LENGTH AND ESTIMATION OF TENTHS 0
in terms of the pitch of the micrometer screw, what the setting of the micrometer should be when A^ is set exactly at 0.1, 0.2, 0.3, etc., ... of the distance between two ad- jacent graduations of the inch scale.
Thus, suppose we take the first small division of Si. If
a = value of 1 inch in millimeters then ~~ = value of one
small division of Si, and tt. X ^ added to the mean value
of the series of settings of N on the first graduation of Si will give the most probable value of the micrometer reading
when A^ stands at — the distance between this division and
the next. In this way compute the settings for 0.2, 0.3 .. . 0.9 of this small division.
Having computed the value of these settings set the mi- crometer at the computed values, and observe carefully the position of A^ on the inch scale. In other words, let the eye see how the various tenths of the division really appear. It is well to train the eye in the following way: Set the mi- crometer, say at 0.3 of the division, and after observing, the position of N carefully, move it to one side, and then set it by the eye as nearly as possible to the original po- sition without looking at the micrometer. Then compare the reading of the micrometer with the computed value. The difference gives a measure of the precision with which the observer is capable of estimating that particular tenth. Since the distance estimated in each case is yV of ^V' or 2^o"= 0.005" the observer must be able to estimate to less than one-half this distance if his estimation is to be correct to the nearest tenth; that is, the estimated settings must differ from the true settings by less than 0.0025" or about 0.063 mm. Students should practice until they can readily estimate tenths to approximately 0.05 mm. When some skill has been acquired record a series of independent estimations from 0.1 to 0.9. A comparison with the computed values should not be made until the estimated series of observa- tions is complete. In order that this may represent fairly the observer's ability to estimate tenths no observation
6 PHYSICAL LABORATORY EXPERIMENTS
should be rejected if found to differ too widely from the computed values.
Computation. — Compute the average deviation of a single eye estimation from the differences between the estimated and computed values. This will give a measure of the pre- cision with which the observer can estimate tenths.
Precision Discussion. — Do the measurements on the length of the inch scale in terms of the micrometer screw indi- cate that the experimental errors are greater or less than the errors inherent in the instrument, e.g., errors due to faulty grad- uation of the scales or cutting of the screw?
To answer this question, we must compare the deviation measure of the final result with the amount by which the result differs from the true value (if this be known) or from that obtained by the use of some other instrument.
If mi and jn^ are the mean settings on the inch scale and 3i = ± iA.D.)i and 62 = ± (A.D.)2, their respective deviation measures, then the deviation in M = mi — m2 due to ob- servational errors alone may be as small as 5i — 62 if the deviations happen to be of such sign as to tend to neutraUze each other, or as large as Si + ^2 if they both happen to effect the result with the same sign. Since their sign is not ' known, and they are equally likely to be plus or minus, it
can be shown that the most probable uncertainty in M is obtained by squaring the deviation which each measurement individually produces in the final result, adding these to- gether, and taking the square root of their sum, i.e., in the above case
A = ± V 5^1 + 52
will be the probable deviation in M. If this value is ap- preciably less than the deviation between the observed and true value of the inch (1" = 25.4005 mm.), it may be con- cluded that the apparatus contains some instrumental error or is used under conditions which are not standard. If, on the other hand, the difference between the true and computed value is of the same order of magnitude or is less than the deviation in the final result, it may be concluded that, within the limit of observational error, constant errors in the appa- ratus are not excessive or may be neglected.
Questions. — 1. Do you think there is a source of con- stant error in tlie method or apparatus used? Why?
VERNIERS
VERNIERS.
Object. — The object of this experiment is to familiarize the student with the theory and reading of various types of verniers.
Discussion. — The vernier, so called from its inventor Pierre Vernier, is a device for reading with accuracy and facility a fraction of a scale division, thereby avoiding the necessity of minute and expensive graduations or of eye estimation. It is universally used on barometers, cathetome- ters, divided circles, and other instruments of precision. The principle of the device is as follows: moving flush and parallel with the scale proper S, which is usually fixed, is a second scale V, the vernier, graduated into a number of equidistant parts n, of such length that the total number of parts n of the vernier is exactly equal in length to n ± 1 smallest divisions of the scale. Thus in A, Fig. 2, ten divisions of the vernier equal nine of the scale; in B, twenty- five of the vernier equal twenty-four of the scale. The actual length of a
vernier division is, therefore, — less or
n
greater than the length of the smallest
division of the scale, as shown by the
following demonstration.
Demonstration. — If n vernier divisions equal n ± 1 scale divisions, and s and v are the lengths of a scale and vernier division respectively, then
nv = (n± l)s, (n±l) n
1
. . S V = - s,
n
- V S
I V ~Z
_ . — I
7 30
or
A. 7.60 B. 30.026
Fie. ».
8 PHYSICAL LABORATORY EXPERIMENTS
This difference, s — v or v — s, between the length of a scale and vernier division, is called the least count of the vernier, and is always found by dividing the value of the smallest scale division by the total number of divisions into which the vernier is divided. Thus in A the smallest scale division is seen to be -^^ of unity or 0.1, and the vernier is divided into 10 parts; hence the vernier will read to (or the least count will be) -^q of 0.1 or 0.01; in B, the smallest scale division is ^q of unity or 0.05, and the vernier is divided into 25 parts; hence it reads to (or its least count is) 23 of 0.05 or 0.002; again, if the smallest scale division is ^ of a degree or 20' and the vernier is divided into 40 parts, it will read to ^\ of 20' or i' = 30".
Consider now vernier A in figure 2. In moving the zero division of the vernier from a coincidence with the scale division 7.6, to a coincidence with the next adjacent division 7.7, it is evident that there will be ten intermediate coinci- dences of the successive vernier divisions with some one of the scale divisions. If the vernier be moved up along the scale from the position in the figure until the first vernier division coincides with the next scale division, namely 7.7, the zero will have been moved through a distance -f^ of the scale division, and the reading of the vernier, i.e., the posi- tion of its zero division, will be the scale reading plus y^ of a scale division, i.e., 7.6 -f 0.01 = 7.61. If it be moved along until the eighth division of the vernier is in coincidence with a scale division, the reading of the vernier will be the scale reading plus -^^ of the smallest scale division, namely 7.68, and so on. Hence to read any vernier, take the reading of the scale division next preceding the zero of the vernier and add m times the least count of the vernier, where m is the di- vision of the vernier in coincidence with the scale. Usually the divisions on the vernier are so numbered that this mental multiplication is unnecessary, the numbers giving directly the fraction of a division to be added. For example, in B the least count was found to be ^h of 0.05 or 0.002, hence the fifth, tenth, fifteenth, etc., divisions of the vernier are numbered 1, 2, 3, etc., respectively, since a coincidence of these divisions with the scale means that the zero is 5 X 0.002=0.01,
VERNIERS 9
10 X 0.002 = 0.02, 15 X 0.002 = 0.03, etc., above the last scale division, respectively. As drawn, the reading of the zero is 30.026.
Verniers are generally so graduated that n divisions of the vernier equal n — 1 of the scale, as in A and B, in which case the vernier and scale graduations are numbered in the same direction. Sometimes however, it is convenient, for reasons of construction, to have the zero of the vernier at the top, in which case n vernier divisions are made to equal n + 1 of the scale. The vernier is then said to be "retro- grade" and inspection of the second vernier provided will show that in this case the scale and vernier are graduated in opposite directions, and coincidences of the vernier are to be found by following along the scale backwards from the zero. The "least count" is found in the same way for both classes of verniers, as is evident from the above demonstration.
A convenient way of regarding a vernier, when its princi- ple is once clearly understood, is to consider the whole vernier as a magnified smallest scale division, divided into n equal parts. A moment's consideration will show that, looked at from this point of view, the reading of the vernier is obtained by adding to the direct scale reading of the division next preceding the zero of the vernier, the direct reading of the coincident line of the vernier itself.
Procedure. — Become perfectly familiar with all of the verniers on the block provided. Set the verniers to read as follows: 1st, 8.03; 2d, 29.89; 3d, 30.874; 4th, 4° 10'; 5th, 0° 17; 6th, 2° 58' 30"; 7th, 48° 52', and have them verified by an instructor. Then set them at random and record read- ings as they stand. These are also to be verified. Students should satisfy themselves that they thoroughly understand and can read readily all of the verniers provided before leaving the experiment, as similar verniers will be met with continually in subsequent work.
10 PHYSICAL LABORATORY EXPERLMENTS
AREA MEASUREMENTS.
GENERAL DISCUSSION.
The methods for the measurement of areas such as occur in laboratory practice may be divided into two groups accord- ing as the area is a geometrical figure, or irregular in form. In the former case the measurements reduce themselves generally to the simple measurement of lengths from which the given area is computed by mensuration. Very large areas are of course measured by methods of triangulation, and are properly discussed in works on surveying.
By far the most common and important areas met with in the laboratory, such as indicator diagrams, hysteresis curves, alternating current wave forms, etc., are irregular in outline, and special methods and instruments have been devised for their measurement. The only method possessing high accu- racy and employed to any extent is that in which a planimeter is used. This instrument is a form of mechanical integrator, by which the area of any figure is given at once by the simple operation of tracing its periphery with the point of the instru- ment. Various forms of planimeter have been invented, and practice with one of the best of them, the Amsler, is afforded in the following experiment. When, however, a planimeter is not available, one of the following methods may often be found of use: —
First. — ^The area may be computed by one of the follow- ing approximation formulae, commonly called Simpson's rules. For the deduction of these rules, see Williamson's Integral Calculus, p. 212.
1. A series of equidistant parallel ordinates is drawn across the area. If the area of the figure be regarded as that of the polygon resulting from connecting the intersection of the ordinates with the curve by straight lines, the area will evidently be the sum of one-half the extreme ordinates plus the whole of the intermediate ordinates, multiplied by the common interval.
2. A closer approximation is obtained by assuming the curve to consist of a series of parabolic arcs, i.e., of second degree curves of the form y = a -{- bx -{- cj(^. Under this
AREA MEASUREMENTS 11
assumption it can be shown that the area may be found as follows: —
Add together the first and last ordinate, twice every second intermediate ordinate, and jour times each remaining ordi- nate, and multiply the sum by one-third the common interval.
3. A still closer approximation is obtained by supposing the curve to be made up of third degree elements of the form y = a -\- bx ■{- cx^ -\- dx^.
Then if the number of intervals be taken a multiple of three, the area is obtamed by the following rule : —
Add together the first and last ordinate, twice every third intermediate ordinate, and three times each remaining ordi- nate, and multiply the sum by f the common interval.
It is evident that in all these cases the accuracy is increased by increasing the number of intervals.
Second. — By weighing.
Cut the area out of a piece of homogeneous cardboard, light bristol board or thin sheet metal, and weigh it. Cut out of the same material an accurately measured area, e.g., one square decimeter, or several square inches, and weigh it. The ratio of the two weights will evidently be the area desired, expressed in the unit of area adopted. The accu- racy of this method depends on the homogeneity of the material weighed, and on the exactness with which the areas are cut out. This method is capable of an accuracy of about one per cent.
Third. — By coordinate paper.
Draw the area on accurately ruled coordinate paper. Count up the number of complete squares contained, and estimate the area of the remaining fractional squares. This method gives of course only approximate results.
Fourth. — In case of irregular polygons the total area may be divided into a number of triangles, the area of each of which may be computed from linear measurements.
Object. — ^The object of this experhnent is to afford practice in the manipulation of the Amsler planimeter, to test the accuracy of the instrument, and to become familiar with the relative values of the units of area in the English and Metric
12 PHYSICAL LABORATORY EXPERIMENTS
systems. The experiment also gives practice in the use of the dividing engine.
Apparatus. — The type of Amsler planimeter provided is shown in Fig. 3. It consists of two arms hinged together, one of which is pivoted at the end by the needle point F, while the other, the tracer-arm, carries a point P, which is caused to trace the outline of the area to be measured. The planimeter rests at Z) on a roller-wheel so mounted on the tracer-arm that it slides without rotation when moved in the direction of the arm A, and rotates without sliding when moved at right angles to this direction. Motion in all other directions is accompanied by sliding and rotation. The
Fig. 3.
number of complete revolutions made by the wheel D is recorded on the dial G, while fractions of a revolution may be noted by means of the vernier E, which reads to 0.001 of a revolution of D. The effective length of the arm A can be varied by clamping the sleeve carrying the hinge and record- ing mechanism at any desired position by the screw S. The exact adjustment at any desired division on A is made by the fine adjustment screw below S. The tracer-arm A is graduated at three points such that when the planimeter is set exactly at these positions, the instrument records areas in square decimeters, ten square inches or tenths of a square foot respectively. The instrument may thus be used to measure areas directly in any one of these units. ^
The planimeter is a delicate and expensive piece of appa- ratus and should be handled carefully. It should be wiped
IFor an elementary explanation of the theory of the instrument see Peabody'8 Manual of the Steam Engine Indicator, pp. 76-80. For the complete mathematical theory see Lanza's Dynamometers, Planimeters, Governors, and Fly-wheels, p. 28.
AREA MEASUREMENTS 13
with chamois skin and returned to its case at the conchision of the experiment.
Procedure. — First. — To measure the area of a figure and to determine the experimental error involved.
Stretch tightly upon a drawing board a sheet of smooth unglazed paper. Tack upon this a sheet of laboratory note- paper, and with a pair of compasses draw in ink a circle of about one and a half inches radius. The line should be drawn very fine. Draw also several diameters extend- ing across to the edge of the paper. Set the planimeter to read in square decimeters by unclamping S and moving the recording mechanism along the arm until the index on it nearly coincides with the division marked. Clamp S and set exactly on the division by means of the fine adjust- ment screw below S. Place the planimeter on the board with reference to the area to be measured as shown in Fig. 4, and see that the tracing point P can be moved freely around the whole circumference without cramping the instrument and without drawing the wheel over the edge of the paper on which the area is drawn. With a needle or pin prick a small
Fig. 4.
14 PHYSICAL LABORATORY EXPERIMENTS
hole on the circumference as a starting point and moye the tracer until it is exactly at this point. The recording wheel may now be adjusted so as to read exactly zero or, better, the reading may be taken on G and D as it stands. With P trace the circumference of the circle moving around clock- wise, until the point P returns exactly to the starting point. This is of the greatest importance, for if the tracer does not start and end at exactly the same point a large error will be introduced. After a little practice it will be found that the tracer can be made to follow the line with surpris- ing closeness. Positive errors introduced by running off outside the line tend to become eliminated by corresponding negative errors resulting from departing from the line on the other side, since both are likely to occur.
Planimeter the circle four times. Next set the planimeter to record in square inches, and planimeter the same area again four times. With the instrument still set to register square inches, planimeter the irregular area furnished by an instruc- tor. Insert the figure and record its area in the notes. Measure the length of several diameters of the circle on the dividing engine, the manipulation of which will be explained by an instructor.
Second. — To determine the constant error of the planimeter.
The errors affecting planimeter measurements are of two kinds. First, indeterminate errors arising from the inability to exactly trace the area owing to the unsteadiness of the hand, slip of the revolving wheel, etc. ; and, second, constant errors arising from inaccurate adjustment or construction of the instrument itself. The former errors should not exceed 0.01 or 0.02 square inch; and the latter, in instruments of the best construction, may be made negligible, i.e., less than the smallest area which the instrument is capable of recording.
For the purpose of determining the instrumental error, a geometrical area (circle or square) is engraved on a metal plate, and its linear dimensions measured with great accuracy on a dividing engine. The computed area is then compared with the area as determined by the planimeter. If the figure is slightly cut into the metal, the tracer of the planimeter can be made to follow the area exactly, and the indetermin-
AREA MEASUREMENTS 15
ate errors due to trembling of the hand, etc., are thus eliminated.
Planimeter the circle on the test plate provided, in square inches, and measure its diameter on the dividing engine.
Computation. — a. Compute the mean, a.d., A.D., and the percentage deviation of a single observation and of the mean, of the two series of observations on the area of the circle. Compare the mean value of the area expressed in square inches with the area calculated from the measurements of the diameter of the circle. What is the percentage difference? How many figures should be retained in tt in the computation ?
b. From the mean value of the area expressed in square inches and square decimeters respectively, compute the value of one square decimeter in square inches, and conversely.
c. Give the area of the irregular figure provided.
d. From a comparison of the observed and computed area of the test plate, calculate the error of the planimeter per square inch.
Precision Discussion. — Suppose it is desired to construct a circular test plate for calibrating a planimeter which shall fulfil the following conditions: first, its true area must be known to 0.1 per cent, and second, it must be sufficiently large so that the smallest value which the planimeter is capable of recording, namely, that corresponding to one vernier division, i.e., 0.01 square inch shall not be greater than 0.1 per cent of the whole area. What must be the diameter of this circle and how closely should it be measured ?
If the allowable deviation in the measured area is to be less than 0.01 square inch, and this not greater than 0.1 per cent of the total area, the area of the circle should be not less than 10 square inches, i.e., the diameter must be
=/^=V^
'" 3.6"
The value of the diameter must be known to J of 0.1 per
cent = 0.05 per cent, since — = 2 - , i.e., the fractional A d
error in the area is twice as great as the fractional error in the diameter. Hence d must be measured to 0.05 per cent of 3.6 inches = 0.0018 inch.
16
PHYSICAL LABORATORY EXPERIMENTS
Problem. — What would be the numerical and percentage error in an area of 8.315 square inches measured with the planimeter used if the correction for the constant error of the instrument, as above determined, were not applied?
MEASUREMENT OF THICKNESS.
THE MICROMETER CALIPER, SPHEROMETER, AND OPTICAL MICROMETER.
Object. — The object of this experiment is to give faciUty in the use of the micrometer caUper, spherometer, and optical micrometer, instruments which are designed to measure the thickness of small objects with accuracy. The discussion of the results illustrates the effect of various sources of error, and affords practice in "weighting" observations.
MICROMETER CALIPER.
Apparatus. — This instrument is shown in Fig. 5. It con- sists of a uniform screw, one end of which is fixed to the inner end of a movable hollow barrel B, by means of which
it can be turned
c A
V^
B
a back and forth I j in a nut N, I / rigidly attached to the body of the instrument. The nut N is graduated to correspond with *'*e. 6. the pitch of the
screw so that one revolution of the screw advances B through just one (or sometimes one-half) division on A^. Fractions of a revolution are determined by the graduations on the edge of the barrel B.
The end of the screw A is finished with a polished plane face at right angles to its length. A similar fixed plane face is attached to the instrument at C. These faces should be plane and parallel so that perfect contact is made between
MEASUREMENT OF THICKNESS
17
them when screwed together. C is so adjusted that when the end of the screw A (called its tooth) is in contact with C the zero graduation on B stands at zero on N. Often this adjustment is not made exactly, in which case the instrument has a "zero error." The difference in reading of the instrument when A and C are in contact (zero reading) and when some object is placed between them gives at once the thickness of the object.
Procedure. — Determine the "zero error" of the calipers (four settings will suffice) and then measure the thickness of the object provided, taking nine independent settings at different parts of the object to insure a measure of its aver- age thickness. In setting the caliper do not strain it by * turning the screw so as to clamp the jaws too tightly. It is best to turn the screw until the fingers just slip off the end of the barrel.
THE SPHEROMETER. ^
Apparatus. — Spherometers are instruments designed to measure the thickness of small objects with a high degree of precision and to measure the radius of curvature of spherical » convex or concave surfaces. It is from this latter use that the instrument derives its name. *
One form of this instrument (due to Perreaux) is shown in • Fig. 6. It consists essentially of a vertical micrometer-screw , moving in a nut fixed at the cen- tre of an equilateral tripod. The • head of the micrometer screw A • is a graduated circle, the position of which is read on a fixed vertical scale B graduated in divisions equal to the pitch of the screw. One com- ( plete revolution of the screw thus ' raises or lowers A through one divi- sion of the fixed scale. Fractions of ^ a revolution are read off on the head $ of the micrometer with reference to this scale. #
The sensitiveness of the instrument »
— • ^ »
,v,...
>
18 PHYSICAL LABORATORY EXPERIMENTS
depends on the precision with which one can detect when the end of the micrometer screw just makes proper "contact" with the object placed beneath it. In the ordinary form where no special device is provided for facilitating this observation, the screw is turned until the instrument just turns as a whole on the end of the screw as a pivot, or until the hand can just detect a slight rocking motion of the mstrument like that of a table, one of whose legs is too long. With these apparently somewhat crude methods of detecting contact, settings to one or two thousandths of a millimeter may readily be made.
To increase the sensitiveness of the instrument, various devices have been invented. The Perreaux multiple lever attachment is that shown in the figure. The micrometer screw is hollow and through it a steel rod passes freely. The upper end of this rod acts upon a system of two multiplying levers, by means of which a slight motion of the rod becomes greatly magnified. C is a fixed reference edge, to which the end of the longer lever may be brought, in which case the pressure exerted on the lower end of the rod will evi- dently be the same for different settings. In this instru- ment the pitch of the screw is 0.5 mm., the head is gradu- ated into 500 parts, and hence reads directly to 0.001 mm., and by estimation to 0.0001 mm.
Procedure. — Measure the thickness of the object previously used as follows: Place the spherometer on a plate glass surface, and turn the screw until the rod passing through it presses upon the plate just sufficiently to bring the end of the long lever up to the reference mark C. Record the position of the head of the micrometer on the vertical scale, estimat- ing by eye, to tenths of the smallest division. Move the spherometer to a new position on the plate and set again. Proceeding in this manner, take nine independent settings over the plate. Next turn the micrometer back through a distance somewhat greater than the thickness of the object to be measured and place the object upon the plate under the micrometer screw. Make nine independent settings as before on different parts of the surface. The difference of the
MEASUREMENT OF THICKNESS 19
means of these two series of settings gives the desired thickness.
Measurement of the radius of curvature of a spherical sur- face by means of a spherometer. — This use of the instrument may be deferred until the experiment on the focal length of lenses is performed.
To measure the curvature of a spherical surface with a spherometer, three measurements are necessary. The spher- ometer is first placed on a plane surface, and the micrometer screw turned until it just makes contact with the surface. Let this reading be denoted by nii. The instrument is then placed on the surface or lens to be measured, and the micrometer turned until it again touches the surface with the same pressure as before. Let the reading be denoted by m^. Then m^ — mi = h is the height of a segment of the sphere whose radius R is to be determined, the base of the segment being a circle passing through the three feet of the spherometer. If r is the radius of this circle, i.e., the dis- tance of the end of the micrometer screw from the feet of the spherometer when all four points lie in the same plane, then
To measure r, make a slight imprint of the ends of the legs of the spherometer and of the central screw on a sheet of paper. Measure the distance of each leg from the axis of the screw, and take the mean; r may also be computed by geometry from the mean distance between the legs of the instrument, since these form an equilateral triangle.
THE OPTICAL MICROMETER.
Apparatus. — This instrument is shown in Fig. 7. It con- sists of a simple micrometer screw of 0.5 millimeter pitch, the head divided into 250 parts, thus reading directly to 0.002 mm. It is used for thickness measurements only. The unique feature in its construction is the beautiful device
20 PHYSICAL LABORATORY EXPERIMENTS
for indicating the moment of "contact" of the end of the screw with the object. This is effected as follows: The base of the instrument is a plate of plane black glass, A. On this rests a small plate of glass, B, the faces of which are approximately plane and parallel. If this plate be pressed upon the glass base, care being taken that no dust particles are between them, Newton's interfer- ence fringes will be produced by the thin air film between the plates. The fringes will appear colored if white light be allowed to fall on the plate, and yellow, if sodium light be used. If, now, the micrometer screw be turned slowly down, the instant that it comes into contact with the plate, a sudden displacement of the fringes will be observed. The instant of contact can thus be detected with the greatest nicety.
Procedure. — Wipe the glass base and plate very carefully and place the latter under the micrometer screw so that the fringes are clearly seen. Monochromatic light is best, but not at all necessary for observing the phenomenon. Turn the screw until it just makes contact with the plate, as indi- cated by the displacement of the fringes, and record the read- ing of the micrometer head. Make, at different portions of the plate, nine settings. Place the object C, whose thickness is to be measured, between the end of the screw and the test plate, and take a second series of settings as before. The difference of the means gives the thickness of the object.
Computation. — Compute the thickness of the object from the measurements made by each of the three instruments. Compute also the deviation measure of each result. (See Notes on Precision of Measurements.)
Questions and Problems. — 1. From a comparison of the results would yoii say that any of the instruments had a constant instrumental error, and why?
THE SPIRIT LEVEL 21
2 {Optional) . — If the deviation measures of the final results obtained with the spherometer and optical micrometer be assumed equal to their precision measures, compute their relative "weights" and their "weighted mean." (See Precision of Measurements, p. 20.)
THE SPIRIT LEVEL.
PART I. — ADJUSTMENT OF LEVEL.
Object. — To adjust a spirit level and to level a surface.
Discussion. — The spirit level is an instrument used for rendering surfaces and parts of apparatus truly horizontal, that is, perpendicular to the direction of the earth's attrac- tion, or to a plumb line set up at the place in question. It
consists essen-
,_ngrp^-^ — ijgg tially of a glass } * 3 tube, A, the up-
' ' 'i per surface of
Fig, 8 which is slight-
ly curved in such a manner that a longitudinal section (through a theo- retically perfect level) would be the arc of a circle of several feet radius. This is effected by carefully grinding the inner surface to the arc of a circle. The greater the radius of curvature, the more sensitive is the level. The tube is filled, except for a small air bubble, with alcohol or a mixture of alcohol and a little sulphuric ether. The latter renders the liquid more mobile and the bubble more reliable and rapid in its action. Pure ether cannot be used on account of its great expansibility, unless the tube is provided at one end with a suitable expansion chamber. Such levels are used only for astronomical or geodetic work where the greatest possible sensitiveness is desired. The tube is firmly mounted in a brass case to protect it from injury, the upper surface of the level alone being exposed to view. This surface is graduated to the right and left of the
22 PHYSICAL LABORATORY EXPERIMENTS
centre in equidistant divisions. The level in its case is mounted on either a flat or on a A-shaped base according as it is to be used for levelling plane or cylindrical surfaces, such as the telescopes of various optical instruments. The case containing the level is provided with an adjusting screw, B, at one end (sometimes at both), so that it may be brought into such a position that the tangent plane through the cen- tre of graduation is parallel to the plane of the base, or, in other words, so that when the level is placed on a horizontal surface, both ends of the bubble will be equally distant from the centre. That this condition must be fulfilled in an adjusted level is evident, since the bubble always assumes such a position that the free surface of the liquid is truly horizontal, that is, perpendicular to the direction of the earth's attraction; hence the plane of the base and the tan- gent plane through the centre of graduation must be parallel, if when the former is horizontal, the bubble shall take up a symmetrical position with respect to the latter,
A level should never be assumed to be in adjustment without testing. This may be done, of course, by placing the level on a horizontal surface, and noting whether the ends of the bubble are at equal distances from the centre of the graduations; if they are not, the adjustment would consist in simply turning the screw B until this was the case. As a horizontal surface is seldom at one's disposal, this pro- cedure is usually impracticable, and the adjustment to be made depends on the following demonstration.
Demonstration. — Let XY represent the plane of the base of the level, which may or may not be horizontal; we will assume that it is not. Suppose first, that the level is in
adjustment, i.e., that AB,
^. K- . r- ' ■ B the direction of the tan-
A - '_^,jg::=--ft' a" — ' -A" gent plane at the centre o
A' --^^ of the graduations, is par-
^'^ ■ allel to XY. The bubble
Fig. 0.
will then take up a posi- tion a h depending on the inclination of XY to the horizontal. If the level be now exactly reversed, A will coincide with
THE SPIRIT LEVEL 23
the former position of B relatively to the plane XY, and the bubble will take up the same position relative to the surface as before; in other words, the end of the bubble towards A will occupy exactly the same position as that previously occupied by the end towards B, and if the new readings be called a2, h^^,, we shall have a = 63 ^^<^^ ^ =^ %• This, then, is the condition which must be fulfilled when the level is in adjustment. If, now, the plane AB is not parallel to XY but is inclined to it as A'B' , the bubble will take up a higher position in the tube a' h' . On now reversing the level, the bubble will again move to the higher end of the tube and assume the position a"h" . In this case a' will no longer coincide with h" nor h' with a"; the difference h' — a" or h" — a! is proportional to the amount the level is out of adjustment. If, therefore, the adjusting screw be turned so that the bubble moves by one-half this difference, the level should be in adjustment. Several trials are usually necessary if the level is very sensitive.
Apparatus. — The apparatus required for adjusting a level besides the level itself is a plane surface mounted on a plate provided with three levelling screws. A carefully ground plate of glass about three-quarters of an inch thick is very satisfactory for this purpose. The ends of the levellmg screws should set in conical supports to prevent any lateral motion of the plate. The linear dimensions of the surface should be somewhat greater than the length of the level, as the latter must never be placed on the plate with one end projecting over the edge. The plate should be placed on a shelf or support as free from vibration as possible.
Procedure. — First. — Taking the level and plate as they stand, both presumably much out of adjustment, the first step is to render the plate approximately horizontal so that both ends of the bubble can be read on reversal. To do this, place the level on the plate parallel to two levelling screws. In general, one end of the bubble will disappear under the end of the level. Note the approximate position of the bubble. Mark with a pencil the exact position of the level on the plate and reverse. The bubble will either remain at the same
24 PHYSICAL LABORATORY EXPERIMENTS
(absolute) end of the level, in which case the plate is approxi- mately horizontal, — since the bubble of an unadjusted level will remain at rest in all positions of the level on a horizon- tal surface, — or it will move across to the opposite end of the level, in which case the plate is more out of adjustment than the level. If the former is found to be the case, the adjust- ing screw of the level should be turned so that the bubble moves out toward the centre. If the latter is the case, the same result should be obtained by turning the levelling screw of the plate. Note the new position of the bubble and reverse again. Repeat the adjustment of the level or plate according as the bubble remains nearly at rest or moves across from one end of the level to the other. A few trials will bring the plate and level into such adjustment that both ends of the bubble will fall on the scale in the direct and reversed positions. No readings need be recorded in this preliminary adjustment. When the adjustment is made, how- ever, read and record the exact positions a' h' of both ends of the bubble to tenths of a division, always recording the readings toward the A end as a's, and those toward the B end as 6's. Reverse the level exactly and call the new read- ings a" and h" . If a' does not equal h" and h' = a", adjust by half the difference. Record the new readings again under a' and b' and test the adjustment by reversal. If it has been correctly made, a' = h" , h' = a" ; if not, repeat until this is the case. An adjustment correct to 0.2 of a division will be considered satisfactory with the levels used. This corre- sponds to about 5" of arc and a finer adjustment often requires an excessive expenditure of time.
The same procedure described above applies equally well to the adjustment of levels parallel to the axis of reading telescopes of cathetometers, spectrometers, etc.
In all manipulations with a sensitive level, both in adjust- ing and in determining the angular value of a division as described below, care must be taken to avoid heating the level by the hand, breath, and in case of very sensitive levels, by radiation from the body, as a variation of temperature causes a change in the length of the bubble.
THE SPIRIT LEVEL 25
Second. — With the now adjusted level, level the plate. This is best done by placing the level on the plate, first, parallel to two of the three screws, and turning one of them until the ends of the bubble are equidistant from the centre, and second, at right angles to this position and levelling by the third screw. If two lines at right angles to each other in the same plane are horizontal, the plane itself is also horizontal. Many instruments are provided with two adjusted levels fixed permanently at right angles to each other on the base of the instrument. In this case, the instrument is levelled by simply turning the levelling screws until each bubble stands at the centre of its respective level.
PART II. — SENSITIVENESS OF LEVEL.
Object. — To determine the angular value of a level division and the radius of curvature of the level.
Discussion. — Although the principal use of a level is to render instruments horizontal, it is also used in astronomy and geodesy for measuring very small vertical angles. For this purpose, it is necessary to know the angular value of the level graduations, that is, the angle through which the level must be turned to cause the bubble to move over one division. This angle defines the sensitiveness of the level, and, in a perfect level, should be the same for all divisions. This, however, is seldom the case, owing to slight irregu- larities in the grinding of the level. For accurate angular measurements, the value of each division must, therefore, be determined.
It is of the greatest importance that the levels accom- panying any instrument be of a sensitiveness corresponding to the grade of work to which the instrument is to be put, and comparable with the degree of accuracy attainable with the other adjustments and reading parts of the instrument. A too sensitive level used in conjunction with an instrument of medium precision, or with one set up in an unsteady position, is almost useless, while to use a level capable of indicating only several minutes of arc on an instrument, the
26 PHYSICAL LABORATORY EXPERLMENTS
circle of which reads to thirty seconds, is to sacrifice com- pletely the precision of the instrument.
Apparatus. — For determining the angular value of a level
division, a level tester is employed. This consists of a massive
bar of iron, one end
of which A rests
_^^ B on two projecting ^ — a C
Iz^
" Fi^Tio^ ^^^^^ points, while
the other end B carries a micrometer screw, the point of which rests on a polished piece of metal or glass. By means of this screw, the end B can be raised or lowered any desired amount. The tester sets on a heavy metal plate C, which should be mounted on a rigid wall or shelf as free from vibration and jarring as possible. The micrometer screw of the instruments provided is of one millimeter pitch and the head is divided into 100 parts, thus permitting thousandths of a millimeter to be obtained by estimation.
Procedure. — Place the level on the bar, preferably near the end, and turn the micrometer screw until one end of the bubble is brought exactly to the first division. Record the reading of the micrometer. Repeat the setting four times, recording the micrometer reading each time. In these set- tings the bubble should always be brought up to the division in the same direction. This precaution is necessary in order to eliminate errors arising from tardy action of the bubble, due to friction against the tube, and to any backlash there may be in the micrometer screw. In the same way, take a series of four settings each on divisions 2, 3, 4, and 5 of the level. Finally remove the level and measure with a steel scale the distance between the point of the micrometer screw and the line connecting the steel points on which the bar rests. Measure also the length of the graduations on the level itself. This is best done by marking the total length of four consecutive divisions on a strip of paper and measur- ing the distance between these marks with a scale; if greater precision is desired, a dividing engine or two microscopes
MEASUREMENT OF MASS 27
may be used. This last measurement is necessary for computing the radius of curvature of the level.
Computation. — From the mean values of the settings taken, compute the average amount the level tester is raised to move the bubble over one division. With this and the length of the tester compute the corresponding angular value in seconds. From the average length of the graduations on the level compute its radius of curvature.
Answer the following questions in a general way by considering the percentage uncertainty in each of the quantities measured.
Questions. — 1. With what precision must the length of the tester be measured in order that the deviation in this measurement may be neghgible in the computed value of the angle?
2. About how precise do you estimate the value of the computed radius of curvature to be, and why?
3. Does this precision represent the accuracy of the re- sult, and why?
MEASUREMENT OF MASS.
GENERAL DISCUSSION.
The determination of the weight or the mass of a body, one of the most frequent and most fundamental of all phys- ical measurements, is almost invariably made with an equal arm balance, and the construction of this instrument has reached a very high degree of perfection. Balances suitable for analytical and most physical work are sensitive to at least 0.1 milligram under a load of 100 grams, i.e., to 1 part in one million, and in some very delicate balances a con- siderably greater precision is attained.
The following description applies, with slight modifications in details, to the several forms of balance used in the labora- tory. The essential parts of a good equal arm balance are shown in Fig. 11.
The balance beam, constructed to combine maximum rigidity with minimum weight, carries at its centre a knife edge A, which, when the balance is in use is lowered upon
1 One radian = 206265".
28
PHYSICAL LABORATORY EXPERIMENTS
a plane on which the beam swings. At the ends of the beam are fixed two other knife edges, B, B, upon which the scale pans are suspended by means of planes. Knife edges and planes are constructed of agate in the best balances. In
M.
(i|i linns
Fig. 11.
cheaper instruments they are made of steel. These three knife edges are adjusted so as to lie very nearly in the same straight line. By the length of the balance arms is meant the distance of each end knife edge from the central one, i.e., the distance AB. Both arms should be made as nearly equal in length as possible, and in good balances a difference not more than one part in 50,000 or one part in 100,000 is attained. At one end (sometimes both ends) of the beam is a fine screw, provided with a small nut, N, by means of which the effective weights of the balance arms may be adjusted equal. The beam is provided with a long pointer swinging over a fixed scale, S, near the base of the balance case. A small adjustable nut, M, on the pointer serves to raise or lower the centre of gravity of the moving parts of the beam, an adjustment which is necessary for reasons pointed out below. In some balances this adjustment is made by a nut attached above the beam.
MEASUREMENT OF MASS 29
When not in use, the beam should always be raised off its support, and the pans off their respective knife edges, to pre- vent dulling the knife edges by jarring. This is effected by means of a frame operated by a milled head at the front of the balance case. This mechanism for raising and lowering the beam should be so constructed that no lateral displacement of the knife edge on its support can occur. To further pro- vide against injury from jarring when lowering the beam and pans, and to prevent the pans from swinging, light stops or rests are provided beneath them. These should always be raised to support the pans, except when a weighing is being made; they are then carefully lowered by pressing the button operating them at the front of the case. The height of these rests is regulated by means of an adjusting screw at the back of the balance. A good balance is also provided with either a plumb bob, or preferably, a spirit level, and with levelling screws.
Theory of the Balance. — Reduced to its elements, an equal arm balance consists of a rigid beam supported horizontally at its centre on a knife edge and loaded at its ends. The various conditions upon which the sensitiveness of a balance depends will be readily seen from the following demonstration:
Let I = length of arms (assumed equal) .
w = weight of beam acting downward through centre
of gravity. r = distance of centre of gravity, CG, below point of support of the beam.
With a load of P grams in each pan, the balance will evi- dently be in equilibrium. If a small additional weight p be added to the right-hand pan, the balance will swing through an angle a and take up a new position of equilibrium as in Fig. 12. The condition for equilibrium will evidently be found by taking moments of the three effective forces, P, P -\- p, and w about an axis passing through the knife edge, i.e.,
PI cos a -\- wV sin a = (P + p) Z cos a,
, sin a Ip f^s
or tana ^= ^ -^,. (1)
cos a Wl
30
PHYSICAL LABORATORY EXPERIMENTS
Fig. 12.
Now the sensitiveness of a balance is measured by the angle through which the beam will rotate when a definite weight, usually one milligram, is added to either pan. As the angle practically coincides with its tangent for very small angles, the expression deduced for tan a may be taken as a measure of the sensitiveness of the balance when p = 1 milligram.
From this it follows that the sensitiveness of a balance is di- rectly proportional to the length of the balance arms; in- versely proportional to the weight of the beam; and inversely proportional to the distance of the centre of gravity below the point of support. It is also evident that the observed scale deflection is proportional to the length of the pointer.
Unfortunately the above conditions for maximum sensi- tiveness conflict with each other in the mechanical construc- tion of a balance. Thus long arms are incompatible with minimum weight. The length of the arms is also practically limited by another condition, namely, the time of swing of the balance, which must not be excessive. Regarding a balance as a compound pendulum, it can be shown that its time of vibration is given by the expression
MEASUREMENT OF MASS 31
t = ,sJW+El (2)
wV g where
k = radius of gyration,
g z= acceleration due to gravity, and w, P, I, and V have the same significance as before.
From this it will be seen that a limit is soon reached beyond which it is undesirable to increase the sensitiveness of a balance by increasing the length of its arms, I, for its time of vibration becomes excessive, and the process of weighing very tedious. A period of 12 to 15 seconds is about the maximum time desirable. Again the sensitive- ness cannot be increased by diminishing I' beyond a certain limit for the same reason. (The centre of gravity, CG, must, of course, always be below the knife edge, otherwise the balance will be in neutral or unstable equilibrium). Makers have, therefore, a considerable choice of conditions which they can vary to obtain a given result, and we find on the market excellent "Long Arm" as well as "Short Arm" balances. Long arm balances, although sensitive, possess the disadvantage of a long time of swing (10 to 15 seconds) which renders weighing with them very tedious. For moderate loads short arm balances constructed with light beams (aluminum or aluminum alloys are employed) are much to be preferred, as they are not only sensitive, but very quick in their action (time of vibration 6 to 10 seconds).
In the above discussion the assumption is made that the points of support of the pans and the knife edge about which the beam swings are in the same straight line. This is an ideal condition which is always departed from more or less in actual instruments, since no beam can be made abso- lutely rigid, and, hence, if the above condition be fulfilled for some one load, it must be further and further departed from with increasing load. To determine the effect of this deviation from the ideal condition, suppose that the lines joining the points of support of the pans with the central knife edge make an angle /3 with the horizontal line through
32
PHYSICAL LABORATORY EXPERIMENTS
the latter. (See Fig. 13.) When the balance is in equilibrium under loads P and P + p we shall have
PI cos (a + /3) + WV sin a— {P + J))l COS (a — /3)
or expanding and solving for tan a we obtain
tana =
wl' „ 2P + P , ^ — r • sec )8 !— ^ • tan fl
(3)
If the points of support of the pans are helow the central knife edge, /3 becomes negative, and for this case
tan
wl' . , 2P4-P , ^
— r . sec /? + !— ^ • tan B
pi V
(4)
The value of /3 which renders both of these expressions for the sensitiveness a maximum is easily seen to be ;8 = 0,
in which case both formula? reduce to tan a = ^ , as
wl
already shown. If the points of support of the pans under zero load are above the horizontal plane through the central
MEASUREMENT OF MASS 33
knife edge, the sensitiveness of the balance will increase with increasing load. If the points of support of the pans are below the horizontal through the central knife edge, the sensitiveness will continually diminish with increasing load. A determination of the sensitiveness of a balance under dif- ferent loads affords, therefore, information regarding this point in the construction of the beam. In the best con- struction the beam is so designed that the maximum sensi- tiveness is reached at about one-half the maximum load which the balance is intended to carry.
It will be seen from (3) that in the case when the points of support of the pans are above the central knife edge it would be theoretically possible to increase the load, P, to such a
, 2P + p wV . , . 1
value that tan /? = — r • sec (i, m which case the
balance would reach a state of unstable equilibrium. The period of vibration of the balance would at the same time be greatly increased.
PRECAUTIONS TO BE OBSERVED IN USING A BALANCE.
1. The balance should be set up on a firm support fixed to a wall or on a pier which is as free from mechanical vibration as possible. A place should be chosen where direct sunlight will not fall upon the balance, as the resulting tempera- ture changes will introduce irregularities and errors in the weighings.
2. To prevent the knife edges (if of steel) and other parts of the balance from tarnishing, an open dish of some drying agent, as concentrated sulphuric acid, may advantageously be kept in the balance case.
3. When not in use, the balance beam should always be raised off the knife edges. Injury to the knife edges from jarring will otherwise result.
4. The beam should always be lowered slowly and carefully. When lowered, the pans should just rest upon the stops or arrests beneath them, and the index should stand at the middle of the scale. The pan arrests when lowered should free both pans simultaneously.
34 PHYSICAL LABORATORY EXPERIMENTS
5. The beam should never rest upon the knife edge while weights or substances are being added to or removed from the pans, except in the case of small fractional weights less than one gram.
6. The beam should be set swinging either by dropping the rider upon the beam or removing it for a moment, or by fanning one pan gently with a motion of the hand. Never hit the pan with the forceps, or attempt to set the beam swinging by suddenly lowering it upon its knife edge.
7. All weighings should be made methodically, not hap- hazard; otherwise an enormous loss of time will result. A little experience is necessary before rapid weighings can be made.
8. Final adjustment with the rider and all observations of the swing of the index should always be made with the balance case closed, to prevent errors arising from air currents.
9. The weight of a substance should be recorded, first, by adding up the weights missing from the box (in which every weight should have its own place), and second, by adding the weights as they are returned to the box. This gives a check which is most desirable, as nothing is easier after having made a correct weighing than to record the weight erroneously.
10. Any substance which can possibly injure the scale pans should never be placed directly upon them, but upon a watch glass or in some other suitable vessel.
11. Substances should never be weighed while hot, as air currents will be produced which will introduce serious errors in the weighing.
12. Care should be taken never to overload a balance by attempting to weigh bodies heavier than the balance is con- structed to carry.
METHODS OF WEIGHING.
The weight of a substance may be obtained with an equal arm balance by any one of the following methods, the relative advantages of each of which are stated below.
MEASUREMENT OF MASS 35
First. — Ordinary Method or Method of Equal Swings. In this method the balance must first be put in perfect adjust- ment, so that under zero load the index swings equal dis- tances to the right and left of the middle scale division, allowance being made for the diminution in amplitude of successive swings resulting from friction at the knife edges, air resistance, etc. It is convenient to call the middle division of the scale, zero, and to reckon deflections to the right and left of it. The body to be weighed is placed in the left-hand pan and weights added to the right-hand pan until, with a final adjustment of the rider, the pointer swings equal distances to the right and left of the middle division. This method is applicable only when the balance is provided with a rider. It is not to be recommended when the extreme pre- cision attainable with the balance is desired. For ordinary weighing it is, however, the method commonly used.
Second. — Method of Swings. In this method the balance need not be initially adjusted to swing equally to the right and left of the zero under no load, and it is the only method applicable to balances not provided with riders. The pro- cedure is as follows: Determine the point of rest under zero load by noting the position of the extreme swing of the pointer, three times to one side and twice to the other side of the position of rest. In this case the divisions on the scale should be numbered from right to left, the middle division being thus numbered 10 instead of 0. Suppose the readings are as follows: —
Left. Right
14.3 4.2
14.1 4.4
4.5
14.20 4.37
14.20
2)18.57
Point of rest = 9.29
The point of rest will then evidently be 9.29. An odd num- ber of readings is always taken to eliminate the effect of air resistance.
36 PHYSICAL LABORATORY EXPERIMENTS
Call the position of rest under zero load as thus found Uq. Next, nearly balance the body by a weight Wi, and deter- mine the point of rest ai as before. (If the balance is provided with a rider, the weight t^i should be adjusted to the nearest 0.001 gram.) Finally add a weight m grams (0.001 gram or 0.002 gram if a rider is provided), so that the body is slightly over balanced, and again find the point of rest a^ under the weight W2 = Wi -{- m. As the point of rest shifts proportional to the change in load, i.e., ^^ w — Wx _W2 — Wi _ Wi + m — Wi
we obtain for w, the true weight which would bring the point of rest back to ao,
€1-2 — ai
For example, if ao = 9.29; ai = 8.27 under a load wi =
14.167 gms.; as = 10.41 under a load 102 = 14.168 gms.,
then
q on c 07
w = 14.167 + 0.0010 iQ 41 _ ^27 = 14.167 + 0.00048 =
14.16748 gms.
This method requires an accurate knowledge of the point of rest under zero load. As this is likely to vary if the balance is jarred or its level changed, it must be frequently determined, unless the balance is set up under exceptionally steady and isolated conditions. When a large number of weighings are to be made, and the extreme precision of the balance is desired, this method is to be preferred to the preceding.
Double Weighing. — Both of the preceding methods are subject to constant instrumental errors arising from inequality in the length of the balance arms. This may be corrected for by a determination of the ratio of the length of the arms (see p. 43), or by either of the two following methods of so-called "double weighing."
Third. — Borda's Method or Method of Taring. The body to be weighed is first balanced or "tared" by an appropriate substance, e.g., shot or fine sand. It is then removed from the pan and weights added in its place, until the taring sub-
MEASUREMENT OF MASS 37
stance is exactly balanced. The weight added is then evidently equal to the weight of the substance whatever be the relative length of the balance arms.
Fourth. — Gauss's Method. The substance is first placed in the left-hand pan and weighed by either of the first two methods. Let the weight thus obtained be Wr. It is then placed in the right-hand pan and weighed again, the weights being added in this case to the left pan. Call the weight thus obtained Wi. If r and I be the lengths of the right and left-hand arms of the beam, respectively, and w the true weight of the body, then
W 1 = WrT
Wil = w r whence eliminating r and I,
1^ = V 'WrWi •
Since Wr and Wi usually dilTer from each other at most by only a very small amount, the arithmetical mean may be taken equal to the geometrical mean without introducing a sensible error, i.e.,
W = ^{Wr + Wi ),
thus simplifying greatly the computation. If, for example, the substance weighs 1 gram and the weights found are Wr = 1.001 and Wi = 1.000, then ^{Wr + Wi) = 1.0005, while ^fi^^ = 1.0004999.
Of these two methods of double weighing, Gauss's Method is to be preferred, as the two weighings afford a check on each other, and the method is, on the whole, the more rapid of the two.
Reduction of apparent weight of a body to its true weight in vacuo. — The weight of a substance as determined by an equal arm balance, after all corrections have been made for instru- mental errors, is only its apparent weight under the given conditions of temperature and pressure at the time of weigh- ing. This arises from the fact that, in general, the volumes
38 PHYSICAL LABORATORY EXPERIMENTS
of air displaced by the body and by the weights are different, owing to the difference in their densities. The buoyant effect of the displaced air is, therefore, different on the body and on the weights, causing the former to apparently weigh too much if it occupies a smaller volume than the weights, and too little, if the converse is the case. Only when the substance weighed has the same density as the weights, is its apparent weight the same as its true weight in vacuo.
Let W = true weight of substance in vacuo. w = apparent weight. 8 = specific gravity of the substance.^ A = '' " " " weights.^
o- = weight of 1 cc. of air under the conditions of temperature, pressure, and humidity at the time of weighing.
W Then evidently the body is buoyed up by a weight of air -^-o"
o
and the weights by the amount —or. We have, therefore, when the substance and weights are in equilibrium.
|
8 "" |
= w - |
w |
|
1- |
"a |
|
|
W = |
w — |
whence
1-?
W = w
1 -f °"( ^ — t] approximately (1).
By this important formula the true weight in vacuo of any substance may be calculated from its apparent weight in air.
That the value of this correction may be very appreciable when 8 and A differ widely, may be seen from the following table which contains the value of the correction term for several common substances. The correction is computed in milligrams for one gram of substance, assuming a = 0.0012,
iNote : In general 6 and A need be known only approximately.
MEASUREMENT OF MASS 39
its mean value for ordinary atmospheric conditions in the laboratory, and A = 8.5 (Becker's brass weights).
|
Substance. |
8 |
in milligrams. |
|
Alcohol |
0.8 |
+ 1.36 |
|
Water |
1.0 |
+ 1.06 |
|
Glass |
2.5 |
+ 0.34 |
|
Brass |
8.5 |
0.00 |
|
Mercury |
13.6 |
— 0.055 |
|
Platinum |
21.4 |
— 0.085 |
Thus a mass of water weighing exactly one gram in air would weigh 1.00106 grams in vacuo. Neglect to apply this correction in the case of water would, therefore, introduce an error of 0.1 per cent. In the case of glass the error would be 0.03 per cent, and so on.
When a precision not exceeding 0.1 per cent is desired, the correction is evidently negligible for substances heavier than water. When, on the other hand, substances like gases or vapors are weighed, the correction is of great importance, and must be computed with great care. The value of o- depends on the temperature of the air in the balance case and on the barometric pressure, both of which should always be recorded when an exact weighing is made. It also depends to a less degree on the humidity of the atmosphere, and the effect of this may usually be neglected. If the balance case is kept dry by sulphuric acid or some other drying agent, this factor is largely eliminated. Table V., Appendix, contains values of the weight of one cubic centimeter of air under various conditions.
In applying the above correction, it is most convenient to increase or diminish the apparent weight w by a certain per cent as determined by the value of the correction term compared with unity (see formula). Thus, if a mass of glass weighs 5.1131 grams, its weight in vacuo will be 1.00034 times greater, i.e., 0.034 per cent greater:
1 per cent of 5.1131 = 0.051; 0.034 per cent = 0.034 X 0.051 = 0.0017
40 PHYSICAL LABORATORY EXPERLMENTS
which gives at once the correction to be added. Note that two significant figures are amply suflScient in this case for computing the correction term, whereas if 5.1131 X 1.00034 is multiplied out nothing is gained in accuracy, and much labor is wasted.
THE BALANCE.— PART I.
METHODS OF WEIGHING.
Before beginning this experiment, study carefully the preceding Gen- eral Discussion and the Theory of the Balance. The manipulation of the balance and procedure to be followed in weighing will be explained and illustrated by an instructor. Careless handling of a delicate balance by an inexperienced person is liable to result in serious injury to the balance.
Object. — The object of this experiment, is first, to give practice in the use of an analytical balance by weighing a substance both by the Ordinary or Equal Swing Method and by the Method of Swings, and, second, to compare the volume of the substance computed from its weight and density with the volume calculated from its linear dimensions, the sub- stance being a true geometrical figure.
Apparatus. — The apparatus provided is a good equal arm balance sensitive to 0.1 milligram and provided with a rider. The substance to be weighed is a carefully turned right cylinder. Vernier calipers reading to 0.02 mm. are also pro- vided for measuring the dimensions of the cylinder.
Procedure.— i^iVs^.— Take the balance as it stands, level it by means of the spirit level or plumb bob, and determine the weight of the substance by the "Method of Swings," described on p. 35.
Second. — Adjust the balance for weighing by the "Ordi- nary Method," p. 35, and weigh the substance again.
Third. — Record the temperature and barometric pressure in the laboratory as given by the self-recording instruments.
Fourth. — Measure the linear dimensions (diameter and length) of the cylinder with vernier or micrometer calipers taking four independent measurements of each.
THE BALANCE 41
Computation. — First. — The density of the substance weighed will be given by an instructor. Reduce the apparent weight of the substance to its true weight in vacuo, taking as the apparent weight the mean of the two determinations made. The value of o- (the density of the air) is to be taken from Table V., Appendix.
Second. — Compute the volume of the substance in cubic centimeters from its true weight and compare with the value calculated from its dimensions.
Precision Discussion. — The volume of a cylinder is F = J ird-h where d is the diameter and h is the height. Suppose the precision measures of h and d are 5/, and 5^ respectively. These may be assumed equal to the A.D.'a of the measurements. What is the precision measure of the final result, Vf
First Solution. — The deviation in V due to the deviation S/t m h alone will be
9
^iwd'd^.
Similarly the deviation in V due to the deviation da in d alone wiU be
A^ = ^(ind%).Sa.
= \Trh. 2d. Sj.
The resultant deviation in V due to the combined effect of both deviations will be
A = VA/;^ + Arf2
= V(i Td2. 5a)2 + (^ ^h. 2d. Sd)2.
Second Solution. — (Always simpler when the function ia a product, quotient or power of the measured quantities).
The fractional deviation of d is -V- a
" " ^ is -r-
n Therefore the fractional deviation in V due to the frac tional deviation in h is, by inspection, — = — (as A enters as first power in a product).
42 PHYSICAL LABORATORY EXPERIMENTS
Similarly the fractional deviation in V due to the frac- tional deviation -^ in d is -^ — 2 -^ (as d enters as a a V a
square).
The resultant fractional deviation in V is, therefore.
and hence
This method evidently involves very much less work than the preceding.
Questions and Problems. — 1. Which method of deter- mining the volume do you consider the more reliable, and why? (The error in the value of the density given may be considered not greater than one part in 5000.)
Optional ProbleMo.
2. From the A.D. of the diameter measurements (5j), and the A.D. of the length measurements (5^), find the percentage deviation of d, h, and of the final result V. What is the corresponding actual deviation of the volume in cubic centimeters?
3. Is it necessary to reduce the weight to vacuo in com- puting the volume, if the result is desired to only 0.1 per cent? to 0.05 per cent?
THE BALANCE.— PART II.
CURVE OF SENSITIVENESS OF A BALANCE AND RATIO OF BALANCE ARMS.
Before performing this experiment, read carefully pp. 27-34 on General Theory of the Balance. Students must also have performed "The Bal- ance, Part I" or already have had practice in the use of a delicate balance.
Object. — The object of this experiment is two-fold: first, to determine the sensitiveness of a balance under various loads; and second, to determine the ratio of its arms.
THE BALANCE 43
Apparatus. — The apparatus provided is a delicate, analyti- cal balance sensitive to 0.1 milligram.
Procedure. — Preliminary adjustments. — First, level the bal- ance by means of the spirit level or plumb bob attached to the balance. Second, determine the period of vibration of the beam by noting the time of several transits of the pointer through the lowest part of its arc. With a short arm balance the time of a single vibration should not be over eight or ten seconds; with a long arm balance it should not exceed twelve or fourteen seconds. In case these limits are sur- passed, speak to an instructor, who will adjust the centre of gravity of the beam to a proper distance below the knife edge.
Sensitiveness. — The sensitiveness of a balance has been defined to be the angle through which the beam rotates under an excess of load of one milligram in either pan. As this angle is proportional to the number of scale divisions over which the pointer moves, the latter is usually taken as a measure of the sensitiveness instead of the angle itself.
The sensitiveness is to be determined with loads of 0, 10, 20, 30, 40 and 50 grams in each pan. The "Method of Swings" is to be employed throughout. The procedure is to determine the point of rest as described on page 35; first, with say, zero grams in each pan, and second, with one milligram weight added with the rider on the right side of the balance beam. The difference of the points of rest under these two conditions is the number of divisions through which the pointer is deflected by an excess of weight of one milligram. Proceed in the same way to determine the sensitiveness for each of the above loads.
Ratio of Arms. — The ratio of the length of the balance arms could be obtained most simply with a set of weights which were exactly adjusted among themselves, or at least to as high a degree of precision as it is desired to ascertain the ratio of the balance arms. With such a set of weights it would simply be necessary to find the weight w^ in the right- hand pan, which would balance a weight Wi in the left-hand
44 PHYSICAL LABORATORY EXPERLMENTS
pan; then, if r and I are the lengths of the right and left arm of the balance, respectively,
WrT ^ Wil
r ivi
Weights cannot be assumed to be adjusted to the degree of precision required for this work without a special com- parison (see p. 45), and hence the ratio of the arms of a balance is best determined as follows : Weigh some substance of weight a, first in the left-hand pan, and then in the right- hand pan. Suppose Wr and Wi are the weights (very nearly the same) required to balance it in these two cases, respec- tively. Then
al= WrT
Wil = ar
Eliminating a we have
r^ Wi
or
I ^ Wr »
rvi Wr
= 1 + -^ — -, approximately.
Note that in this case the same weights may be used for Wj and Wr, the slight difference between them being added or subtracted by means of the rider. The errors of the set of weights are thus eliminated. As only two or, at most,
three significant figures are retained in — ^r — -, iv may be
taken as either Wi or Wr .
Determine in this way the ratio of the arms of the balance using for a a weight about one-half the maximum load for which the balance is designed.
Computation. — Plot a curve of sensitiveness of the balance with values of the sensitiveness as ordinatos and the corre- sponding loads as abscissa?. In choosing scales, be careful
THE BALANCE 45
that the scale of ordinates does not exceed ten times the precision of the data, otherwise the plot will be much dis- torted, and it will be found difficult to draw a "best repre- sentative line."
Questions and Problems. — 1. What conclusion can be drawn from the curve of sensitiveness regarding the con- struction of the balance beam? (Compare p. 33.)
2. To what precision may the balance be used without correcting for the ratio of its arms? What is the minimum weight which would be affected to one milligram by this correction? To one-tenth milligram?
THE BALANCE.— PART III.
CALIBRATION OF A SET OF WEIGHTS.
Object — To determine the errors of a set of weights.
A set of weights furnished by the best makers cannot as a rule be relied upon closer than to one milligram, and hence for exact work in absolute weighing it is necessary to determine not only the relative errors of the weights among themselves, but also the absolute error of the set by comparison with a standard. By the calibration of a set of weights is meant the determination of the ratio of each single weight to the true combined weight of all together, as determined by a comparison with a standard.
Apparatus. — The apparatus required is a delicate balance sensitive to 0.1 or 0.05 milligram, a set of weights and a standard weight.
Procedure. — The procedure is as follows: All weighings are to be made by the "Method of Swings" (p. 35). It will be assumed that the balance arms are equal to within the precision which it is desired to calibrate the weights. If this is not the case, the ratio of the arms must be first determined as described in Part II, and each weighing corrected by this amount.
Suppose the set of weights is made up as follows: 50, 20, 10(1), 10(3)> 5, 2(1), 2(2), 1. The fractional platinum weights will
46 PHYSICAL LABORATORY EXPERIMENTS
not be considered. The subscripts (1) (2) denote different weights of the same denomination. The weights are then to be compared in the following order:
Left-hand pan. Right-hand pan.
5H-2a)+2(2)+l =10ai +« 10(2)=10a) +b 20 =10(1) +10(2)+c 50 =20+10a,+10(2)+5+2(i,+2(2)+l+d
a, h, c, d being the small weights which it is found necessary to add to or subtract from the right-hand side of the balance in order to establish equilibrium. Suppose further that the combined weight w of all the weights compared with a standard 100 gram weight is found to be
w; = 50 + 20 + 10(1) + • • • . + 1 = 100 + e
where e may, of course, be plus or minus.
If each weight is now expressed in terms of some one weight, as' 10(i) taken for reference, we have
5 + 2(1) + 2(2) + 1 =10(1) + a 10(1) = 10(1) 10(2) = 10(1) + b
20 = 2X10(1) + 6 + c 50 = 5X10(1) + a + 26 + c + (i
and hence adding these equations,
w = 10X 10(1) + 2a + 46 + 2c + d = 100 + e
This may be written,
10 [10(1) + TO (2a + 46 + 2c + d — e)] = 100
or
10(1) + T (5 (2a + 46 + 2c + d — e) = W" = 10 grms. (standard) .
Hence if S = iV (2a + 46 + 2c + d — e),
lOn, = 10 — 8.
SPECIFIC GRAVITY DETERMINATIONS 47
The true value of each of the weights in terms of the standard will therefore be
5 + 2(i) + 2(2) + l =10- S + a 10(1) = 10— 8 10(2) = 10— S + 6 20 = 20 — 28 + 6 + c 50 = 50 — 58 + a + 26 + c + <^
from which the numerical correction to be applied to each weight is readily determined.
The correction for the 5, 2^■^^, 2(2), and 1 gram weights and fractional weights can, of course, be determined in like manner. If the weights are to be used for relative weigh- ings only, as in most chemical work, the comparison with the standard may be omitted and the value of e in the above formulae taken as 0.
SPECIFIC GRAVITY DETERMINATIONS.
GENERAL DISCUSSION.
The density of a body is the mass or quantity of matter contained in the unit of volume. The numerical value of the density depends, therefore, on the system of units in which the mass is measured. Thus in the English system of weights and measures the density of water is 62.5, the mass of one cubic foot (unit of volume) being 62.5 pounds. In the centimeter-gram-second (c.g.s.) system, the only system in scientific use, the density of water is unity, since the unit of mass, the gram (the one-thousandth part of the mass of the standard kilogram), is practically equal to the mass of one cubic centimeter of water.
The specific gravity of a substance is the ratio of the weight of a given volume of the substance to the weight of an equal volume of water at 4° C, the temperature of its maximum
48 PHYSICAL LABORATORY EXPERIMENTS
density. Specific gravity is, therefore, a pure number inde- pendent of the system of units employed. From the above relation between the units of mass in the c.g.s. system and the mass of one cubic centimeter of water, it follows that the numerical value of the density of a body in the c.g.s. sys- tem is identical with its specific gravity. This is not abso- lutely true however, as it has been found that the mass of one cubic centimeter of water at 4° C. is not exactly equal to one gram. The difference is so slight that for most pur- poses it may be neglected.
The specific gravity of a substance is sometimes erroneously referred to water at some other temperature than 4° C. This is frequently done with liquids which are compared with water at the same temperature as the liquids themselves. Thus sp. gr. benzene 0.808 ^"72o° means that the weight of equal volumes of benzene and water, both at 20°, are in the ratio 0.808 to 1. The true specific gravity of benzene at 20°, as defined above, is somewhat less, and should be expressed thus: 0.808 ^°74°- The temperature, both of the substance and of the water to which it is referred, should therefore always be stated to avoid ambiguity.
The efTect of changes of barometric pressure on the specific gravity of liquids and solids is negligible. This is not true, however, in the case of gases and vapors. Unless otherwise specifically stated, these are always reduced to " normal con- ditions," i.e., to 0° C, and 760 mm. pressure. Moreover, the standard substance to which they are referred is dry air or hydrogen, under the same conditions of temperature and pressure, instead of water. The specific gravity of a vapor, as distinct from a gas, is often spoken of as its "vapor den- sity," although no reason other than the custom warrants the use of the expression.
The specific volume of a substance is the volume occupied by a unit mass of it under standard conditions of tempera- ture and pressure or, in other words, the reciprocal of its density. In the c.g.s. system it may be defined as the volume occupied by one gram of the substance under stand- ard conditions.
SPECIFIC GRAVITY OF SOLIDS 49
The molecular and atomic volume of a substance, quantities of frequent use and much importance in chemistry, are the volumes occupied by one molecular mass and one atomic mass of the substance, respectively. They are obtained by multiplying the specific volume by the molecular and atomic mass (weight), of the substance, respectively.
The various methods in common use for determining spe- cific gravity all depend on determining, first, the weight of the substance itself; and second, the weight of an equal volume of the standard substance water (or air) . The princi- pal methods in use for determining the specific gravity of solids and liquids are described below. For gases and vapors Ostwald's Physico-Chemical Measurements, pp. 100-106, may be consulted.
SPECIFIC GRAVITY OF SOLIDS.— I.
Object. — ^To determine to 0.1 per cent the specific gravity of a solid insoluble in water, with a specific gravity flask. The computation illustrates the application of temperature corrections and reduction of weighings to vacuo.
Discussion. — In this very convenient and accurate method, the weight of a volume of water equal to the volume of the substance is obtained by determining the weight of water displaced from a small flask or bottle, when the substance is introduced. This is done by weighing the flask, first, when completely filled with water or with the water adjusted to some reference mark; and second, when the flask contains both substance and water, the total volume being again adjusted the same as before. From these two weights, to- gether with that of the substance dry, the weight of the water displaced by the immersed solid can be at once deter- mined. The accuracy of the determination depends on the precision with which the contents of the flask can be adjusted successively to the same volume.
50
PHYSICAL LABORATORY EXPERLMENTS
In Fig. 14 are shown three of the best forms of specific gravity bottles in general use. All are provided with care- fully ground stoppers. The stopper of A has a capillary bore and the water is adjusted so as to com- pletely fill the flask and stopper. In B and C the liquid is adjusted to a reference mark etched on the narrow stem. C is provided in addition with a sensitive thermometer, ground to fit the neck of the flask, for indi- cating the temperature of the liquid.
Apparatus. — The apparatus provided is some form of specific gravity flask illustrated above and a good analytical balance.
Procedure.^Weigh the substance dry. Clean the specific gravity flask with bichromate cleaning solution if necessary, rinse with distilled water and then with alcohol, and dry by allowing a current of warm air to blow into the bottle.^ Obtain the weight of the bottle to 0.01 gram (for reasons
1 A special arrangement for drying bottles is provided at the sink.
SPECIFIC GRAVITY OF SOLIDS 51
given below). Next fill the bottle with distilled water, mak- ing sure that no air bubbles adhere to the sides, and care- fully insert the glass stopper firmly but not tightly, without enclosing any air. If the glass be thin, special care must be taken not to force the stopper in and so distort and change volume of the flask. Adjust the liquid by means of filter paper to the reference mark if either flask B or C is used. Dry the outside carefully, particularly around the neck of the stopper, and weigh. Immediately afterwards take the temperature ti of the water, by inserting the bulb of the thermometer into the bottle. Next drop the substance into the flask, refilling with more water if necessary, and replace stopper as before. See that no air bubbles are caught between the pieces of substance introduced. Again weigh, and record the temperature t^ of the water. Record also the barometer height and the temperature of the air in the bal- ance case.
If the flask with its contents is adjusted both times at the same temperature by immersing it in a constant temperature water bath, the corrections for expansion of glass and change of density of the water given below may be eliminated.
Computation. — The specific gravity of the substance at 1^° referred to water at 4° C. is to be computed, all corrections being applied which affect the final result to 0.1 per cent.
Let w = apparent weight of substance dry;
Wi= " " " bottle plus water at ti°;
W2= " " " " water and substance at f 2° J
6 =: " " " bottle dry.
The approximate specific gravity of the substance is
w .-.
s = (1)
The value of s thus obtained is likely to be in error by a per cent or more, due to the omission of the following corrections :
First, for change of volume of the bottle due to the change of temperature from ti° to 1^°;
Second, for change in density of the water due to the same temperature change;
Third, for buoyancy of the air.
52 PHYSICAL LABORATORY EXPERIMENTS
The first two corrections may often be made negligible, as pointed out above, by making t° and t^ sufficiently near the same. The third correction must always be applied unless shown to be smaller than the experimental error; this is often the case when the specific gravity of the substance is nearly unity. The above three corrections are made as follows : —
Correction for change of temperature. — We wish to find what the weight w^ would have been, had the bottle plus water been weighed at t2° instead of ti°. The weight of water filling the fiask at ti° is w'l — b. The volume of the flask
at ^1° is iVj = ^^^ — , where D,^ is the density of water
at ^1°. The volume of the flask at ^2° is, therefore,
vt, = i\ [1 + A: (/2 - ^1)],
where k = 0.000026 is the mean coefficient of cubical ex- pansion of the glass between t-^° and 1^°. Hence the weight of water of density Dt^ filling this volume at t<^ will be
= {w,-h)[l^-k{t,-t,)]^.
Hence the weight of the bottle plus water at ^2° would be
w^= h + {w, - 6) [1 + A; (^2 - ^1)] ^
r= h + {ii\ — h)[\^-k (^2 — ^1) + Dt„^ — ^t^ approx.
=■ w, + {w, - b) [k (^2 - ^1) + A, - D,;\. (2)
The weight of the water displaced by the solid is, therefore,
w' = W2 -\-w — w^. (3)
Discussion. — It is to be noticed that the value of the sec- ond term in formula (2) is a small correction to be added to or subtracted from the observed weight w-^. If ^^'i is weighed to 0.001 gram, the correction is negligible if loss than 0.0005 gram; if to 0.0001 gram, it should be computed to the
SPECIFIC GRAVITY OF SOLIDS 53
nearest 0.1 milligram. In general two significant figures in each of the factors are sufficient, as the whole correc- tion is seldom greater than a few milligrams. It is for this reason that the weight of the bottle b need only be known approximately. To illustrate the magnitude of the cor- rection involved, suppose the capacity of the bottle is about 30 cc, that tj° = 20°.0 and tr° = 21°.0 C. Then w^ — 6 = 30 approximately, and b need only be weighed to 0.1 gram. The value,
A, — A, = 0.99804 — 0.99825 = — 0.00021 (see Appendix, Table VIII). Hence the correction per degree at 20° to be applied to Wi is
30 X [0.000026 — 0.00021] = — 0.0055 grams. (4)
The correction due to change in density of the water is seen to be considerably greater and in the opposite direction to that due to the expansion of the glass. For a bottle of even 10 cc. capacity the above correction is not negligible for a variation of one degree, if the weighings are carried to milligrams. For a larger bottle the correction would be proportionately greater.
Reduction to vacuo. — If W and W be the true weights in vacuo of the solid and of the displaced water respectively Csee p. 38),
rjr W w
W o- = W — -a-,
W — — -o- = W -0-.
Dividing one equation by the other, we obtain W(l-^) w(l-^)
S A
(1——) W W Dt. n) (Df — a
s
54 PHYSICAL LABORATORY EXPERIMENTS
W W
but :f^ = — = true volume of substance Dt^ s
W s
Hence s = — {D, — a) -\- <t. (5)
w ^
The value of s thus computed is the value of the specific gravity of the substance at the temperature ^2° referred to water at 4° C. If it is further desired to compute the specific gravity of the substance at another temperature, say t°, the mean coefficient of cubical expansion of the substance must be known. If this be fi, then
V = V, [1+ /8 (^2° - t)l
since the specific gravity is inversely proportional to the volume of a given mass. The exact equation for this reduc- tion is
where /3i and ^2 ^re the mean coefficients of expansion between, say 0° and t° and 0° and t^° respectively. Prac- tically, however, p^ = /?2 for most substances, but unless ^ be known with sufficient accuracy, a greater error may be introduced in the resulting value of s than occurs in the actual experimental work.
Problem. — 1. With a flask of 25 c.c. capacity, what is the greatest allowable difference which the temperatures h and <2 may have and the correction for expansion of the glass and density of water in formula (2) be negligible, first, if the weighings are assumed accurate to one milligram, and, second, to one-tenth miUigram? [Solve approximately: see equation (4).]
SPECIFIC GRAVITY OF SOLIDS 65
SPECIFIC GRAVITY OF SOLIDS.— II.
Object. — To determine the specific gravity of a solid sub- stance by the apphcation of Archimedes' Principle.
Discussion. — If a substance is weighed first, in air, and, second, when suspended in a liquid in which it is insoluble, the loss in weight is, by Archimedes' Principle, equal to the weight of the liquid displaced by the substance. If the liquid be water, the weight of a volume of water equal to that of the substance is thus determined. Water is always chosen as the liquid unless the substance is soluble in it, in which case benzene, or some other organic liquid is used. The method may evidently be employed to determine the spe- cific gravity of the liquid instead of that of the solid, if the specific gravity of the latter is known.
Apparatus. — The apparatus required is a delicate analyti- cal balance provided with a table for supporting a beaker of water over one pan, some very fine wire or fibre for sus- pending the substance, and a thermometer.
Procedure. — Find the weight w of the substance dry. Next suspend it by a very fine wire of such a length that the substance will be completely immersed in a beaker of water which sets upon a little table placed over the balance pan. Let the weight of the substance when thus freely suspended in distilled water be Wi. Owing to the great damping effect of the water, the weighings must be made by noting the position where the pointer of the balance comes to rest, and not by the usual method of equal swings. The method of swings may be employed when the damping is not too great. Record the temperature of the water immediately after weighing. Finally remove the solid from the wire support- ing it and weigh the latter with its lower end immersed in the water to the same extent as during the preceding weigh- ing. Let the weight of the wire weighed in this manner be W2. Then w' = w -\- iv^ — ''^'i will be the weight of the water displaced by the substance alone, the error due to immersed wire being thus eliminated. If, on the other hand, the wire be weighed dry and not partially immersed, a cor-
56 PHYSICAL LABORATORY EXPERIMENTS
rection for the buoyant effect of the water on that portion immersed when the substance is suspended must be made, or shown to be neghgible. This correction may usually be made negligible by using a sufficiently fine suspension. Record the barometer reading, and the temperature of air in balance case.
The above method is capable of a high degree of accuracy when all details are properly attended to. The suspending fibre, or wire, should be as fine as possible in order to reduce to a minimum the error arising from capillarity at the point where the fibre leaves the surface of the liquid. A cocoon fibre has been found, when practicable, to give excellent results. The substance must hang freely in the liquid and not approach too near the sides of the beaker. If several pieces are suspended together, great care must be taken to prevent air bubbles from remaining enclosed between them. Air bubbles may usually be removed by placing the beaker and contents under the receiver of an air pump and exhausting.
Computation. — The specific gravity of the substance at f referred to water at 4° C. is, approximately,
w' '
when Dt is the density of water at t°. The only correction to be applied to the above expression is that for the reduction of the weights w and w' to vacuo. For this reduction, see formula (1), page 38.
SPECIFIC GRAVITY OF LIQUIDS.— I.
Object. — To determine the specific gravity of a liquid to 0.1 per cent by means of a specific gravity bottle.
Apparatus. — Flasks of the form shown in Fig. 14, p. 50, may be used for this purpose; when intended for liquids the necks of the flasks are usually made considerably nar- rower than when intended for solids.
Procedure. — Clean thoroughly, dry and weigh the flask empty. Fill completely with the liquid whose specific gravity
SPECIFIC GRAVITY OF LIQUIDS 57
is to be determined, or adjust the liquid by means of filter paper exactly to the reference mark. Weigh, and immedi- ately after record the temperature ^2 of the liquid. Thor- oughly rinse the flask with distilled water, and then fill with distilled water previously boiled to remove dissolved air. Adjust again, weigh, and record the temperature t^ of the water; or adjust the flask with water at the same temperature ^2 as before, in order to eliminate temperature corrections.
Computation. — Compute the specific gravity of the liquid at ^2° referred to water at 4°, applying all corrections affect- ing the result to more than 0.1 per cent. Let b = weight of flask empty;
u'l = " " " filled with water at ^1°; w^= " " " " " liquid at ^2°; Dt^=^ density of water at ti°; o- = " " air in balance case at time of weighing. The approximate specific gravity of the liquid is
^_w^ — b
Wi — b
This value is, however, in error for the same causes as those discussed under specific gravity of solids, p. 52.
Correction for temperature. — The weight of water which would have filled the flask at ^2° is (see p. 52),
w^ = (w,-b)[l + k{t,-t,)]^.
h
Hence
and
U'2 — 6
referred to water at t°, iV2 — b -D<i
w,-b i + k(t^ — t,y
referred to water at 4° C.
Reduction to vacuo. — If both the weight of the liquid and f the water are further reduced to vacuo, we have (see p. 54)
58 PHYSICAL LABORATORY EXPERIMENTS
This last reduction is usually unnecessary; for, writing the equation in the form,
iV2 — b
— ' A, + cr(l
u'o — 6,
it is seen that, if
w^ — h
2, the value of the correction
term o- (1 ^^-, — ) is only — 0.0012; hence neglecting
the correction entirely will not introduce an error greater than one part in 2000, or 0.05 per cent.
SPECIFIC GRAVITY OF LIQUIDS.— II.
Object. — To determine the specific gravity of a liquid to 0.1 per cent by means of a Sprengel-Ostwald picnometer.
Apparatus. — A much more convenient form of apparatus for accurate specific gravity determinations of liquids is the Sprengel picnometer as modified by Ostwald and shown in Fig. 15. It is con- structed so that when filled with water or with the liquid the adjustment may be readily made at the same tem- perature by suspending the whole instrument, except its capillary ends, in a bath kept at constant temperature. All tem- perature corrections are thus eliminated. The capacity varies from 10 cc. to 50 cc, according to the accuracy of the results desired. The picnometer is most readily filled (or emptied) by sucking (or blowing) through a small rubber tube connected to the end A.
SPECIFIC GRAVITY OF LIQUIDS 59
Procedure. — Thoroughly clean the picnometer with a solu- tion of potassium bichromate in sulphuric acid if necessary, rinse with distilled water, and finally with clean alcohol. Warm in a drying closet or by a drying blast, and dry by drawing through it a current of air.
When the picnometer has cooled to the room temperature, weigh it. Next fill with the liquid whose specific gravity is to be determined, and suspend the picnometer up to the horizontal ends in a water bath, the temperature of which is
carefully adjusted, by the addition of hot or cold water, to the temperature at which the gravity deter- mination is to be made, which is 20° C. The fill- ing is most readily accom- plished by drawing in the liquid at B by sucking through a small rubber pj jg tube connected to the end
A, as shown in Fig. 16. After about five minutes, bring the meniscus of the liquid to the reference mark B, by carefully touching the point A with a bit of filter paper. If the meniscus then remains per- fectly stationary at the mark B, the picnometer and its con- tents have assumed the temperature of the surrounding bath. If not, the adjustment should be repeated. A little liquid may be added without removing the picnometer from the bath by touching a drop on the end of a stirring rod to the capillary end A.
When the adjustment has been made, dry the picnometer, and weigh it. In case the room temperature is much above that of the bath, precautions must be taken to prevent the liquid from expanding beyond the ends of the picnometer while weighing. This may be prevented by cooling the liquid (not below the dew-point however), before weighing. To reduce the time of weighing to a minimum, it is advanta- geous to have the approximate weight on the balance pan beforehand. For very expansive and volatile liquids, small
60 PHYSICAL LABORATORY EXPERIMENTS
ground glass cups are sometimes placed over the ends of the picnometer to prevent loss during weighing.
In the same way as described above, find the weight of the picnometer filled with distilled water, rinsing the pic- nometer thoroughly three or four times before filling.
Computation. — Since the weight of equal volumes of liq- uid and water are here compared at the same temperature, the corrections for expansion of glass and change of density of water are eliminated. If iv is the weight of the liquid, and Wi and Dt the weight and density of the water respec- tively at t°, the corrected specific gravity of the liquid at t° referred to water at 4° is (see equation (4), page 54).
It will readily be seen that the reduction to vacuo is here negligible if the specific gravity of the liquid does not vary much from unity. The specific gravity of the liquid at 20° C. is to be referred both to water at 20° C. and to 4° C.
Problems. — 1. If the coefficient of expansion of glass is 0.000026 per degree and the density of water diminishes 0.00021 per degree at 20°, how closely should the tempera- ture of the water bath be adjusted in filling the picnometer in order that the combined error due to expansion of glass and change of density of the water may not be greater than 0.1 milligram. Assume volume of the picnometer to be 10 cubic centimeters.
2. Is it necessary in general to reduce weighings to vacuo in determining the specific gravity of liquids? Why?
THE MOHR-WESTPHAL BALANCE.
Object. — This experiment gives practice in the manipula- tion of a Mohr-Westphal balance, by means of which the specific gravity of liquids may be determined with great ease and rapidity to about 0.1 or 0.2 per cent.
Apparatus. — The principle of this balance is as follows: At the end of a balance arm, which is divided into ten parts, is suspended a thermometer in the form of a glass
THE MOHR-WESTPHAL BALANCE 61
sinker. When suspended freely in air, this is exactly bal- anced by a fixed counterpoise at the other end of the balance beam. In the usual type of balance the volume of the sinker is such that when completely immersed in distilled water at 15° C, it displaces five grams of water. Hence, if immersed in distilled water, it will require the addition of a weight A hung from the end of the beam, equal to the weight of this volume of water, in order to bring the balance again to equilibrium. The large weights A are adjusted therefore to equal the weight of water displaced by the sinker at the standard temperature chosen, — 15° C. A series of weights B, C, and D equal respectively to y^ A, iIq A, and j-J^qq A are also provided. If, therefore, the sinker is suspended in any other liquid than water whose specific gravity is to be determined, and it is found that, in order to bring the bal- ance to equilibrium, the weights A, B, C, D have to be placed on the beam at position a, b, c, d respectively, these being expressed in tenths of the length of the beam, the specific gravity will be given at once as O.ahcd. For,
weight of displaced liquid
weight of equal volume of water a . , h c d
a.bA.cA.d A
10 "^ To 10 ~*~ To 100 "^ To 1000
= O.abcd.
Procedure. — Determine the specific gravity of the same liquid which was used with the Sprengel-Ostwald picnometer. Adjust the balance by turning the screw at the base so that the sinker is exactly balanced in air. Suspend the sinker in distilled water and see that the weight A equals the weight of water displaced at 15°. Then measure the specific gravity of the solution provided, at 20° C.
Computation. — Refer the specific gravity of liquid at 20° to water at 4° C. and compare result with that obtained with the picnometer.
62
PHYSICAL LABORATORY EXPERIMENTS
THE JOLLY BALANCE.
Object. — This experiment affords practice in the manipula- tion of a Jolly Spring Balance, an experimental investigation of Hooke's Law, and illustrates a ready method of determin- ing specific gravities without the use of an equal arm balance or set of weights.
Discussion. — By Hooke's Law the elongation of a spring is directly proportional to the force applied; or if a coiled spring be suspended at one end and weighted at the other, the elongation of the spring will be proportional to the weight applied. This relation is true as long as the elastic limit of the spring is not approached. If the elongation correspond- ing to unit weight, one gram, is determined once for all, a measurement of the elongation pro- duced by any substance suspended by the spring affords all necessary data for deducing its weight. The elongation per unit weight. A", is a constant for a given spring except in so far as the elasticity of the spring varies with the tempera- ture. The constant depends, of course, on the dimensions and elasticity of each individual spring, and may be varied within very wide limits. In certain in- struments springs of extra- ordinary sensitiveness are often employed.
If the ratio of two weights, Ti'i and K'2, is desired, as in specific gravity determina- tions, and fli and aj are the respective elongations of the spring which they produce,
we
have —
W2
Wi kai «i
I
kao
«2
Fig. 17. hence, whenever relative measure-
ments only are involved, the value of the constant k need not be known.
THE JOLLY BALANCE 63
Apparatus.— The form of apparatus provided is shown in Fig. 17. A spring, S, is suspended from the top of a tube, A, which can be raised or lowered by means of an endless chain enclosed within the tube, B, and operated by the screw, C. The tube A is graduated in millimeters, and its height is accurately determined to 0.1 millimeter by means of a fixed vernier, V. From the lower end of the spring is suspended a light aluminum index I, and a double pan. The pans and their suspending wires are made as light as possible. The upper is made of aluminum, and the lower, which is immersed in a beaker of water, is of glass or mica. The index is suspended, as shown in Fig. 18, within a glass tube, on which are etched three reference lines. A reference mark is also scratched on the index itself. The tube carry- ing the index is adjustable in height as well as the support T, for carrying the beaker.
Procedure. — Lower the column A, by the screw C, as far as possible so that all weight is removed from the spring. Clamp the support T, at the lower end of B, and adjust / at such a height that the lower pan is suspended near the bottom of a beaker of distilled water placed on T. Raise the spring gradually until the index / is raised off the top of the glass support, and then level the whole apparatus by the screws at the base until the index is freely suspended at the centre of the glass cylinder. The apparatus is then ready for use.
If the spring oscillates excessively, it can be quickly brought to rest by lowering it until the index just rests on the glass support, and then very gradually raising it again until the index swings freely. The water, in which the lower pan should always be immersed, also acts as a very effective damp- ing medium to the oscillations of the freely suspended spring.
First. — To study the sensitiveness of the apparatus and to test Hooke's Law.
Raise the spring until the reference mark on the index is exactly at the height of the middle circle etched on the glass cylinder. As these reference marks are etched completely around the cylinder, the error due to parallax is easily avoided by bringing the eye to such a height that the mark on the
64 PHYSICAL LABORATORY EXPERIMENTS
index coincides with the plane of the etched circle. Record the reading of the vernier V. Make four independent seU tings and take the mean ttiq as the setting for zero load. Next place a one-gram weight in the upper pan, and raise A until the index again comes to the same reference mark as before, and read the vernier. Make four independent settings and take the mean mj. The difference mj — m© is the elongation of the spring for one gram. Determine the elongation with two, three, and four grams in the upper pan, making four settings with each.
Second. — To determine the specific gravity of a solid sub- stance.
See that the beaker is filled with fresh distilled water. Adjust the spring so that the mdex is exactly at the reference mark with no load in either pan, and call the vernier reading Oq. Place the substance whose specific gravity is to be determined in the upper pan and adjust the index again to the reference mark, calling the reading a^. Finally place the substance in the lower pan so that it is completely immersed in the water and call the reading of the vernier when the index is again brought to the reference mark a^. Be care- ful that no air bubbles adhere to the substance when immersed. The temperature, t°, of the water should be taken immediately after the experiment. Make a duplicate determination.
The specific gravity of the substance will evidently be given by the expression.
«i — «o
Dt,
where D, is the density of the water at t°.
Third. — To determine the specific gravity of a liquid.
Replace the beaker of water by a beaker containing the liquid to be investigated, and determine the difference in elongation a\ — a'2 of the spring, when the same substance previously used is placed in the upper and lower pan respect- ively. Record the temperature t°i of the liquid. The differ- ence a\ — a'2 is proportional to the loss of weight of the sub- stance in the given liquid, and hence, dividing this by the
THE JOLLY BALANCE 65
elongation proportional to the loss of weight of the same substance in water, a^ — a^, we obtain the specific gravity of the liquid at fi referred to water at f. If the specific gravity of the solid substance is known, a single determination of the elongation of the spring when the substance is weighed first in air and second when immersed in the given liquid, gives of course, all data necessary for computing the specific gravity of the latter.
Computation. — a. To ascertain whether the apparatus fol- lows Hooke's Law within the experimental error involved in the manipulation. — Compute the mean and a.d. of each series of four settings with 0, 1, 2, 3, and 4 grams load. Find the elongations A-q = nii — m^ ; k-^ = r??2 — Wj, etc., per gram for the several loads and compute their mean and a.d. This last a.d. gives the amount by which the elongation of the spring per gram is likely to differ from the mean value of the elongations for loads from one to four grams. It depends both on the error in setting the instrument and on the deviations of the elongation of the spring from strict proportionality to the weights applied under different loads, i.e., deviations from Hooke's Law.
To determine the error due to setting alone we may com- pute the indeterminate error of each value of k and take their mean, or proceed as follows :
Take the arithmetical mean of all deviations of setting, i.e.,
a.d.Q -\- a.d. I . . . -|- a.d. a — j
^—^ ^ ■ ■* — a.d.
This gives a fair estimate, from twenty settings under various conditions, of the accuracy with which the experi- menter can adjust the index to the reference mark. Hence the average deviation of any elongation computed from the difference of tw^o such settings w^ill lie between the extreme values 2 a.d. and 0 according as the deviations in the two settings happen to be of the same or opposite sign. Since, however, the sign of accidental deviations is never known and is equally likely to be plus or minus,
66 PHYSICAL LABORATOEY EXPERIMENTS
the most probable value of the deviation in the elongation will be
(See Precision of Measurements, p. 29.)
Compare this result with the average deviation of the elongations A'o, A;i, etc., and state whether the instrument follows Hooke's Law within the experimental error or not. Another method of investigating this point would be to plot values of h as ordinates and corresponding total loads as abscissae and note whether the points deviate regularly or not from a line parallel to the axis of X.
h. Compute the specific gravity of the solid, referring it to water at 4° C.
c. Compute the specific gravity of the liquid at fi, refer- ring it to water at 4° C.
BAROMETRIC MEASUREMENTS 67
BAROMETRIC MEASUREMENTS.
GENERAL DISCUSSION.
One of the quantities, the value of which is very frequently required in physical, chemical, and engineering work, is the pressure of the atmosphere at the time and place at which an observation is made. This quantity is measured by some form of barometer, and is usually spoken of as the barometric pressure. A number of different forms of barometer are in common use, the most important of which may be classed as either mercurial or aneroid barometers. Primary instru- ments are of the former type; the latter are always second- ary, i.e., they must be calibrated by a comparison with a standard instrument. Both types include forms of self- registering instruments.
By the barometric height is meant the height of a column of pure mercury at 0° C, which just balances the pressure of the atmosphere at the time and place of the observation. In order that measurements taken by different instruments and at different places may be comparable, they must be reduced to standard conditions. This involves the application to the observed height of several corrections, which will be dis- cussed below.
The standard or normal barometric pressure is defined as the pressure of a column of pure mercury 76 cm. high at 0° C. As this pressure varies with the latitude and altitude owing to the variation in gravity, it is necessary to further stipulate in what latitude and at what altitude the pressure shall be considered standard. Latitude 45° and sea level have been so chosen, under which conditions g = 980.6 ''^'/aec* The normal barometric pressure may therefore be expressed in any of the following ways: the pressure per square centi- meter of a column of pure mercury 76 cm. high at 0° C, latitude 45° and sea level, or the pressure per square centi-
68 PHYSICAL LABORATORY EXPERIMENTS
meter of 76 X 13.596 = 1033 grams weight in latitude 45° and sea level, or the pressure per square centimeter of 1033 X 980.6 = 1013200 dynes. This pressure is also called the normal atinosphere, and is very frequently used as the unit of pressure in work on the mechanics of gases.
Corrections. — The corrections which must be applied to the observed height of a barometer in order to reduce it to standard conditions are the following:
First. — Correction for temperature of ?nercury and scale.
Let h = observed height of barometer at t° C. If the scale on which the height is measured be standard at t°, or if it be of a material the coefficient of expansion of which is so small that the changes in its length from the standard are negligible for ordinary temperature changes, then the reduc- tion of the barometric height to 0° C. involves simply the computation of the contraction of the mercury column in cooling from t° to 0°. This may be calculated from the formula,
from which ho = K
hf = ho (l-\-at), 1
1 +ar
= hf (l — a t) approximately, (1)
= lit — tt iht ,
where a is the mean coefficient of expansion of mercury between 0° and t° C. The value of a from Regnault's ex- periments as reduced by Broch is a = 0.0001818 between 0° and 100° C. The correction is therefore, 0.0001818 t Ik and will be subtractive above 0° and additive below this temperature. For ordinary laboratory conditions, say / = 20* a,nd h = 760 mm., the value is about 2.76 mm. If the barometer is read only to the nearest 0.1 mm., it is usually sufficient to use the formula
0.0001818 X 760 ^ = 0.138 <, which gives the correction to two significant figures with
BAROMETRIC MEASUREMENTS 59
sufRcient accuracy for pressures in the neighborhood of 760 mm.
If however, as is usually the case, the value of the scale unit is not standard at the temperature of the measurement, the above value K must be further corrected for the expansion of the scale.
Let yS = the linear coefficient of expansion of the mate- rial of which the scale is constructed; t = observed temperature of the scale at time of measurement (assumed the same as tempera- ture of the mercury); f = temperature at which scale is standard ; hj = the value of ho reduced to standard units of length. Then
h/ = ho[l+p{t — f)].
If the scale is standard at 0°, this becomes
h/ = K(l +^t).
Combining this with (1) we have
hj = ht[l — (a — j8) i] approximately.
= ht—{a — p) tht , (2)
an expression in which the effect of the expansion of the scale and of the mercury are combined into one correction term. The value of /? varies with the nature of the material of which the scale is constructed. The values for some materials in common use are given below:
Brass 0.0000184 to 0.000019
Glass 0.0000085 to 0.0000088
Steel 0.0000245
Pine wood along grain 0.0000037
" " across " 0.000058
The scales on most barometers are of brass. For such barometers the correction for any observed height h, and temperature t, is therefore given by the expression (0.0001818-0.0000184) tht, or 0.0001G34 thf
70 PHYSICAL LABORATORY EXPERIMENTS
The value of this correction computed for temperatures ranging from 0° C. to 35.8° C. and for pressures from 440 to 795 millimeters of mercury may be found in the Smithsonian Meteorological Tables, pp. 34-56. Table IX, appendix, is taken from this source. For any temperature below 0°, the value of the correction will evidently be numerically equal but of opposite sign to that for the same interval above 0°. For barometers reading in English units, the mercury column is reduced to its height at 32° F. and the scale is referred to its length at 62° F., the temperature at which the standard yard is standard. The corrections for barome- ters with brass scales have been computed for every 0.5° F. from 0° F. to 100° F. and for pressures varying from 19 to 31.6 inches of mercury, and will be found in Table 10 of the Smithsonian Tables and in Table X of the appendix.
Second. — Correction for Capillarity.
The readings of all cistern barometers in which the cross section of the barometer tube is less than a certain amount, require a correction for the depression of the mercury column, due to capillarity. The magnitude of the depression depends on the internal diameter of the barometer tube and on the angle which the mercury surface makes with the wall of the tube. As the depression is by no means constant for a given tube, but varies with the second factor mentioned above, the height of the meniscus must be known in every case as well as the diameter of the tube. The correction is an uncertain one at best. Values for the correction, interpolated from observations of Mendelejeff and Gutkowsky, are given in Table XIV. For tubes 20 mm. in diameter the correction amounts to only 0.025 mm., and for tubes 25 mm. or more in diameter the correction becomes entirely negligible. It is therefore theoretically much better to construct barometers with large tubes at least 25 mm. in diameter, and thus eliminate this source of error, instead of attempting to cor- rect for it. Barometers so constructed are called normal barometers. Such barometers are not very portable owing to the liability of breakage arising from the large quantity of mercury contained in the tube.
BAROMETRIC MEASUREMENTS 71
Third. — Correction for the tension of mercury vapor.
This correction is very small except at high temperatures. It is negligible when an accuracy not greater than 0.1 mm. in the barometric pressure is desired. Values of the cor- rection at various temperatures are given in Table XIII, appendix.
Fourth. — Correction for variation of gravity with the lati- tude.
The pressure of a given column of mercury at any place is proportional to the force of gravity obtaining at that place. As stated above, latitude 45° and sea level are the conditions chosen as standard. The value of g in latitude 45° and at sea level is
,,, = 980.6^,
In any other latitude, </>, at sea level,
g = ^45(1 — 0.00259 cos 2 <f>).
The reduced height of the barometer must therefore be mul- tiplied by the ratio — to reduce it to latitude 45°.
Fifth. — Correction for variation of gravity with the alti- tude.
The variation of the value of g with the altitude H above sea level is given, for moderate altitudes, by the expression
9' = gd - ^) = 9(1 - 0.00000020 H),
where R the radius of the earth and H are expressed in meters. The last two corrections for variation of gravity with latitude and altitude may be combined into one formula as follows:
g' = g^^(i — 0.0026 cos 2 <^ — 0.0000002 H).
Types of Barometers.— For the procedure to be followed in filling a barometer, see Wiillner " Lehrbuch der Experiment-
72
PHYSICAL LABORATORY EXPERIMENTS
alphysik," I, p. 403. For description of numerous types of this instrument, see Deschanel's "Natural Philosophy," Part I, Chap, xvii; Ganot's "Physics," pp. 151-160. Only the types used in the laboratory are here described.
A barometer in its simplest form consists of a glass tube about 85 cm. long closed at one end, which after being com- pletely filled with pure mercury is inverted in a trough of mercury as shown in A, Fig. 19. The mercury sinks in the tube to such a height that the weight of the column of mer- cury above the level of the mercury in the trough is exactly equal to the weight of the atmosphere. Since the difference in height between the top of the mercury column and the sur- face of the mercury in the trough varies from day to day, with the variations in atmospheric pressure, some device is necessary for always measuring the height from the level of the surface in the cistern. A convenient arrangement for this purpose is a steel rod C of known length which can be moved parallel with the barometer tube by a rack and pinion and set so that its lower end just makes contact with the surface of the mercury in the cis- tern. The height of the barometer under these conditions may then be found by adding the length of the rod to the difference in height of the top of the rod and the top of the meniscus, as read on a verti- cal scale. These readings may be conveniently and accurately made with a reading telescope. A cathe- tometer may also be used for meas- uring the height. B, Fig. 19, represents a second form of barometer tube enlarged at the top in order to diminish or eliminate the effect of capillarity.
BAROMETRIC MEASUREMENTS
73
The Fortin Barometer. — This well-known and very prac- tical form of barometer is shown in Fig. 20. The barometer tube, which tapers to a small opening at its lower end, pro- jects well into a cistern, the construction of which is the characteristic feature of the in- strument. It is usually made of a cylinder of boxwood, to the top of which is fixed a cylinder of glass to permit the mercury surface being seen. The lower end of the cistern is made adjustable in height by being closed by a flexible bottom of leather, or leather lined with pure rubber. The adjustment is made by means of a screw S, which presses against a bearing attached to the bottom of the leather pouch. The glass cylinder is closed at the top by a cover through which passes the ba- rometer tube, as shown in the figure. The connection between the tube and cover is made by chamois leather, which permits the air to pass freely between the cistern and the outside atmosphere, but which at the same time prevents mer- cury from leaking out when the cistern is full. The cover is of metal, but lined with boxwood, or other non-metallic coating to prevent contact of the mercury with the metal.
From the top of the cover an ivory point, P, projects downward, the position of the lower end of which is taken as the zero point for adjusting the scale on which the height of the barometer is read. This scale is attached to the upper part of the tube, as shown in Fig. 20, and is provided with a vernier which is set by means of a rack and pinion. The whole barometer is encased in metal, except at the top and
Fig. 20.
74 PHYSICAL LABORATORY EXPERIMENTS
bottom, where it is cut away to permit the mercury surfaces being seen. It is provided with a thermometer for indicat- ing the temperature of the mercury and scale, the bulb being placed inside the metal case next to the barometer tube.
To set the barometer, the screw, S, at the bottom of the instrument is first turned down until the surface of the mercury in the cistern is well below the end of the reference point, P. The mercury is then raised slowly until the end of the ivory point just coincides with its image reflected in the surface of the mercury or until it makes a minute dimple in the mercury surface. The latter method is, in general, more accurate. This procedure always insures a rising meniscus in the barometer tube. The height of the barometer is then read by first raising the vernier above the mercury meniscus and then gradually bringing it down until the lower edge is in a plane exactly tangent with the top of the menis- cus. This is accomplished by the aid of a second plate similar to the vernier and moving with it at the back of the barom- eter tube. The lower edge of this plate is in a plane with the lower edge of the vernier and perpendicular to the tube. If the eye be brought just in line with these two edges, the line of sight will be at right angles to the tube ; hence, in setting the vernier on the meniscus, the eye should be kept in such a position that both edges simultaneously appear to be tan- gent to the highest point of the meniscus. This adjustment is very important for the correct reading of the instrument.
The two chief advantages of this type of instrument are the ease and accuracy with which the surface of the mercury in the cistern may be adjusted to the same point, and its portability. The proportions are such that when the bottom of the cistern is screwed up the mercury will fill not only the cistern, but also the barometer tube. Under these conditions the barometer may be transported in any position without danger of air entering the tube.
Kew Barometer. — This is another form of cistern barome- ter, in which the area of the cross-section of the barometer tube bears a known ratio to that of the cistern. The scale
BAROMETRIC MEASUREMENTS
75
is then so graduated that the divisions are shortened by an amount which just compensates for the change in the level of the mercury in the cistern, thus making unnecessary the adjustment of the mercury in the cistern to a reference point before each reading as in barometers of the Fortin type. One reading of the position of the mercury column gives the barometric height at once. The instrument though very convenient is less reliable than the Fortin.
Siphon Barometer. — The siphon barometer in its simplest
form is shown in A, Fig. 21. It consists of a bent tube 85
to 90 cm. long, the longer end of which is closed, and the
shorter open to the atmosphere. When completely filled
with mercury and mounted in
the position shown, the column
of mercury in the closed arm is
just balanced by the weight of
mercury in the open arm plus
the pressure of the atmosphere.
To bring the meniscus in the
upper arm vertically over that
in the lower, the tube is usually
bent as shown in B.
The height of the mercury column may be read either by means of a cathetometer or by means of a telescope and scale. In some instruments the scale is made adjustable so that its zero may be brought to the level of the mercury in the open arm; in others the scale is fixed, and the whole barometer is adjust- able in the same way to the scale; in both of these cases the B Q reading of the position of the
riK- 21. upper meniscus gives the de-
sired height. More frequently
76 PHYSICAL LABORATORY EXPERIMENTS
however, the barometer is read by two verniers of similar construction to those described above. In this case it is usual to graduate the scale only near the ends of the barometer. The zero of the scale is at the centre of the tube and the graduations numbered up and down from this point so that the difference in height of the mercury in the open and closed arms is given by the sum of the two vernier readings. These correspond to the position of the lower edge of each vernier when set tangent to the meniscus in each arm of the barometer respectively.
One of the obvious advantages of this type of barometer is the elimination of the correction for capillary depression, for if the diameter of the two arms where the meniscus stands is the same, the depression should be the same in each. To ensure this however, it is necessary to tap the barometer tube sharply before each reading.
In order to make barometers of this type portable, various devices have been introduced into their construction.^ One form, due to Gay-Lussac, is shown in B, Fig. 21. The two arms are connected by a capillary tube bent as shown in the figure. The short arm is sealed except for a capillary hole, 0. To transport the barometer without admitting air into the longer arm, the barometer is tilted until the mercury completely fills the long arm, when it is inverted as in C, the excess of mercury falling to the bottom of the short arm under the opening, 0. It cannot escape through this minute opening owing to capillarity. The longer arm, being com- pletely filled, can suffer no jar from a motion of the mer- cury, neither can air easily enter it through the capillary tube.
Standard Barometer. — One excellent form of standard barometer, constructed on the siphon principle, is shown in Fig. 22. The barometer tube proper A, at least 25 mm. in diameter, is drawn out to a tapering end which reaches nearly to the bottom of a steel cistern provided with two
iSee Wullner's Lehrbuch der Experimentalphysik, I, p. 412.
BAROMETRIC MEASUREMENTS
77
lateral arms. The tube is held air tight in this cistern by means of a rubber stop- per, which is clamped be- tween two brass plates the lower of whi(;h forms the top of the cistern. One arm of the cistern is pro- vided with a steel stop- cock, D, which in turn is connected to a reservoir of mercury, C, by means of heavy pressure tubing. This reservoir can be raised or lowered and clamped at any desired height. The other arm of the steel cis- tern ends in a right angle elbow, and communicates with the open arm, B, of the barometer which is carefully chosen of the same cross- section as the tube, A.
The whole instrument is mounted on a rigid board pivoted above and below. The mounting itself is rendered vertical by means of a plumb bob, F, and levelling screws in the base. Slots several inches long are cut in the board behind the mercury meniscus in A and B respectively, and adjust- able mirrors are provided for illuminating the meniscus from behind. A fine thermometer, T, is mounted along the barometer column to record the temperature of the mercury. Provision is made for mounting a standard meter scale between A and B in the plane of the mercury columns if it is desired to read the barometric height by means of a read- ing telescope instead of a cathetometer.
To adjust this barometer, first level the instrument by means of the levelling screws in the base. Clamp the reser- voir a little higher than the top of the meniscus in B, and
Fig. 3!J.
78
PHYSICAL LABORATORY EXPERIMENTS
carefully open the cock, D. The mercury in B will rise slowly. With the cock still open lower C until the mercury in B begins to fall, and continue to allow the mercury to flow back into C until the mercury in A stands about ten centimeters below the top of the barometer, A. Close the stop-cock. Now raise C and clamp it at such a height that the mercury will rise in both arms A and B when the stop- cock is opened. To set the barometer, open the stop-cock and allow the mercury to slowly rise a few millimeters in A and B. Then close it. This procedure insures a rising meniscus in each arm of the barometer, and if both are of equal cross-section the error due to capillarity becomes elim- inated. By choosing the cross-section of the tubes, 25 mm. or more in diameter, this error is rendered practically negli- gible in any case. The height of the barometer may then be read either by means of a cathetometer, or a telescope and scale. The latter method is to be preferred. A second and third setting should be made by opening D and allowing the mercury to take up a new position in both arms.
Aneroid Barometers. — Barometers of this type are con- structed without the use of any liquid (whence their name). Numerous forms are in common use, but all are based on the same principle. One form, known as the Richards record-
rig. 83.
ing barometer or barograph, is shown in Fig. 23. The essential part of the instrument consists of a series of flat- tened cylindrical boxes of thin metal placed on top of each other and separated by small pieces of metal attached to the centre of each. The boxes are corrugated so as to render them
BAROMETERS 79
more flexible and yielding to change of pressure. Each box is partially exhausted and hermetically sealed. An increase in the external pressure therefore tends to flatten each element. By connecting them all in a series as shown, the displace- ments due to this deformation are added and transmitted by means of a system of multiplying levers, B, C, D, to a long index arm, E. This is made to move over a scale graduated to give the barometric pressure at once in inches or milli- meters; or a pen, P, is attached to its end by means of which a record of the barometric changes is traced on a revolving drum as shown in the figure. The drum revolves by clock- work, making one revolution a week.
Aneroid barometers may be made exceedingly sensitive to slight changes of pressure ; — a variation in the pressure of the atmosphere for a change in altitude of only a few feet may be detected. They may also be made portable and no larger than an ordinary watch. Such pocket barometers are very convenient for approximate altitude determinations. The chief disadvantages of the instrument arise from the effect of changes of temperature on its indications and from its liability to get out of order. Frequent comparisons with a standard barometer are necessary if its indications are to be relied upon.
BAROMETERS.
Before performins; this experiment, the general discussioa of barometric measurements, pp. 67-79, should be carefully studied.
Object. — The object of this experiment is to familiarize the student with the construction and manipulation of vari- ous types of barometers, and with the exact reduction of barometric pressures to standard conditions. The principles involved in this reduction are of fundamental importance in all pressure measurements. Practice in the use of standard meteorological tables, and the computation of altitudes, is also afforded.
80 PHYSICAL LABORATORY EXPERIMENTS
Apparatus. — The barometers to be studied are several forms of simple cistern barometers, the Fortin barometer, the Kew barometer, the siphon barometer, the standard barome- ter, and the aneroid barometer.
Procedure. — Study the construction of each barometer and its method of use. Set and read each barometer at least twice, recording the temperature of the attached thermome- ter in each case. The standard barometer is not to be read (simply inspected), unless the instructor so dh^ects.
Computation. — Reduce the mean reading of each barome- ter to standard conditions applying all corrections which affect the result to 0.1 mm. Data on the internal diameter of the several barometer tubes will be given by an instructor. English barometers are to be reduced to metric units for comparison with the metric barometers. Tabulate all results.
Problems. — 1. Deduce the formula for the reduced height of an Enghsh barometer at t° = 32° F., the scale being of brass and standard at 62° F. Show why the cor- rection is subtractive above 28°. 5 F. and additive below this temperature (see p. 70).
2. As an exercise in the use of barometric tables for computing altitudes, the student is recommended to solve the following problem, complete directions for which will be found in the Smithsonian Meteorological Tables, 1S96, pp. xxix to xxxi. Given: —
Barometric pressure reduced to 0° C. at upper station
== 635.8 mm. Barometric pressure " " " " lower "
= 767.3 mm. Mean temperature of air column between stations
= 14°.3 C. Mean vapor pressure = S mm. Latitude = 44°.
Lower station is 27 meters above sea level, (a) Find the altitude of upper station. (6) Compute the altitude also by Babinet's formula, (p. xxxii,
Smithsonian Tables), and compare results.
METEOROLOGICAL INSTRUMENTS 81
METEOROLOGICAL INSTRUMENTS.
Object. — The object of this exercise is to familiarize the student with the principle, construction and use of the more common instruments employed in meteorological observa- tions, i.e., in determining the pressure, temperature and humidity of the atmosphere, the velocity of the wind, etc. The reduction of the observations gives practice in the use of standard Meteorological Tables.
Apparatus. — The collection of instruments includes the following : —
First. — Barometers and barographs. See preceding exper- iment on Barometric Measurements.
Second. — Thermometers.
1. Six's maximum and minimum thermometer.
2. Rutherford's maximum thermometer, U. S. Weather Bureau type.
3. Rutherford's minimum thermometer, U. S. Weather Bureau type.
4. Richard's recording thermograph. Third. — Hygrometric apparatus.
1. Negretti and Zambra wet and dry bulb thermometer or psychrometer.
2. Lambrecht poly meter.
3. Hygrodeik.
Fourth. — Wind recording apparatus.
1. Cup anemometer for measuring velocity of wind.
2. Vane anemometer for measuring velocity of draughts in flues.
Procedure. — First. — Record the reading of the Fortin barometer and its attached thermometer, as these data are necessary in later computations. A study of the various types of barometers is to be taken as a separate exercise.
Second. — Inspect the several self-registering thermonu>- ters and see that the principle of their construction is thor- oughly understood. A description will be found in the ref- erence books placed with the apparatus.
82 PHYSICAL LABORATORY EXPERIMENTS
Third. — Determination of the hygrometric conditions of the atmosphere.
This includes the determination of (a) the pressure of the water vapor present in the atmosphere, (6) the dew point, (c) the humidity or relative humidity, and (d) the weight of vapor in the unit of volume.
The quantity of water present in the atmosphere may be determined in a number of ways, the most direct and accu- rate of which is to pass a known volume of air through dry- ing tubes and weighing the amount of water absorbed in the tubes. This is the fundamental method by which are ob- tained the data on which psychrometric tables are based. As this procedure is very tedious other simpler though less direct methods are usually employed. Of these the most reliable is that based on the indications of a wet and dry bulb thermometer.
The principle of this instrument is based upon the fact that water at the temperature of the air evaporates less rapidly the greater the amount of water vapor present; also that, in evaporating, the water absorbs heat from bodies with which it is in contact. Hence the drier the air the more rapid will be the evaporation at the bulb of the wet thermometer, and consequently the greater the difference in reading between it and the dry bulb thermometer. In absolutely saturated air both thermometers will read the same. The simplest rela- tion which has been found to hold for the difference in tem- perature of a wet and dry bulb thermometer and the pressure of the vapor in the air is
f = f,-AB{t-t,), in which
t = temperature of the air (dry bulb) ;
ti = temperature of wet bulb thermometer;
/ = pressure of aqueous vapor in the air at i°;
fi = pressure of saturated vapor at t°',
B = barometric pressure ;
A = a, constant.
The value of A depends to a certain extent on the form and size of the wet bulb thermometer and velocity of ventilation.
METEOROLOGICAL INSTRUMENTS 83
It has been found, however, that if the air in the neighbor- hood of the wet bulb moves at a rate of about ten feet or more per second, a constant maximum depression is reached, and the effect of the form of instrument is eliminated. Observations should therefore be made in a current of air whose velocity is such that a maximum depression is obtained, or the thermometer should be swung so as to give it a cor- responding ventilation, as is done in the U. S. Weather Bureau observations.
In the apparatus as arranged, ventilation is secured by- means of a small fan driven by an electric motor. Read the two thermometers under this condition taking care not to warm either during the observation.
From these observations the following data expressing the condition of the atmosphere are to be looked out in the hygrometrical tables of the Smithsonian Collection (see pp. xxxv to xlii, last edition, 1896). These tables are com- puted from direct measurements of the humidity of the air and the corresponding differences in readings of a wet and dry bulb thermometer, taken under the conditions described above.
1st. Temperature. — Given by dry bulb thermometer.
2d. Vapor Pressure. — To find the pressure / of aqueous vapor turn to Table 40, p. 134, and with t^ as argument find /i, the pressure of saturated vapor at the temperature of the wet bulb thermometer. This gives the first term in the formula for /. The value of the second term is found in Table 41, by taking the barometric pressure B, in inches, and the observed temperature difference t — fj as arguments,
/ = table 40 — table 41.
3d. Dew-point. — The "dew-point," i.e., the temperature at which the aqueous vapor in the atmosphere would begin to be precipitated, or in other words, the temperature at which the actual vapor present would be just sufficient to saturate the air, is evidently the value of t in Table 40 correspond- ing to the value of / just found.
4th. The relative humidity, or simply humidity, of the air
84 PHYSICAL LABORATORY EXPERIMENTS
is the ratio of the quantity of vapor actually present to that which would be present if the air were saturated at the tem- perature of observation; i.e., relative humidity = -^, where
F is the value for saturated vapor at t°, the temperature of the dry thermometer. The value is to be found in Table 40. Humidity is usually expressed in percentages. Thus, relative humidity = 75 per cent, or 0.75 denotes that the air contains three-fourths of the amount of moisture required to saturate
it at the temperature of observation. This ratio, !=, may
r
also be defined as the ratio of the maximum vapor pressure
at the dew-point d, to that at the observed temperature t,
and corresponding to this definition, values of y, are given
r
in Table 42 for arguments t — d (horizontal column) and i (vertical column). Corresponding tables are also given for metric units and centigrade degrees.
5th. Weight of Vapor. — The amount of moisture in the air is also sometimes expressed as the actual weight of water which the air contains as vapor in the unit of volume; e.g., as grains per cubic foot or grams per cubic meter. To find the weight of vapor present, turn to Table 38, which contains the weight of saturated vapor in air at different temperatures, and look out the value corresponding to the dew-point d, since at this temperature the vapor present would just satu- rate the air.
The hygrodeik is an empirically graduated instrument based on the indications of a wet and dry bulb thermometer. The hygrometric data are plotted on a chart in such a way that if an index is adjusted to the reading of the wet bulb thermometer the corresponding humidity, dew-point, etc., may be read off the chart at once. The readings of this instrument (Lloyd's) will be found to agree very well with the results interpolated from the tables. It is a very con- venient instrument for ordinary work.
The polymcter (Lambrecht's) is another secondar}^ instru- ment for indicating hygrometric data, i.e., relative humidity, dew-point and vapor pressure. It is based on the principle
METEOROLOGICAL INSTRUMENTS 85
of the hair hygrometer. The indications of this instrument are, under ordinary conditions, much less rehable than those of the preceding instruments. For details of construction and method of reading the instrument, see descriptive pamphlet with the apparatus.
The readings of all three instruments are to be recorded and results tabulated for comparison.
Anemometers. — ^These instruments are designed for meas- uring the velocity of the wind or of air currents in general.
The "cup anemometer" used by the U. S. Signal Service, is that most commonly used for measuring wind velocities. It consists of four arms at right angles to each other at the ends of which are attached hemispherical cups. A current of air striking these cups meets with less resistance from the convex than from the concave side, and hence causes a rota- tion in the direction of least resistance. The number of rotations is recorded by a suitable mechanism. Wind veloci- ties are usually expressed in miles per hour.
The vane anemometer is a much more delicate instrument and is adapted for measurements of such air currents as occur in the ventilating system of buildings. It consists of a very delicately pivoted wheel carrying a number of inclined alu- minum vanes; when placed in a current of air the vanes revolve with a velocity depending on the velocity of the air current. The wheel transmits its motion to a series of dials which record the number of revolutions. The recording mechanism may be thrown in or out of gear at will, while the vanes are revolving, by a small lever on the side of the instru- ment. These instruments require careful calibration in a current of air of known velocity. The corrections to be applied vary greatly with the velocity owing to friction of the bearings, and the accurate calibration of one of these instruments is not a simple matter. For small velocities an approximate correction can be obtained by walking with the anemometer at a constant rate for a known distance, a hun- dred feet or more, in a room where the air is at rest, and noting the time.
The instruments provided are to be inspected carefully. The velocity of the incoming or outgoing air in one of the
86
PHYSICAL LABORATORY EXPERLMENTS
flues of the laboratory is to be measured with the vane anemometer.
m\\ ^ iii'ic
BOYLE'S LAW.— I.
Object. — ^This experiment is designed to illustrate the re- lation between the pressure and volume of a gas, dry air, at room temperature, for pressure less than one atmosphere. The precision aimed at is about 0.3 per cent. The compu- tation illustrates the graphical method of deducing the em- pirical law connecting two quantities, a series of values for which has been determined by experiment.
Apparatus. — The apparatus consists of a heavy walled glass cistern A, nearly filled with mercury, into which a glass tube B, sealed at the top, can be raised or lowered at will. This tube is about one meter in length and is graduated from the top downwards in centimeters and millimeters. The exact vol- ume in cubic centimeters corresponding to any division is determined once for all by calibrating the tube with known volumes of mercury before the apparatus is set up. The corrections thus obtained, which are to be added to the linear readings in order to convert them into the corresponding volumes in cubic centimeters, are given on the calibration curve provided with the apparatus. The tube B is initially com- pletely filled with pure mercury and inverted in the cistern. The apparatus then con- stitutes a simple barometer, the height of the mercury column in B above the level of the mercury in the cistern being just bal- anced by the atmospheric pressure. In the apparatus as set up a little dry air has been allowed to enter the vacuum above the mer- Fi«. »4. cury column, so that when the tube B is
raised to the stop C, the air is under a pres- sure of about half an atmosphere. The tube is held in any
boyle's law — I. 87
desired position by a clamp D. To facilitate reading the height of the mercury column above the variable height of the surface of the mercury in the cistern, a steel rod, S, is provided, which may be raised or lowered parallel with the tube B, by a rack and pinion. The cistern is covered with a glass plate to prevent dust settling on the mercury surface.
Procedure. — Before beginning the experiment proper, re- cord the height of one of the barometers (the siphon), and also the reading of the thermometer attached to the apparatus. Raise the tube B up to the stop C, and read the position of the top of the mercury column. In setting the tube for this and all subsequent measurements, the precaution should always be taken to tap the tube or to set with a rising men- iscus so as to insure that the meniscus is convex. The reading is best taken by clasping a strip of black paper around the manometer and bringing it down until its lower edge is in a plane tangent to the top of the meniscus. Next adjust the steel rod, S, so that the lower end coincides with its image in the mercury of the cistern. Record the position of the upper end of the rod S on the manometer. Care must be taken in making this observation to bring the eye exactly on a level with the end of the pointer, otherwise a serious error due to parallax will be introduced. This error may be avoided if the eye be moved until the top of the pointer appears to coincide with its own reflection in the mercury in the manometer. Repeat these observations at seven or eight different heights of the tube, taking them over as wide a range as possible. In setting the tube great care should be taken not to warm the upper portion containing the air, by the hand or otherwise. The tube is best handled with a handkerchief at the lower portion containing the mercury, as a slight change in temperature of this produces little effect on the resulting pressure or volume of the gas. The success of the experiment depends on maintaining the gas at as nearly a constant temperature throughout all meas- urements as possible. Sudden changes of volume are also to be avoided, as they produce adiabatic heating or cooling
IFor Method of Calibration, see Stewart and Gee's Practical Physica, vol. i, p. 109.
88 PHYSICAL LABORATORY EXPERIMENTS
of the gas. Finally remove the pointer S and measure its length on a steel scale. The barometer and temperature of the surroundings should also be again recorded.
Computation. — The pressure on the gas at any position of the tube is evidently equal to the atmospheric pressure expressed in centimeters of mercury, diminished by the height of the mercury column in the tube B above the level of the mercury in the cistern, expressed in the same units. This is easily computed from the observed data. The correspond- ing volume occupied by the gas is obtained by adding to the observed reading, the proper correction interpolated from the calibration plot accompanying the apparatus.
Tabulate the values of the pressure, p, expressed in centime- ters of mercury and of the corresponding volumes, v, in cubic centimeters; in a third column tabulate also the values of
-, for reasons explained below. (Look out these values in
V
a table of reciprocals, or use a slide rule.) To determine the form of the function between the quantities p and v, suppos- ing it unknown, first, make a direct plot with values of p as ordinates, and v as abscissae. The points will be found to lie along a curved line which by inspection, from its sym- metry with respect to the axes, might be suspected of being an hyperbola referred to its asymptotes as axes. The equa- tion of such a curve is xy = constant, or y = c (-). This
X
equation may be transformed into the equation of a straight line by changing the variable x to x' = — . Making this sub- stitution we obtain y = ex', the equation of a straight line passing through the origin and making an angle whose tan- gent is c, with the axis of x. If, therefore, a second plot be constructed on the same sheet with the same values oi p = y
as ordinates, and with the reciprocals of the volumes — = — = x' as abscissa:^, the points should lie along a straight line
X
passing through the origin if the gas follows Boyle's Law.
boyle's law — I. 89
This will be found to be the case if the data have been accurately taken. Draw the best representative line, and determine its equation. Follow directions for reciprocal functions discussed under Graphical Methods, p. 50.
The relation may also be tested by computing the product pv for each pair of values of p and v and discussing the results analytically or graphically. Plotting the values of pv as ordinates and values of p as abscissa?, the Une should be parallel to the axis of abscissae for a perfect gas.
Precision Discussion. — What would be the percentage change produced iu the product pv hy a change of tempera- ture of 1°C.?
pv =z poVo (1+ a 0,
= ^^^^(273 + 1),
= RT.
Let X =pv — RT;
then -^-^'
or the fractional change in x is equal to the fractional change in the absolute temperature. If the temperature at the time of the experiment is 18° C, T = 273 + 18 = 291°, and
|
5r = 1°, therefore, |
|
|
A 1 T~291 |
|
|
or 100 A X |
= 100^0 3 291 |
0.34 per cent.
Hence a change in temperature of 1° at 18° will introduce a change of 0.34 per cent in the value of the product pv.
Questions and Problems — 1. How many significant fig- ures should be retained in — if four figures are properly
V
retained in v ?
2. What is the percentage deviation in pv due to the maxi- mum observed change in temperature of the surroundings during the experiment?
90
PHYSICAL LABORATORY EXPERLMENTS
BOYLE'S LAW.— II.
Object. — In this experiment the relation between the pres- sure and volume of a gas maintained at constant temperature is to be investigated at pressures greater than one atmos- phere. A nearly perfect gas, dry air, and some imperfect gas, as ammonia or carbon dioxide, are those chosen for in- vestigation. Practice is afforded in use of an open mano- meter for measuring pressures.
Apparatus. — The gases to be investigated are permanently sealed in the glass tubes A and B which are tightly screwed into a rectangular block of iron, soft stoppers being used for packing. An open manometer, C, is likewise fastened into
a third opening in the block, all three tubes being freely connected with each other, as shown in the figure, and with a steel stop-cock, S, which connects the iron base with a reservoir of mercury, R. This is connected to a force pump by means of heavy pressure tubing, T.
The glass tubes A and B are graduated from the top downward in linear centime- ters and millimeters. The height of the mercury in the manometer is read off on a boxwood scale also graduated in millimeters. To facilitate reading the mercury column, the scale is grooved to fit the manometer tube. Accompa- nying the apparatus is a cali- bration curve for each of the tubes A, B, by means of which the volume of the gas in each tube can be obtained at once for any height of the mercury column.
Fig. 25.
boyle's law — II. 91
Procedure.— Open the stop-cock S and slowly pump the mercury from the reservoir R into the system of tubes A, B, C, until it nearly reaches the top of the manometer C. Close the stop-cock S. This operation should be performed slowly, to prevent a sudden compression of the gases in A and B which will raise their temperature. The success of the experiment depends largely on maintaining the tempera- ture of the gases as nearly constant as possible throughout. For this reason great care must be observed not to warm the portions of the tubes A and B containing the gas with the hands, breath, or otherwise during the experiment. Some temperature changes will undoubtedly occur in the gases owing to the change in volume to which they are subjected. It is therefore necessary to wait after each setting, particu- larly after the first compression, until the gas has assumed the temperature of the surrounding air, as indicated by a con- stant reading of the mercury columns in A and B. Usually it will be found that the gas has been heated during the first compression, and the pressure will be too great. As the gas then cools to the room temperature, the pressure will fall slightly as indicated by the rise of the mercury in the closed tubes. Final readings should only be taken after the mer- cury columns have assumed a constant position. When this is the case, read the height of the mercury columns in A, B, and C. Then open S and allow the mercury to flow back into the reservoir until the pressure has fallen in the open manometer about 30 cm. Close S and read A, B, and C as before. Repeat this procedure for not less than six different pressures, extending over as great a range as the apparatus will permit. Record also the reading of the barometer at the beginning and end of the experiment and of the ther- mometer suspended by the apparatus. Before leaving the experiment record also the "zero points" (see below) of the three tubes and obtain from the calibration curve the cor- rections necessary for reducing the observed readings to volumes.
Computation. — To com.pute the pressure p. The pressure which is exerted on the gases in the tubes A and B is evi-
92 PHYSICAL LABORATORY EXPERIMENTS
dently equal to the difference in height of the mercury columns in them and in C respectively, plus the pressure of the atmosphere exerted on the open end of C. The scales of A, B, and C, are all graduated from the top downwards, and if they were so adjusted that corresponding divisions were exactly in the same horizontal line, the difference in height of the columns would be given directly by the differ- ence of the respective readings of the open and closed tubes. This condition is impractical and, indeed, unnecessary to fulfill as the result is readily obtained if the corresponding readings on A, B, and C at any point of the tubes are known. This may be obtained, once for all, by means of an accu- rately adjusted spirit level or a cathetometer, or better (and the way actually employed), by reading the height of the mercury columns in all three tubes when freely open to the air. Under these circumstances the mercury stands, of course, at exactly the same height in each tube, all being of the same internal diameter to eliminate effects of capil- larity. Let the zero readings be Qq, 6q, Cq for the tubes A, B, C respectively. Then the difference in height of the mercury columns in A and C, for example, corresponding to readings a, c respectively, will be
(ro — c) — (ao — a).
The actual pressure on the gas in A will be therefore,
p = H -^ (cq — c) — {Qq — a),
where H is the height of the barometer expressed in the same units as a and c. As
// -h Cq — (io=^ k, a constant,
the expression p z= k -\-a — c
is more convenient to use when several pressures are to be computed.
To find the volume v. — If the tube containing the gas were of uniform cross-section, and tlic zero graduation properly
boyle's law — II. 93
placed, the graduations would give the volumes directly in units depending on the volume corresponding to unit length of the tube. Neither of these conditions is exactly fulfilled so that a calibration of the tube is necessary. This is made once for all before setting up the apparatus. The cross-sec- tion of the particular tubes used is so chosen that the vol- ume corresponding to one linear centimeter is approxi- mately equal to one cubic centimeter. In the calibration curves accompanying the apparatus, the corrections to be added to the observed readings, expressed in centimeters, are plotted as ordinates, and the corresponding readings as abscissae.
Compute and tabulate the value of p and of v for each set of observations, and also their product pv. For gases which strictly follow Boyle's Law, this product should be constant, and a plot made with values of pv as ordinates, and p as abscissae should be a straight line parallel to OX. Inspection of the computed values of pv, will show that the law is approximately followed in the case of air, but quite widely deviated from in the case of ammonia. To bring out the deviations more clearly, plot values of pv as ordinates and p as abscissae. The curves best representing the data will be found to be straight lines having negative tangents, but as the quantities of air and ammonia in the two tubes are not the same, a comparison of the tangents does not give a true measure of the relative deviations of the gases from a perfect gas. Find from the plot the equation of the line represent- ing the data on ammonia gas.
Problems. — 1. If the deviation in p is Sp 0.5 mm., and the deviation in v is Sv = 0.02 cc, what is the resultant deviation in the product pv for the first observation of p and of V taken on air?
2. What is the resultant percentage deviation in this case?
3. How much will a rise of temperature of 2° C. affect the product pv at 20° C? Compare Precision Discussion, p.89.
4. If the air is suddenly compressed adiabatically to one- third its volume, its initial temperature being 20° C, how much will its temperature be raised?
94 PHYSICAL LABORATORY EXPERLMENTS
Note: For adiabatic compression or expansion,
where k = 1.41 for permanent gases, and T = absolute temperature.
LAW OF THE PENDULUM.
Object. — The object of this simple experiment is to deter- mine by the graphical method the law connecting the time of vibration and the length of a simple pendulum, and to determine the value of g. The experiment gives practice in the use of a stop watch in timing vibrations, and the com- putation affords an excellent precision discussion.
Apparatus. — The apparatus consists of a spherical ball suspended by a very fine steel wire, the length of which can be varied by means of a windlass to which its upper end is attached. When the pendulum has been adjusted to the length desired, the wire is tightly held by a rigidly supported brass clamp. The length of the pendulum is then the distance from the lower edge of the support to the centre of oscillation of the ball. A stop watch registering 0.2 seconds is used for determining the time of vibration, and scales graduated in millimeters and calipers are pro- vided for measuring the length of the pendulum and the diameter of the ball respectively.
Procedure. — Determine the time of vibration of the pendu- lum for five or six different lengths, varying from about one-half a meter to a meter and a half. This may be done with sufficient precision by counting the number of single vibrations for a period of not less than three minutes, using a stop watch reading to 0.2 seconds. Note that in counting the vibrations, the swing at the instant the watch is started is to be counted as zero, not as one. The swings should be timed when the pendulum is swinging with its maximum velocity, i.e., as it swings through the lowest part of its arc. This time can be estimated with much greater precision ^han when the pendulum is at the end of the arc, i.e., moving
LAW OF THE PENDULUM 95
with zero velocity. Care should also be taken that the pen- dulum is set swinging in a plane, and not in an ellipse, and through an arc of not more than 10° or 15°. The length of the pendulum is to be measured with a millimeter scale, from the lower edge of the jaws clamping the wire, to the top of the ball. The diameter of the ball is to be measured with calipers. The measurements should in general be made as near as possible to tenths of a millimeter, except in the case of lengths of a meter or over, where the nearest half millimeter will suffice, for reasons shown under the precision discussion.
Computation. — Compute and tabulate the values of the time of vibration t, corresponding to each length I, of the pendulum. The length of the pendulum is to be computed by the expression,
2 r2
where h is the length of the wire from the point of support to the top of the ball, and r is the radius of the ball. The last term of the formula is the distance of the centre of oscillation below the centre of the ball. The computation is to be carried at each stage of the work to the nearest tenth of a millimeter.
First. — To deduce the law connecting t and I.
Suppose it is desired to deduce the functional relationship existing between the values of t and I, i.e., the form of the function t = f{l). This can be done most conveniently by means of a graphical solution. If a "direct plot," ^ be made with values of t as ordinates and I as abscissse, the data will be found to lie along a curved line suggesting an exponential relationship of the general form t = W, when k and n are constants. If such an equation holds, the values of the constants k and n can always be determined by the "logarithmic method."^ Construct therefore a logarithmic plot, by plotting logarithms of t and I as ordinates and ab- scissae respectively' on rectangular coordinate paper, and de-
1 See Graphical Methods, p. 41.
'Ibid., p. 52.
• Express I in meters.
96 PHYSICAL LABORATORY EXPERIMENTS
termine the equation of the resulting best representative line. From it deduce the desired constants in the equation
t = /x-r.
Second. — To find the value of g from the plotted data. The value of g may be computed at once from the value
of k, as found above, if we assume the formula t = tt\ -
^ g
for the pendulum to be known. For this may be written t = TT=l^, which by comparison with the general equation,
t = kr, shows that n = ^ and k = -7==, or g = ^.
V g k^
Compute g in this manner.
Precision Discussion. — How precise should t and I be measured in order that the directly computed value of g shall be reliable to 0.2 per cent?
Solving by equal effects we have at once for the equation
- = - = ^:.- - = -4= X 0.002 = 0.0014. 9 9 \AY g V 2
But ^ = -' and ^ = 2 '-'^
9 I 9 i
Hence — = 0.0014 or I must be measured to 0.14 per cent, and -' = 0.0007 or t " " " " 0.07 per cent
If the pendulum is a seconds pendulum, i.e., one meter in length, the time of one vibration must be known to 0.07 per cent of one second or 0.0007 second; and the computed length I must be known to 0.14 per cent of 1000 millime- ters or 1.4 mm. For a pendulum only one-quarter of a meter in length and beating therefore half-seconds, the required precision will be 0.14 per cent of 250 mm. or 0.35 mm. in the length, and 0.07 per cent of 0.5 second or 0.00035 second in the time. This precision can be attained without great difficulty with the apparatus pro\idod.
Questions and Problems. — 1. About what do you esti- mate the precision of the constants k and n as determined from the plot to be?
THE METHOD OF COINCIDENCES 97
2. Do you think your data would warrant further discus- sion b}' means of a "residual plot"? (See Notes on Graphi- cal Methods, p. 60). Why?
2 7'2
3. Is the correction term - — -; — negligible in the
5 h + r
computation of any values of I? Why?
THE METHOD OF COINCIDENCES.
GENERAL DISCUSSION.
The two following experiments involve a more accurate determination of the time of vibration of a pendulum than in the preceding experiment, and the method used, known as the "Method of Coincidences," is of great importance and wide application. The following discussion should therefore be carefully studied before performing either of the experi- ments in question.
We have given two pendulums A and P, of approximately the same time of swing, that of A being unknown, while that of P is known. It is desired to find the rate of A in terms of P.
Suppose first, that both pendulums A and P have exactly the same time of vibration. If both are started swinging in the same phase, they will continue in phase, both passing through the lowest point of their respective arcs simultane- ously. If at these points contact devices, for example mer- cury cups, are placed by which an electric circuit is closed when both pendulums make contact simultaneously, the time of passage will be indicated by the stroke of a bell, sounder, or other recording device placed in the cu-cuit. If the pendulums are started swinging in exactly opposite phase, they will also continuously pass through the lowest part of their arcs at the same instant. If they start in any- thing except the same or opposite phase, they will never come into phase or coincidence, and the circuit will never be closed. Suppose however, that the time of vibration of A is slightly greater than that of P and they are started swing-
98 PHYSICAL LABORATORY EXPERIMENTS
ing in the same phase. A will immediately begin to lag behind P, and hence both pendulums will, after the first swing, no longer pass through their respective mercury cups at the same instant; the circuit will therefore not be closed, and no signal will be given. The amount by which A lags behind P will increase continually at each swing until it is just one single vibration behind P, i.e., until it comes into coincidence but in opposite phase with P, when again both pendulums will simultaneously close the circuit at the mer- cury cups, and the event will be indicated by the recording device. If the number of vibrations of the standard pendu- lum P, elapsing between these two successive "coincidences" be n, then while P has made n vibrations, A will have made n — 1 vibrations; consequently if P is a seconds pendu-
71
lum, the time of vibration of A will be t seconds.
Conversely if the pendulum A has a shorter time of vibration than P, A will make n -f 1 vibrations while P makes n
vibrations, and the rate of A will be — r-^ seconds.
If the rate of the standard pendulum P is not exactly known, it must be determined by a comparison with a standard clock, chronometer, or watch. If its rate, i.e., its time of vibration in true seconds is found to be r, then the rate of ^, as determined above, must be multiplied by r.
Precision Discussion. — From the preceding discussion, it follows that the time of vibration determined by the method of coincidences is given by the expression t = ,
where n is the time elapsing between successive coinciden- ces. Let us find what deviation A is introduced in t by a deviation (^n in n. Differentiating the expression for t with respect to n, we have
_ n ± 1 — n
Hence ^1 = ^ ^ ^,
t n±l n
THE PHYSICAL PENDULUM
that is, the fractional or percentage deviation in t is only
the ^ part of that in n. Thus if the deviation meas- n± 1
ure of the time of coincidence is one per cent, the resulting value of t will be uncertain by only of one per cent.
It is evident, therefore, that the method gains in precision the greater the time of coincidence, i.e., the smaller the difference in the rates of the two beating pendulums.
To illustrate: Suppose n = 100 seconds and ('n =1
second; then —^ = — , i.e., n is uncertain by one per n 100
cent, while t, on the other hand, is uncertain by only of one per cent.
99
100 ±1
If n = 50 seconds and ^n = l second, the percentage deviation in n will be two per cent, but the percentage
deviation in the resulting value of t will be only
50 ± 1
of two per cent or 0.04 per cent. The method is thus seen
to be one of extreme precision.
THE PHYSICAL PENDULUM.
Object.— This experiment is designed to measure the value of g to about 0.1 per cent. It illustrates the "method of coincidences" in measuring time, and also a method of meas- uring lengths greater than the length of the measuring scale with the aid of two microscopes. The experiment consists of two parts : first, the determination of the time of vibration of the pendulum; and second, the measurement of its length.
I, — DETERMINATION OF TIME OF VIBRATION.
Apparatus. — The pendulum consists of a brass sphere A fixed near the lower end of a brass rod, which is supported on a steel knife edge, A^. Above the knife edge the rod carries a smaller brass sphere, B. The whole constitutes of course a metronome pendulum; but the position of the sphere
100
PHYSICAL LABORATORY EXPERIMENTS
A is so adjusted once for all, that the time of vibration of the pendulum as a whole, is equal to that of an equivalent simple
pendulum, consisting of the ball A sus- pended by a weight- less thread at the same distance below the knife edge. This is accomplished as follows: The ball A being removed from the rod, the time of vibration of the latter is accurately deter- mined, the smaller ball B being so adjusted above the knife edge that the time of vibra- tion is approximately one second. The ball A is then replaced on the rod and adjusted by trial until a posi- tion is found such that the time of vibra- tion is the same as with A removed. A is fixed in this posi- tion and the whole pendulum may then be regarded as a simple pendulum, the length of which is the distance of the centre of oscillation of the ball A below the knife edge. This ad- justment has been carefully made and is not to be disturbed. The rod of the pendulum terminates in a platinum point which swings through a mercury globule Mi. By the side of the pendulum is mounted a clock C beating seconds, the pendulum of which is also provided with a platinum point which swings through an adjustable mercury cup il/2. The two pendulums are connected in series with a battery D and an electric bell S (or sounder), so that when c(Hitact is made at Mj and M^ simultaneously, the circuit is closed and the bell (or sounder) responds.
Fig. 36.
THE PHYSICAL PENDULUM 101
Procedure. — With P at rest and making contact through M2, adjust the mercury contact M^ until the bell strikes regularly at every swing of A. The adjustment should be so made that the platinum point just cuts through the top of the mercury drop. Next, with A at rest, adjust M^ so that when P is set swinging, contact is made regularly at each vibration. Finally set both pendulums swinging through an arc of about 10°, taking care that they swing in a plane and not an ellipse. The pendulum A is best set swinging in a plane by drawing it to one side with a thread and burning the thread. The clock pendulum can be set swinging with sufficient accuracy with the hand. If the adjustments have been well made the bell or sounder should strike not more than two consecutive times when the pendu- lums come into coincidence. The reason why more than one coincidence is often observed is that one pendulum gains or loses on the other so little in one or even two vibrations, that both pendulums are able to touch some portion of the mer- cury contacts, which are of sensible width, at the same instant for several consecutive beats. When the pendulums are swinging properly, note hy the clock the time of coincidence, recording the time of first and last coincidence in case the bell strikes more than once. The mean is to be taken as the true time of coincidence. The time of about nine consecutive coincidences should be recorded. Then start the pendulums swinging again and take a second similar series.
To complete the time observations it remains only to deter- mine the rate of the clock which should not be assumed to beat seconds without further test. The rate may be obtained with sufficient accuracy by making a fifteen minute compari- son with a watch. Wind up the clock and with watch in hand (which must be provided with a seconds hand), note the exact time in seconds of the watch corresponding to the time in seconds of the clock. This is best done by keeping the clock time in mind by listening to the tick, while observ- ing at the same time the seconds hand of the watch. Record the second of the clock and watch when both tick together. No further attention need then be paid to the clock for about
102 PHYSICAL LABORATORY EXPERIMENTS
fifteen minutes. At the end of this time note as before the time of the clock and keep time mentally by counting its beats while looking at the seconds hand of the watch. At the end of exactly fifteen minutes by the clock note the second and fraction of a second, if possible, of the watch. If, for example, it is found that fifteen minutes by the clock is equal to fifteen minutes plus one and one-half seconds by the watch, the rate of the clock is
901.5 , ^^,^ r = — — = 1.0017 seconds.
II. — MEASUREMENT OF LENGTH OF PENDULUM.
Apparatus. — The apparatus required for measuring the length of the pendulum consists of a horizontal frame for holding the pendulum, a steel scale graduated in millimeters
Fig. 27.
which can be adjusted in the plane of the knife edge of the pendulum, and two reading microscopes.
Procedure. — Remove the pendulum from its support and place it in the frame, taking care in carrying it not to bend the pendulum rod. Carefully focus the microscopes A and B, so that the intersection of the cross hairs comes exactly on the knife edge, and on the point where the rod enters the top of the ball respectively. The length it is desired to measure is then the distance between the cross wires of the two microscopes; the greatest care must therefore be taken not to displace them after once focused. Next move the frame to one side so that the scale is seen under the micro-
THE PHYSICAL PENDULUM 103
scopes instead of the pendulum. It will, in general, be found to be too short to be seen under both at the same time. If it is not in good focus, raise or lower it by means of the adjusting screws S, at ends of the frame until the divi- sions can be clearly seen without parallax in either micro- scope.
When the scale is focused, move it along until some division a, exactly coincides with the cross hairs in microscope A. Then, without moving the scale, move microscope A towards B, and set it so that the intersection of the cross hairs comes on some division h. It has been moved towards B through a distance a — b. Now leaving A fixed in position, slide the scale along towards B until a third division, c, coincides with the cross hairs in A, and finally read the position of the cross hairs in B on the scale to tenths of a millimeter, and call this reading d. The length of the pendulum from the knife edge to the top of the ball is then
h=: (a — h)-\-(c — d).
Repeat the measurement of h and take the mean. The radius r of the ball is to be found from the mean of several measurements of its diameter, taken with calipers. All measurements should be made to tenths of a millimeter.
Computation. — The distance of the knife edge to the centre of oscillation of the pendulum is to be computed from the above measurements by the formula,
2 r2
/ = /i+r +
5/i + r
Compute the mean time of coincidence and its percentage deviation. Calculate the time of vibration t, of the pendulum, its percentage deviation and its numerical deviation in sec- onds. (See Precision Discussion, p. 98.) With this value of t (corrected for rate of clock) and the value of I, com- pute g.
Questions and Problems. — 1 . How closely should t and I be measured if it is desired to obtain the value of g to 0.1 per cent ?
104 PHYSICAL LABORATORY EXPERLMENTS
2. What is the percentage deviation of your computed value of t? What is its deviation in seconds?
3. Assuming (5;= 0.2 mm. and (^t = the value com- puted in Problem 2, find the resultant deviation in the value of g and also its percentage deviation.
METRONOME PENDULUM.
The discussion of the "Method of Coincidences," p. 97, should be read before performing this experiment.
Object. — This experiment affords practice in determining the time of vibration of a pendulum by the method of coinci- dences, the coincidences being observed by a reflected beam of light, in the use of a telescope and scale for measuring vertical distances, and in the use of the vernier caliper. The data serve to verify the formula for the time of vibration of a metronome pendulum.
Apparatus. — The pendulum consists of two brass spheres, A and B, Fig. 28, the smaller, A, being fixed near the upper end of a solid brass rod, and the larger, B, near the lower end. The pendulum swings on a steel knife edge C, adjusted at the centre of gravity of the rod. Just above the knife edge a small plane mirror M is attached to the rod. By the side of the pendulum a steel scale not shown in the figure is suspentled for measuring its length. The apparatus for observing the time of vibration of the pendulum is shown in Fig. 29. T is a reading telescope provided with cross hairs at the focus of the eye-piece. It is adjustable to any height and inclination. The apparatus shown above the telescope is a device for sending a flash of light to the mirror on the pendulum once every second. The light from any brilliant source, e.g., an incandescent lamp, is reflected by a mirror N through the lens L by means of which it is focused on a horizontal slit Si. Behind S^ is placed a second horizontal slit S^, attached to the end of a lever connected with the armature of an electromagnet. The slits are so con- structed and adjusted that each time the armature is attracted
METRONOME PENDULUM
105
to the magnet, the slit S2 uncovers the slit Si and allows a flash of light to pass to the pendulum beyond. This is made to occur once every second by connecting the electromagnet in series with the pendulum of a standard clock beating seconds, so that the circuit is made once a second. The flash is observed by focusing the telescope on the slit S^, as
i A
i
I
%i
clM
Flff. 38.
Fig. 39.
seen reflected in the mirror M of the pendulum. If the pendulum is at rest and the magnet set in operation, a flash will be seen in the telescope once every second. If the pendulum is swinging, the flash will occur only when the pendulum and the clock come into "coincidence."
Procedure. — Adjustment. — Set up the telescope about one meter from the pendulum and clamp the slit a little above, and the telescope T a little below the height of the mirror M. Focus the eye-piece carefully on the cross hairs. Then
106 PHYSICAL LABORATORY EXPERLMENTS
with the pendulum at rest, turn the telescope towards the mirror M and focus on the slit S as seen in the mirror. This is most readily done by holding a lighted match or candle in front of S and adjusting the slit and the telescope until the former is clearly seen. It is then easy to complete the adjustment so that the image of the slit is seen on the horizontal cross hair of the eye-piece. When the telescope is thus focused, close the circuit through the electromagnet, and adjust the mirror A^ and the lens L until a bright flash of light is seen in the telescope at each beat of the clock. The apparatus is then in adjustment.
Measurement of time of coincidence. — Set the pendulum swinging through a very small arc, and look for the flash in the eye-piece of the telescope, when the clock and pendu- lum come into coincidence. Several flashes will be seen al- ternately above and below the central cross wire but always approaching it until the point of coincidence is reached. The time when the flash occurs on the cross hair is to be taken as the true time of coincidence. The time elapsing between successive coincidences is easily counted by the clicks of the armature of the electromagnet. Record the time of about nine consecutive coincidences, then start a new series. The rate of the clock may be assumed zero.
Measurement of the dimensions of the pendulum. — It is necessary to measure the distance of each ball above and below the knife edge C, the diameter of each ball, and the length and diameter of the rod. Suspend a steel scale beside the pendulum in the plane of the knife edge. Raise the slit apparatus out of the way to the top of the telescope stand. Level the telescope and focus it on the top of the rod. The telescope should be placed at such a distance from the pendulum that both rod and scale appear in the field of view at the same time. Adjust until the horizontal cross hair is coincident with the top of the rod and read its height on the scale to 0.1 millimeter by estimation. In the same way take readings of the height of the top and bottom of both balls and of the knife edge. Repeat the measurements if time permits.
Measure the diameter of the rod with the vernier calipers
METRONOME PENDULUM 107
provided. The masses of the various parts of the pendulum are given with the apparatus.
Computation. — First. — Compute the true time of vibration of the metronome pendulum from the data on coincidences, as explained on p. 98.
Second. — Compute from the dimensions and mass of the pendulum the length of the equivalent simple pendulum having the same time of vibration. The formula for this reduction is
sum of moments of inertia of moving parts about C sum of statical moments about C
Let li = distance of centre of gravity of A above C; l^= " " " " " " 5 below 0; nil = mass of A; ma = " " B; m= " "rod; I = half the length of the rod; r = radius of the rod. Then the value of I becomes
^ p m^l^ + rn^l.^ -\-m{--\-~)
~~ mill + ^2^2
The statical moment of the rod which ordinarily enters in the denominator is here zero as the knife edge C passes through the centre of gravity of the rod. For a demonstra- tion of above formula, see Lanza's Applied Mechanics.
With the value of I computed by this formula, compute the corresponding time of vibration of the equivalent simple pendulum assuming g = 980.4. Compare with the value experimentally found, and state the percentage deviation between them.
Question. — 1. State all sources of error which you rec- ognize in this experiment.
108
PHYSICAL LABORATORY EXPERIMENTS
LAW OF FREELY FALLING BODIES.
TUNING-FORK CHRONOGRAPH.
Object. — The object of this experiment is to give practice in the measurement of small intervals of time (a second or less) by means of the tuning-fork chronograph. The use of this instrument is illustrated by the determination of g, the acceleration due to gravity, by a freely falling body, i.e., by measuring the time required for a body to fall through a given distance. The discussion of the results gives practice in the use of the logarithmic method of plotting, and incidentally the experiment illustrates the method of recording the occurrence of an event by an electric spark, and the method of estimating tenths of an harmonic vibration.
Apparatus.— The general arrangement of the apparatus is shown in Fig. 30. A is an electro- magnet placed on the wall about sixteen feet above the floor. This when magnetized holds up a steel ball, which is released on breaking the cir- cuit by a knife switch S. The ball in its descent BUPH falls through a trap B, which may be adjusted at
any desired C distance be-
low A, the distance AB being read off on a fixed box- wood scale graduated in millimeters. The trap B is so constructed (see below), that a second circuit is broken the instant the ball passes through. The apparatus for recording the interval of time elapsing between the breaking of the circuit at A and at B, consists
r=L
LAW OF frp:ely falling bodies
109
first, of an induction coil C, the prinicary coil of which is con- nected in series first with A, and immediately afterward with B. When the circuit through A or 5 is broken, the current induced in the secondary of the coil causes a spark to pass across its terminals. These sparks are recorded in the man- ner described below on a revolving drum D. Since the sparks occur practically simultaneously with the releasing of the ball and the opening of the trap, a measurement of the time interval between them gives the time desired. This method of recording two events electrically has the great advantage that the simultaneously occurring secondary event, the sparking, can be made to occur at any place convenient for measuring or recording, while the primary event itself may take place at any remote distance.
The interval elapsing between two consecutive sparks may be easily and accurately recorded as follows: A cylindri- cal metallic drum is so mounted on a horizontal screw, that on turning the latter, the drum is given a combiiled rotary and lateral motion.
The drum is covered with a sheet of thin paper which is evenly coated with lampblack. One terminal of the sec- ondary of the induction coil is connected through the bear-
rig. 31.
ings to the cylinder, the other terminal is connected to a metallic style resting on the paper. A spark, on passing between the style and the drum, punctures the paper and "splashes" the lampblack at the point of contact of the style, in the form of a white dot. If the drum be rotated 'under
110
PHYSICAL LABORATORY EXPERIMENTS
the style, a straight line will be traced on the lampblack, and sparks or events will be recorded as dots on a smooth line. If the drum were rotated uniformly at a known rate, the time between any two events could be determined. This condition would be very difficult to realize, but may be got around by causing a tuning-fork of known pitch to simulta- neously record its vibrations on the drum. This may con- veniently be done by fixing the style (a piece of very thin spring), to one prong of a tuning-fork, as in Fig. 31. The
fork being now con- nected to one terminal of the coil and set into vigorous vibration, and the drum rotated, a sin- usoidal curve will be traced on the drum, and the events marked by the sparks will now be recorded on a sine curve, each wave of which rep- resents a known interval of time. The time be- tween two events is found by counting the number of vibrations and fractions of a vibration of the fork between the corresponding dots. The result is independent of the velocity of rotation of the drum, a greater velocity serving merely to draw out the waves to a greater extent, but in no way altering their number. It is convenient to have the fork driven electrically by a small electromagnet placed between the prongs and run on an independent battery circuit, as shown in Fig. 31. The amplitude of vibration may thus be maintained constant and vigorous for any length of time.
The trap B (Fig. 30), consists of a flat strip of iron A (Fig. 32), hinged at one end and held up horizontally in the path of the falling ball by a strong permanent magnet B. The current passes across from B through A to C, so that the instant the ball falls upon A the circuit is broken. The electrical connections will be easily seen from Fig. 30.
Fig. 32.
LAW OF FREELY FALLING BODIES 111
When the circuit through the electromagnet is closed, the current passes from the positive terminal of the storage bat- tery to 1, a, A, a', C, back to the negative terminal. On throwing the jack-knife switch from 1 to 2, the circuit is broken, but instantly afterward closed again through the trap B, the current then passing from the positive terminal to 2, h, up the metallic rail / to B, across the trap to the other rail r" down to h' , a', C, and so back to the negative terminal.
Procedure. — Preparation of lampblack cylinder. — Fasten a sheet of the special chronograph paper tightly over the drum. See that the edges of the paper overlap in such a manner that on rotating the drum the style passes from the upper to the under, otherwise it is likely to catch and spoil the record. Hold a piece of burning camphor in the spoon provided close under the cylinder, and revolve the cylinder back and forth over the smoky flame until uniformly covered with a thin coating of soot. The deposit shoidd not he too thick to secure the best results. An even brown deposit is much better than a heavy black one, as the spark records are much better defined on the former, and it is, moreover, much easier to "fix," as explained below.
Preliminary trial. — First: Close the battery circuit and switch /S on 1; place the ball at A; see that B is closed and placed at some convenient distance below A. Rest the point of the style lightly against the drum, and while rotating the latter slowly, quickly pull open the switch S. Inspection will show that this is so constructed that the one act of pulling it open, instantly after closes the circuit through B. Two sparks should leave their records upon the drum. If they do not, look over the connections carefully, and repeat until this result is obtained.
Second. — Close the auxiliary battery circuit through the tuning-fork, and adjust the contact so that on striking the fork it continues to vibrate vigorously. Adjust it so that the style rests lightly on and at right angles to the drum, and rotate the latter so as to determine the speed at which the vibrations are drawn out into a nearly normal sinusoid.
112 PHYSICAL LABORATORY EXPERIMENTS
Third. — Repeat the first adjustment with fork vibrating. If the two sparks are clearly recorded on the sine wave, the apparatus is ready for use.
Experiment proper. — Determine the time it takes the ball to fall through distances about 1, 1^, 2, 3, and 4 meters, making three determinations of the time for each distance. It is best to locate with a pencil the position of the two sparks immediately after each experiment, marking the first 1 and the second 2. To facilitate the subsequent estimation of tenths of a vibration, a straight line should be traced through the centre of the sinusoid with the tuning-fork at rest. When the cylinder has become covered with records, they may be "fixed" with a fixatif sprayed from an atomizer, and the paper removed for future reduction, or they may be "reduced" at once.
It is interesting to use several balls of different diameters in this experiment to verify Galileo's experiment on falling bodies.
Measure the diameter of the ball with calipers.
Record data given with the apparatus on the position of the electromagnet A and the true rate of the fork. This has been accurately rated while being driven electrically under the conditions of the experiment against a standard fork.
Computation. — The distance s, through which the ball falls, is the difference in height as read off on the vertical scale, of the lower end of the core of the electromagnet and the upper surface of the trap, minus the diameter of the ball. The height of the former has been accurately determined once for all, and is given on the apparatus.
Reduction of time. — The interval of time elapsing between the two events may be determined to one vibration of the fork by counting the number of complete vibrations between the two recorded points. In general, the two events do not occur in the same phase of the vibration, so that to obtain the accuracy of which the method is capable, it is necessary to estimate fractions of a vibration. As the motion of the fork is simple harmonic, equal distances on the sinusoid do not correspond to equal differences of time. The estimation
LAW OF FREELY FALLING BODIES
113
or measurement of tenths of a vibration must therefore be made as follows: —
Fig. 33.
Suppose ah represents the amplitude and actual path of vibration of the point of the style, o being its position of rest. It vibrates with simple harmonic motion in this line, and traces on the drum revolving at right angles to it the sinu- soidal curve a a' a".
If the fork be considered to start vibrating from its extreme position a, its position at any time t will be found by project- ing on ah, the position of an imaginary point revolving in the
circle ach with a uniform angular velocity ^ , where T is the
time of one single vibration of the fork. At times
lOT
etc., the imaginary point and the style will be
2 TT 3 TT
lof' Tot'
at points 1, 2, 3, etc., on the circle ach and line ab respect- ively, while the corresponding points on the curve traced by the style will be the dots indicated. Thus it is evident that equal intervals of time mark off different distances on the sine curve, and that the estimation of tenths of a vibration in the ordinary manner would lead to entirely false results. The estimation of