EXCHANGE 
 
H!l 18 I'M' 
 
 THE MEASUREMENT OF THE COEFFICIENT 
 
 OF VISCOSITY BY MEANS OF THE 
 FORCED VIBRATIONS OF A SPHERE 
 
 BY 
 
 GEORGE FRANCIS McEWEN 
 
 A.B., Stanford University, 1908 
 
 A THESIS 
 
 Submitted in Partial Fulfillment of the Requirements for the Degree of 
 
 Doctor of Philosophy in Physics at 
 The Leland Stanford Junior University, 1911 
 
 mss OP 
 
 THE NEW ERA PR'KTtNO COMPANY 
 LAftCASTCR. PA 
 
THE MEASUREMENT OF THE COEFFICIENT 
 
 OF VISCOSITY BY MEANS OF THE 
 FORCED VIBRATIONS OF A SPHERE 
 
 BY 
 
 GEORGE FRANCIS McEWEN 
 
 A.B., Stanford University, 1908 
 
 A THESIS 
 
 Submitted in Partial Fulfillment of the Requirements for the Degree of 
 
 Doctor of Philosophy in Physics at 
 The Leland Stanford Junior University, 1911 
 
 PRCSS OF 
 
 THE NEW ERA PP'NTINO COMPANY 
 LANCASTER. PA 
 
 IQTI 
 

 
(Reprinted from the PHYSICAL REVIEW, Vol. XXXIII., No. 6, December, 1911.1 
 
 THE MEASUREMENT OF THE FRICTIONAL FORCE EX- 
 ERTED ON A SPHERE BY A VISCOUS FLUID, WHEN THE 
 CENTER OF THE SPHERE PERFORMS SMALL PERIODIC 
 OSCILLATIONS ALONG A STRAIGHT LINE. 
 
 BY GEORGE F. McEwEN. 
 
 ^HE purpose of the following investigation was to devise a method 
 by which Stokes's law of the frictional resistance of a fluid to 
 the motion of a sphere whose center performs small periodic oscilla- 
 tions could be applied to the measurement of the coefficient of viscosity 
 of fluids. The two main objects were: first, to obtain as close an agree- 
 ment as possible between the actual working conditions and those de- 
 manded by theory; second, to devise a process of measuring the force 
 acting on the sphere, even for fluids having a very large coefficient of 
 viscosity. 
 
 The contents of this paper fall under the following five heads: 
 
 I. The effect of the internal friction of fluids on the motion of pendu- 
 lums, from Sir G. G. Stokes's Math, and Phys. Papers, Cambridge, 1880 
 and 1901, Vols. I. and III. 
 
 II. An account of the method adopted in the present investigation 
 to overcome the difficulties mentioned by Stokes, and to more nearly 
 realize in the experimental work the ideal condition assumed in the 
 theory from which Stokes's law was deduced. 
 
 III. Experimental tests of the present method. 
 
 IV. Suggestions for future research. 
 
 V. Summary of the paper. 
 
 I. THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS ON THE 
 
 MOTION OF PENDULUMS. 
 I. Observations on the Motion of Pendulums. 
 
 An account of the experiments made by Bessel, Baily, Dubuat, and 
 Sabine, and the theoretical results obtained by Poisson, Challis and 
 Plana is given in Stokes's Math, and Phys. Papers, Vol. III., pp. 1-7. 
 
 The effect of the surrounding fluid on the time of vibration of a pendu 
 lum was computed by Poisson, Challis, Green, and Plana from the 
 hydrodynamical theory of a frictionless fluid. A fair agreement with 
 the observations was found in some cases, but in many cases, especially 
 
493 GEORGE F. McEWEN. [VOL. XXXIII. 
 
 where the dimensions were small, the theory failed entirely to account 
 for the experimental results. 
 
 Because of this failure, Stokes was led to apply the equations of 
 motion 1 of a viscous fluid to pendulum problems. 
 
 2. Stokes' s Deduction of the Law of Resistance. 2 
 
 In 1850 Stokes completed the solution of the following problem: 
 The center of a sphere performs small periodic oscillations along a 
 straight line; the sphere having a motion of translation only; it is re- 
 quired to determine from the ordinary hydrodynamic equations of 
 motion of a viscous fluid, the motion of the surrounding fluid, and the 
 force exerted on the sphere. 
 
 He assumed the velocities to be so small that their squares could be 
 neglected, that there was no slipping of the fluid along the surface of 
 the solid in contact with it. and that the amplitude of vibration of the 
 sphere was very small and remained constant. 
 His result for the force is 
 
 p A d " y R dy 
 
 F - A dt*- B dt> 
 
 where y is the displacement and t is the time, and the coefficients A and 
 B have the following values : 
 
 where Mi = the mass of the fluid displaced by the sphere, 
 r = the radius of the sphere, 
 
 a = , where T is the period of oscillation, X = ^1 , , and 
 
 fji f = coefficient of viscosity -5- density. 
 
 From the same assumptions and equations, the same result has been 
 obtained by a different method. 3 
 
 For a cylinder oscillating in a direction perpendicular to its axis, 
 Stokes deduced the following expression : 
 
 where F = the force acting upon unit length of the cylinder, m<l the 
 
 1 Stokes's Math, and Phys. Papers, Vol. I., pp. 75-105. 
 2 Stokes's Math, and Phys. Papers, Vol. III., pp. 11-36. 
 8 Lamb's Hydrodynamics, edition 3, pp. 583-584. 
 
No. 6.] MEASUREMENTS OF FRICTION A L FORCE. 494 
 
 mass of the fluid displaced by unit length of the cylinder, and K and KJ' 
 are constants depending on the radius of the cylinder, the period of 
 oscillation and /x'. 
 
 No simple expressions for K 2 ' and K*" were found that covered all 
 cases, and they are best determined by special series and tables which 
 are given by Stokes in his Math, and Phys. Papers, Vol. III., pp. 47-54. 
 In the same volume he gives a discussion of the conditions upon which 
 the theory from which the above expression was deduced depends. 
 
 In the case of the cylinder as well as the sphere, the first coefficient 
 shows that the fluid has the same effect as an increase of the inertia of 
 the system, and the second coefficient shows that the fluid opposes the 
 motion of the system by a force proportional to the velocity. 
 
 From observations on the time of vibration the quantity A can be 
 computed, and B can be computed from observations on the arc. Both 
 A and B can be calculated from theory with the aid of Stokes's equations. 1 
 3. The Application of Stokes's Law to the Pendulum Observations of Bess el, 
 
 Baily and Dubuat. 2 
 
 By choosing a constant value of ju' Stokes calculated from his theory 
 the periods of a variety of pendulums swinging in air, and the agreement 
 with the values observed by Bessel and Baily was very satisfactory. 
 Only a few observations on the arc of vibration were available, and these 
 were only approximate, in these cases he calculated the decrement of 
 the arc of vibration from his theory, and found as good an agreement 
 with the experiments as could be expected considering the character of 
 the observations. Though most of the observations made on pendulums 
 oscillating in water were also in fair agreement with his theory, there 
 were a number of discrepancies, which he attributed to the experimental 
 methods used. 
 
 When a pendulum oscillates in water, or a more viscous liquid, the 
 arc of oscillation rapidly decreases; this diminution forms, in fact, the 
 greatest difficulty in experiments of this kind. This difficulty, which 
 made it impossible to test his theory or to determine the coefficient of 
 viscosity of water or of more viscous liquids by means of a vibrating 
 body, suggested the present investigation. 
 
 II. THE METHOD ADOPTED IN THE PRESENT INVESTIGATION FOR 
 
 MEASURING THE FORCE ACTING ON AN OSCILLATING SPHERE. 
 I. Description of the Apparatus. 
 A light rigid bar 3 FED is secured to a knife-edge at , which rests 
 
 1 Stokes's Math, and Phys. Papers, Vol. III., pp. 1-141. 
 Stokes's Math, and Phys. Papers, Vol. III., pp. 76-140. 
 8 See Fig. i for a diagram of the apparatus. 
 
495 
 
 GEORGE F. McEWEN. 
 
 [VOL. XXXIII. 
 
 upon a fixed horizontal support. A second bar MN, perpendicular to 
 the first one, carries two movable weights Ws and PF" 4 ; the upper one can 
 be moved to or from E by turning it about the screw EN, and a scale 
 fastened to MN is provided for recording the distance of Wz from E. 
 The lower weight W* can be slid along EM and clamped in any desired 
 
 position. At Fand D (points in 
 line with JE) two scale-pans W\ 
 and Wz are attached by means 
 of very thin strips of steel. A 
 second light rigid bar ABC is 
 attached by means of similar 
 strips of steel at C and A , to the 
 pan Wz, and to a support above 
 A , which can be given a vertical 
 periodic motion of small ampli- 
 tude. This support does not 
 touch the apparatus, except at 
 A . The sphere is suspended in 
 the fluid by means of a fine wire, 
 as shown; the wire being at- 
 tached by a thin steel strip, to 
 a small support B, which can be 
 fastened to the bar AC, in any 
 position between A and C, but 
 is always in line with A and C. 
 
 By adding weights to the 
 scale-pans, the system can be 
 brought to a state of equilibrium 
 when FD is horizontal. By ad- 
 justing the weights W* and W^ 
 the natural period of oscillation 
 can be varied, as this adjustment changes the height of the center of 
 gravity of the system rigidly attached to the knife-edge. 
 
 The oscillations will, from the construction of the apparatus, cause 
 the sphere to move in a vertical line, and if the support of A is fixed, 
 these oscillations will subside because of the friction. But if the support 
 of A is given a vertical periodic motion, the system will be set in forced 
 vibration, and the amplitude, phase and period of this forced vibration 
 will depend upon the amplitude, phase and period of the motion of the 
 support, for any given adjustment of the vibrating system. 
 
 * h 
 
 Fig. i. 
 
No. 6.] MEASUREMENTS OF FRICTIONAL FORCE. 496 
 
 2. The Theory of the Forced Vibrations of the Above System. 
 
 Explanation of the Symbols: 
 
 M" equals the mass rigidly connected to the knife-edge E. 
 
 I c " equals the moment of inertia of M" about C\. 
 
 IE" equals the moment of inertia of M" about E. 
 
 Ci is the center of mass of M" '. 
 
 ECi equals /. 
 
 m s ' equals the mass of the lower bar. 
 
 7 3 ' equals the moment of inertia of m 3 f about its center of mass. 
 
 3> 3 equals the displacement of the center of mass of m-l from the position 
 of equilibrium. 
 
 Mi equals the mass of the sphere and its suspending wire. 
 
 6 equals the angular displacement of FD from its equilibrium position 
 which is assumed to be horizontal. 
 
 Mi equals the mass of the fluid displaced by the sphere. 
 
 g equals the acceleration of gravity 
 
 2 
 
 5 = s + s 3 ; h = -- s 2 . 
 
 do equals the angular displacement of the heavy pendulum used to 
 maintain the vibrations. 
 
 /i0 equals yo, the linear displacement of the point A. 
 
 |8 equals the change in torque due to the elasticity of one suspending 
 strip when bent through unit angle. 
 
 &\y equals the change in the force on the wire holding the sphere, due 
 to the elasticity of the surface film of the liquid, and to the varying length 
 of the wire immersed. 
 
 The variable force exerted on the sphere by the fluid equals 
 
 Denote the variable force exerted on the wire by the fluid by: 
 
 < equals the angular displacement of the lower bar from the equilibrium 
 position. 
 
 d<f> 
 kz ~r = the torque on the lower bar due to the friction of the air 
 
 and the suspending wires. 
 
 ki -,- = the torque on the upper bar due to the friction of the air, 
 the suspending wires, and the knife-edge. 
 
497 GEORGE F. McEWEN. [VOL. XXXIII. 
 
 Fg equals the horizontal force at E acting on the knife-edge. 
 F K equals the vertical force at E acting on the knife-edge. 
 F A equals the vertical force at A acting on the lower bar. 
 
 F B equals the vertical force at B acting on the lower bar. 
 F c equals the vertical force at C acting on the lower bar. 
 
 F D equals the vertical force at D acting on the upper bar. 
 
 F equals the vertical force at F acting on the upper p*. 
 
 m s equals the mass of each scale-pan and contents, and wires between 
 C and D. 
 
 m equals the mass of each scale pan and contents, and wires attached 
 at F. 
 
 The meaning of the other symbols used is indicated on the diagram 
 of the apparatus. 
 
 Fundamental Dynamical Relations: 
 
 Mi'g -(A+A') 2 -(B + B') 
 
 -B 
 
 ~ = - F B + F A + F c - m, f g t 
 
 Io"^ = - Fj>Si + FS* - F E 'L cos 6 - F E L sin 6 - k^ - 2/30 
 + FL sin 6 + /y sin 0, 
 
 M" ^ = K" I (cos 0) (^ )V (sin 0) ^ J / = F E - F - ^ - M"g, 
 
 c , . - ^ . , , de 
 
 01 1 r c 
 
 {?2>3 
 (cos0)^-(sin 
 
 f r / do \ 2 d*d i 1 
 
 /(sin e I M"l [ (cos 0) ( d - ) + (sin 0) ^ - J + M"g | 
 
No. 6.] MEASUREMENTS OF FRICTIONAL FORCE. 498 
 
 Fundamental Kinemalical Relations: 
 x and y are the coordinates of C\, 
 
 x = I sin 6, y I cos 9, 
 
 dx d9 dy ^0 
 
 dt =l(c s6} dt< dt = l(sm ^dt' 
 
 d$ 
 
 o fl dy*_ o^ <Py< 
 " - S ' 9 ' dt - ~ St dt' df 
 
 y = 
 
 9 <? 7 
 
 s lS2 di*~ S3ll W 
 
 y _ if ~<_, <h\ 
 d? "~ 2 \ '* 2 "'' dp)' 
 
 d<i> 
 
 dt ~ 
 
 d? ' 6l 
 
 The above dynamical and kinematical relations can be so combined 
 as to give the following ordinary differential equation : 
 
499 GEORGE F. McEWEN. [VOL. XXXIII. 
 
 in which the coefficients are constants and have the following values: 
 P = [/*" + m*S? + w 3 Si 2 + Sl ^ (M, + A + A') + S ~m, f + ^'~], 
 
 Assume 
 
 jo = ce~ a ' sin at = lido, 
 
 and substitute in the second member of the differential equation; the 
 result will be : 
 
 c s 
 
 where 
 
 P " = (P 2 _ [ a s _ o2]p" _ ap') f and P ' = o(P' - 
 
 Assume that the following ordinary equation is a particular integral 
 of the differential equation : 
 
 6 = Atf~ at sin (at - 61) + B,e~ at cos (at - 2 ), 
 substitute in the equation and solve for the quantities: 
 
 Ai, 0i, jBi, and 2 - 
 
 The following values of the constants will make the assumed form for 
 0, a solution: 
 
 P 2 + P[ 2 - a 2 ] - Pi + (P\a - 2aaP) tan t ' 
 
 P 2 + P[a 2 - a 2 ] - aPi + (Pia - 2ao:P) tan 2 ' 
 tan 0i = - 
 
No. 6.] MEASUREMENTS OF FRICTION A L FORCE. 5OO 
 
 To obtain the complete integral of the equation, the complementary 
 function, Aae~ Plt/zp sin (&/ + 3 ) would have to be added to the above 
 value of 0. A$ and 3 are arbitrary constants, depending for their values 
 on how the system is started, and 
 
 = 
 
 2P 
 
 In the present method of measurement, no observations are taken 
 until the complementary function has a negligible value, so only the 
 first or particular integral will be retained. 
 
 In practical applications, the ratio of the maximum value of to the 
 maximum value of 0o is determined by experiment. The relation of the 
 coefficient B to that ratio, which is required for determining B will now 
 be derived. The following equation for 0o is assumed: 
 
 0o = r e~ at sin at. 
 l\ 
 
 Then the expression for can be reduced to the form 
 
 = A 2 e~ at sin (at - 0i + 3 ) 
 where 
 
 + B i 2 , and tan 3 = - 
 
 A l P"' 
 
 Let t\ be the time at which has a maximum value, and k be the time 
 at which 0o first reaches its maximum value after t\. Let k ti = A/. 
 
 By taking the first derivative of and 0o, the maximum values, 0' 
 and 0o' given below are obtained : 
 
 0' = A 2 e 
 
 Dividing the first equation by the second gives 
 
 - 
 
 9J 
 
 This ratio is obtained experimentally by first observing the maximum 
 value of 0, then observing the next maximum value of 0o, and dividing 
 the first by the second. Substituting the value of A and B in the expres- 
 sion for A gives 
 
 2 + (A") 2 
 
 a 2 ] 
 
5<DI . GEORGE F. McEWEN. [VOL. XXXIII. 
 
 and 
 
 v/(P 2 + P[<* 2 - a 2 ] - aP l ) 2 ~+~(JP l a^2aaP^ ' 
 
 This expression shows that R' is independent of the amplitude of the 
 vibration, but is a function of the constant coefficients of one of the 
 preceding differential equations. 
 
 3. The Deduction of a Practical Working Formula from the Above Theory. 
 It appears that P 2 is the constant by which 6 is multiplied to obtain 
 that part of the torque which opposes the displacement of the system, 
 but does not depend on the friction. In the expression 
 
 / is the distance above E of the center of mass of the system rigidly 
 attached to the knife-edge. As will be seen by referring to the expression 
 for R f , for the values of the constants in the expression for R', the quan- 
 tity / appears only in the expression for P 2 . Therefore, by shifting the 
 weight Wz, I and consequently M"gl can be changed without affecting the 
 remainder of the expression for R'. 1 
 
 Let 
 
 *-< '#.. _, 2 + (Po") 2 
 
 f P[a 2 - a 2 ] - aPO 2 H 
 
 now if R is measured for three different values of M"gl, there will be 
 three equations from which PI can be computed by eliminating the 
 other unknown quantities. 
 
 Suppose M"gl is changed by raising or lowering the weight WB, a 
 known amount; the amount due to one complete turn, for example. 
 Denote this change by */&. Then 
 
 where / is the distance of the center of the mass of M" from E, / is the 
 distance of the center of mass of (M" W$) from , and k is the distance 
 of the center of mass of W* from E. Let the change in / 3 be A/ 3 , and 
 denote the corresponding change in / by A/, then / will remain the same 
 as it depends only on that part of the mass whose position with reference 
 
 1 Changing I in this way would produce a known small change in P, which can be easily 
 compensated for by changing the weights in the scale-pans Wi and Wz. It will be assumed 
 that this is done, and therefore P will be treated as a constant. 
 
No. 6.] MEASUREMENTS OF FRICTION AL FORCE. 502 
 
 to E is fixed. Therefore, 
 
 M"(l + A/) = (M" - Wi)l + Wz(h + A/ 3 ), 
 
 and 
 
 In the equation for R, denote the numerator by v/ 23> denote 
 (aPi-2aP) 2 by zi, denote (P 2 + P[ 2 - a 2 ] - aPi) by H^ZZ - z', 1 and let 
 
 i - = Wm Then 
 
 A / .. . 
 
 (P 2 + P[<* 2 - fl 2 ] - c 
 and the equation for R takes the following simplified form : 
 
 2 z 8 
 
 Now by varying n, the number of complete turns of W s , R n will vary 
 but 22, z 3 , and z\ will remain constant, and are positive, w is a con- 
 stant, positive or negative, and depends on the position of PF 3 when n is 
 assumed to be zero. 
 
 The maximum value of R, denoted by R w is : 
 
 R w = 
 
 therefore R w is a function of z\ and z 3 only, and corresponds to the special 
 case when there is resonance. The above equation involving R n can be 
 written in the following forms: 
 
 (n - w) = WF - I = 
 
 -71 
 
 (n - 
 
 where V = . 
 
 22 
 
 An inspection of the above equations shows that if R n and R w are both 
 multiplied by an arbitrary constant, the roots of the equation will be 
 
 1 Assume that A/j corresponds to one turn of Ws, and denote by n the number of turns 
 from a given position near the top of the screw. Then n and P increase as Wi is lowered. 
 
503 GEORGE F. McEWEN. [VOL. XXXIII. 
 
 unaltered. Therefore the ratios used need only be proportional to the 
 true values of the ratios. Also, if (A) has a constant amplitude, only 
 the readings for 6 need to be taken. 
 
 From three observations, three correspondeing values of n and R n 
 can be obtained, and the two quantities R w and w, which will be the same 
 in all cases can be eliminated, thus giving the value of ^ V. 
 
 Now, 
 
 V^ = y/~V<S~zz = aPi - 2aaP, 
 and therefore 
 
 also 
 
 and 
 
 where K is a positive constant, and depends only on s, $1, s*, a, P and 
 the friction not due to the sphere. Therefore, if two different spheres 
 of radii r\ and r z are used, 
 
 and 
 
 where Ki K\ is practically zero. Therefore, subtracting the first from 
 the second gives 
 
 From this equation, the difference between 5 2 and BI can be found, 
 since all the quantities in the second member can be measured. From 
 the theory given by Stokes, this difference has the following value : 
 
 - n) ( 
 
 ft -I- 
 
 and from this equation /x the coefficient of viscosity can be computed. 
 But the vessel containing the liquid must be large enough, compared 
 to the sphere used so that the assumption on which the equations were 
 
No. 6.[ MEASUREMENTS OF FRICTION A L FORCE. 504 
 
 derived are justified, or a correction for the effect of the containing vessel 
 must be made. 
 
 Let Bi and B 2 ' be the values of BI and Bz corresponding to a vessel 
 of infinite radius. Let Xi and X 2 be coefficients depending on a, \L , and 
 the ratio of the radius of the sphere to the containing vessel, so that 
 
 \iBi = BI, 
 and 
 
 \2-O2 = B%, 
 
 then 
 
 Let Xi" and X2 /; correspond to the cases where the radius of the vessel is 
 R" and R f respectively; the same sphere being used. Then 
 
 If this result is zero, then the radius is large enough, and no correction 
 is necessary. 
 
 III. EXPERIMENTAL RESULTS. 
 
 A gravity pendulum consisting of a wrought iron rod 1.5 cm. in diam- 
 eter, and 1 80 cm. in length; and two cylindrical weights, each being of 
 5 kg. mass, was mounted on a frame supported by two concrete piers. 
 By adjusting the position of the weights, any value of the period, from 
 about 2 to 25 seconds could be secured. By means of a mirror attached 
 to the knife-edge, and a vertical scale and horizontal telescope supported 
 on one of the piers, 115 cm. from the mirror, measurements of the angular 
 displacements were made. The scale was divided into mm., and readings 
 were taken to o.i mm. Within the limits of observational errors, the 
 displacement of the pendulum agreed with the following formula: 
 
 *=(-;,) 
 
 r I e at sin at. 
 
 The measuring apparatus, already described, was supported with its 
 knife-edge E parallel to that of the large pendulum, 1 and about 8 cm. 
 above it. A mirror was attached to E above the mirror of the large 
 pendulum, and a second horizontal telescope was mounted above that 
 used for the large pendulum. The same scale was used for both. The 
 apparatus was not enclosed nor shielded from air currents. 
 
 A short rigid bar, secured to the large pendulum in a position perpen- 
 dicular to its knife-edge and pendulum rod, supported at the distance /i 
 
 1 See Fig. 2 for diagram. 
 
505 
 
 GEORGE F. McEWEN. 
 
 [VOL. XXXIII. 
 
 from the knife-edge, by means of a fine wire, the end A of the lower beam 
 of the measuring apparatus. Because of this arrangement, the end A 
 of the beam AC was constrained to have the vertical displacement: 
 
 yo = /i0o = ce~ at sin at. 
 
 In order to test the capacity of the apparatus for measuring large 
 forces acting on the sphere, dark filtered cylinder oil, grade "N," was 
 used at temperatures from 18 to 22 Cent. At these temperatures, 
 the coefficient of viscosity of the oil is several thousand times as great 
 as that of water. 
 
 The numerical values of the constants in the experiment on oil are 
 tabulated below: 
 
 Weight of the measuring apparatus, about 400 grams. 
 
 Weight in each scale-pan, about 
 Weight of each scale-pan, about 
 
 W 3 = 13.52 gr. 
 
 A/3 = 00.0794 cm - 
 Si = 5.0 cm. 
 5 3 = 7.5 cm, 
 a = 1.702 (sec.)" 1 . 
 
 80 grams. 
 15 grams. 
 
 M" = 76.84 gr. 
 
 ^z 2 = 1,052 dyne cm, 
 
 s 2 = 2.5 cm. 
 
 5 = 10.0 cm. 
 
 a = 0.0007 (sec.)" 1 , 
 
 The radius of the cylinder containing the 
 liquid was 5.5 cm.; its depth was 18 cm. 
 The cylinder was practically full of the liquid, 
 and in all cases the top of the sphere was 6.5 
 cm. below the liquid surface. The maximum 
 velocity of the sphere was less than o.i cm./ 
 sec., and itsmaximum displacement was less 
 than 0.07 cm. 
 
 The maximum deflections 6' and ' were 
 determined alternately, and the value for 0' 
 was divided by the mean of the two nearest 
 values of 0o' for a single determination of the 
 ratio R'. The following set of readings and 
 results is typical of the accuracy of the work. In the first and second 
 columns are the average of the differences of the scale readings for 
 determining ' and respectively. The last column contains the average 
 value of R f minus each value. 
 
 P. 2 
 
No. 6.1 
 
 MEASUREMENTS OF FRICTIONAL FORCE. 
 
 506 
 
 
 
 R' 
 
 .569*-*' 
 
 
 
 R' 
 
 .569*-*' 
 
 2.590 
 2.575 
 2.560 
 2.540 
 2.515 
 2.480 
 
 1.560 
 1.520 
 1.510 
 1.505 
 1.495 
 1.485 
 
 .6025 
 .5910 
 .5900 
 .5925 
 .5945 
 .5990 
 
 -.0063 
 .0048 
 .0062 
 .0037 
 .0017 
 -.0028 
 
 2.445 
 2.425 
 2.400 
 2.370 
 
 1.460 
 1.460 
 1.430 
 1.420 
 
 .5970 
 .6062 
 .5960 
 .5980 
 
 -.0008 
 -.0058 
 .0002 
 -.0018 
 
 .5962 
 
 Tabulation of Results. The temperature of the liquid is denoted by /. 
 Under V V are the results of substituting the given values of w, w, R n , 
 and R w in the formula: 
 
 *<-> 
 
 / = 18.1 n = .495 cm. RJ = 8.30 w = 1.384. 
 
 n 
 
 *n 
 
 *n 
 
 1*\ 
 
 i+A"=39S.7^i 
 
 1 
 
 2.612 
 
 6.822 
 
 2.973 
 
 
 7 
 
 1.5513 
 
 2.4065 
 
 2.950 
 
 
 11 
 
 .9375 
 
 .8789 
 
 2.965 
 
 
 17 
 
 .5709 
 
 .3259 
 
 2.954 
 
 
 
 
 
 2.960 
 
 1,171 
 
 = 18.05 r 2 = .792 cm. R w * = 10.12 w = -1.700. 
 
 n 
 
 * 
 
 *n 
 
 <v* 
 
 *+-**: 
 
 
 
 3.123 
 
 9.755 
 
 8.79 
 
 
 6 
 
 2.392 
 
 5.722 
 
 8.78 
 
 
 18 
 
 1.296 
 
 1.680 
 
 8.79 
 
 
 
 
 I 8.787 
 
 3,477 
 
 22.3 
 
 .495 cm. RJ = 11.30 w = 2.98. 
 
 
 
 * 
 
 
 
 VFi 
 
 *M 
 
 1 
 
 1.9817 
 
 3.927 
 
 1.445 
 
 
 7 
 
 1.1335 
 
 1.2848 
 
 1.436 
 
 
 11 
 
 .5792 
 
 .33507 
 
 1.405 
 
 
 17 
 
 .3423 
 
 .11717 
 
 1.436 
 
 
 
 
 
 1.431 
 
 566 
 
 22.15 n = .792 cm. l? w = 9.34 w - 1.50. 
 
 
 
 /? 
 
 * 
 
 g 
 
 WMT 
 
 
 
 2.755 
 
 7.590 
 
 3.125 
 
 
 6 
 
 1.748 
 
 3.055 
 
 3.137 
 
 
 12 
 
 .9001 
 
 .8102 
 
 3.237 
 
 
 18 
 
 .5701 
 
 .3255 
 
 3.136 
 
 
 
 1 
 
 3.159 
 
 1,249 
 
507 
 
 GEORGE F. McEWEN. 
 
 [VOL. XXXIII 
 
 t = 22.10 
 
 .792 cm. /?,, 2 = 8.63 w = 1.73. 
 
 n 
 
 ^n 
 
 R n * 
 
 V ^ 
 
 B^K 
 
 1 
 
 2.862 
 
 8.191 
 
 3.155 
 
 
 7 
 
 1.513 
 
 2.289 
 
 3.142 
 
 
 11 
 
 .9500 
 
 .9025 
 
 3.160 
 
 
 17 
 
 .5962 
 
 .3555 
 
 3.162 
 
 
 
 1 i 3.155 
 
 1,248 
 
 The coefficient of viscosity was not computed from the preceding 
 data, because the cylinder was not large enough compared to the spheres 
 for the assumption of a cylinder of infinite radius to be valid for a fluid 
 having such a large coefficient of viscosity. However, the above trials 
 show that the method adopted can be successfully applied, even to the 
 measurement of very large frictional forces. 
 
 The apparatus, used in the experiment just described, was not well 
 adapted to the measurement of small forces acting on the sphere, because 
 of its crudeness, and because the friction in the apparatus itself was com- 
 paratively large. But in order to test the method through a wide range 
 of conditions, an attempt was made to determine the coefficient of 
 viscosity of water. In this experiment the sphere was suspended close 
 to the point C of the lower bar, in order to increase the effect of the fric- 
 tional force. Also, since the quantity z\ was now much smaller than 
 before, the series of values of R was obtained by correspondingly smaller 
 changes in the expression (P 2 + P[a 2 a 2 ] aPi). The changes were 
 made by adding equal weights to each scale-pan, thus, varying P instead 
 of Pz as was done before; and by a method similar to that already 
 described the following formula was derived : 
 
 in which n is the number of multiples of a given weight W added to 
 each scale-pan, v/22 equals the change in the moment of inertia of the 
 system when one of these weights is added to each pan. That is: 
 
 In this case a decrease in n has the same effect as an increase in the 
 previous case, but this formula can be used in the same way as the 
 previous one, assuming the symbols to have the values indicated above. 
 
 The numerical values of the constants in the following experiment 
 are tabulated below: 
 
 Weight of the measuring apparatus, about 400 grams. 
 
No. 6.] 
 
 MEASUREMENTS OF FRICTIONAL FORCE. 
 
 508 
 
 Weight in each scale-pan, about 50 grams. 
 
 Weight of each scale-pan, about 15 grams. 
 
 W\ i.oo grams. ^z 2 = 146.5 dyne cm. 
 
 Si = 5.0 cm. 
 
 = I.I cm. 
 
 p = i.ooo. 
 
 = 8.90 cm. 
 
 = 10.0 cm. 
 
 a = 1.711 (sec.)" 1 . 
 
 The radius of the cylinder containing the liquid was 8 cm.; its depth 
 was 20 cm. The cylinder was practically full of water, and in each case 
 the top of the sphere was 6.5 cm. below the surface of the water. The 
 maximum velocity of the sphere was less than .15 cm./sec., and its 
 maximum displacement was less than .09 cm. 
 
 The amplitude of the large pendulum was maintained constant by 
 means of an electro-magnet and the scale readings for determining 6 
 were taken as before. In this case they should be constant. The fol- 
 lowing set of readings and results is typical of the accuracy of the work. 
 The differences between successive scale readings, denoted by R, are given 
 in the first column. The last column contains the average value of R 
 minus each value. 
 
 R 
 
 2.719-;? 
 
 R 
 
 2.719-;? 
 
 R 
 
 2.719;? 
 
 2.67 
 
 + .049 
 
 2.74 
 
 -.021 
 
 2.68 
 
 + .039 
 
 2.70 
 
 + .019 
 
 2.73 
 
 -.011 
 
 2.67 
 
 + .049 
 
 2.68 
 
 + .039 
 
 2.68 
 
 + .039 
 
 2.68 
 
 +.039 
 
 2.69 
 
 + .029 
 
 2.69 
 
 + .029 
 
 2.71 
 
 + .009 
 
 2.68 
 
 +.039 
 
 2.72 
 
 -.001 
 
 2.76 
 
 -.041 
 
 2.72 
 
 -.001 
 
 2.73 
 
 .011 
 
 2.76 
 
 -.041 
 
 2.79 
 
 -.071 
 
 2.69 
 
 +.029 
 
 2.78 
 
 -.061 
 
 2.77 
 
 -.051 
 
 2.72 
 
 .001 
 
 2.71 
 
 +.009 
 
 2.78 
 
 -.061 
 
 2.71 
 
 +.009 
 
 2.75 
 
 -.031 
 
 2.77 
 
 -.051 
 
 2.71 
 
 + .009 
 
 2.70 
 
 + .019 
 
 2.72 
 
 -.001 
 
 2.70 
 
 +.019 
 
 Aver. R 2.719 
 
 
 2.72 
 
 -.001 
 
 
 
 
 
 TABULATION OF RESULTS. 
 
 The temperature of the liquid is denoted by /. Under ^ V are given 
 the results of substituting the values of n, w, R nt and R w in the formula: 
 
 " - R 
 
509 
 
 GEORGE F. McEWEN. 
 t = 16.5 r = .495 cm. RJ = 85.8 w = 3.474. 
 
 [VOL. XXXIII. 
 
 n 
 
 *. 
 
 JW 
 
 VVi 
 
 *+**MI^ 
 
 4 
 
 6.994 
 
 48.95 
 
 .6057 
 
 
 2 
 
 3.525 
 
 12.42 
 
 .6065 
 
 
 
 
 1.593 
 
 2.538 
 
 .6065 
 
 
 
 
 
 .606 
 
 1.530 
 
 t = 16.5 r 2 = .952 cm. R tv * = 42.80 w = 4.34. 
 
 
 
 * 
 
 Jtf 
 
 V V* 
 
 *+A-=*S*/* 
 
 8 
 
 1.850 
 
 3.422 
 
 1.079 
 
 
 4 
 
 6.238 
 
 38.92 
 
 1.080 
 
 
 2 
 
 2.721 
 
 7.405 
 
 1.070 
 
 
 
 
 1.556 
 
 2.420 
 
 1.064 
 
 
 
 
 
 1.073 
 
 2.710 
 
 B 2 Bi = 2.710 1.530 = 1.18. 
 The expression (J?2 -Si) can be transformed into the following 
 
 - B l = 
 
 - n) 
 
 from which after substituting the values from the above experiment, the 
 following equation for determining ju is obtained : 
 
 or 
 
 1.18 = STT ^3.422 \V{ (457) (1447 + 1. 
 M' -f I-338 ^M 7 - -1369 = o. 
 
 The solution of this equation gives ju = .0091, while the value .01 1 1 is 
 given by the formula: 1 
 
 .0178 
 
 ~ I + .0337/ + .00022 It 2 ' 
 
 Thus there is a fair agreement with other methods, considering the un- 
 favorable working conditions. 
 
 IV. SUGGESTIONS FOR FUTURE RESEARCH. 2 
 
 In the measuring apparatus used in the experiments just described, 
 the force which opposed the displacement of the system but did not 
 
 1 This formula is based on Poiseuilles' capillary tube experiment. See Lamb's Hydro- 
 dynamics, 3d edition, p. 536. 
 
 8 Further suggestions for future research are given in Stokes's Math, and Phys. Papers, 
 Vol. III., pp. 123-127. 
 
No. 9.1 MEASUREMENTS OF FRICTION AL FORCE. 510 
 
 depend upon the friction, was due partly to the elasticity of the steel 
 strips and partly to gravity. If, instead of steel strips knife-edges were 
 used, the return force would depend upon gravity only and could be 
 made so small, without loss of stability, that a given natural period could 
 be secured by a much lighter apparatus. If, in addition to reducing the 
 weight, agate bearings were provided, the friction of the apparatus would 
 be greatly reduced, and therefore greater accuracy would result. It 
 would also be desirable to protect the apparatus from air currents and 
 dust. 
 
 In making measurements, the weight W$ should be so adjusted that 
 two values of the ratio R are roughly one half the maximum value, one 
 corresponding to the case where (P 2 + P[a 2 # 2 ] aPi) = v' ' Zi(n w) is 
 positive and the other when it is negative, and the third value should be 
 near the maximum. The three values of R thus obtained are sufficient 
 for computing the force, but it is desirable to check the work by addi- 
 tional values of R between the first two. 
 
 The same apparatus can be used for the measurement of the force 
 acting on any vibrating body, if the force opposes the displacement and 
 is proportional to the velocity, and it is well adapted to the case in 
 which a very small maximum velocity and displacement is required. 
 
 The advantages of measuring the coefficient of viscosity by the oscil- 
 lating sphere method can only be settled by a series of experiments in 
 which the force is measured when the working conditions are in agree- 
 ment with those required by the theory. 
 
 V. SUMMARY OF THE PAPER. 
 
 The hydrodynamical theory of a frictionless fluid failed to account 
 for the experimental results of the pendulum observations of Bessel, 
 Baily and Dubuat, and this fact led Stokes to apply the theory when the 
 fluid friction was taken into account. 
 
 The predictions based on this theory agreed well with the experiments 
 on pendulums vibrating in gases, but when the fluid was water, instead 
 of gas, the agreement was not very satisfactory. This disagreement 
 was due to the difficulty of making the observations, rather than to the 
 theory. 
 
 The fact that the inertia correction could be determined experimentally 
 and that it had been shown how it depended on the coefficient of vis- 
 cosity led Stokes to suggest that his theory might be employed to cal- 
 culate the coefficient of viscosity of gases from observations on a pendu- 
 lum, consisting either of a sphere attached to a fine wire, or of a cylinder. 
 
 He also showed how to compute the frictional force acting on a sphere 
 
511 GEORGE F. McEWEN. [VOL. XXXI II 
 
 and a cylinder, vibrating in a viscous fluid. From this result the coef- 
 ficient of viscosity can be computed from observations which give the 
 retarding force. For this purpose he proposed that the decrement of 
 the arc of oscillation be used. This does not require as accurate a value 
 of the period as the other method. 
 
 The greatest difficulty in each method when the coefficient of viscosity 
 is as large as that of water, or larger, is the rapidity with which the 
 oscillations diminish. 
 
 The present investigation was undertaken to overcome the difficulties 
 mentioned by Stokes when applying the oscillating sphere method to the 
 measurement of large coefficients of viscosity. This object was accom- 
 plished by employing a forced vibration method in which the sphere was 
 suspended by a fine wire, and had a slow motion of translation only 
 along a vertical line, and the oscillations were small, and either diminished 
 very slowly, or were maintained constant for any desired length of time. 
 
 The above method was applied to the measurement of the force acting 
 on a sphere in water and in a very viscous oil. In each case a satisfactory 
 measurement of the force was obtained. 
 
 It is assumed in the hydrodynamical theory that the velocities and 
 displacements are very small, that the amplitude remains constant, and 
 that there is no rotation of the sphere. In measuring the coefficient of 
 viscosity it is necessary that the apparatus can be easily cleaned, that 
 the required quantities can be accurately measured, and that the condi- 
 tions can be readily reproduced. 
 
 These requirements are fulfilled by the method just described, but 
 the advantages of measuring the coefficient of viscosity by the oscillating 
 sphere method can only be determined by further experiments in which 
 the force is measured when the conditions agree with those required by 
 the theory. 
 
 9 
 
 University, 
 rtl , 11/1. 
 
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