In one of our laboratory classes recently, in connection with a routine experiment with Kater\u27s pendulum, certain of the students were confronted with the situation in which, although the periods of vibration from each of the two knife edges were practically identical, the distance between the knife edges was by no means equal to the length of the equivalent simple pendulum. None of the treatises on dynamics available offered any help in their difficulty. In all the discussions it was virtually stated that when the periods from the two knife edges were equal the distance between knife edges was equal to the length of the simple pendulum of equal period

Sieg, L. P.

English

University of Northern Iowa

Proceedings of the Iowa Academy of Science Volume 28 Annual Issue Article 19 1921 A Note on Kater's Reversible Pendulum L. P. Sieg State University of Iowa Copyright ©1921 Iowa Academy of Science, Inc. Follow this and additional works at: https://scholarworks.uni.edu/pias Recommended Citation Sieg, L. P. (1921) "A Note on Kater's Reversible Pendulum," Proceedings of the Iowa Academy of Science, 28(1), 98-102. Available at: https://scholarworks.uni.edu/pias/vol28/iss1/19 This Research is brought to you for free and open access by the Iowa Academy of Science at UNI ScholarWorks. It has been accepted for inclusion in Proceedings of the Iowa Academy of Science by an authorized editor of UNI ScholarWorks. For more information, please contact scholarworks@uni.edu. A NOTE ON KATER'S REVERSIBLE PENDULUM L. P. SIEG In one of our laboratory classes recently, in connection with a routine experiment with Kater's pendulum, certain of the stu-dents were confronted with the situation in which, although the periods of vibration from each of the two knife edges were prac• tically identical, the distance between the knife edges was by no means equal to the length of the equivalent simple pendulum. None of the treatises on dynamics available offered any help in their difficulty. In all the discussions it was virtually stated that when the periods from the two knife edges were equal the distance between knife edges was equal to the length of the simple pendu-lum of equal period. When their difficulty was presented to the present writer he was at once reminded of a study he had made some twenty years ago, but which at that time he had deemed unimportant enough for publication. In this study, a paper read before a Sigma Xi meeting, the writer pointed out that the shifting of a knife edge on any compound pendulum causes a variation in the period in which in certain cases it can pass through a minimum. In other words there are two positions of a knife edge with respect to the center of gravity of the system in which the periods of vibration will be the same. Some time later an article by Tatnall 1 appeared and covered almost identically the same ground. As neither of these treatments specifically deals with the present case it is thought worth while to publish a note on the question. In Fig. 20-A, let C denote the center of gravity (in future ab-breviated to e.g.) of the system, and 0 1 and 0 2 the two points of suspension. Further let h1 and h2 be respectively the distances of the e.g. of the whole system from 0 1 and 0 2 , and let T1 and T 2 be respectively the corresponding periods of vibration. Then we have the well-known expressions, T1=2r I I =211~K2+h12 .................... (1) '\JMgh1 gh1 T. = 2 r.JK2 + h,2 ........................... (2) g h. 1 R. R. Tatnall, Phys; Rev., 17, p. 460, 1903. 1Sieg: A Note on Kater's Reversible PendulumPublished by UNI ScholarWorks, 1921KATER'S REVERSIBLE PENDULUM 99 r r ···O, ·--c, ):. c t t:. _Lo,. -02 A B Fig. 20. Kater's pendulum. A, with weightless knife edges; B, with actual knife edges. where I denotes the moment of inertia of the system about C, M is the total mass, g the acceleration of gravity, and K the radius of gyration of the system about C. Denoting by Li and L2 , re-spectively the lengths of the equivalent simple pendulums in the two cases, we have K2 + hi2 Li= hi .......................... , . (3) K2 + h,2 L.= h. .. .......................... (4) Let us first, to avoid confusion, make the assumption that the knife edges have no mass, so that their movement will not affect the location of the center of gravity of the system. Later this assumption will be avoided. Let Oi be at such a distance, .i-; from C that we have again the same length of the equivalent simple pendulum. Then from ( 3), Solving for .i-, we have the two values K2 . x=h. x= hi-.; ...... .- ............. (5) Thus the knife edge Oi can be either.at a distance hi from C, as in K2 the figure, or -- from C, and we shall have in both cases the hi 2Proceedings of the Iowa Academy of Science, Vol. 28 [1921], No. 1, Art. 19https://scholarworks.uni.edu/pias/vol28/iss1/19100 IOWA ~CADEMY OF SCIENCE Vor.. XXVIII, 1921 same period of vibration.· Similarly, from the other point of sus-pension, 0 2, if y is the distance of 0 2 from C we have two other 1 . d f 'b . ·'f h K 2 equa per10 s o v1 ration 1 y = 2 , or Ii;' We can imagine then that our pendulum (Fig. 20-A) has four weightless knife edges, two on each side of the center, so adjusted · with respect to the masses on the pendulum rod that when it is hung in turn from the four, the periods will be the same. The possible distances between knife edges, .x + y, will then be seen to be, K2 K2 (I I J h, +h., hi+ JI;• h; + h2, and K2 h. + h. In general these four distances will be different, and only one of them corresponds' to the length of the equivalent simple pendu-lum. In practice there is no difficulty, for one can by rough cal-culation determine if the knife edges are in the proper positions. Again, the laboratory forms of Kater's pendulum are usually so constructed that one can not easily attain on the actual pendulum the wrong positions of the knife edges, but in the case referred t? in this article, the students had succeeded in doing just this thing. It is evident that if there are two values, .x, of the distances from 0 1 to C that yield the same period of vibration, there must be some .x beween them which is the only distance corresponding to a single period, and that a maximum or a minimum. Our function of .x. to be studied is K 2 + .x2 , and a simple examination shows that .x this becomes a minimum for .x = + K. Only the plus value of K need be considered, since the other value relates to the other support. The relation becomes more understandable on platting a typical curve connecting .x and T. We have T=2.,,.~K2g:x2 ...................... (6) or y2 = c [ K2 ~ .i-2] .................... (7) where C is a constant. In figure 21 we have such a graph for the special case where C = 1, and K = 2 (not supposed at all to rep~. resent the facts, but merely to represent the nature of the func-tion). On the curve the point D represents the minimum. It is the only value of .x corresponding to the period represented by the ordinate at that point. From A we draw the line ABC parallel to the X-axis. The two intersections B and C represent the two 3Sieg: A Note on Kater's Reversible PendulumPublished by UNI ScholarWorks, 1921T 0 KATER'S REVERSIBLE PENDULUM 5 C= I K=2 x JO IS 101 2.0 Fig. 21. Variation of the period of vibration, I, with the distance, x, from knife edge to center of gravity of the whole pendulUm. values of z corresponding to the period A. To revert fpr a mo-ment to the first paragraph of this article, my students had chanced ,in· their experiment on the distance AB, when they should· have employed the greater distance AC for h1• ' Let us assume now that the knife edges have mass, and let us refer to Fig. 20-B. We can assume, without any loss of gener-ality, that we can attain equality of periods by moving solely 0 1 • Hence the mass of 0 2 can be considered as merged with the re-mainder of the pendulum mass. · Let M =the mass of all the pendulum excepting 01" m =the mass of 0 1 • Cx =the center of mass of the whole. C0 =the center of mass of M. cl= the center of mass of ml (assume cl to be coincident with the near .edge of 0 1 • The error will be only a small constant correction) . z = the distance of 0 1 from Cx. Then we have Ti=2w~ I ·=2w~(M+m)K2 +(M+m)x2 (M+m)gx (M+m)gx = 2 w · /K2 + x2 .......................... '(8) ~ gx ·. wh~re K is the radius of gyration about Cx, and I is the moment of inertia about 0 1 • The only difference between equations (8) and (6) is that in the former K is not a constant, but is a function of z. Now let CoCx = z', then mx=Mx' .......................... (9) 4Proceedings of the Iowa Academy of Science, Vol. 28 [1921], No. 1, Art. 19https://scholarworks.uni.edu/pias/vol28/iss1/19102 IOWA ACADEMY OF SCIENCE Vo1.. XXVIII, 1921 and , mx x = M ............................ (10) Let K 0 be the radius of gyration of M about C0 , and k0 that of m about Oi- Then the moment of inertia about Cx is given by ! 0 = (M + m)k2 = Mk0 2 + Mx'2 + mk0 2 + mx2 ...... (11) Substituting for x' from ( 10) we have MK,2 + mk,2 + m2x2/M + mx2 K2 = M + m . . ............ (12) Let then M = aM ........ : ................. (13) Ko2 + ako2 K2.= l+a + ax2 ...................... (14) or where K 2 = B + ax2 ••••••••••••••••••••••• (15) Ko2 + ako2 B = l +a ......................... (16) Substituting (15) in (8) we have T, = 271' ~B + ax2 + x 2 = 271' ~B + (1 + a)x2 ......... (17) gx gx . Again for every value of T 1 there are two values for x, except at the minimum point. Determining from ( 17) the value of x to make T 1 a minimum, we find x = ~1 ! a This readily reduces to the former value x = K, if m = 0 (and hence a= 0). _ The form of ( 17) is seen to be the same as that of ( 1), and so here too we have the possibility in the most general case of four positions of the knife edges which would yield the same period of vibration. In conclusion then we can state the following: The equality of periods, when a pendulum is suspended in turn from two knife edges, is a necessary condition that the length between the knife edges may be equal to the equivalent simple pendulum, but it can not be said to be a sufficient condition. STATE UNIVERSITY OF IowA. 5Sieg: A Note on Kater's Reversible PendulumPublished by UNI ScholarWorks, 1921

A Note on Kater\u27s Reversible Pendulum

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A Note on Kater\u27s Reversible Pendulum

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