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TRANS. ST. LOUIS ACAD. SCIENCE. 



Isodynamic Surf aces of the Compound Pendulum.* 



By Francis E. Nipher. 



In discussing the compound peudulum, the statement is some- 

 times made, that particles near and below the axis of suspension 

 are retarded, and that those near the bottom of the pendulum are 

 accelerated by reason of, their connection with the system. The 

 series of particles forming the axis of oscillation are neither accel- 

 erated or retarded. 



In a general way, so far as it concerns the time of a complete 

 oscillation, this is all true, but it is not true in any compound pen- 

 dulum that the particles near the bottom continually exert a 

 retarding effect upon the system. At any given instant certain 

 particles in the system tend to diminish the actual acceleration, 

 while others tend to increase it. These two tendencies always 

 balance, although the value of each continually varies. These 

 two groups of particles are separated by a surface, each particle 

 of which has no tendency to change the acceleration of the sys- 

 tem at that instant. The axis of oscillation always lies in this 

 surface. On either side of this neutral surface there must be 

 surfaces of equal tendency, those on one side having a plus, and 

 those on the other side a minus sign. It is required to find the 

 loci of these isodynamic surfaces at any given instant. This can 

 be done by means of well known equations for the pendulum, 

 which are first given. 



In Fig. 1, let O represent the axis of 

 oscillation, G the center of gravity, and S 

 the axis of suspension. Call S G = K, 



5 O = /, and let r be the distance of any 

 element of mass dm from the axis S. Let 



6 = the angle V S O, and o. the angle be- 

 tween the lines / and r, V S being the ver- 

 tical plane containing the axis S. 



The entire mass of the pendulum may 

 be supposed condensed on the vertical 

 plane passing through G, and at right an- 

 gles to the axes O and S, each element of 

 mass being supposed to be projected along 



* Read Oct. 19th, 1885. 



