80] COMPOUND PENDULUM. 251 



But this is the equation of motion of a simple pendulum of length 



4) I- K --* 2 



~m "' 



The body, which is often called a compound pendulum, accord- 

 ingly moves like a simple pendulum of length I. This is called the 

 equivalent simple pendulum. It is to be noticed that in virtue of 

 the constraint of rigidity, points at distances from the axis less 

 than I move more slowly than they would if moving alone in the 

 same paths, while those at greater distances move faster, and those 

 at distance I move just as they would if free to move in the same 

 circular paths. 



Let kg be the radius of gyration of the body about a parallel 

 axis through the center of gravity. Then by 70, 



4) * = ftJ + A, Z = ? + ft, 



ft 



so that I is always greater than h. If we take a parallel axis 0' at 

 a distance h' = I h beyond the center of mass 6r, so that it, 6r, 

 and the original axis are in the same plane, we have 



5) hh' = 1$. 



If now the axis 0' be made the axis of suspension, the equi- 

 valent simple pendulum has a length 



fc' 2 2 _L h '* 



6) V --hT = -~F~ = h + h ' = L 



The axis 0' is called the axis of oscillation, and we have the 

 theorem that the axes of suspension and oscillation are interchangeable 

 and separated by the distance equal to the length of the equivalent 

 simple pendulum. This is the principle of Eater's reversible pen- 

 dulum, used to determine the acceleration of gravity. The pendulum 

 is furnished with two knife edges, so that it may be swung with 

 either end down. Movable masses attached to the pendulum are so 

 adjusted that the time of vibration is the same in both positions, 

 and then the distance between the knife-edges gives the length I 



from which g = -mr" The present example also includes the metro- 

 nome and the beam of the ordinary balance. The masses of the 

 pans may be regarded as concentrated at the knife-edges. 



If the fixed axis is not horizontal, the modification in the result 

 is very simple. Suppose the axis makes an angle K with the vertical. 

 Let us take two sets of fixed axes, Z' vertical, Z the axis of rota- 

 tion, T 1 horizontal in the plane of Z and Z', Y in the same plane, 



