GRAVITATIONAL METHODS 



271 



a function of /, g, and the azimuthal angle from which the pendulum was 

 initially set into oscillation. It is given by the formula : 



r = 2«- V- [1 + (3^)2 sin2|+ (1.3/2.4)2 sin^§ + • . . ] 



g 



(22) 



where Xo is the angle made by the string with the vertical at the start of 

 the oscillations. For small amplitudes the period formula becomes 



T, 



o = 2nyl- 



9 



(23) 



Compound Pendulum: Reversible Type 



A compound pendulum is a rigid body of any shape suspended by a 

 horizontal axis and swinging through a small angle with negligible friction. 

 Figure 133a shows a vertical section through the center of mass C and 

 perpendicular to the axis of suspension S. The gravitational moment tend- 



ing to rotate the pendulum about 5^ is mgJii sin i. The acceleration is -j-^. 



Hence, from Newton's second law of motion 



dH 



where /, is the moment of inertia about the axis through S and m is the 

 mass of the pendulum. This equation of motion for a compound pendulum 



rf3 





CL 



Fig. 133. — Schematic diagrams of reversible type, com- 

 pound pendulum. 



is similar to that for a simple pendulum. Hence, for small values of i the 

 solution is a periodic function with period Ti, where 



' -111 It 



mliig 



A comparison of this expression with the corresponding expression for a 

 simple pendulum (Equation 23) shows that the compound pendulum is 



