256, 257] 



OSCILLATIONS OF KIGID BODIES. 



291 



where the right-hand member is -mga0ta.ua, and this equation shows that 

 the motion in 6 is the same as for small oscillations of a simple pendulum 

 of length 



a cot a Q- + cot 2 a). 



II. A uniform rod is supported at its ends by two equal vertical cords 

 suspended from fixed points. To find the small oscillations when the middle 

 point moves vertically and the rod, remaining horizontal, turns round its middle 

 point. 



Let 2a be the length of the rod, 

 I the length of either cord, z the dis 

 tance through which the middle point 

 has risen at time t, 6 the angle through 

 which the rod has turned in the same 

 time. The depth of either end A or 

 B below the corresponding point of 

 support is l z, and the distance 

 AM or BN of an end from the vertical 

 plane through the points of support is 

 2a sin \6. Hence we have 



this equation shows that when z and 

 6 are small z=^(a 2 jl }6 2 to the second 

 order, and 1=0 to the first order. 



Now, if m is the mass of the rod, 

 the kinetic energy in any position is 



and the potential energy is 



mgz, 



Fig. 77. 



the lowest position being the standard position. 



Hence in the small oscillations the kinetic energy is, with sufficient 

 approximation, 



and the potential energy is, with sufficient approximation, 



The motion in 6 is therefore the same as for small oscillations of a simple 

 pendulum of length ^l. 



*257. Examples. 



1. A uniform rod of length 2a rests in a smooth bowl in the form of a 

 surface of revolution whose axis is vertical, so that the ends are at points 

 where the radius of curvature of the meridian curve is p and the normal 

 makes an angle a with the vertical. Prove that the length of the simple 



192 



