260-262] ENERGY AND MOMENTUM. 297 



3. A small ring can slide on a smooth circular wire of radius a which 

 rotates uniformly about a vertical diameter. Prove that, if in the position 

 of relative equilibrium the radius through the ring makes an angle a with 

 the vertical, the period of small oscillations about the state of steady motion 

 is the same as for a simple pendulum of length 



a sec a (c + a sin a)/(c + a sin 3 a). 



4. An elastic circular ring of mass m and modulus of elasticity X rotates 

 uniformly in its own plane about its centre under no external forces. Prove 

 that, if a is the radius in steady motion, and I is the radius when the ring is 

 unstrained, the period of the small oscillations about the state of steady 

 motion is 



*262. Illustrative problem. In further illustration of the principles of 

 Energy and Momentum consider the following problem : 



A uniform rod and a particle are connected by an inextensible thread 

 attached to one end of the rod, the system is laid out straight, and the 

 particle is projected at right angles to the thread. To find the motion when 

 there are no forces. 



Let 2a be the length of the rod, I the length of the thread, ^ the angle 

 the thread makes with the line of the rod produced at time t. Consider first 



Fig. 79. 



the motion of the particle P relative to the centre of inertia M of the rod 

 AB. 



Let 6 be the angle which AB makes at time t with its initial direction. 

 Then the velocity of B relative to M is aB at right angles to AB, and, since 

 BP makes an angle 0+x W ^ Q a ^ me nxe( ^ m tne plane of motion, the velocity 

 of P relative to B is I (6 + x) perpendicular to BP. The velocity of P relative 

 to M is the resultant of these two velocities. Its resolved parts along and 

 perpendicular to A B are accordingly 



-l(0+x)sinx and a6+l(0 + x) cos^. 



Now the centre of inertia G is always at the point dividing MP in the 

 ratio of the masses of the particle and the rod, and, if these masses are p 

 and m respectively, the velocity of M relative to G has components 



and - {a 8 + l(B+x) cos*} 



