MOMENT OF MOMENTUM 299 



The equation of motion for a simple pendulum of length I is 



so that we see on comparison that the motion is the same as that 

 of a simple pendulum of length I = tf/h. 



For instance, the complete period of small oscillations is 



ILLUSTRATIVE EXAMPLE 



A ring (e.g. a dinner napkin ring] stands vertically on a table, and a gradually 

 increasing pressure is applied by a finger to one point of the ring in such a way 

 that equilibrium is broken by the point of contact with the table slipping along the 

 table. Find the subsequent motion of the ring. 



We have seen in example 2, p. 109, that it is 

 possible to apply pressure in the manner described. 



Let us suppose that when the ring leaves the 

 finger it is observed to be moving with a velocity 

 V forward and a rotation 12 in the direction oppo- 

 site to that in which it would rotate if it were 

 rolling without sliding. Let r, w be the values of 

 the velocity and rotation at any instant, measured 

 in the same directions as V and Q. 



Let a be the radius of the ring and m its mass. The forces acting on it are 



(a) its weight mg ; 



(b) the 'vertical component of the reaction with the table, which is equal to 

 mg since the center of gravity of the ring has no vertical acceleration ; 



(c) the f rictional reaction at the lowest point of the ring, which is equal to 

 mg /* so long as sliding takes place. 



By the theorem of 180 we have 



FlQ 



We can obtain a second equation from the theorem of 243. Let us take as 

 axis the axis of the ring at instant t. The moment of inertia at this instant is 

 ma 2 . To obtain the moment of momentum we regard the whole motion as com- 

 pounded of a motion of translation of the center of gravity (velocity u), and a 



