300 , MOTION OF RIGID BODIES 



motion of rotation about an axis through the center of gravity (velocity w). 

 The former contributes nothing to the moment of momentum, so that the whole 

 moment of momentum is 



At the end of a small interval dt the ring will have moved forward a distance 

 vdt, so that we are now considering the moment of inertia about an axis which 

 is distant v dt from the center of gravity of the ring. The moment of inertia 

 after an interval dt is, accordingly, by 235, 



We may, however, neglect the small quantity of the second order (dt) 2 and 

 treat the moment of inertia as though it remained constant and equal to ma 2 . 



The rate of increase of the moment of momentum is, accordingly, ma 2 



dt 



The moment of the external forces, measured about the same axis and in the 

 same direction, is 



mg jta, 



du 

 so that we have the equation ma 2 = mg /tta, (6) 



or, simplified, a -^ = - w, (c) 



dt 



while equation (a) reduces to = p.g. (d) 



These relations give the rates of decrease of v and u so long as sliding is 

 taking place. Sliding clearly ceases as soon as we have v + wa = 0, for v + ua 

 is the forward velocity of the lowest point of the ring. From equations (c) and 

 (d) we have 



(7) + wa) = -2/tgr, 



and initially the value of v + wa is V+ fla. The time required to reduce v + wa 

 to zero is, accordingly, 



T+fla 



After this interval sliding ceases. The velocity of the ring at this instant is 

 given by 



so that the motion may be either forwards or backwards according as we had 

 initially V > or < fla. After sliding has once ceased there is no force tending 

 to start it afresh, so that the ring simply rolls on with uniform velocity v. If 

 V > fla, it rolls farther from its point of projection ; while if F< fia, it will 

 return to the point of projection. 



