308 



VII. DYNAMICS OF ROTATING BODIES. 



describe circles. In this case we have for the moment about the 

 center of mass of the forces tending to increase &, by 82), 



191) P^ = 



The forces which act to change # are, the weight of the hoop, 

 which has zero moment about the center of mass, and the reaction 

 of the table. Let i?, Fig. 109, (an edge view of 

 the hoop) represent the vertical reaction, F the 

 horizontal component due to friction, which is 

 normal to the path of the point of contact, the 

 tangential component disappearing on account of 

 the assumed constancy of the velocity of rolling, 

 as in the case of the rolling sphere. We accord- 

 ingly have, taking moments, 



192) P<> = Fa sin # - Ea cos #, 



F a being the radius of the hoop. But considering 

 rig. 109. the motion of the center of mass, which is uniform 



circular motion, and supposing all the forces there 

 applied, since there is no vertical motion, the resultant vertical 

 component vanishes, or E = Mg and the horizontal component balances 

 the centrifugal force, so that 



193) F = 



where b is the radius of the circle described by the center of mass. 

 Beside the dynamical equation we have the equation of constraint 

 describing the rolling. Since there is no slipping, the rate at which 

 the center of mass advances in its path is 



194) ar = a(y 1 + ^' cos#). 



But this is also, from the circular motion, equal to biff'. From 

 the equation of constraint, 



195) a(<p'+ ^'cos-fr) = 6^', 



we may express <p' in terms of ^'. Doing this, and inserting in 

 191), 192), we obtain, 



or 



196) 



Ma 2 ) 



as the equation for the steady motion, connecting the inclination of 

 the hoop, the radius of the path, and the velocity of its description. 

 In order to make the hoop roll in this manner, the proper velocity 

 of pivoting ^ p , as well as that of rolling qp', must be initially imparted 



