310 VII. DYNAMICS OF ROTATING BODIES. 



199) 



/dv g \ 



M (-^ + p Vy q v x ) = E Z Mg cos #. 



For the rotation we have, as in 96, 165), since p=p Q , # = # > 



200) A f - (Cr - 



dr 



We have finally, as the conditions for rolling and pivoting, the 

 equations stating that the velocity of the point of contact with the 

 plane (whose coordinates are x,y, 8) is at rest. 



v x 4- qz = 0, 



201) v y + rx - pz = 0, 



v z qx = 0. 



The coordinates x, s of the point of contact are obtained as known 

 functions of & from the equation of the meridian of the hody. We 

 have accordingly the eighteen equations 197 201) between the 

 eighteen quantities 



v x , v y , v z , p, q, r, p Q , g , r , X, Y, Z, #, ^, qp, E x , E y , E 2 , 



or just enough to determine them. The differential equations are all 

 of the first order. 



The reactions may be at once eliminated from the equations 

 199), 200). By differentiating 201) we may eliminate the derivatives 

 of v X9 v y , v z from 199). In doing this, however, we introduce the 

 derivatives of x, 2, which are functions of #, so that in general the 

 equations become complicated. We shall therefore confine ourselves 

 to the case of a body rolling on a sharp edge, like a circular 

 cylinder with a plane bottom, or a hoop or disk. We then have 

 x, z constants, 



x = a, z = c, 



where a is the radius of the circular edge, c the distance of the 

 center of mass from its plane, which is zero in the case of the hoop. 



