306 VH. DYNAMICS OF ROTATING BODIES. 



Integrating we have 



184) ^ = const. = ^- 



Thus we find that F makes a constant angle with the axes of 

 coordinates , and since it has the constant magnitude pMg the center 

 of the sphere experiences a constant acceleration, and describes a 

 parabola. 



If the center of the sphere starts to move with the velocities 

 V x , V y and with a "twist", whose components are ^> , g , r , we have, 

 integrating 179), since X, Y are constant, 



185 ) q = qo - a ^t, 







r = r . 



Integrating the equations for the center of mass 



Inserting in 181) we find for v x , v y 



i Xt > 



~V 2 



187) M 



V - 



^" =r = = T 



Accordingly, 



X V x -aq 



188) ^^-g, -/X 2 +r 2 = ^, 



F ^-o 



X = - ^M^r 



189) 



Since v X9 v y are linearly decreasing functions of the time, whose 

 ratio is constant, they vanish at the same time 



t 



