BOOK OF MATHEMATICAL PROBLEMS. 



the string will vanish when the length of the string unwound is 

 that which initially subtended at A an angle 6, such that 



8p(0 tan0 + 1) cos0 + 3w = 0; 

 and that the angular velocity of the disc is then 



1539. A rough sphere, radius a, moves on the concave surface 

 of a vertical circular cylinder, radius a + 6, and the centre of the 

 sphere initially moves horizontally with a velocity v : prove that 

 the depth of the centre below its initial position after a time t is 



Prove that, in order that perfect rolling may be maintained, 

 the coefficient of friction must be not less than ^ 8 



1540. A right circular cylinder is fixed with its axis hori- 

 zontal and a rough sphere is projected so as to move in contact 

 with the cylinder, being initially at the lowest point and its 

 centre moving in a direction making an angle a with the axis of 

 the cylinder : prove that, in order that the sphere may reach the 

 highest point, its initial velocity must be not less than 



s P here an <l cylinder 



V/ (V 



respectively. 



ProTe the equations of motion 

 d<f>\' u 2 sin a a 



dz 



