250 MOTION OF A RIGID BODY IN TWO DIMENSIONS. [CHAP. XI. 



19. A homogeneous sphere of radius a is initially at rest on a horizontal 

 plane and the plane is made to move backwards and forwards horizontally so 

 that its displacement at time t is b cos nt. Prove that, if /A the coefficient 

 of friction <%bn 2 /g, the changes from rolling to sliding take place at times 

 (nr a)/?i, where r is a positive integer and a is the least positive root of the 

 equation cosa=7/i<7/2&ft 2 ; prove also that the changes from sliding to rolling 

 (except the first) take place at times (rn+y}ln, where y is the least positive 

 root of the equation 



sin y + sin a = 7/i<7 (y + a 



20. A uniform sphere of mass M rests on a rough plank of mass N' 

 which is on a rough horizontal plane ; the plank is suddenly set in motion 

 along its length with velocity V. Prove that the sphere will first slide and 

 then roll on the plank and that the whole system will come to rest after a time 

 M'Vj^g (M+M'} from the beginning of the motion, where \i is the coefficient 

 of friction at each of the places of contact. 



21. On a smooth table rests a board of mass M, having its upper surface 

 rough and the lower smooth. A sphere of mass m is projected on the upper 

 surface of the board so that the direction of projection passes through the 

 centre of inertia of the board ; the velocity of projection is V and the 

 sphere has an angular velocity Q. about a horizontal axis perpendicular to 

 the plane of projection. Prove that after a time 



the motion will become uniform, and that the velocity of the board will 

 then be 



22. A reel of mass M and radius a rests on a rough floor, p. being the 

 coefficient of friction. Fine thread is coiled on the reel so as to lie on a 

 cylinder of radius b (< a) and coaxal with the reel. The free end of the 

 thread is carried in a vertical line over a smooth peg at a height h above 

 the centre of the reel and supports a body of mass m. Prove that if either 



p<mb/(M-m)a or if M<m[l -6 2 (l+a/A-a 2 M)/( 2 + ^ 2 )], 

 the thread will be unwound from the reel. 



23. A garden roller, in which the mass of the handle may be neglected, 

 is pulled with a force P in a direction making an angle a with the horizontal 

 plane on which it rests. Show that it will not roll unless 



P (sin a sin -f cos a cos & 2 /(a 2 + 2 )}< TFsin 0, 



where a, &, W are the radius, the radius of gyration about the axis, and the 

 weight of the roller, and is the angle of friction between it and the ground. 



24. Two rough cylinders of radii r lt r 2 are put on a rough table and on 

 them is placed a rough plank. Prove that, under certain conditions, the 



