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



At any later stage of the motion let U be the velocity. Then so long as 

 aw &amp;gt; U the friction F acts in the same sense, and we have 



whence U increases and o&amp;gt; diminishes according to the equation 



Fig. 59. 

 When U becomes equal to ao&amp;gt; the value of either is 



and at this instant the cylinder is rolling on the plane. Thereafter the 

 cylinder rolls on the plane uniformly. 



It is to be noticed that, in this problem, so long as the cylinder slips, the 

 friction is constantly equal to pMg, where p is the coefficient of friction 

 between the cylinder and the plane. 



Examples. 



1. In the problem just considered prove that the time from the beginning 

 of the motion until the motion becomes uniform is ^ 



2. A homogeneous cylinder of mass M and radius a is free to turn about 

 its axis which is horizontal, and a particle of mass m is placed upon it close 

 to the highest generator. Prove that when the particle begins to slip, the 

 angle 6 which the radius through it makes with the vertical is given by the 

 equation 



/i {(M+ 6m) cos 6 - 4m} = M sin 6, 



where /* is the coefficient of friction between the particle and the cylinder. 



3. A uniform thin circular hoop of radius a spinning in a vertical plane 

 about its centre with angular velocity o&amp;gt; is gently placed on a rough plane of 

 inclination a equal to the angle of friction between the hoop and the plane 

 so that the sense of rotation is that for which the slipping at the point of 

 contact is down a line of greatest slope. Prove that the hoop will remain 

 stationary for a time au/g sin a before descending with acceleration \g sin a. 



