224, 225] FRICTION. 239 



Thus the acceleration with which m descends is 



It appears that the effect of the inertia of the pulley is equivalent to an 

 increase of each of the masses in the simple problem (where the pulley is 

 regarded as smooth) by J/ 2 /a 2 . 



II. Rolling and sliding. We take the problem presented by a uniform 

 cylinder of mass M and radius a which is set rolling and sliding on a rough 

 horizontal plane, the angular velocity being initially such that the points on 

 the lowest generator have the greatest velocity. 



Fig. 58. 



Let V be the velocity of the axis, and &amp;lt;B the angular velocity at time t, 

 the senses being those shown in the figure. 



The system of kinetic reactions reduces to M V horizontally through the 

 centre of inertia, in the sense of F, and a couple Mk*d&amp;gt; in the sense of o&amp;gt;, 

 where k is the radius of gyration about the axis of the cylinder. 



Taking moments about the point of contact we have 



Now let F be the friction between the cylinder and the plane. The 

 particles on the lowest generator have velocity V+ ao&amp;gt; in the sense of F, and 

 therefore F has the opposite sense. 



Resolving horizontally we have 



MV=-F, 

 where F is positive. Hence V is negative and o&amp;gt; is also negative. 



The velocity V diminishes and the angular velocity o&amp;gt; also diminishes 

 according to the equation 



where F and o&amp;gt; are the values of F and o&amp;gt; in the beginning of the motion. 

 We shall proceed with the case where F &amp;lt; o&amp;gt; F/a. Then there must come 

 an instant at which F vanishes, and at this instant o&amp;gt; has the value 

 o&amp;gt; F /P. At this instant the lowest point has velocity ao&amp;gt; - F a 2 /& 2 in 

 the same sense as before, the friction is still finite and in the same sense as 

 before, and a velocity in the opposite sense begins to be generated. 



