225] 



ROLLING. 



241 



III. Kinematic condition of rolling. Formation of equations of motion. 

 A cylinder of radius b rolls on a cylinder of radius a, which rolls on a 

 horizontal plane. 



Let m and m be the masses, A and B the centres, F the horizontal 

 velocity of m, Q the angular velocity of m, 6 the angle AB makes with the 

 vertical, to the angular velocity of m . 



The condition that m rolls on the plane is F=aQ (1). 



Fig. 60. 



The velocity of B relative to A is (a + b) B at right angles to AB, and its 

 velocity is therefore compounded of this velocity and V horizontally. 



The velocity of P (considered as a point of m } relative to B is bo&amp;gt; at right 

 angles to AB, in the sense of (a + b) 0. 



The velocity of P (considered as a point of m) relative to A is aQ. at right 

 angles to AB, but in the opposite sense. 



The condition of rolling is that the particles of m and m that are at P 

 have the same velocity along the common tangent to the two circles. 



We therefore have (a + b)6 + ba&amp;gt;= -aQ (2). 



In the diagram of accelerations (Fig. 61) we have introduced the value of 

 Ffrom equation (1). 



Since B describes a circle relative to A with angular velocity 6, the accele 

 ration of B relative to A is compounded of (a + 6) 6 at right angles to AB, 

 and (a + b) 2 in BA. This gives us the diagram. 



Now, to form the equations of motion, take moments about P for m , and 

 about for the system. We have 



- m b (a + b)6 + m aQ, b cos 6 + m k z a&amp;gt; = m gb sin 6 (3), 



and 



ma^Q. + m aQ, {a+(a + b) cos 6} + m K ) 



(a + b) 6(a+ b + acos6) + m (a + b)d 2 asm6= m cf(a + b)sm8) &quot; ^ * 

 L. 16 



