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, o> the angular velocity of m'. 



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



Fig. 60. 



The velocity of B relative to A is (a + b)0 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 ba> 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 a& at right 

 angles to A B, 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)0 + ba> = -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 0, the accele- 

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

 and (a + b) 0' 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) 0+m'a&bcos0+m'k' 2 u = -m'gbsin0 (3), 



and 



mk 2 Q, + ma 2 Q + m'aQ, {a -f (a + 6) cos 6} + m'k' 2 u> ) . . 



b')sm0) "" ^ '* 

 16 



