!^ 2 KINEMATICS. [274. 



is therefore proportional to the distance r of this point from 

 the axis, so that the accelerations of all points can be found 

 as soon as that of any one point is known. 



274. We proceed next to the investigation of the accelera- 

 tions of the various points of a rigid body having plane motion. 

 The motion is determined by that of a plane section of the body 

 parallel to the plane of motion, and this consists in the rolling 

 of the body centrode over the space centrode (Art. 22). 



During any element of time dt, every point P of the plane 

 section rotates with angular velocity co about the instantaneous 

 centre of rotation C which is the point of contact of the two 

 centrodes. During the next element of time dt, the angular 

 velocity is co + dco, and the centre of rotation has changed to the 

 infinitely near point C^ on the space centrode, which has now 

 become the point of contact of the two centrodes. The accel- 

 eration of a point P at the distance r from C evidently depends 

 on this distance r, the angular velocity co, the angular accelera- 

 tion cx.=da)/dt, and the element CC l = dcr of the space centrode. 

 This element divided by dt may be regarded as a velocity, 

 u = d(T/dt, viz. the velocity with which the instantaneous centre 

 changes its position. We may call it the velocity of rolling of 

 the body centrode. The change in the state of motion during 

 two consecutive elements of time depends on a and u. 



275. The relation of the velocity of rolling u to the angular 

 velocity co depends on the relative curvature of the centrodes. 

 c,c'. 



To fix the ideas imagine these curves to lie on the same side 

 of their common tangent ; let da, da! be their angles of contin- 

 gence, and let p, p' be their radii of curvature (Fig. 70). 



The rotation about C brings the second element of c' to co- 

 incidence with the second element of c. The angle dO of this 

 rotation is therefore equal to the difference of the angles of 

 contingence of the two curves, i.e. 



