I54 KINEMATICS. [276. 



276. To determine the components of the acceleration of any 

 point P of the body, it will be convenient to imagine the angular 

 velocities represented by their rotors : the velocity <a about C by 

 a line of length o> erected at C at right angles to the plane of 

 motion, on that side of this plane from which the rotation 

 appears counter-clockwise; similarly the angular velocity w + da 

 by a parallel line of length o> + da erected at C v 



The rotor a) + dco through 6\ can be replaced by a parallel 

 rotor of the same magnitude and sense through C, in combination 

 with a rotor-couple whose moment is (co + da))- CC l = codo- (see 

 Arts. 255, 256). This couple being equivalent to a vector wda- 

 at right angles to the plane of the couple produces an infinitesi- 

 mal velocity of translation. 



Thus the body, during the first element of time dt, rotates 

 about the axis through C with angular velocity o> ; and during 

 the second element of time dt, it can be regarded as having the 

 angular velocity co + dco about the same axis, and at the same 

 time a velocity of translation codo- at right angles to the tangent 

 at C. The change in the state of motion consists, therefore, in 

 the angular acceleration d(o/dt=a and in the linear acceleration 

 wdo-/dt=wu, the former being due to the change in the magni- 

 tude of the acceleration, the latter to the change in the position 

 of the axis of rotation. 



While the acceleration of translation cou is the same for all 

 points of the figure, the angular acceleration a produces in 



every point P (Fig. 71) a linear 

 acceleration proportional to its dis- 

 tance r=CP from the centre C, 

 just as in the case of rotation 

 about a fixed axis (Art. 273). 

 Resolving this acceleration into 

 c i c its tangential and normal compo- 



nents we have for the acceleration 



of P the following three components : ar at right angles to CP t 

 coV along PC, and atu at right angles to CC^. 



