68 KINEMATICS. L J 34- 



and when the initial velocities are V Q and &> , respectively : 



FOR TRANSLATION : FOR ROTATION : 



134. Let a point P, whose perpendicular distance from the 

 axis of rotation is OP=r, rotate about the axis with the angular 

 velocity w = dB/dt. In the element of time, dt, it will describe 

 an element of arc ds=rd8=r&dt. Its velocity v = ds/dt (fre- 

 quently called its linear velocity in contradistinction to the 

 angular velocity) is therefore related to the angular velocity of 

 rotation by the equation 



v=wr. . (8) 



135. The radius vector OP=r sweeps over a circular sector 

 which in uniform rotation has an area S=^0r^ = ^a)^, while in 

 variable rotation the infinitesimal sector described during the 

 element of time dt is dS^r^dQ^^dt. 



The quotients 



: ' _ |=l^ = la,^ (9) 



for uniform rotation, and 



for variable rotation, represent, therefore, the sectorial, or areal, 

 velocity, i.e. the rate of increase of area with the time. 

 The rate of change of this velocity with the time, 



, . 



dt*dt ' 



is called the sectorial, or areal, acceleration. 



