ANGULAB VELOCITY 289 



Thus if a/ is- represented by PQ', then H will, on the same scale, 

 be represented by PR. 



We have now proved the following : 



The resultant of two angular velocities represented by the edges 

 PQ, PQ' of a parallelogram is an angular velocity represented by 

 the diagonal PR of the parallelogram. 



Thus angular velocity is a vector, and possesses the properties 

 which have been proved to be true of all vectors. 



232. It follows that an angular velocity fl about an axis of 

 rotation of which the direction cosines are I, ra, n may be replaced 

 by three angular velocities m lf a> 2 , o> 3 about the axes of coordinates, 

 such that 



o> 3 = nQ. (118) 



Squaring and adding, we find that 



ft 2 = cof + o> 2 2 + o> 3 2 . (119) 



We now see that the motion of a rigid body is given when we know 



(a) u, v, w, the components of velocity of the point P; 



(b) (o lf o> 2 , ew 3 , the components of angular velocity. 



KINETIC ENERGY OF KOTATION 



233. Suppose that at any instant a rigid body is rotating about 

 an axis of rotation PQ with 

 angular velocity O. 



Let L be any particle of 

 the body, its mass being m, 

 and let LN, the perpendicu- 

 lar distance from L to PQ, 

 be denoted by p. Then the 

 velocity of the particle L is 

 pl, and its kinetic energy p^ 140 



is Jra/n 2 . 



On summation, the kinetic energy of the whole body is seen to be 



