ANGULAR VELOCITY 287 



through an angle d6. Then the limit, when dt is made to vanish, 



of the rate is called the angular velocity of the body, it 

 dt 



measures the angle turned through per unit time. 



Thus to have a full knowledge of the motion of a rigid body at 

 any instant we must know 



(a) the direction and magnitude of the velocity of the point P 

 which has been selected to give a frame of reference ; 



(5) the direction of the axis of rotation through P ; 



(c) the magnitude of the angular velocity about the axis of 

 rotation. 



231. The angular velocity has associated with it a direction 

 the axis of rotation and a magnitude. Thus it may be repre- 

 sented by a line. We shall now prove that it is a vector, i.e. 

 that angular velocities may be compounded according to the 

 parallelogram law. 



Let a rigid body have a ro- 

 tation about P compounded 

 of (a) a rotation of angular 

 velocity o> about an axis PQ, 

 and (6) a rotation of angular 

 velocity o>' about a second 

 axis PQ'. Let the lengths 

 PQ, PQ' be taken proportional to co, co', so that the lines PQ, PQ f 

 will represent the directions and magnitudes of the angular 

 velocities on the same scale. 



Let the parallelogram PQRQ* be completed, and let L be any 

 point on the diagonal PR. Let LN y LN 1 be drawn perpendicular 

 to PQ,PQ' respectively. 



In time dt there is, from the first angular velocity, a rotation of 

 the rigid body through an angle CD dt about PQ. The effect of this 

 rotation is to move the particle of the body which originally coin- 

 cided with L through a distance LN- codt at right angles to the 

 plane PLN. Similarly the effect of the rotation about PQ' is to 

 move the same particle through a distance LN' co'dt at right 



