38 VELOCITY AND ACCELERATION. [CHAP. III. 



37. Angular velocity and acceleration. Suppose a line, 

 for example the line joining the positions at any time of two 

 moving points, moves so as always to be in the same plane with 

 reference to any frame. To fix ideas we shall take the plane 

 to be the coordinate plane of (#, y}. Suppose the line makes 

 an angle 0* with the axis x at time t, and an angle 6 + A0 

 with the same axis at time t + A. Then A0 is the measure of 

 the angle turned through by the line in the interval measured 

 by A*, and the limit of the ratio of these two numbers is 0, 

 the differential coefficient of 6 with respect to t. This number, 0, 

 is called the angular velocity of the line. In the same way 6 

 is called the angular acceleration of the line. 



The definition of the angular velocity of a line which does not 

 remain in one plane is deferred. 



38. Uniform circular motion. Suppose that, relative to 

 any frame, the path of a moving point is a circle which occupies a 

 fixed position relative to the frame, and suppose that the speed 

 is uniform. 



Let time be measured from the instant when the point was 

 at A on the circle, and let arc AP, = s, and 

 Z AGP, =6, be the arc described by the 

 point, and the angle described by the radius 

 vector, in the interval, t, from the instant 

 when the point was at A. 



Let a be the radius of the circle. Then 

 s = aO, and therefore s = a0. 



Hence, the speed s being assumed con 

 stant, the angular velocity 6 is also constant. 

 Now the acceleration parallel to the tangent, being the rate of 

 increase of s per unit of time, is zero. 



The acceleration parallel to the normal is - where p is the 



radius of curvature. For a circle of radius a, p = a, and we have 

 seen that s = a6. Hence the acceleration of a point describing 

 a circle with uniform speed is directed to the centre, and its 

 magnitude is a0 2 , where a is the radius, and 6 is the angular 

 velocity of the radius joining the centre to the point. 



* We shall generally take the angle to be measured in circular measure. 



