I5 8 KINETICS OF A RIGID BODY. [294. 



must be used. Now, for rotation of angular velocity co about 

 the axis of #, we have ^= (*>y, y = wx. Hence 



^m (xy yx) w^m (x* -fj/ 2 ) = w'Snir* = &>/. 

 The equation assumes, therefore, the form (i). 



294. The reactions of the fixed axis do not enter into the 

 composition of the resulting moment H. As they intersect the 

 axis, their moments about this axis are zero. 



The student should notice the close analogy between equa- 

 tion ^ and the equation for the rectilinear motion of a particle, 



dv_F 

 j. > 

 at m 



where v is the velocity and F the resultant of all the forces act- 

 ing on the particle. 



The expression for the kinetic energy of a body rotating about 

 a fixed axis is 



T=S%mv*=2%mrfr* = %fco*, (2) 



and has also a form similar to that for the kinetic energy of a 

 particle m moving with velocity v in a straight line, viz. 





295. Let us denote the angle of rotation by 0, so that 

 <*> = d6/dt, d<ti/dt=d 2 d/dt\ If the resulting moment be con- 

 stant or a given function of 6, say //=/(#), the equation of 

 motion 



can be integrated once, and gives 



where &> is the angular velocity corresponding to the angle . 



This is the equation of kinetic energy. It might have been 

 derived directly, according to Art. 234, by expressing that the 

 increase of the kinetic energy equals the work of the forces. 



