290 MOTION OF RIGID BODIES 



. The quantity V/7i> 2 is called the moment of inertia about the 

 axis PQ. 



If we introduce a quantity k, denned by 



so that k* is the mean value of p 2 averaged over all the particles of 

 the body, then k is called the radius of gyration about the axis PQ. 

 The kinetic energy can now be written in the form 



so that the energy is the same as if the whole mass were concen- 

 trated in a single particle at a distance k from the axis of rotation. 



KINETIC ENERGY OF A EIGID BODY 



234. The point P is at our disposal : let us suppose it to be the 

 center of gravity of the body. Then the most general motion may 

 be compounded of a motion of translation, this being identical with 

 that of the center of gravity, and a motion of rotation about an 

 axis through the center of gravity. 



Let V be the velocity of the center of gravity, let H be the 

 angular velocity, and let k be the radius of gyration about the 

 axis of rotation through the center of gravity. Let M be the total 

 mass, Vm, of the body. 



By the theorem of 186, the total kinetic energy of the body is 

 the sum of two parts : , 



(a) the kinetic energy of a single particle of mass M moving 

 with the center of gravity of the body ; 



(b) the kinetic energy of motion relative to the center of gravity. 



The value of part (a) is J. M V z ; that of part (b) is \ 

 Hence we have for the total kinetic energy 



This expression is of extreme importance in itself, but is also of 

 interest because it enables us to prove the following theorem. 



