218, 219] MOMENTUM OF RIGID BODY. 233 



of a particle of mass equal to the mass of the body placed at the 

 centre of inertia and moving with it. (Article 100.) 



The moment of momentum of the body about an axis through 

 the centre of inertia perpendicular to the plane of motion is 



2m [x f (v + cox') -y' (u coy')} 



= coZm (x"> + y'*) = M fcX 

 where k is the radius of gyration about the axis. 



The moment of momentum about any parallel axis is the 

 moment about that axis of the momentum of the whole mass 

 placed at the centre of inertia and moving with it together with 

 the moment Mk-a> (Article 102). Thus the momentum of the 

 rigid body is specified by the resultant and couple of a system of 

 vectors localised in lines. The resultant has resolved parts Mu, 

 Mv in the two chosen directions, and the moment of the couple is 



Again, the kinetic energy of the body is 



2ra {(u - coy') 12 + (v + cox')*} 



which is the kinetic energy of the whole mass moving with the 

 centre of inertia together with the kinetic energy of the rotation 

 about the centre of inertia (Article 104). 



The formulae for the velocity of a point show that at each 

 instant the point whose coordinates relative to G are v/to and 

 u/co has zero velocity, so that the motion of the body at the 

 instant is a motion of rotation about an axis through this point 

 perpendicular to the plane of motion. The point is called the 

 instantaneous centre of no velocity, or frequently " the instantaneous 

 centre." The fact that the motion of the body at each instant is 

 equivalent to rotation about a point is of importance in many 

 geometrical investigations. 



219. Kinetic Reaction of rigid body. With the notation 

 of the last Article, the point P moves relatively to G in a 

 circle of radius r with angular velocity equal to co at time t ; its 

 acceleration relative to G may therefore be resolved into rco at 

 right angles to GP, and r<w 2 along PG. Hence the resolved parts 



