232 MOTION OF A RIGID BODY IN TWO DIMENSIONS. [CHAP. XI. 



218. Velocity and Momentum of rigid body. 



Fig. 55. 



Let G be the centre of inertia of a rigid body moving in two 

 dimensions, and let u and v be resolved parts of the velocity of G 

 parallel to the axes of x and y. Let P be any other particle of the 

 body, r its distance from G, and x', y' its coordinates relative to G at 

 time t. Then the line GP is turning with the angular velocity &> 

 of the rigid body, and the velocity of P relative to G is rco at right 

 angles to GP ; the resolved parts of this relative velocity parallel to 

 the axes are coy' and MX', since the line GP makes with the axis 

 of x an angle whose cosine is af/r and whose sine is y'jr. 



Hence the resolved velocities of P parallel to the axes are 

 u wy' and v + cox'. 



Let m be the mass of the particle at P. Then the linear 

 momentum of the body parallel to the axis of x is 



2m (u ay') = Mu, 



where M, 2m, is the mass of the body. Similarly the linear 

 momentum of the body parallel to the axis of y is Mv. Thus the 

 linear momentum of the body in any direction is the same as that 



