236, 237] RIGID BODIES UNDER IMPULSES. 269 



their relative velocity before the impulse, and 6, 6' the angles which the 

 directions of the relative velocity, before and after, make with the line of 

 centres, prove that 



where M is the harmonic mean of the masses. 



12. Two small bodies of equal mass are attached to the ends of a rod of 

 negligible mass; the rod is turning uniformly about its centre, which is 

 supported, so that each of the bodies is describing a horizontal circle, when 

 one of the bodies is struck by a vertical blow equal in magnitude to twice its 

 momentum. Prove that the direction of motion of each of the bodies is in- 

 stantaneously deflected through half a right angle. 



*237. Impulsive motion of rigid bodies. The theory already 

 explained in this Chapter and the theory of the momentum of a 

 rigid body considered in Article- 218 are sufficient for the discussion 

 of the impulsive motion of rigid bodies in two dimensions. 



For each body we have three equations of impulsive motion 

 expressing that the change of momentum of the body is equivalent 

 to the impulses exerted upon it. 



The momentum of the body was shown to be equivalent to 

 a single vector localised in a line through the centre of inertia, 

 and equal to the momentum of the mass of the body moving with 

 the centre of inertia, and a couple of amount equal to the product 

 of the angular velocity of the body and the moment of inertia 

 about an axis through the centre of inertia perpendicular to the 

 plane of motion. 



Let ra be the mass of the body U, V the resolved velocities of 

 the centre of inertia in two rectangular directions in the plane of 

 motion, and H the angular velocity before impact ; let u, v be the 

 resolved velocities of the centre of inertia in the same two directions 

 after impact, and co the angular velocity ; also let k be the radius 

 of gyration of the body about an axis through the centre of inertia 

 perpendicular to the plane. 



Then the change of momentum of the system can be expressed 

 as a vector localised in a line through the centre of inertia, whose 

 resolved parts in the two specified directions are m (u U) and 

 m(v V) ; and a couple, in the plane of motion, of moment 



mk? (o> - fl). 



