237-240] RIGID BODIES UNDER IMPULSES. 271 



products of the external impulses and the arithmetic means of the 

 velocities of their points of application resolved in their directions 

 before and after. 



Now the theorem of Article 155 asserts that the change of 

 kinetic energy is equal to the value of the like sum for all the 

 impulses internal and external. It follows that the internal 

 impulses between the parts of a rigid body, which undergoes 

 a sudden change of motion, contribute nothing to this sum. 



*239. Examples. 



1. A uniform rod at rest is struck at one end by an impulse at right 

 angles to its length. Prove that, if the rod is free, it begins to turn about 

 the point of it which is distant one-third of its length from the other end, 

 and that the kinetic energy generated is greater than it would be if the other 

 end were fixed in the ratio 4 : 3. 



2. A free rigid body is rotating about an axis through its centre of inertia 

 for which the radius of gyration is k when a parallel axis at a distance c 

 becomes fixed. Prove that the angular velocity of the body is suddenly 

 diminished in the ratio & 2 



3. An elliptic disc is rotating in its plane about one end P of a diameter 

 PP', when P' is suddenly fixed. Find the impulse at P and the angular 

 velocity about it, and prove that, if the eccentricity exceeds </> the diameter 

 PP' may be so chosen that the disc is reduced to rest. 



4. A uniform rod of length 2a and mass m is constrained to move with 

 its ends on two smooth fixed straight wires at right angles to each other, and 

 is set in motion by an impulse of magnitude mV. Prove that the kinetic 

 energy generated is f ra F 2 jo 2 /a 2 , where p is the perpendicular from the inter- 

 section of the fixed wires on a line parallel to the line of the impulse and 

 such that the centre of inertia is midway between the two parallels. 



*240. Rigid bodies with restitution. Let two rigid bodies moving in 

 the same plane come into contact at a point P and suppose the bodies to be 

 smooth at P. Let R be the impulsive pressure between the bodies at P. 

 The direction of R is the common normal at P to the two surfaces. Let the 

 axis of x be taken in this direction, the axis of y being any fixed line in a 

 perpendicular direction. 



Let m and m' be the masses of the bodies, U, F, Q, the velocity system of 

 m before impact, u, v, to corresponding quantities after impact, and let 

 accented letters denote similar quantities for m'. Also let #, y be the co- 

 ordinates of the centre of inertia of m and x', y' those of m' at the instant 

 of impact, and let , rj be coordinates of P at the same instant. Also suppose 

 that, as acting on m, the sense of R is the negative sense of the axis of x. 



