230-232] 



GENERALISED NEWTON S RULE. 



261 



the &quot; generalised Newton s rule/ and we shall apply it instead of 

 Poisson s hypothesis. We shall show later on that its equivalence 

 with this hypothesis is not limited to the cases of smooth bodies or 

 of spheres. 



231. Elastic systems. The method followed in applying this rule is to 

 treat the impact as instantaneous, and the impinging bodies as rigid both 

 before and after it. This method is adequate for the discussion of many 

 questions. It cannot however be applied to the transmission of stress in 

 elastic systems capable of large deformations. In such systems no internal 

 stress is developed except after a finite deformation has taken place, so that 

 at the beginning of a motion impulsively produced some part of the system 

 yields at once, and starts to move with a finite velocity ; after a finite time a 

 finite strain is produced, and is opposed by a finite elastic stress, this stress 

 continuing as long as there is any strain. This statement may conveniently 

 be summed up in the proposition: An elastic system cannot support an 

 impulse. It is now clear that the method founded on Newton s result is of 

 the nature of a compromise, the whole time of the action in which the 

 elasticity of the bodies is concerned being treated as infinitesimal. An 

 example of the statement that an elastic system cannot support an impulse 

 will be found in the action of elastic strings attached to rigid bodies whose 

 motion is altered suddenly. There is no impulsive tension in such a string, 

 and the motion of the body immediately after the impulse is exactly the 

 same as if the string were not attached to it (cf. Article 258). On the other 

 hand, an inelastic string is conceived as capable of supporting an impulsive 

 tension. 



232. Impact of smooth spheres. We return to the problem 

 presented by the collision of 

 two smooth uniform spheres, 

 and we shall show how to ap 

 ply the generalised Newton s 

 rule and the equations of 

 momentum to determine the 

 whole motion, and shall es 

 timate the loss of kinetic 

 The notation is the 



energy 



same as that in Article 230. 



Fig. 67 (bis). 



The generalised Newton s Rule gives the equation 

 u-u =-e(U-U). 



The equation of conservation of momentum parallel to the line 

 of centres is 



mu + tn u = m U + m U . 



