76 H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. 



Substituting this value of P in equation (3), 

 And consequently 



m(u — v) = — Mv. 



' 35 



m 

 v = — — — . u (4). 



m + — M 



35 



From (2) it appears that the total vis viva after impact or 



•■I 



f#|Mf_17/M B H w 



f 35 f 35 



and the total vis viva of the beam and ball together is 



mv" + — Mv 2 = (from 4) — — — mu? (5). 



35 v ' 17 K ' 



m + —M 



35 



Adopting, then, the elastic curve to represent the initial velocities of the several parts of 

 the system, effecting the integration and supplying the numerical calculations, we find ulti- 

 mately that rather less than one half the inertia of the beam may be supposed to act initially 

 to resist the ball ; or, to speak more precisely, that at the instant after impact the impinging 

 ball loses as much of its motion as it would have done if it had impinged on another free ball 

 having 17-35ths of the mass of the beam. From this conclusion it is easy to infer, as in the 

 accompanying formula, the total vis viva of the system after impact. 



(2) The second part of the problem consists in determining the effect of the elastic forces 

 developed in the beam by deflection. At the end of the deflection the whole system is sup- 

 posed to be brought to rest simultaneously, and by the principle of the conservation of vis viva, 

 the whole work done by the elastic forces is equal to half the vis viva destroyed. 



It seems safe to assume that the elastic forces are functions of the distances between the 

 particles of the beam and not of their velocities. This assumption is made in investigations of 

 vibrating cords and rods, of which the results are confirmed by experiment. 



If, then, the elastic forces of the beam vary as the extension and compression directly, the 

 work done in bending the beam into a particular form will be the same whether the particles 

 move intermediately with a greater or less velocity. Now when the beam is deflected statically 

 through a certain distance at its centre, the deflecting pressure is to the distance of deflection 

 in a nearly constant ratio which is usually determined by ascertaining experimentally the 

 number of pounds weight which will statically maintain a deflection of one inch. 



Let a be that weight, f the deflection in inches, af is the pressure necessary to maintain, 

 and ^a/ 2 the work necessary to produce the deflection. 



Consequently 1 a/ 2 will be the work done in deflecting the beam after impulse, if / be the 

 central deflection, and the final form of the beam be that which it would statically assume. 

 Therefore from (5) by the principle of vis viva as explained 



^ .m« 2 = a/ 2 (6). 



ft M+m 



