H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. 77 



Now u is the velocity of the ball before impact : if this velocity be produced by a ver- 

 tical descent h, u* - 2gh, where g is the force of gravity. Whence (6) becomes 



m 



2 ^-ip>^ =a/2 (7) - 



Also by the geometrical properties of the circle, if r be the radius and c the chord, 



c 1 

 h ■ — . So that 

 8r 



»»£• - . rim- = «/ ........ (8). 



The principal mathematical formula arrived at by the above methods may be enunciated 

 as follows. Divide the weight of the ball by itself + ^yths of the weight of the beam. 

 Multiply the resulting fraction by twice the product of the weight of the ball in pounds by 

 the vertical distance of descent ; the result is equal to the square of the deflection multiplied 

 by the number of pounds which statically maintain one inch deflection. 



Hence we conclude that for a beam of assigned mass and elasticity struck by a ball of 

 given weight the deflection varies as 



1st. The velocity of impact directly ; from (6), 



2nd. The square root of the vertical distance of the ball's descent ; from (7), 



3rd. The chord of impact directly ; from (8). 



All these results are confirmed by the experiments above referred to. The comparison of 

 theory and observation in the accompanying table is extremely satisfactory, and has been made 

 for beams and balls of very different dimensions, and the agreement of the results under widely- 

 varying circumstances is so close as to leave nothing to be desired. 



When the beam is very flexible and subjected to great velocity of impact, parts of it will 

 recede with the blow and parts move in contrary directions. In this case the above investiga- 

 tions do not apply, and the problem becomes excessively difficult : but the difficulty is the less 

 to be regretted because in practice beams of great rigidity are always employed. 



