IMPACT UPON BEAMS. 
103 
qj £ 
e : p : : e: q*, for then q = -—. Putting this for q gives 
, (w + r) (e — e') f p , 
h - - h -1 7rS e -e)-u 
.(C.) 
where e — e' is the deflection from impact, and ~ is constant 
in impacts upon the same beam. 
To obtain the value of h in terms of the weights, instead of 
6 ( T) — Q ) 
the deflections ; substituting for e — e’ its value, ——-—. we 
p 
obtain 
h ~ ( W + ^ ~ ^ . ( D 0 
When the beam is uniform, r — q nearly, as appears from 
Cor. 2. Prob. 2. 
If the beam be very light, q = o, r = o, and 
h = e (JL-i) 
\2 w ) 
Prob. 4. To find the weight of that beam which will bear the 
greatest impact from a given body falling upon it, the strength 
and flexibility of the beams being the same. 
From the last problem 
h ~ + r ) (P~9)(P~9 ~ 2 w )> 
and the question is to find q when h is a maximum. 
In beams of the same form the inertia will bear a constant 
ratio to the weight; and, therefore, in similar beams, we may 
substitute for r, in terms of q , in the preceding formula. Sup¬ 
pose a q = r, then 
h = (w + a ^ ( p-q-2w) 
maximum. 
Neglecting the constant multiplier, and calling the other part 
y, we have 
y = (w + a q) (p — q) (p — q — 2 w) = maximum. 
* This assumption is not strictly true: the deflection from the weight of the 
beam is a little greater than that due to a weight = q ; but the error is no 
greater than in the supposition that the deflection e in elastic beams is always 
as p, which is only the case in horizontal pressures. 
