IMPACT UPON BEAMS* 
97 
to that of the striking 1 body, is nearly equal to one third of the 
weight which would break the beam by pressure. 
This is shown to be the ease in Corollary to Problem 4; and 
in the impacts upon bodies suspended by wires (see remarks 
after our experiments upon them further on,) the maximum re¬ 
sults differ only from the conclusions of that Problem in giving 
the weight of the body struck a little higher. In this 7th Con¬ 
clusion, the weight of the striking body is assumed to be less 
than one third of what would break the beam by pressure. 
Other conclusions will be deduced from the problems and 
theorems which follow, and which are introduced, mostly, to 
compare their results with those of experiment. 
Prob. I. If a ball, or other body, be suspended like a pendu¬ 
lum, and made to strike horizontally, at its lowest point, against 
the middle of a beam, A B, supported at the ends : to find the 
quantity of recoil of the ball, and the time of straightening of 
the beam. 
We shall here suppose, as 
mentioned before, that the beam 
and ball have moved together 
as one mass from the time of 
the first contact to that of separation, when the beam has re¬ 
covered its original form A C B. 
* Let C D, the whole deflection of the beam caused by the im¬ 
pact, = b, any distance D E in the recoil = x, the correspond¬ 
ing velocity — v 9 the time = t , the inertia of the beam = r 9 
the weight of the ball — w, the chord of the arc ascended by 
the ball = c, the radius or length of the pendulum = /, the 
force of gravity = g. And let p be a pressure which, applied 
in the middle of the beam, would bend it through a distance e. 
Then -— = pressure of the beam at E, 
e 1 
And since vj -f- r is the mass moved, 
^ ^ = the accelerating force* 
e (tv + r) 
But by mechanics, 
d v _ gp (h — x) 
- 1 -. --— ■ -o 
a x e (w + r) 
v 
Integrating, 
tr 
hr 
gp 
(h x 
e(iv -f r) 
H 
1835 . 
