If R1 = R2, which is the case when the resistance, as well as the effort, arises from the mutual actions of the two bodies, the above becomes,
E : E1 : E2
:: W1 + W2 : W2 : W1,
(81)
that is to say, the energy is exerted on the bodies in shares inversely proportional to their weights; and they receive accelerations inversely proportional to their weights, according to the principle of dynamics, already quoted in a note to § 110, that the mutual actions of a system of bodies do not affect the motion of their common centre of gravity.
For example, if the weight of a gun be 160 times that of its ball 160⁄161 of the energy exerted by the powder in exploding will be employed in propelling the ball, and 1⁄161 in producing the recoil of the gun, provided the gun up to the instant of the ball’s quitting the muzzle meets with no resistance to its recoil except the friction of the ball.
§ 124. Centre of Percussion.—It is obviously desirable that the deviations or changes of motion of oscillating pieces in machinery should, as far as possible, be effected by forces applied at their centres of percussion.
If the deviation be a translation—that is, an equal change of motion of all the particles of the body—the centre of percussion is obviously the centre of gravity itself; and, according to the second law of motion, if dv be the deviation of velocity to be produced in the interval dt, and W the weight of the body, then
| P = | W | · | dv |
| g | dt |
(82)
is the unbalanced effort required.