98

PENDULUM.

set of experiments; and the value of this term being found, the formula is extremelysimple and adapted to logarithmic computation. Since 2 sin £ a — chord of a, thevelocity is directly proportional to the chord of the arc of semi-vibration, and inverselyproportional to the product b i.

2. By the gun pendulum .—The same notation being employed as for the ballisticpendulum, the moment of the quantity of motion of the pendulum is

phi _ , ph /—r

w -— =2sin|o -— v g l-9 9

As the ball and wad leave the muzzle of the gun together, their quantity of motion is

(b + w 'l ” , w being the weight of the wad. The expansive force of the gunpowder

which produces this quantity of motion may be considered as acting on an area of agreat circle of the ball; and as it acts with equal intensity on the annulus betweenthe ball and the bore, driving out a portion of the elastic fluid past the ball, thequantity of motion will be increased by this circumstance in the proportion of thearea of the cross section of the bore to that of a great circle of the ball, or in theproportion of the square of the diameter of the bore to the square of the diameter ofthe ball. The quantity of motion, therefore, of the ball and wad, and of the fluidwhich escapes past the ball, all taken together, is

(b + w)v D 2

- x jT’

9 d 2

and its moment about the axis of suspension is

(b + w)vi D 29 d 2 ’

D and d being the diameters of the bore and ball respectively, and « being now thedistance from the axis of suspension to the axis of the gun.

Again, if c' be the weight of the cartridge, including the bag, and the elastic fluidbehind the ball be assumed to have a mean velocity equal to half that of the ball atthe moment the ball leaves the gun,* the quantity of motion of the inflamed powder

and of the cartridge-bag is represented approximately by i C —!L, and its moment

9

with respect to the axis of suspension by J

c'vi

9

After the ball has left the gun, the elastic fluid still continues to expand, and, inconsequence of the resistance of the air, to re-act on the pendulum and increase itsrecoil. The quantity of motion due to this cause may be considered proportional tothe quantity of powder in the charge, and may therefore be represented approximatelyc m

by c being the weight of the powder, and m a constant multiplier to be deter-mined by experiment. The moment, then, of this quantity of motion about the axisc mi

of suspension is --

The sum of the moments of all the quantities of motion of the ball and the chargeis therefore

{b + w)vi9

D 2 c'vi cmi

and this must be equal to the moment of the quantity of motion of the pendulum;so that

* Hutton, 37th Tract. Prob. 19.