THE FORCE OF

76

y^ 4 • 3 ^ ~F. ^—_£. 32. j 8 ^ (Philos, Instit.) which mul-

tiplied by the weight of the pendulum and ball, since this

does not rebound, i. e. by P ad+qd i , will give - x P ad+ qd *

X \/ 32. 18^ for the quantity of motion. Let « be theuniform velocity with with the ball impinged on the pendu-lum ; as from the moment it entered the pendulum, it fol-lowed the direction of its motion, uqd % will express thequantity of the. motion of the ball: thus in a state of

equilibrium uqd ’

— — X P ad + qd 3b

v/32. 18 d, and

will be the velocity sought.

e X P a + qd Z2- 1

bqd

162. In the second place, let the ball strike a point D, notin the center of oscillation: (Fig. 18) then if f express theperpendicular distance from the point D to the axis of mo-tion, and x the uniform velocity of this point, the quantityof motion of the point D will be *xP ad+qf*, and if theball impinged on the pendulum with the velocity », its quan-tity of motion will be uqf*. Then the equation will be

- uq f 2,

xx Pad + qf*= uqf x , and x — + Now as the

motion of a pendulum is as great* as if its whole mass wereunited to the centre of oscillation, and as this centre changeswhenever the ball impinges on any other point; by express-ing its distance from the axis of motion by the vis inertiæPad T qf 1 divided by P a + qf., the distance of C G fromthe point G taken as a new centre of oscillation to the axis

of motion will be C G = —T^TT* But the radii CD,

Pa + qf '

C G, are proportional to the velocities of the points D, G,

since they describe the archs DO, G H in the fame time ;

then CD : CG :/: ^— T"qf ~ ^ velocity of the

point D = # = _ u lf__ -js to the velocity of the point

^ Pad+qf*

G =-UJL-

''P7+ qf^^' 36 Gi

versed sine of the arch G H.

where G I

expresses theTo