ssagwsp

TRACT 38.

PROBLEMS.

343-

tance descended 99 being the same in both. For, though theweight l moved in this latter case, be double of what it wasin the former, the moving force x is also double, becausehere the one end of the chain shortens as much as the otherend lengthens, so that the space descended x is doubled,and becomes x ; and hence the accelerative force ~ orfisthe same in both; and of course the velocity and time thesame for the same distance'descended.

PROBLEM XIX.

To find the Number of Vibrations made by two Weights,connected by a very fine Thread, passing freely over a Tackor a Pulley, while the less Weight is drawn up to it by the de-scent of the heavier Weight at the other End.

Suppose the motion to commence at equal dis~ Atances below the pulley at b ; and that the weightsare 1 and 2 pounds.

Put a — ab, half the length of the thread ;

b zz 39-J-inc. or 3g-|feet, the second’s pend.x — b w = bw, any space passed over;z = the number of vibrations.

Then —f ~ -j is tb e accelerating force. And hence

But, by the

V*sf*

v or >/±gfs = flgfo, and t or ~ =

nature of pendulums, */(a±x ) : Vb : : 1 vibr. : V-^r x thevibrations per second made by either weight, namely, thelonger or shorter, according as the upper or under sign isused, if the threads were to continue of that length for 1

second. Hence then, as

„ , b . * , 1 . b x

1 ” : t: : — X - . ,

the fluxion of the number of vibrations.

Now when the upper sign + takes place, the fluent is

* = vA x I v --' i T Ln -- = * i. g - + - 2y + 2V(g3r + a3 l

v igf iga a

*&T

H

/