350

miscellaneous

TRACT 38.

PROBLEM XXIII.

It is proposed to determine the Velocity, and the Time of Vi-bration, of a Fluid in the Arms of a Canal or bent Tube.

Let the tube abcdef have its twobranches ac, ge vertical, and the lowerpart cdk in any position whatever, thewhole being of a uniform diameter orwidth throughout. Let water, or quick-silver, or any other fluid, be poured in,till it stand in equilibrio, at any hori-zontal line BF. Then let one surface be pressed or pusheddown by shaking, from b to c, and the other will ascendthrough the equal space FG; after which let them be per-mitted freely to return. The surfaces will then continuallyvibrate in equal times between ac and f.g. The velocity andtimes of which oscillations are therefore required.

When the surfaces are any where out of a horizontal line,as at p and q, the parts of the fluid in odr, on each side,below qr, will balance each other ; and the weight of thepart in pr, which is equal to 2pf, gives motion to the whole.So that the weight of the part 2pf is the motive force by

which the whole fluid is urged, and therefore ° f ~ FF is the

° whole wt.

accelerative force. Which weights being proportional totheir lengths, if l be the length of the whole fluid, or axis ofthe tube filled, and a = fg or bc ; then is the accelera-tive force. Putting theref. t=gp any variable distance, v thevelocity, and t the time; then pf = a — x, and L - ■ =: f

the accelerative force ; hence w or 2 gfs =. (ax — xx) ;the fluents of which give .if = ~(2ax — x~), and v :=f(4g x '—— l ——) is the general expression for the velocity

at any term. And when x — a, it becomes v = 2a f ~ forthe greatest velocity at b and f.