344

MISCELLANEOUS

TRACT 38. v

PROBLEM XXI.

To find the Velocity and Time of Vibration of a smallWeight, fixed to the middle of a Line, or fine Thread void ofGravity, and stretched by a given Tension, the Extent of theVibration being very small.

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Let l — Ac half the length of the thread ;a — cd the extent of the vibration ;x = ce any variable distance from c;w — \vt. of the small body fixed to the middle ;

W — a wt. which, hung at each end of the thread, willbe equal to the constant tension at each end, act-ing in the direction of the thread.

Now, by the nature of forces, ae : ce : : w the force indirection EA : the force in direction ec. Or, because AC isnearly = ae, the vibration being very small, taking ac in-stead of ae, it is ac : ce : w : the force in ec arising

from the tension in ea. Which will be also the same forthat in eb. Therefore the sum is = the whole motiveforce in ec arising from the tensions on both sides. Conse-quently = f the accelerative force there. Hence the

equation of the fluxions vv or 2 gfs = ; and the flus.

i _ Vv* g iit w j ien x _ a tiji s _ 4?w . “ . and should

lw lw

a? — x 9

be = 0; theref. the correct fluents are v z = 4 gw x —y— >and hence v — ^/(4^w X —j~~) the velocity of the little

body w at any point e. And when x — 0, it is v = 2a V—for the greatest velocity at the point c.