Vibrations and Waves · Problem 3-17

Forced vibrations and resonance — A.P. French

Physics — Waves · u-tube-oscillation

Problem

A U-tube has vertical arms of radii rr and 2r2r, connected by a horizontal tube of length ll whose radius increases linearly from rr to 2r2r. The U-tube contains liquid up to a height hh in each arm. The liquid is set oscillating, and at a given instant the liquid in the narrower arm is a distance yy above the equilibrium level. (a) Show that the potential energy of the liquid is given by U=58gρπr2y2U = \frac{5}{8}g\rho\pi r^2 y^2. (b) Show that the kinetic energy of a small slice of liquid in the horizontal arm (see the diagram) is given by dK=12ρπr2 dx(1+x/l)2(dydt)2dK = \tfrac{1}{2}\rho \frac{\pi r^2\, dx}{(1 + x/l)^2} \left(\frac{dy}{dt}\right)^2 (Note that, if liquid is not to pile up anywhere, the product velocity ×\times cross section must have the same value everywhere along the tube.) (c) Using the result of part (b), show that the total kinetic energy of all the moving liquid is given by K=14ρπr2(l+52h)(dydt)2K = \tfrac{1}{4}\rho\pi r^2(l + \tfrac{5}{2}h)\left(\frac{dy}{dt}\right)^2 (Ignore any nastiness at the corners.) (d) From (a) and (c), calculate period of oscillations if l=5h/2l = 5h/2.

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