A long cylindrical tube of length L, and density ?t, is very slightly weighted on one end so that it floats upright in a fluid having density ?f. Assume the tube is constant density and the slight weighting on one end is very small (negligible), with the only purpose to stablize the cylinder vertically in the fluid. It is pushed down a distance x from its equilibrium position and released.
(a) Show that it will execute simple harmonic motion if the resistive effects of the fluid are neglected, and
(b) determine the period of the oscillation in terms of L, t, and f.