Question

A cylinder of length \(L\) and density \(\rho\) is floating in a large tank with water (with density \(\rho_{\mathrm{w}}\) ) and oil (with density \(\rho_{\mathrm{o}}\) ). The cylinder is initially at rest. It is then pushed downward slightly and released, such that it undergoes a vertical oscillatory motion around the initial position. Assume that the fluids are incompressible and inviscid. Throughout the motion, the cylinder is always partially in the water, and partially in the oil.

(a) Express the acceleration of the cylinder in terms of the vertical displacement and the parameters.

(b) Write down the solution of the simple harmonic motion (SHM).

(c) What is the period of the oscillation?

(d) What is the period in the limit \(\rho_{\mathrm{w}} \rightarrow \rho_{\mathrm{o}}\) ? Explain the meaning of the limit.

Answer 1

1

So @ Now; mg = {wAh + & A (L-h) Now, It is displaced I downward then Frut = &WA (hty) + 8A (L-b-y) - mg Fuet - Sway Bay Fnet - Ay (Par &] (in verrical direction) Pret - - Ay (Pw-P) a = - Ay (8w-lo) SO a = PAL -Y (dw-8) IL w? - Pw-lo so, we PL so= ib so, W = √ PE A amplitude

Time period 2 vk T=21 SA2 Alw-8) PL T= 20 View to -) If Pwl then TD 1+ will not oscillates It will Sol semain stationary