Question

A liquid of density 2p is filled in a cylindrical vessel whose cross sectional area is 2A. A wooden cylinder of height H cross section area A and density p is floating in the liquid at equilibrium with its axis vertical. The cylinder is pushed down by a small distance x from its equilibrium position and released. Find its initial acceleration



Answer 1

After the displacement the volume of the part of the cylinder under the initial water level increased by

$$\delta V = A \cdot x$$

thus the water level rose up for compensation:

$$x_{\mathrm{water}} = \frac{\delta V}{2A - A} = x$$

so the total volume under the water is 2Ax. Due to Archimedes principle, the force upward is

$$F = \rho_{\mathrm{water}} g (2Ax)$$

so the acceleration is

$$a = \frac{ \rho_{\mathrm{water}} g \cdot 2Ax }{m} = \frac{ 2p g \cdot 2Ax }{pAH} = 4g \frac{x}{H}$$