A cylindrical block of mass M and area of cross section A is floating in a liquid of density $$\rho$$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ___

If the block is pushed down by a small distance x,

extra volume submerged is

So extra buoyant force upward is

$$F_b=\rho g(Ax)$$

This acts as restoring force:

$$F=-\rho gAx$$

which is of SHM form

$$F=−kx$$

with effective spring constant

$$k=ρgA$$

For SHM,

$$T=2\pi\sqrt{\frac{M}{k}}$$

Substituting $$k=ρgA$$

we get

$$2\pi \sqrt{\frac{M}{\rho Ag}}$$

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