A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $$\sigma$$ at equilibrium position. When the cylinder is given a downward push and released, it starts oscillating vertically with a small amplitude. The time period T of the oscillations of the cylinder will be :

The forces balancing the cylinder at rest are $$Mg = kx_0 + F_{b0}$$

When pushed down by a small distance $$x$$, both the spring and buoyancy act to push it back up:

Extra spring force: $$-kx$$ and Extra buoyant force: $$-A\sigma gx$$

Total restoring force: $$F = -(k + A\sigma g)x$$

Using $$F = Ma$$, the motion fits standard SHM ($$a = -\omega^2 x$$):

$$\omega = \sqrt{\frac{k + A\sigma g}{M}} \implies T_{\text{ideal}} = 2\pi \left[ \frac{M}{k + A\sigma g} \right]^{1/2}$$

In a real liquid, viscous drag and the added mass of the surrounding fluid accelerated by the cylinder both reduce the frequency of oscillation. This makes the actual time period longer than the ideal case:

$$T_{\text{actual}} > 2\pi \left[ \frac{M}{k + A\sigma g} \right]^{1/2}$$