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 (rho) at equilibrium position. When the cylinder is given a small downward push and released it starts oscillating vertically with small amplitude. If the force constant of the spring is (k), the prequency of oscillation of the cylindcer is.

Explanation

`l` : extension in spring
The cylinder is in equilibrium
`kl + sigma A (L)/(2) g = Mg` …(i)
Let at some instant `((L)/(2) + y)` length of cylinder is inside liquid
Restoring force
`F = -[k(l + y) + sigma A((L)/(2) + y) g - Mg]`
=`-(k + sigma Ag)y`
Acceleration : `a = (F)/(M)`
`a = -((k + sigma Ag)/(M))y`
`a prop-y` (case of SHM)
Time period : `T = 2pi sqrt((M)/(k + sigma Ag))`
Frequency : `f = (1)/(T) = (1)/(2 pi) sqrt((k + sigma Ag)/(M))`.

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