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 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

(C) Larger than 2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}

(D) 2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}

JEE Mains · Physics · STD 11 - Oscillations

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

Pick an option to check your answer.

Verified Solution

Correct Answer

(A) Smaller than \(2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\)

Detailed explanation

If \(x\) is the displacement then,

\(M{\omega ^2}x = \left[ {\rho Ag + k} \right]x\)

\( \Rightarrow \omega = {\left[ {\frac{{\rho Ag + k}}{m}} \right]^{1/2}}\)

\( \Rightarrow T=2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\)

It Should be less than \(2\pi {\left[ {\frac{M}{{\left( {k + A\sigma g} \right)}}} \right]^{1/2}}\)

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