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