Question

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

• A. $\frac{1}{2\pi}\bigg(\frac{k-A\rho g}{M}\bigg)$
• B. $\frac{1}{2\pi}\bigg(\frac{k+A\rho g}{M}\bigg)^{1/2}$
• C. $\frac{1}{2\pi}\bigg(\frac{k+\rho gL^2}{M}\bigg)^{1/2}$
• D. $\frac{1}{2\pi}\bigg(\frac{k+A\rho g}{A\rho g}\bigg)^{1/2}$

Answer 1

Answer: (B)

$\frac{1}{2\pi}\bigg(\frac{k+A\rho g}{M}\bigg)^{1/2}$

Answer 2

When cylinder is displaced by an amount x from its mean
position, spring force and upthrust both will increase.
Hence,
Net restoring force = extra spring force + extra upthrust
or $F=-(kx+Ax\rho g)$
or $a=-\bigg(\frac{k+\rho Ag}{M}\bigg)x$
Now, $f=\frac{1}{2\pi}\sqrt{\bigg|\frac{a}{x}}\bigg|$
$=\frac{1}{2\pi}\sqrt{\frac{k+\rho Ag}{M}}$
$\therefore$ Correct optionis (b)