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

Answer 1

Answer: (B)

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

Answer 2

F = – {kY + AYrg} Ž Ma = – {k + Arg} Y

a = – $\left\{ \frac{k + A\rho g}{M} \right\} Y$ Ž f = $\frac { 1 } { 2 \pi }$ $\sqrt{(K + A\rho g)/M}$