Question

A uniform cylinder of length L and mass M having cross – sectional area A is suspended with its vertical length, from a fixed point by a massless spring, such that it is half – submerged in a liquid of density d 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.
• B.
• C.
• D.

Answer 1

The correct option is C

Let the spring is further extended by y when the cylinder is given small downward push. Then the restoring force on the spring are,
(i) Ky due to elastic properties of spring
(ii) Up thrust = yAdg
$\therefore$ Total restoring force
F = (K + Adg)y
$=M\times a=-(K+Adg)y$
Comparing with $F=-{\omega }^{2}y$ we get
${\omega }^2=\left(\frac{K+Adg}{M}\right)$ or $\omega =\sqrt{\frac{K+Adg}{M}}$
$f=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{K+Adg}{M}}$