Question

A U-tube is of non uniform cross-section. The area of cross sections of two sides of tube are $A$ and $2A$ (see fig.). It contains non-viscous liquid of mass $m$. The liquid is displaced slightly and free to oscillate. Its time period of oscillations is

• A. $T=2\pi \sqrt{\frac{m}{3\rho gA}}$
• B. $T=2\pi \sqrt{\frac{m}{2\rho gA}}$
• C. $T=2\pi \sqrt{\frac{m}{\rho gA}}$
• D. None of these

Answer 1

Answer: (A)

Suppose the liquid in left side limb is displaced slightly by $y$, the liquid in right limb will increase by $y/2$. The restoring force
$F=-PA$
$=-\rho g\left(\frac{3y}{2}\right)\times 2A=3\rho gA(-y)$
$a=\frac{F}{m}=3\rho gA(-y)/m$
On comparing with, $a=-{\omega }^{2}y$, we get
$\omega =\sqrt{\frac{3\rho gA}{m}}\text{ and }T=2\pi \sqrt{\frac{m}{3\rho gA}}$