Question

A U-tube is of non uniform cross-section. The area of cross sections of two sides of tube are $A$ and $2 A$ (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. $T=2 \pi \sqrt{\frac{m}{3 \rho g A}}$
• B. $T=2 \pi \sqrt{\frac{m}{2 \rho g A}}$
• C. $T=2 \pi \sqrt{\frac{m}{\rho g A}}$
• 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
$F =-P A$
$=-\rho g\left(\frac{3 y}{2}\right) \times 2 A=3 \rho g A(-y)$
$a=\frac{F}{m}=3 \rho g A(-y) / m$
On comparing with, $a=-\omega^2 y$, we get
$\omega=\sqrt{\frac{3 \rho g A}{m}} \text { and } T=2 \pi \sqrt{\frac{m}{3 \rho g A}}$