Question

A $u$-tube of uniform cross-section is kept in a vertical position. A liquid of mass $m$ and density $d$ is filled in one of its limbs. This liquid will oscillate in this tube with time period $T$ given by

• A. $T=2\pi \sqrt{\frac{m}{gda}}$
• B. $T=2\pi \sqrt{\frac{ma}{gd}}$
• C. $T=2\pi \sqrt{\frac{m}{g}}$
• D. $T=2\pi \sqrt{\frac{m}{2agd}}$

Answer 1

Answer: (D)

$T=2\pi \sqrt{\frac{m}{2agd}}$

Equation of motion for the liquid can be given by

$m\frac{d^{2}x}{d{t}^2}=-2xadg$, where a is the cross section of the u tube.

$\Rightarrow \frac{d^{2}x}{d{t}^2}+\frac{2adg}{m}x=0$

$\Rightarrow \frac{d^{2}x}{d{t}^2}+{\omega }_0^{2}x=0$,

where ${\omega }_0^2=\frac{2adg}{m}\Rightarrow \frac{2\pi }{T}=\sqrt{\frac{2adg}{m}}\Rightarrow T=2\pi \sqrt{\frac{m}{2adg}}$

Ans: D