Question

A U tube held vertically contains liquid column of height $L$. If the liquid in one of the limbs is depressed and released, it oscillates with a period of

• A. $2\pi \sqrt{\frac{L}{g}}$
• B. $2\pi \sqrt{\frac{3L}{g}}$
• C. $2\pi \sqrt{\frac{2L}{g}}$
• D. $2\pi \sqrt{\frac{g}{2L}}$

Answer 1

The correct option is A $2\pi \sqrt{\frac{L}{g}}$

$Pressure=-h\rho g$

$Force=$ pressure $\times$ cross section area

$F=-h\rho g\times A$---(1)

Mass of liquid column

$m=L\times A\times \rho$

$\therefore F=m\times a=LA\rho a$ ---(2)

from (1)and (2)

$-h\rho gA=LA\rho a$

$a=\frac{-gh}{L}$

$so,{\omega }^2=\frac{g}{L}$ (comparing with $a=-{\omega }^{2}x$)

and $T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{L}{g}}$