Question

A uniform U-tube kept vertical contains a very slightly mobile liquid of length $l$. If the liquid in one limb is depressed by $x$ and then released, find the period of oscillation of the liquid in the tube.

Answer 1

This problem can be solved by energy method

$0=\frac{dE}{dt}$

Where $E$,

$\frac{1}{2}\rho AI{V}^2+\rho Ag{x}^2$ [$V$-Velocity of liquid]

$\frac{dE}{dt}=\rho A[V\frac{dV}{dt}+2\rho Agx\frac{dx}{dt}]V\rho A(l\frac{d^{2}x}{d{t}^2}+2gx)=0$

General soution to this equation is

$x\left(t\right)=B\cos \left({\omega }_{0}t\right)+C\sin \left({\omega }_{0}t\right){\omega }_0=\sqrt{\frac{2g}{L}}T=\frac{2\pi }{\omega }T=2\pi \sqrt{\frac{L}{2g}}$