A vertical `U-tube` contains a liquid. The total length of the liquid column inside the tibe is `1`. When the liquid is in equilibrium, the liquid surface in one of the arms of the `U-tube` is pushed down slightly and released. The entire liquid column will undergo a periodic motion.

Options

A. The motion is not simple harmonic motion
B. The motion is simple harmonic motion
C. If it undergoes simple harmonic motion, the time period will be `2 pi sqrt(l//g)`.
D. If it undergoes simple harmonic motion, the time period will be `2 pi sqrt(l//2 g)`.

Answer

D

Explanation

Let `a` : area of cross-section,
`rho` : density of liquid
Mass of liquid in the tube `m = rho al`
Suppose liquid in left column be depressed by `y`, level
in right column increases `y` as shown
Restoring force `F = -rho a. 2 y g`
Acceleration of mass `a = (F)/(m) = -(2 rho ag y)/(rho al)`
`a = - (2 g)/(l) y = - omega^2 y` (case of SHM)
Time period `T = 2 pi sqrt((l)/(2g))`.