Ncert Exercises & Solutions

Motion of an oscillating liquid column in a U-tube is

1.	periodic but not simple harmonic
2.	non-periodic
3.	simple harmonic and time period is independent of the density of the liquid
4.	simple harmonic and time period is directly proportional to the density of the liquid

(3) Hint: Motion of an oscillating liquid column in a U-tube is simple harmonic and the time period is independent of the density of the liquid.

Step 1: Find the time period of the liquid.

Consider a U-tube filled with a liquid of density

ρ

up to height h.

When the liquid is raised by height y in one arm, it goes down by a height y in another arm.

The pressure difference in the two arms =

ρg (2 y)

The weight of the liquid =

ρgA (2 y)

So, the restoring force = ma = -

ρgA (2 y)

\Rightarrow a = - \frac{ρgA (2 y)}{m} = - \frac{ρgA (2 y)}{ρA (2 L)} = - \frac{g}{L} y

So, the time period,

T = 2 π \sqrt{\frac{L}{g}}