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

$ρ$ 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)

$ρg (2 y)$

The weight of the liquid =

ρgA (2 y)

$ρgA (2 y)$

So, the restoring force = ma = -

ρgA (2 y)

$ρgA (2 y)$

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

$T = 2 π \sqrt{\frac{L}{g}}$