A tube of uniform bore of cross-sectional area ' \(A\) ' has been set up vertically with open end facing up. Now 'M' gram of a liquid of density ' \(d\) ' is poured into it. The column of liquid in this tube will oscillate with a period ' T ', which is equal to \([\mathrm{g}=\) acceleration due to gravity]

For a depression of y cm on one side, the level of liquid will be 2 y cm higher on the other side.
\(\therefore \quad\) Weight of extra liquid on the P -side \(=2 \mathrm{Aydg}\) The above weight acts as the restoring force acting on mass M .
\(\therefore \quad\) Restoring acceleration \(=-\frac{2 \mathrm{Aydg}}{\mathrm{M}} \ldots(-\mathrm{ve} \cdot\) sign indicates the acceleration and force are opposite to each other) ... (i)
We know for simple harmonic motion,
\(a=-\left(\frac{k}{m}\right) x\)
Comparing (i) and (ii), we see the motion is simple harmonic.
\(\begin{aligned}
\therefore \quad \mathrm{T} & =2 \pi \sqrt{\frac{\text { Displacement }}{\text { Acceleration }}}=2 \pi \sqrt{\frac{\mathrm{y}}{\frac{2 \mathrm{Aydg}}{\mathrm{M}}}} \\
& =2 \pi \sqrt{\frac{\mathrm{M}}{2 \mathrm{Adg}}}
\end{aligned}\)