A rectangular block of mass ' \(\mathrm{m}\) ' and crosssectional area A, floats on a liquid of density ' \(\rho\) '. It is given a small vertical displacement from equilibrium, it starts oscillating with frequency ' \(n\) ' equal to ( \(g=\) acceleration 'due to gravity)

The formula for the time period is given as
\(\mathrm{T}=2 \pi \sqrt{\frac{l}{\mathrm{~g}}}\)
The mass of displaced fluid is
\(\text { Mass }=\text { density } \times \text { volume }\)
\(\mathrm{m}=\rho \times \mathrm{Al}\)
At equilibrium,
Weight of the block = Weight of the displaced liquid
\(\begin{aligned}
& \therefore \quad \mathrm{mg}=\mathrm{A} / \mathrm{pg} \\
& \therefore \quad l=\frac{\mathrm{m}}{\mathrm{Ap}}
\end{aligned}\)
Substituting the values in the equation
\(\begin{aligned}
& \mathrm{T}=2 \pi \sqrt{\frac{l}{\mathrm{~g}}} \\
& \mathrm{~T}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{A} \rho \mathrm{g}}}
\end{aligned}\)
The frequency \(f=\frac{1}{T}\)
\(\therefore \quad \mathrm{f}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{Apg}}{\mathrm{m}}}\)