Physics - SHM Question with Solution | TestHub

Let $m$ be the mass of the cork and $A$ be its cross-sectional area.

When the cork is depressed by a small distance $x$ in a liquid of density ${\rho }_L$, the extra buoyant force provides the restoring force:

$F=-A{\rho }_{L}gx$

The acceleration of the cork is:

$a=-\frac{A{\rho }_{L}g}{m}x$

Comparing this with the standard equation of SHM $a=-{\omega }^{2}x$, we get:

$\omega =\sqrt{\frac{A{\rho }_{L}g}{m}}$

The time period of oscillation is:

$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{A{\rho }_{L}g}}$

This shows that $T\propto \frac{1}{\sqrt{{\rho }_L}}$.

For the first liquid of density ${\rho }_1$:

$T\propto \frac{1}{\sqrt{{\rho }_1}}$

For the second liquid of density ${\rho }_2$:

$2T\propto \frac{1}{\sqrt{{\rho }_2}}$

Dividing the two equations:

$\frac{2T}{T}=\sqrt{\frac{{\rho }_1}{{\rho }_2}}$

$2=\sqrt{\frac{{\rho }_1}{{\rho }_2}}$

Squaring both sides:

$4=\frac{{\rho }_1}{{\rho }_2}$

$\frac{{\rho }_2}{{\rho }_1}=\frac{1}{4}$

Answer: $1/4$