Question

When a block of iron floats in mercury at 0${}^0$ C, a fraction k${}_1$ of its volume is submerged, while at the temperature 60${}^0$C, a fraction k${}_2$ is seen to be submerged. If the coefficient of volume expansion of iron is ${\gamma }_{Fe}$ and that of mercury is ${\gamma }_{Hg}$, the ratio k${}_1$/k${}_2$ can be expressed as:

• A. $\frac{1+60{\gamma }_{Fe}}{1+60{\gamma }_{Hg}}$
• B. $\frac{1-60{\gamma }_{Fe}}{1+60{\gamma }_{Hg}}$
• C. $\frac{1+60{\gamma }_{Fe}}{1-60{\gamma }_{Hg}}$
• D. $\frac{1+60{\gamma }_{Hg}}{1+60{\gamma }_{Fe}}$

Answer 1

The correct option is A $\frac{1+60{\gamma }_{Fe}}{1+60{\gamma }_{Hg}}$

For $k_1$ fraction of iron block to be submerged in mercury at $0^{\circ }C$,

Buoyant force=Weight of the block

$⟹k_{1}V({\rho }_{Hg}{)}_{0}g=V({\rho }_{Fe}{)}_{0}g$

For $k_2$ fraction of iron block to be submerged in mercury at ${60}^{\circ }C$,

Buoyant force $=$ Weight of the block

$⟹k_{2}V\left({\rho }_{Hg}\right)g=V\left({\rho }_{Fe}\right)g$

$⟹k_{2}V\frac{({\rho }_{Hg}{)}_0}{1+60{\gamma }_{Hg}}g=V\frac{({\rho }_{Fe}{)}_0}{1+60{\gamma }_{Fe}}g$

Hence, $\frac{k_1}{k_2}=\frac{1+60{\gamma }_{Fe}}{1+60{\gamma }_{Hg}}$.