Question

A piece of metal floats in mercury. The coefficients of volume expansion of the metal and mercury are $\gamma _{1}$ and $\gamma _{2}$ , respectively. If the temperatures of both mercury and the metal are increased by $\Delta \text{T}$ , the fraction of the volume of the metal submerged in mercury changes by the factor of

• A. $\frac{1 + \gamma _{2} \Delta \text{T}}{1 + \gamma _{1} \Delta \text{T}}$
• B. $1 + \gamma _{2} \Delta \text{T}$
• C. $1 + \gamma _{1} \Delta \text{T}$
• D. $\frac{1 + \gamma _{2} \Delta \text{T}}{1 - \gamma _{1} \Delta \text{T}}$

Answer 1

Answer: (A)

As, $ \gamma_1=\frac{\Delta V _1}{ V _1 \Delta T }$ and $\gamma_2=\frac{\Delta V _2}{ V _2 \Delta T }$
$\Rightarrow \Delta V _1=\gamma_1 V _1 \Delta T$
and $\Delta V _2=\gamma_2 V _2 \Delta T$
Fraction of volume submerged before temperature is raised is given by $f=\rho_1 / \rho_2$. Fraction of volume submerged after the temperature is raised is given by $f^{\prime}=\rho_1^{\prime} / \rho_2^{\prime}$.
$\alpha^{\prime} =\frac{\rho_1}{1+\gamma_1 \Delta T } \frac{1+\gamma_2 \Delta T }{\rho_2}=\frac{\rho_1}{\rho_2} \frac{1+\gamma_2 \Delta T }{1+\gamma_1 \Delta T } $
$\Rightarrow \alpha^{\prime} =f \frac{1+\gamma_2 \Delta T }{1+\gamma_1 \Delta T } $
$\frac{\alpha^{\prime}}{f} =\frac{1+\gamma_2 \Delta T }{1+\gamma_1 \Delta T }$