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 $f^{\prime }={\rho }_1^{\prime }/{\rho }_2^{\prime }$.
${\alpha }^{\prime }=\frac{{\rho }_1}{1+{\gamma }_1\Delta T}\frac{1+{\gamma }_2\Delta T}{{\rho }_2}=\frac{{\rho }_1}{{\rho }_2}\frac{1+{\gamma }_2\Delta T}{1+{\gamma }_1\Delta T}$
$\Rightarrow {\alpha }^{\prime }=f\frac{1+{\gamma }_2\Delta T}{1+{\gamma }_1\Delta T}$
$\frac{{\alpha }^{\prime }}{f}=\frac{1+{\gamma }_2\Delta T}{1+{\gamma }_1\Delta T}$