Question

A piece of metal floats on Hg. The coefficient of expansion of metal and Hg and γ₁ and γ₂ respectively. If the temperature of both Hg and metal are increased by an amount ∆T, by what factor the fraction of the volume of metal submerged in mercury changes?

• A. (a) (γ₂ - γ₁)∆T
• A. (b) $\left( \frac{\gamma_{2} + \gamma_{1}}{2} \right)\Delta T$
• A. (c) $\frac{2\gamma_{1}\gamma_{2}}{\gamma_{1} + \gamma_{2}}$∆T
• A. (d) $\frac{\gamma_{1}\gamma_{2}}{\gamma_{1} + \gamma_{2}}$∆T

Answer 1

(a)

f_in = $\frac{V_{in}}{V} = \frac{\rho}{\sigma}$ where ρ is density of metal and σ is density of

Hg. $\frac{\Delta f}{f} = \frac{f'_{in} - f_{in}}{f_{in}} = \frac{f'_{in}}{f_{in}} - 1$

= $\frac{\frac{\rho}{\sigma}\left( \frac{1 + \gamma_{2}\Delta T}{1 + \gamma_{1}\Delta T} \right)}{\frac{\rho}{\sigma}} - 1$

= (γ₂ - γ₁)∆T

(using Binomial theorem)