Question

A piece of metal floats on Hg. The coefficient of expansion of metal and Hg are g₁ and g₂ respectively. If the temperature of both Hg and metal are increased by an amount DT, by what factor the fraction of the volume of metal submerged in mercury changes?

• A. (g₂ – g₁) DT
• B. $\left( \frac{\gamma_{2} + \gamma_{1}}{2} \right)$DT
• C. $\frac{2\gamma_{1}\gamma_{2}}{\gamma_{1} + \gamma_{2}}$DT
• D. $\frac{\gamma_{1}\gamma_{2}}{\gamma_{1} + \gamma_{2}}$DT

Answer 1

Answer: (A)

(g₂ – g₁) DT

Answer 2

f_in = $\frac{V_{in}}{V}$ = $\frac{\Delta f}{f}$ = $\frac{f'_{in}–f_{in}}{f_{in}}$

= $\frac{f'_{in}}{f_{in}}$ – 1

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

= (g₂ – g₁) DT (using Binomial theorem)