Question

A piece of metal floats on mercury. The coefficient of volume expansion of metal and mercury are

γ_{1}

and

γ_{2}

respectively. If the temperature of both mercury and metal are increased by an amount

Δ T

, by what factor does the fraction of the volume of the metal submerged in mercury changes?

Accepted Answer

On increasing temperature, the density $(ρ)$ of material decreases.

The relation between density and the coefficient $(γ)$ of volumetric expansion and increase in temperature $(∆ T)$ is $ρ' = ρ (1 - γ ∆ T)$

Let $V$ be the total volume and $V_{l}$ be the volume of metal immersed in mercury.

By using archimedes principle $ρ_{m e t a l} \times V \times g = ρ_{m e r c u r y} \times V_{l} \times g$

The fraction of the volume immersed inside the mercury is given by, $f = \frac{V_{l}}{V} = \frac{ρ_{metal}}{ρ_{mercury}}$

On the increasing the temperature, the new fraction of the volume immersed in the mercury will be

$f' = \frac{ρ'_{metal}}{ρ'_{mercury}}$

$= \frac{ρ_{metal} (1 - γ_{1} ∆ T)}{ρ_{mercury} (1 - γ_{2} ∆ T)}$

The ratio of the fractions,

$\Rightarrow \frac{f'}{f} = \frac{ρ_{metal} (1 - γ_{1} ∆ T)}{ρ_{mercury} (1 - γ_{2} ∆ T)} \times \frac{ρ_{mercury}}{ρ_{metal}}$

$= \frac{(1 - γ_{1} ∆ T)}{(1 - γ_{2} ∆ T)}$

$= (1 - γ_{1} ∆ T) {(1 - γ_{2} ∆ T)}^{- 1}$

$= (1 - γ_{1} ∆ T) (1 + γ_{2} ∆ T)$

$∵$ As ${(1 - γ_{2} ∆ T)}^{- 1} \approx (1 + γ_{2} ∆ T)$

$= 1 + γ_{2} ∆ T - γ_{1} ∆ T - γ_{1} γ_{2} {(∆ T)}^{2}$

$= 1 + (γ_{2} - γ_{1}) (∆ T)$

$∵ γ_{1} γ_{2} {(∆ T)}^{2}$ is negligible.

So the fractional change in volume,

$\Rightarrow \frac{f'}{f} - 1 = \frac{f' - f}{f} = (γ_{2} - γ_{1}) (∆ T)$

D. C. Pandey Solutions for Chapter: Thermometry, Thermal Expansion and Kinetic Theory of Gases, Exercise 2: INTRODUCTORY EXERCISE 20.2

D. C. Pandey Physics Solutions for Exercise - D. C. Pandey Solutions for Chapter: Thermometry, Thermal Expansion and Kinetic Theory of Gases, Exercise 2: INTRODUCTORY EXERCISE 20.2

Questions from D. C. Pandey Solutions for Chapter: Thermometry, Thermal Expansion and Kinetic Theory of Gases, Exercise 2: INTRODUCTORY EXERCISE 20.2 with Hints & Solutions

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