First slide

Thermal expansion

Question

A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are $γ_{1} and γ_{2}$ respectively. If the temperature of both mercury and metal are increased by an amount $ΔT$ , the fraction of the volume of the metal submerged in mercury changes by the factor

Moderate

A

$(γ_{2} / γ_{1}) ΔT$
B

$(γ_{1} - γ_{2}) ΔT$
C

$(γ_{2} - γ_{1}) ΔT$
D

$(γ_{1} / γ_{2}) ΔT$

Solution

Let f be the fraction of the volume of metal submerged in mercury
$∴ f = \frac{density of metal}{density of mercury} = \frac{ρ_{metal}}{ρ_{mercury}}$
When temperature is increased by $Δt$ , then fraction f'
$\begin{array}{l} f^{'} = \frac{ρ_{metal} (1 - γ_{1} Δt)}{ρ_{mercury} (1 - γ_{2} ΔT)} \\ = f (1 - γ_{1} ΔT) {(1 - γ_{2} ΔT)}^{- 1} \\ = f [1 + (γ_{2} - γ_{1}) ΔT] \end{array}$
Fractional change or fraction immersed
$\frac{f^{'} - f}{f} = (γ_{2} - γ_{1}) ΔT$

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