First slide

Thermal expansion

Question

A solid whose volume does not change with temperature, floats in a liquid. Fractions $f_{1} a n d f_{2}$ of its volume remain submerged for two different temperatures $t_{1} a n d t_{2}$ of liquid. The volume coefficient of expansion of liquid is

Moderate

A

$\frac{f_{1} + f_{2}}{f_{1} t_{1} + f_{2} t_{2}}$
B

$\frac{f_{1} + f_{2}}{f_{1} t_{2} + f_{2} t_{1}}$
C

$\frac{f_{1} - f_{2}}{f_{1} t_{1} + f_{2} t_{2}}$
D

$\frac{f_{1} - f_{2}}{f_{2} t_{1} - f_{1} t_{2}}$

Solution

For floating body B = W
ρ_t.V.g = σ.V_S.g
$\large \Rightarrow \frac {\rho_g}{1+\gamma(t-0)}.V=\sigma.V_S$
$\large \Rightarrow f=\frac {V}{V_S}=\frac {\sigma}{\rho_0}(1+\gamma t)$
$\large \Rightarrow f\propto(1+\gamma t)$
$\large \therefore \frac {f_1}{f_2}=\frac {1+\gamma t_1}{1+\gamma t_2}\Rightarrow \frac {(f_1-f_2)}{(f_2t_1-f_1t_2)}$

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