# When a block of iron floats in mercury at 0°C, a fraction k1 of its volume is submerged, while at the temperature 60°C…

Q: When a block of iron floats in mercury at 0°C, a fraction k1 of its volume is submerged, while at the temperature 60°C, a fraction k2 is seen to be submerged. If the coefficient of volume expansion of iron is γFe and that of mercury is γHg, then the ratio k1/k2 can be expressed as

(a) $\displaystyle \frac{1 + 60 \gamma_Fe}{1 + 60 \gamma_Hg}$

(b) $\displaystyle \frac{1 – 60 \gamma_Fe}{1 + 60 \gamma_Hg}$

(c) $\displaystyle \frac{1 + 60 \gamma_Fe}{1 – 60 \gamma_Hg}$

(d) $\displaystyle \frac{1 + 60 \gamma_Hg}{1 + 60 \gamma_Fe}$

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Ans: (a)
Sol: $\displaystyle W = k_1 (\rho_0 V_0 g ) = k_2 (\rho_{60} V_{60} g )$

$\displaystyle \frac{k_1}{k_2} = \frac{\rho_{60}}{\rho_0}.\frac{V_{60}}{V_0}$

$\displaystyle \frac{k_1}{k_2} = \frac{\rho_0 /(1+\gamma_{Hg} \times 60)}{\rho_0} \frac{V_0(1+\gamma_{Fe}\times 60)}{V_0}$

$\displaystyle \frac{k_1}{k_2} = \frac{1+60 \gamma_{Fe}}{1+60 \gamma_{Hg}}$