IIT-JEE 2008 · Paper 1 · Physics Q40

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Paragraph for Question Nos. 38 to 40

A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac53\right)$ is trapped inside a liquid of density $\rho_\ell$ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains $n$ moles of gas. The temperature of the gas when the bubble is at the bottom is $T_0$ , the height of the liquid is $H$ and the atmospheric pressure is $P_0$ (Neglect surface tension).

A container of liquid of height H open to the atmosphere (pressure P₀ at the top); a small gas bubble is at height y above the bottom.

The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)

\rho_\ell nRgT_0\dfrac{(P_0+\rho_\ell gH)^{2/5}}{(P_0+\rho_\ell gy)^{7/5}}

\dfrac{\rho_\ell nRgT_0}{(P_0+\rho_\ell gH)^{2/5}\,[P_0+\rho_\ell g(H-y)]^{3/5}}

\rho_\ell nRgT_0\dfrac{(P_0+\rho_\ell gH)^{3/5}}{(P_0+\rho_\ell gy)^{8/5}}

\dfrac{\rho_\ell nRgT_0}{(P_0+\rho_\ell gH)^{3/5}\,[P_0+\rho_\ell g(H-y)]^{2/5}}

Answer and worked solution

Official answer: B

Volume. $V=\dfrac{nRT}P$ with $T$ from the previous part and $P=P_0+\rho_\ell g(H-y)$ :
$V=\frac{nRT_0}{(P_0+\rho_\ell gH)^{2/5}\,[P_0+\rho_\ell g(H-y)]^{3/5}}.$

Buoyancy. $F=\rho_\ell gV$ :
$F=\frac{\rho_\ell nRgT_0}{(P_0+\rho_\ell gH)^{2/5}\,[P_0+\rho_\ell g(H-y)]^{3/5}}.$

(B)

Topics:Buoyancy and Archimedes' principle Ideal gas equation

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IIT-JEE 2008 · Paper 1, Physics question 40