Paragraph: A small spherical monoatomic ideal gas bubble (γ=(5)/(3)) is trapped inside a liquid of density _1 (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles

Correct answer: B. _l n R g T_0 ( _0+ _l g H)^ (2)/(5) [ _0+ _l g(H-y)]^ (3)/(5) Buoyancy force Here, and aligned F & =( volume of bubble ) ( _l) g & = ((n R T_2)/(p_2))p_l g T_2 & =T_0 [(p_0+p_l g(H-y))/(p_0+ _l g H)] p_2 & =p_0+p_l(H-y) aligned Substituting the values we get, F= _n n g T_0 (p_0+ _l g H)^ 2 / 5 [p_0+ _l g(H-y)]^ 3 / 5 correct option is (b).

JEE Advanced 2008Paper 1

Exam: JEE Advanced
Year: 2008
Paper: Paper 1
Subject: Physics
Chapter: Mechanical Properties of Fluids

JEE Advanced 2008 · Paper 1 · Physics · Mechanical Properties of Fluids

Paragraph: A small spherical monoatomic ideal gas bubble...

Paragraph: A small spherical monoatomic ideal gas bubble

\left(\gamma=\frac{5}{3}\right)

is trapped inside a liquid of density

\rho_1

(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). Paragraph: A small spherical monoatomic ideal gas bubble (γ=(5)/(3)) is trapped inside a liquid of density _1 (see

Question: The buoyancy force acting on the gas bubble is (Assume

R

is the universal gas constant)

Options

A $\rho_l n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}}$ { $\left(\rho_0+\rho_l g y\right)^{\frac{7}{5}}}$
B $\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}$
C $\rho n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}}$ { $\left(\rho_0+\rho_l g y\right)^{\frac{8}{5}}}$
D $\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{2}{5}}}$

Correct answer

B. $\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}$

Step-by-step solution

Buoyancy force Here, and

\begin{aligned} F & =(\text{ volume\text{ of }\text{ bubble } }) \left(\rho_l\right) g \\ & = \left(\frac{n R T_2}{p_2}\right)p_l g \\ T_2 & =T_0 \left[\frac{p_0+p_l g(H-y)}{p_0+\rho_l g H}\right]\\ p_2 & =p_0+p_l(H-y) \end{aligned}

Substituting$ the values we get, F=\frac{\rho_n n g T_0}{\left(p_0+\rho_l g H\right)^{2 / 5}\left[p_0+\rho_l g(H-y)\right]^{3 / 5}}\therefore correct option is (b).

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