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Question #42139

Difficulty: 8/10 | 🟢 Free

The Question

A tall cylindrical tank of depth $H$ is completely filled with a liquid. The temperature of the liquid varies linearly with depth $y$ (measured downwards from the free surface) as $T(y) = T_0 + c y$ , where $T_0$ is the surface temperature and $c$ is a positive constant. The density of the liquid varies with temperature as $\rho(T) = \frac{\rho_0}{1 + \gamma(T - T_0)}$ , where $\rho_0$ is the density at $T_0$ and $\gamma$ is the coefficient of volume expansion. Assuming acceleration due to gravity $g$ is constant and atmospheric pressure is $P_0$ , the absolute pressure at the bottom of the tank is:

Options

A.
$P_0 + \frac{\rho_0 g}{\gamma c} \ln(1 + \gamma c H)$
B.
$P_0 + \rho_0 g H \left(1 - \frac{\gamma c H}{2}\right)$
C.
$P_0 + \frac{\rho_0 g}{\gamma c} (e^{\gamma c H} - 1)$
D.
$P_0 + \rho_0 g H$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Variation of Density with Temperature|Atmospheric Pressure Reference|Hydrostatics

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to move beyond the standard hydrostatic pressure formula ( $P = P_0 + \rho g h$ ) when dealing with a non-isothermal fluid column.

The primary trap is assuming a constant density or incorrectly applying a linear approximation for density ( $\rho \approx \rho_0(1-\gamma \Delta T)$ ) which leads to a polynomial pressure profile instead of the exact logarithmic one.

Students must recognize that density is a local state variable dependent on temperature, which in turn depends on depth, necessitating the setup and integration of the differential equation $dP = \rho(y) g dy$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Variation of Density with Temperature
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free