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Question #42229

Difficulty: 8/10 | 🟢 Free

The Question

A large open cylindrical tank of height $H$ is filled with a liquid. The temperature of the liquid varies linearly with depth $y$ from the surface, being $T_0$ at the surface ( $y=0$ ) and $T_0 + \Delta T$ at the bottom ( $y=H$ ). The density of the liquid varies with temperature as $\rho(T) = \rho_0 [1 - \gamma (T - T_0)]$ , where $\rho_0$ is the density at $T_0$ and $\gamma$ is the coefficient of volume expansion. A small hole is punched at the bottom of the tank. Assuming the fluid in the tank is practically at rest initially, what is the initial velocity of efflux $v$ from the hole? (Atmospheric pressure is $P_0$ )

Options

A.
$\sqrt{gH \left( \frac{2 - \gamma \Delta T}{1 - \gamma \Delta T} \right)}$
B.
$\sqrt{2gH}$
C.
$\sqrt{gH \left( \frac{2 - 2\gamma \Delta T}{2 - \gamma \Delta T} \right)}$
D.
$\sqrt{gH \left( \frac{2 + \gamma \Delta T}{1 + \gamma \Delta T} \right)}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Pascal’s Law Limitations|Variation of Density with Temperature|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize when standard hydrostatic formulas ( $P = P_0 + \rho g h$ ) and Torricelli's law ( $v = \sqrt{2gh}$ ) break down.

By introducing a temperature gradient, the fluid density becomes a function of depth, requiring integration to find the true pressure at the bottom.

Furthermore, it tests the precise application of Bernoulli's principle at the exit hole, where the local density of the exiting fluid ( $\rho_{bottom}$ ) must be used for the kinetic energy term, rather than the average density or surface density.

Students who blindly plug into Torricelli's law will incorrectly choose $\sqrt{2gH}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Pascal’s Law Limitations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free