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Question #42225

Difficulty: 8/10 | 🟢 Free

The Question

A non-cylindrical tank has a height-dependent cross-sectional area $A(h) = A_0 \sqrt{\frac{h}{H}}$ , where $H$ is the initial height of the incompressible liquid (density $\rho$ ) and $h$ is the instantaneous height. A small hole of area $a$ ( $a \ll A_0$ ) at the bottom is open to the atmosphere at a constant pressure $P_0$ . The tank is sealed at the top, and a computer-controlled valve maintains the gas pressure above the liquid at $P(t) = P_0 + \rho g (H - h) + \frac{1}{2}\rho \gamma t$ , where $\gamma = \frac{a}{A_0} (2g)^{3/2} \sqrt{H}$ and $g$ is the acceleration due to gravity. Assuming quasi-steady flow and negligible gas density, find the time $T$ taken to completely drain the tank.

Options

A.
$(2^{2/3} - 1) \frac{A_0}{a} \sqrt{\frac{H}{2g}}$
B.
$(2^{3/2} - 1) \frac{A_0}{a} \sqrt{\frac{H}{2g}}$
C.
$(2^{2/3} - 1) \frac{A_0}{a} \sqrt{\frac{2H}{g}}$
D.
$(2^{1/3} - 1) \frac{A_0}{a} \sqrt{\frac{H}{g}}$

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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

Atmospheric Pressure Reference|Effusion and Diffusion|Calculus Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the generalized Bernoulli principle with a time-varying atmospheric reference pressure and a height-dependent cross-sectional area.

The primary trap lies in the apparent complexity of the pressure function P(t), which couples the instantaneous height h(t) with an explicit time dependence.

Students often fail to realize that substituting this specific P(t) into Bernoulli's equation exactly cancels the hydrostatic pressure term, decoupling the efflux velocity from the liquid height.

Consequently, the non-linear differential equation becomes perfectly separable.

Failing to identify this cancellation leads to an intractable integration, testing the student's algebraic foresight and trust in fundamental conservation laws over blind formula application.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Atmospheric Pressure Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free