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Question #77790

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The Question

A water tank has a non-uniform cross-sectional area given by $A(h) = k \sqrt{h}$ , where $h$ is the height of the water level from the bottom and $k$ is a positive constant with appropriate SI units. A small hole of area $a$ is opened at the bottom ( $h=0$ ). A student calculates the time $T_0$ required for the water level to drop from $h = H$ to $h = H/2$ by assuming the efflux velocity is exactly given by Torricelli's law, $v = \sqrt{2gh}$ . However, the true time $T$ differs because the downward velocity of the free surface is not strictly zero. Assuming $a \ll k\sqrt{H}$ , which of the following expressions best represents the fractional error $\frac{T_0 - T}{T_0}$ in the student's calculation?

Options

A.
$\frac{a^2 \ln 2}{k^2 H}$
B.
$\frac{a^2}{2 k^2 H}$
C.
$\frac{a^2 \ln 2}{2 k^2 H}$
D.
$\frac{2 a^2 \ln 2}{k^2 H}$

Check Your Answer

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

Torricelli’s Efflux Velocity|Relative Error Propagation in Powers|Equation of Continuity

Why This Question Is Tricky

Examiner Levers

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Common Mistakes

This question tests the limits of Torricelli's law and the propagation of approximations through a differential equation.

Students often memorize $v = \sqrt{2gh}$ without recognizing it assumes an infinitely large tank cross-section ( $A \gg a$ ).

When the tank has a tapering profile $A(h) = k\sqrt{h}$ , the finite velocity of the free surface introduces a height-dependent correction term to the efflux velocity.

The trap lies in attempting to apply a static error percentage at the end of the calculation rather than correctly applying the binomial expansion $(1 - x)^{1/2} \approx 1 - x/2$ to the differential equation $-A(h)dh = av(h)dt$ before integrating.

This highlights the non-linear coupling between the dynamic cross-sectional area and the fluid continuity equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Relative Error Propagation in Powers
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free