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Question #42239

Difficulty: 8/10 | 🟢 Free

The Question

A rectangular tank of length $L$ and width $b$ is partially filled with an ideal, incompressible liquid of density $\rho$ to a depth $H$ . The tank is forced to oscillate horizontally such that its position is given by $x(t) = A \sin(\omega t)$ . Assume the frequency $\omega$ is low enough that the free surface remains planar (quasi-static approximation) and the liquid does not spill. Which of the following expressions represents the maximum rate of change of the gravitational potential energy of the liquid?

Options

A.
$\frac{\rho b L^3 A^2 \omega^5}{24 g}$
B.
$\frac{\rho b L^3 A^2 \omega^5}{12 g}$
C.
$\frac{\rho b L^3 A^2 \omega^5}{48 g}$
D.
$\frac{\rho b L^3 A^2 \omega^5}{6 g}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Accelerated Frame Hydrostatics|Boundary Condition Reflection|Average vs. Instantaneous Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link non-inertial fluid statics (Euler's equation) with time-varying energy states.

A lethal trap is assuming the rate of change of potential energy is simply the negative of the rate of change of kinetic energy, which violates the work-energy theorem since the oscillating tank walls do external work on the fluid.

Furthermore, students often incorrectly integrate the triangular wedge of displaced fluid by forgetting the squared dependence on the surface slope, missing the $\sin^2(\omega t)$ term that leads to the $\sin(2\omega t)$ frequency doubling in the power output.

Quick Facts About This Question

Question Information

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