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Question #42240

Difficulty: 8/10 | 🟢 Free

The Question

An open rectangular tank containing an ideal liquid slides down a fixed rough inclined plane of inclination $\theta$ . The tank starts from rest at $x = 0$ . The coefficient of kinetic friction between the tank and the incline varies with the distance $x$ moved down the incline as $\mu = kx$ , where $k$ is a positive constant. Assuming the tank is sufficiently long and deep so that the liquid does not spill, what is the maximum angle $\phi$ that the free surface of the liquid makes with the inclined plane during the downward motion before the tank momentarily comes to rest?

Options

A.
$\tan^{-1}(2\tan\theta)$
B.
$\tan^{-1}(\tan\theta)$
C.
$\tan^{-1}\left(\frac{1}{2}\tan\theta\right)$
D.
$\tan^{-1}(k\tan\theta)$

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

Accelerated Frame Hydrostatics|Work Done by Friction|Vector Decomposition Ambiguity

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of D'Alembert's principle with the Work-Energy Theorem under non-uniform acceleration.

A common lethal trap is assuming the maximum fluid inclination occurs when the tank reaches its maximum velocity (where acceleration is zero, leading to the incorrect answer $\phi = \theta$ ).

Students fail to recognize that during the deceleration phase, the pseudo-force continues to increase the effective gravity component along the incline.

The true maximum inclination occurs at the maximum displacement, requiring the application of the Work-Energy Theorem to find the stopping distance rather than simply analyzing forces at equilibrium.

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