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Question #42242

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The Question

A thin-walled hollow cylinder of mass $M$ , radius $R$ , and length $L$ is completely filled with an inviscid, incompressible fluid of density $\rho$ . The total mass of the fluid is $m$ . The cylinder rests on an inclined plane of angle $\theta$ . The coefficient of kinetic friction between the cylinder and the plane is $\mu_k$ . A horizontal impulsive force $J$ is applied at the center of mass of the cylinder, directed such that it pushes the cylinder into the incline. Assume $\cot\theta > \mu_k$ so that the cylinder slips down the incline during the impulse. Let $\Delta \Pi$ be the magnitude of the difference in pressure impulse between the point in the fluid furthest down the incline and the point furthest up the incline. Let $\Delta \omega$ be the angular velocity acquired by the cylinder just after the impulse. Which of the following expressions gives the ratio $\frac{\Delta \Pi}{R \Delta \omega}$ ?

Options

A.
$\frac{2 \rho R M}{(M+m)\mu_k} (\cot\theta - \mu_k)$
B.
$\frac{\rho R (2M+m)}{(M+m)\mu_k} (\cot\theta - \mu_k)$
C.
$\frac{2 \rho R (M+m)}{M \mu_k} (\cot\theta - \mu_k)$
D.
$\frac{\rho R M}{(M+m)\mu_k} (\cot\theta - \mu_k)$

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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|Impulsive Torque and Angular Momentum|Parallel and Perpendicular Axes

Why This Question Is Tricky

Common Mistakes

This question tests the ability to decouple the translational and rotational dynamics of a fluid-filled container under impulsive loading.

A common trap is assuming the inviscid fluid rotates with the cylinder, which incorrectly alters the effective moment of inertia.

Furthermore, students often fail to recognize that the transient pressure impulse in the fluid is driven purely by the non-inertial linear acceleration (Euler's equation for impulsive motion, $\nabla \Pi = -\rho \Delta \vec{v}$ ), independent of the shell's angular acceleration.

The horizontal impulse also tests the resolution of impulsive normal and frictional forces on an incline.

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