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Question #42122

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid sphere of mass $m$ and radius $R$ is initially at rest in a large tank of viscous fluid. At $t=0$ , an impulsive torque delivers an angular impulse $J$ to the sphere about its central axis. The fluid exerts a viscous retarding torque $\tau = -k\omega$ , where $\omega$ is the instantaneous angular velocity and $k$ is a positive constant. What is the total angular displacement $\Delta \theta$ of the sphere before it comes to rest, and the total heat energy $H$ dissipated by the viscous fluid?

Options

A.
$\Delta \theta = \frac{J}{k}, \quad H = \frac{5J^2}{4mR^2}$
B.
$\Delta \theta = \frac{5J}{2k}, \quad H = \frac{J^2}{2mR^2}$
C.
$\Delta \theta = \frac{J}{k}, \quad H = \frac{J^2}{mR^2}$
D.
$\Delta \theta = \frac{2mR^2 J}{5k}, \quad H = \frac{5J^2}{4mR^2}$

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Correct Answer

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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

Viscosity and Stokes' Law|Impulsive Torque and Angular Momentum|Rigid Body Constraints

Why This Question Is Tricky

Common Mistakes

This question tests the transition from impulsive dynamics to non-conservative continuous damping (rotational Stokes' Law).

A common trap is assuming the viscous torque acts during the impulse or misapplying kinematic equations for constant angular acceleration.

Students often incorrectly assume $\Delta \theta$ depends explicitly on the moment of inertia rather than recognizing that the differential equation $I d\omega = -k d\theta$ directly links the angular impulse $J = I\omega_0$ to the damping constant $k$ .

Furthermore, calculating the dissipated energy requires recognizing that all initial rotational kinetic energy, imparted instantaneously via $J = I\omega_0$ , is entirely dissipated by the non-conservative viscous coupling, preventing the misuse of static conservation laws.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Viscosity and Stokes' Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free