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Question #42606

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid disk of mass $m$ and radius $R$ is pivoted at a frictionless hinge located on its rim. The disk is initially at rest in its stable equilibrium position. It is then given an initial angular velocity $\omega_0 = \sqrt{\frac{2g}{R}}$ . Let $K(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - k^2 \sin^2\phi}}$ denote the complete elliptic integral of the first kind. Which of the following expressions represents the exact time period $T$ of the resulting anharmonic oscillation?

Options

A.
$4 \sqrt{\frac{3R}{2g}} K\left(\frac{\sqrt{3}}{2}\right)$
B.
$2\pi \sqrt{\frac{3R}{2g}}$
C.
$4 \sqrt{\frac{3R}{2g}} K\left(\frac{1}{2}\right)$
D.
$2 \sqrt{\frac{3R}{g}} K\left(\frac{\sqrt{3}}{2}\right)$

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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

Energy of Vibrational Modes|Compound Pendulum Period|Dimensional Independence vs. Scalar/Vector Nature

Why This Question Is Tricky

Common Mistakes

This question tests the breakdown of the small-angle approximation in compound pendulums at high energy states.

Students commonly fall into the trap of blindly applying the standard simple harmonic motion formula $T = 2\pi\sqrt{I/mgd}$ (Option B), ignoring that the initial kinetic energy pushes the pendulum into a highly non-linear potential energy regime ( $\theta_0 = 120^\circ$ ).

By forcing the calculation of the exact maximum amplitude via energy conservation, the problem requires the application of the exact anharmonic time period solution involving the complete elliptic integral of the first kind, thereby testing the transition from restorative potential energy to kinetic rotational dominance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Energy of Vibrational Modes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free