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Question #42374

Difficulty: 8/10 | 🟢 Free

The Question

A uniform rod of mass $M$ and length $L$ is pivoted at a frictionless hinge $O$ located at a distance $L/4$ from its center of mass. The rod is released from rest at an initial angular displacement $\theta_0$ from the vertical downward position. Since $\theta_0$ is large, the small-angle approximation ( $\sin\theta \approx \theta$ ) is invalid. Let $g$ be the acceleration due to gravity. Which of the following expressions represents the exact time period $T$ of the resulting oscillations?

Options

A.
$\sqrt{\frac{14L}{3g}} \int_0^{\theta_0} \frac{\mathrm{d}\theta}{\sqrt{\cos\theta - \cos\theta_0}}$
B.
$\sqrt{\frac{7L}{6g}} \int_0^{\theta_0} \frac{\mathrm{d}\theta}{\sqrt{\cos\theta - \cos\theta_0}}$
C.
$\sqrt{\frac{16L}{3g}} \int_0^{\theta_0} \frac{\mathrm{d}\theta}{\sqrt{\cos\theta - \cos\theta_0}}$
D.
$\sqrt{\frac{8L}{3g}} \int_0^{\theta_0} \frac{\mathrm{d}\theta}{\sqrt{\cos\theta - \cos\theta_0}}$

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

Moment of Inertia (MOI) Symmetry|Compound Pendulum Period|SHM Definition via Differential Equation

Why This Question Is Tricky

Common Mistakes

This question tests the transition from standard Simple Harmonic Motion to non-linear large-angle dynamics.

Students often incorrectly apply the small-angle formula or fail to re-evaluate the moment of inertia about the new pivot using the Parallel Axis Theorem (leading to Option D).

Furthermore, it requires setting up the exact differential equation via Energy Conservation rather than the torque equation, recognizing that the restoring torque cannot be linearized, which naturally leads to an elliptic integral for the exact time period.

Option B traps students who forget to multiply the quarter-period integral by 4.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Moment of Inertia (MOI) Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free