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Question #43863

Difficulty: 8/10 | 🟢 Free

The Question

A uniform rigid non-conducting rod of mass $m$ and length $L$ has a total charge $q$ uniformly distributed along its length. The rod is hinged at one end at the origin $(0,0)$ of a vertical $xy$ -plane and can rotate freely in this plane. Acceleration due to gravity is $g$ acting in the negative $y$ -direction. An electric field $\vec{E} = E_0 \left( \frac{x}{L} \hat{i} - \frac{y}{L} \hat{j} \right)$ exists in the region. Given that $q E_0 = \frac{3}{2} m g$ , the rod is displaced slightly from its stable equilibrium position and released. Find the angular frequency $\omega$ of its small oscillations.

Options

A.
$\frac{3}{2} \sqrt{\frac{g}{L}}$
B.
$\frac{\sqrt{3}}{2} \sqrt{\frac{g}{L}}$
C.
$\sqrt{\frac{3}{2}} \sqrt{\frac{g}{L}}$
D.
$\frac{\sqrt{5}}{2} \sqrt{\frac{g}{L}}$

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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

Electric Field Symmetry|Physical Pendulum Inertia|SHM Definition via Differential Equation|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the dynamics of a rigid body in a non-uniform electric field.

The primary trap is incorrectly assuming the net electric force acts at the center of mass (Option D), which is only true for uniform fields.

Students must integrate the position-dependent torque $d\tau = \vec{r} \times d\vec{F}$ to find the correct electric torque $\tau_E \propto \sin(2\theta)$ .

Furthermore, the symmetry of the field shifts the stable equilibrium away from the vertical.

Failing to linearize the torque about this new equilibrium $\theta_0 = \pi/3$ (Option C) or using the simple pendulum inertia $I=mL^2$ instead of the physical pendulum inertia $I=mL^2/3$ (Option B) leads to lethal collisions with the incorrect options.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Electric Field Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free