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Question #42625

Difficulty: 8/10 | 🟢 Free

The Question

A rigid insulating rod of length $L = 0.5 \text{ m}$ and mass $M$ has a charge $Q = 3 \mu\text{C}$ uniformly distributed along its length. The rod is pivoted at one end at the origin and is free to rotate in the $xy$ -plane. The system is placed in a conservative electrostatic field whose potential is given by $V(x,y) = -E_0 \frac{x^2 - y^2}{L}$ , where $E_0 = 2 \times 10^4 \text{ V/m}$ . The rod is initially at rest at its stable equilibrium position. An external agent slowly rotates the rod to an angle $\theta = \pi/4 \text{ rad}$ from the equilibrium position. Calculate the work done by the external agent.

Options

A.
10 \text{ mJ}
B.
$1.25 \pi^2 \text{ mJ}$
C.
5 \text{ mJ}
D.
$2.5 \pi \text{ mJ}$

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

Torsional Oscillations|Work against Conservative Force|Conservative Field Path Independence|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to calculate work done against a conservative field by evaluating the exact potential energy function rather than relying on small-angle approximations.

The 'Lethal Collision' occurs when students reflexively linearize the restoring torque to find an effective torsional constant ( $k = 4QE_0L/3$ ) and incorrectly apply the simple harmonic motion energy formula $W = \frac{1}{2}k\theta^2$ .

This trap exposes the misconception of treating non-linear potential gradients as constant mechanical torsion fibers for large angular displacements, leading directly to the incorrect option $1.25 \pi^2 \text{ mJ}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Torsional Oscillations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free