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Question #42378

Difficulty: 8/10 | 🟢 Free

The Question

A rigid square loop of uniform thin wire has mass $m$ , side length $a$ , and self-inductance $L$ . Its electrical resistance is negligible. The loop is initially at rest in the $xy$ -plane in a gravity-free space. A uniform magnetic field $\mathbf{B} = B\hat{k}$ is present in the region. The loop is constrained to rotate about a fixed axis lying in the $xy$ -plane. This axis is parallel to one of the loop's diagonals and passes through one of its vertices. An impulsive torque is applied, giving the loop an initial angular velocity $\omega_0$ . Assuming the initial current in the loop is zero, what is the minimum value of $\omega_0$ required for the loop to complete a full rotation?

Options

A.
$Ba \sqrt{\frac{6}{mL}}$
B.
$4Ba \sqrt{\frac{3}{7mL}}$
C.
$2Ba \sqrt{\frac{6}{7mL}}$
D.
$Ba \sqrt{\frac{3}{2mL}}$

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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

Parallel and Perpendicular Axes|Energy Stored in Inductor|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of rigid body dynamics (Parallel Axis Theorem) and electromagnetism (Lenz's Law and energy in an inductor).

The primary trap lies in calculating the moment of inertia: students frequently confuse a square loop with a solid square plate, incorrectly using $I_{cm} = ma^2/12$ instead of $ma^2/6$ .

Furthermore, they may miscalculate the perpendicular distance between the diagonal and the vertex as $a$ instead of $a/\sqrt{2}$ when applying the Parallel Axis Theorem.

Finally, a conceptual trap exists in evaluating the maximum inductive energy; students might mistakenly use the average flux change or forget that the current peaks at $\theta = \pi$ , leading to a missing factor of 4 in the energy barrier.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Parallel and Perpendicular Axes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free