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Question #92011

Difficulty: 5/10 | 🟢 Free

The Question

A symmetric glass prism of mass $M$ , refracting angle $A$ , and refractive index $\mu$ is mounted on a frictionless vertical axis passing through its apex. The moment of inertia of the prism about this axis is $I$ . A continuous laser beam of power $P$ from a fixed laboratory source is incident horizontally on the prism such that the perpendicular distance from the apex to the incident beam is $b$ . The prism is initially at rest in the position of minimum deviation, where the angle of incidence is $i_0$ . If the prism is given a small angular displacement $\theta$ from this equilibrium position and released, it executes simple harmonic motion. What is the angular frequency $\omega$ of these oscillations? (Assume the speed of light is $c$ , and neglect any absorption or reflection losses at the prism faces).

Options

A.
$\sqrt{\frac{2 P b \tan i_0}{I c}}$
B.
$\sqrt{\frac{P b \tan i_0}{I c}}$
C.
$\sqrt{\frac{2 P b \sin i_0}{I c}}$
D.
$0$ (The prism will not oscillate as the restoring torque is zero)

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Prism Minimum Deviation|Impulsive Torque and Angular Momentum|Velocity Vector Direction

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the subtle distinction between angular deviation and lateral shift in optics, coupled with radiation pressure.

A common trap is assuming that because the deviation angle $\delta$ is stationary at minimum deviation ( $d\delta/di = 0$ ), the emergent ray is completely unaffected by small rotations of the prism.

Students falling for this will conclude there is no change in angular momentum flux and thus no restoring torque (Option D).

However, while the emergent ray's direction is constant to first order, its point of emergence shifts laterally along the prism face.

This lateral shift changes the impact parameter of the emergent photon stream, creating a net restoring torque.

The problem elegantly links the kinematic change in the incident angle (relative to the rotating prism frame) to rigid body dynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Prism Minimum Deviation
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free