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Question #43745

Difficulty: 8/10 | 🟢 Free

The Question

A solid cylindrical pulley of mass $M$ and radius $R$ is suspended by a light inextensible string. One end of the string is attached to a rigid ceiling. The string passes under the pulley and its other end is attached to a vertical ideal spring of spring constant $k$ , which is also suspended from the ceiling. A linear polarizer is mounted on the flat circular face of the pulley such that it rotates with the pulley. Unpolarized light of intensity $I_0$ travels horizontally, parallel to the pulley's axis. It first passes through a fixed stationary polarizer with a vertical transmission axis, and then through the rotating polarizer on the pulley. At the equilibrium position of the pulley, the transmission axis of the rotating polarizer is also vertical. The pulley is displaced vertically downwards by a small distance $A$ from its equilibrium position and released from rest at $t=0$ . Assuming the string does not slip on the pulley, which of the following expressions represents the intensity of the transmitted light as a function of time $t$ ?

Options

A.
$\frac{I_0}{2} \cos^2\left[ \frac{A}{R} \cos\left( \sqrt{\frac{2k}{3M}} t \right) \right]$
B.
$\frac{I_0}{2} \cos^2\left[ \frac{A}{R} \cos\left( \sqrt{\frac{8k}{3M}} t \right) \right]$
C.
$I_0 \cos^2\left[ \frac{A}{R} \cos\left( \sqrt{\frac{8k}{3M}} t \right) \right]$
D.
$\frac{I_0}{2} \cos^2\left[ \frac{2A}{R} \cos\left( \sqrt{\frac{4k}{M}} t \right) \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Malus's Law|Dimensional Consistency|Effective Spring Constant|Vertical Oscillations

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of rotational kinematics, effective spring constants, and wave optics (Malus's Law).

The primary trap lies in the miscalculation of the effective spring constant and effective mass.

Students often incorrectly assume the spring extends by $x$ instead of $2x$ , leading to a missing factor of 4 in the restoring force ( $k_{eff} = k$ instead of $4k$ ).

Additionally, they may neglect the rotational kinetic energy of the pulley, using $M$ instead of $3M/2$ .

Finally, a secondary trap exists in forgetting that the first polarizer halves the intensity of the unpolarized light, testing their rigorous application of Malus's Law from the source to the observer.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Malus's Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free