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Question #42470

Difficulty: 8/10 | 🟢 Free

The Question

A smooth wedge of mass $M$ and inclination angle $\theta$ is free to slide on a smooth horizontal floor. A small block of mass $m$ rests on the inclined surface of the wedge. The block is constrained by a fixed, smooth vertical guide to move only in the vertical direction. An ideal light spring of force constant $k$ is attached at one end to the top edge of the wedge and at the other end to the block, such that the spring always lies straight along the inclined surface. If the system is slightly displaced from its stable equilibrium position and released, what is the angular frequency $\omega$ of the resulting small oscillations?

Options

A.
$\sqrt{\frac{k}{M \cos^2 \theta + m \sin^2 \theta}}$
B.
$\sqrt{\frac{k \cos^2 \theta}{M \cos^2 \theta + m \sin^2 \theta}}$
C.
$\sqrt{\frac{k \cos^4 \theta}{M \cos^2 \theta + m \sin^2 \theta}}$
D.
$\sqrt{\frac{k}{M \sin^2 \theta + m \cos^2 \theta}}$

Check Your Answer

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

Wedge Constraint Logic|Effective Spring Constant|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply kinematic constraints to both kinetic and potential energy in a coupled oscillator system.

The primary trap lies in the 'Effective Spring Constant'.

Students often mistakenly assume the restoring force constant remains $k$ (leading to Option B) or incorrectly project the force horizontally as $k \cos^2 \theta$ (leading to Option C).

However, because the spring is aligned with the incline, a horizontal displacement $x$ of the wedge geometrically amplifies the spring's extension to $x \sec \theta$ .

Since potential energy scales with the square of extension, the effective spring constant becomes $k \sec^2 \theta$ .

Recognizing this geometric transformation of the restoring force is a hallmark of JEE Advanced Level 3 mechanics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Wedge Constraint Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free