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Question #43854

Difficulty: 8/10 | 🟢 Free

The Question

A uniform square plate of mass $M$ and side length $a$ is suspended from a frictionless pivot at one of its corners. The plate lies in a vertical plane and is free to oscillate under gravity. It is displaced by a small angle $\theta_0$ (in radians) from its stable equilibrium position and released from rest. The gravitational restoring torque can be modeled as an effective torsional spring with a rotational spring constant $k_{eff}$ . Assuming the acceleration due to gravity is $g$ , what is the value of $k_{eff}$ and the maximum angular velocity $\omega_{max}$ of the plate?

Options

A.
$k_{eff} = \frac{Mga}{\sqrt{2}}$ , $\omega_{max} = \theta_0 \sqrt{\frac{3g}{2\sqrt{2}a}}$
B.
$k_{eff} = \frac{Mga}{\sqrt{2}}$ , $\omega_{max} = \theta_0 \sqrt{\frac{3\sqrt{2}g}{a}}$
C.
$k_{eff} = Mga$ , $\omega_{max} = \theta_0 \sqrt{\frac{3g}{2a}}$
D.
$k_{eff} = \frac{Mga}{2}$ , $\omega_{max} = \theta_0 \sqrt{\frac{3g}{4a}}$

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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

Physical Pendulum Inertia|Effective Spring Constant|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map the gravitational potential energy of a physical pendulum to an equivalent elastic potential energy using an effective rotational spring constant ( $k_{eff}$ ).

The primary trap lies in the calculation of the moment of inertia.

Students frequently forget to apply the Parallel Axis Theorem, incorrectly using the center-of-mass moment of inertia ( $I_{cm} = Ma^2/6$ ) instead of the moment of inertia about the pivot ( $I = 2Ma^2/3$ ).

Additionally, students may misidentify the distance from the pivot to the center of mass, confusing the side length $a$ or half-side $a/2$ with the correct diagonal distance $a/\sqrt{2}$ , leading to an incorrect $k_{eff}$ .

Quick Facts About This Question

Question Information

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