Before You Begin

Question #43545

Difficulty: 8/10 | 🟢 Free

The Question

A rigid, thin rod of length $L$ is suspended from a frictionless pivot at one of its ends ( $x=0$ ). The linear mass density of the rod varies along its length as $\lambda(x) = \alpha x^2$ , where $\alpha$ is a positive constant and $x$ is the distance from the pivot. If the rod is displaced by a small angle $\theta$ from the vertical and released, what is the time period $T$ of its resulting simple harmonic motion? (Assume acceleration due to gravity is $g$ and air resistance is negligible).

Options

A.
$2\pi \sqrt{\frac{6L}{5g}}$
B.
$2\pi \sqrt{\frac{2L}{3g}}$
C.
$2\pi \sqrt{\frac{4L}{5g}}$
D.
$2\pi \sqrt{\frac{4L}{9g}}$

Check Your Answer

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

SHM Definition via Differential Equation|Physical Pendulum Inertia

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous derivation of the SHM differential equation for a physical pendulum, specifically targeting the trap of misapplying standard uniform-body formulas.

Students often memorize the physical pendulum formula $T = 2\pi\sqrt{I/mgd}$ but fail to correctly evaluate $d$ (the distance to the center of mass) for a non-uniform density distribution.

A common mistake is calculating the correct moment of inertia $I$ via integration but incorrectly assuming the center of mass remains at $L/2$ (yielding Option A), or calculating the correct COM but using the standard uniform rod inertia $ML^2/3$ (yielding Option D).

By introducing a power-law density distribution, the problem forces students to independently integrate both the restoring torque's lever arm and the rotational inertia, filtering out formula-memorizers.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	SHM Definition via Differential Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free