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Question #42627

Difficulty: 8/10 | 🟢 Free

The Question

A uniform cylindrical wire of initial length $L_0$ and radius $r_0$ has a shear modulus $\eta$ and Poisson's ratio $\nu$ . A heavy disc of moment of inertia $I$ is attached to its lower end, causing a small longitudinal strain $\epsilon$ (where $\epsilon \ll 1$ ) in the equilibrium position. The disc is then given a small angular twist and released, executing torsional oscillations. Let $T_0$ be the hypothetical time period of torsional oscillations if the wire were completely un-stretched, and $T$ be the actual time period under the load. The fractional change in the time period, $\frac{T - T_0}{T_0}$ , is approximately:

Options

A.
$\frac{1}{2}(1 + 4\nu)\epsilon$
B.
$\frac{1}{2}(1 - 4\nu)\epsilon$
C.
$\frac{1}{2}(1 + 2\nu)\epsilon$
D.
$\frac{1}{2}\epsilon$

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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

Torsional Oscillations|Poisson's Ratio Limits|Stress vs. Strain Definition|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the limits of small deformation assumptions by coupling axial strain with torsional rigidity.

Students commonly assume the torsion constant $C$ only changes due to the length extension $L$ (yielding $\frac{1}{2}\epsilon$ ) or incorrectly assume $C$ is proportional to the cross-sectional area $r^2$ instead of $r^4$ .

By incorporating Poisson's ratio $\nu$ , the radius reduction $r = r_0(1 - \nu \epsilon)$ must be raised to the fourth power, creating a dominant $4\nu \epsilon$ term.

This highlights the critical geometric dependency of shear stress distribution in torsional oscillators under extreme loading.

Quick Facts About This Question

Question Information

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