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Question #42623

Difficulty: 8/10 | 🟢 Free

The Question

A torsion pendulum consists of a uniform solid disc of mass $M$ and linear expansion coefficient $\alpha_d$ , suspended at its center by a uniform wire of length $L$ , radius $r$ , and linear expansion coefficient $\alpha_w$ . The shear modulus of the wire material varies with temperature as $\eta = \eta_0 (1 - \gamma \Delta T)$ , where $\gamma$ is a positive constant and $\Delta T$ is the rise in temperature. If the angular frequency of torsional oscillations remains unchanged for small variations in temperature, which of the following relations is correct?

Options

A.
$\gamma = 3\alpha_w - 2\alpha_d$
B.
$\gamma = 4\alpha_w - 2\alpha_d$
C.
$\gamma = 2\alpha_d - 3\alpha_w$
D.
$\gamma = 2\alpha_d + \alpha_w$

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

Torsional Oscillations|Variation of Density with Temperature|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to simultaneously apply thermal expansion to multiple geometric and material properties in a dynamic system.

A common trap is assuming the torsion constant $C$ only depends on the shear modulus $\eta$ changing with temperature, or incorrectly applying the linear expansion coefficient to the radius $r$ without accounting for the $r^4$ dependence in $C = \frac{\pi \eta r^4}{2L}$ .

Students who memorize the simple pendulum clock formula will fail here, as they must derive the fractional change from first principles, recognizing that the wire's length, radius, and shear modulus all independently contribute to the shift in the restoring torque.

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