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Question #42250

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The Question

A physical pendulum consists of a uniform thin rod of mass $m$ and length $R$ , and a uniform solid circular disc of mass $m$ and radius $R$ . The disc is rigidly attached at its center to one end of the rod, such that they lie in the same plane. The system is suspended from a smooth horizontal pivot passing through the other end of the rod, perpendicular to their plane, and performs small oscillations. The coefficient of linear expansion of the rod is $\alpha_1$ and that of the disc is $\alpha_2$ . If the temperature of the system is increased by a small amount $\Delta\theta$ , the fractional change in the time period $\frac{\Delta T}{T}$ is:

Options

A.
$\left( \frac{5\alpha_1 + 6\alpha_2}{22} \right) \Delta\theta$
B.
$\left( \frac{8\alpha_1 + 3\alpha_2}{11} \right) \Delta\theta$
C.
$\left( \frac{16\alpha_1 + 5\alpha_2}{22} \right) \Delta\theta$
D.
$\left( \frac{\alpha_1 + \alpha_2}{2} \right) \Delta\theta$

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Correct Answer

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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

Compound Pendulum Period|Thermal Expansion of Area and Volume|Calculus-based Moment of Inertia

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of error analysis and small-change calculus to rigid body dynamics, specifically targeting the thermal expansion of composite bodies.

Students often blindly apply the standard result $\Delta T/T = \frac{1}{2}\alpha\Delta\theta$ or fail to correctly decouple the thermal expansion of the moment of inertia ( $I$ ) and the center of mass distance ( $d$ ).

A lethal trap here is assuming the parallel axis theorem term ( $mL^2$ ) for the disc expands with the disc's coefficient ( $\alpha_2$ ) rather than the rod's ( $\alpha_1$ ), or forgetting to subtract the $\Delta d/d$ term entirely.

It forces students to recognize that the disc's CM position is dictated solely by the rod's expansion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Compound Pendulum Period
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free