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Question #42206

Difficulty: 8/10 | 🟢 Free

The Question

A uniform thin circular ring of mass $m$ and radius $R$ is rigidly attached to a massless, non-expanding rod of length $R$ . The rod is collinear with a diameter of the ring, connecting the ring's circumference to a frictionless pivot $P$ . The system oscillates as a physical pendulum in a vertical plane with a small time period $T_0$ . The ring is made of a material with a coefficient of linear thermal expansion $\alpha$ . If the temperature of the ring is increased by $\Delta T$ (while the rod's length remains unaffected), the new time period becomes $T$ . Assuming $\alpha \Delta T \ll 1$ , the fractional change in the time period $\frac{T - T_0}{T_0}$ is:

Options

A.
$\frac{7}{20} \alpha \Delta T$
B.
$\frac{1}{2} \alpha \Delta T$
C.
$\frac{3}{5} \alpha \Delta T$
D.
$\frac{3}{20} \alpha \Delta T$

Check Your Answer

Correct Answer

Step-by-Step Solution

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

Thermal Stress Calculation|Rigid Body Constraints|Parallel and Perpendicular Axes

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle rigid body constraints coupled with thermal expansion.

The primary trap is the assumption that the distance from the pivot to the center of mass ( $d$ ) remains constant because the connecting rod does not expand, or conversely, that the entire system scales uniformly.

Because the ring is pinned at its circumference, its radial thermal expansion shifts its center of mass relative to the non-expanding rod.

Students must correctly apply the Parallel Axis Theorem with a temperature-dependent $d$ and independently calculate the fractional changes in both the moment of inertia and the center of mass distance to find the true variation in the pendulum's time period.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Thermal Stress Calculation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free