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Question #83242

Difficulty: 8/10 | 🟢 Free

The Question

A uniform thin circular ring of mass $m$ , initial radius $R_0$ , uniform cross-sectional area $A$ , Young's modulus $Y$ , and coefficient of linear expansion $\alpha$ is suspended from a fixed frictionless pivot $P$ located exactly on its circumference. The ring is initially at temperature $T_0$ and performs small oscillations in its own vertical plane. The temperature of the ring is then uniformly increased by $\Delta T$ (where $\alpha \Delta T \ll 1$ ). Let $f_I = \frac{\Delta I}{I_0}$ be the fractional change in the moment of inertia of the ring about the pivot $P$ , $f_\tau = \frac{\Delta \tau}{\tau_0}$ be the fractional change in its time period of small oscillations, and $\sigma$ be the thermal stress developed in the ring due to the temperature change. Which of the following represents the correct tuple $(f_I, f_\tau, \sigma)$ ?

Options

A.
$(2\alpha \Delta T, \frac{1}{2}\alpha \Delta T, 0)$
B.
$(\alpha \Delta T, \frac{1}{2}\alpha \Delta T, Y\alpha \Delta T)$
C.
$(2\alpha \Delta T, \alpha \Delta T, Y\alpha \Delta T)$
D.
$(\alpha \Delta T, \alpha \Delta T, 0)$

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

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Common Mistakes

This question tests the synthesis of rigid body dynamics and thermal expansion while introducing two major traps.

The primary trap exploits the misapplication of the Parallel Axis Theorem ( $I = I_{cm} + md^2$ ): students often correctly expand the radius in $I_{cm}$ but incorrectly treat the center-of-mass distance $d$ as a constant $R_0$ , leading to an erroneous fractional change of $\alpha \Delta T$ instead of $2\alpha \Delta T$ .

The secondary trap uses extraneous data (Young's modulus $Y$ and area $A$ ) to bait students into assuming a thermal stress of $Y\alpha \Delta T$ , ignoring the fundamental principle that thermal stress only develops when expansion is geometrically constrained.

Since the ring is pivoted at a single point, it expands freely, yielding zero thermal stress.

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	170 sec
Access	Free