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Question #43923

Difficulty: 8/10 | 🟢 Free

The Question

A thin bimetallic strip of length $L$ and constant flexural rigidity $EI$ is clamped at the origin. Its tangent at the origin points along the positive $x$ -axis. A point mass $m$ is attached to its free end. At temperature $T_0$ , the strip is perfectly straight. When heated to $T_0 + \Delta T$ , thermal stresses cause the strip to bend into a perfect semicircle in the upper half of the $xy$ -plane ( $y \ge 0$ ). A rigid, massless guide rod is pivoted at the origin and attached to the mass $m$ , constraining it to move only along a circular path centered at the origin. The mass $m$ is given a small angular displacement and undergoes rotational oscillations. Assuming the bimetallic strip provides the only restoring force, what is the angular frequency $\omega$ of these oscillations?

Options

A.
$\sqrt{\frac{2 \pi^4 EI}{m L^3 (3\pi^2 - 16)}}$
B.
$\sqrt{\frac{2 \pi^2 EI}{3 m L^3}}$
C.
$\sqrt{\frac{3 \pi^2 EI}{4 m L^3}}$
D.
$\sqrt{\frac{3 EI}{m L^3}}$

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

Effective Spring Constant|Bimetallic Strip Curvature|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the ability to model the effective torsional stiffness of a thermally deformed structure.

Students often fall into the trap of using the straight-beam stiffness ( $3EI/L$ ) or ignoring the radial constraint force ( $P$ ) imposed by the guide rod.

The thermal expansion ( $\Delta T$ ) fundamentally alters the geometry, changing the torque arm to $2L/\pi$ and requiring Castigliano's theorem on a curved beam to find the true compliance matrix.

Ignoring the constraint force $P$ leads to an incorrect, overly compliant system (Option B), while ignoring the geometric shift entirely leads to Options C or D.

Quick Facts About This Question

Question Information

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