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Question #43547

Difficulty: 8/10 | 🟢 Free

The Question

A uniform massless elastic rod of natural length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ is suspended vertically from a rigid ceiling. A block of mass $M$ is attached to its lower end. The system is initially in equilibrium at a temperature $T_0$ . The temperature of the rod is then uniformly increased to $T_0 + \Delta T$ . Let $\alpha$ be the coefficient of linear expansion of the material of the rod. Assuming the cross-sectional area $A$ and Young's modulus $Y$ remain constant with temperature, and $g$ is the acceleration due to gravity, which of the following expressions correctly gives the downward shift in the equilibrium position of the block ( $\Delta y$ ) and the new angular frequency of small vertical oscillations ( $\omega'$ ) at the higher temperature?

Options

A.
$\Delta y = L \alpha \Delta T \left( 1 + \frac{Mg}{YA} \right), \; \omega' = \sqrt{\frac{YA}{ML(1 + \alpha \Delta T)}}$
B.
$\Delta y = L \alpha \Delta T, \; \omega' = \sqrt{\frac{YA}{ML}}$
C.
$\Delta y = L \alpha \Delta T \left( 1 + \frac{Mg}{YA} \right), \; \omega' = \sqrt{\frac{YA}{ML}}$
D.
$\Delta y = L \alpha \Delta T, \; \omega' = \sqrt{\frac{YA}{ML(1 + \alpha \Delta T)}}$

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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

Vertical Oscillations|Thermal Stress in Constrained Rods|Interatomic Force Curve

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple thermal expansion with Hooke's Law in a dynamic system.

The primary trap lies in assuming that the downward shift of the mass is solely due to the thermal expansion of the rod's natural length ( $\Delta y = L\alpha\Delta T$ ).

Students often fail to recognize that the spring constant $k = YA/L$ is inversely proportional to the natural length.

As the rod expands thermally, its stiffness decreases, causing an additional mechanical extension under the same gravitational load.

Consequently, the total shift includes both the thermal expansion of the unstretched rod and the increased elastic stretch.

Furthermore, this softened spring constant directly reduces the angular frequency of oscillation, testing the precise definition of $k$ in continuous elastic media.

Quick Facts About This Question

Question Information

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