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Question #42427

Difficulty: 8/10 | 🟢 Free

The Question

A block is placed on a smooth horizontal floor and is attached to a relaxed horizontal ideal spring of stiffness $k$ . The other end of the spring is fixed to a vertical wall. A light inextensible string is tied to the block, passes over a small smooth fixed pulley located at a height $h$ above the floor, and hangs vertically downwards. The lower end of the string is attached to the free bottom end of a uniform vertical metallic rod of length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ . The upper end of the rod is rigidly fixed to a horizontal ceiling. Initially, the block is at a horizontal distance $x$ from the vertical line passing through the pulley, the string is just taut, and the rod is at temperature $T_0$ . The temperature of the rod is slowly increased by $\Delta T$ . Assuming the block's displacement is small, what is the work done by the string on the block?

Options

A.
$\frac{k x^2 (x^2+h^2) (A Y L \alpha \Delta T)^2}{2 [A Y x^2 + k L (x^2+h^2)]^2}$
B.
$\frac{k (A Y L \alpha \Delta T)^2}{2 (A Y + k L)^2}$
C.
$\frac{k (x^2+h^2) (L \alpha \Delta T)^2}{2 x^2}$
D.
$\frac{k x^2 (x^2+h^2) (A Y L \alpha \Delta T)^2}{2 [A Y (x^2+h^2) + k L x^2]^2}$

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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

Constraint Relations via String Length|Thermal Stress in Constrained Rods|Calculus/Derivatives

Why This Question Is Tricky

Common Mistakes

This problem tests the coupling of a geometric kinematic constraint with a thermal stress constraint.

A common trap is treating the string constraint as 1:1 ( $\Delta x = \delta$ ) or ignoring the elasticity of the rod (assuming it acts as an infinitely stiff actuator, $\delta = L \alpha \Delta T$ ).

Students must realize that the thermal expansion acts as a moving boundary condition that is resisted by the effective stiffness of the constrained block-spring system.

By differentiating the geometric constraint $s^2 = x^2 + h^2$ , they must find the effective stiffness $K = k(x^2+h^2)/x^2$ and equate the tension to the compressive stress in the rod to find the true displacement.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Constraint Relations via String Length
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free