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Question #42453

Difficulty: 8/10 | 🟢 Free

The Question

A rigid block of mass $M$ rests on a weighing scale on the floor of an elevator. The block is also attached to the rigid ceiling of the elevator by a vertical metallic wire. At an initial temperature $T_0$ , the elevator is at rest, the wire has its exact unstretched length $L_0$ , cross-sectional area $A_0$ , and Young's modulus $Y_0$ . The tension in the wire is initially zero. The elevator then accelerates upwards with a constant acceleration $a$ . Simultaneously, the temperature of the wire is decreased by $\Delta T$ . The coefficient of linear expansion of the wire is $\alpha$ , and its Young's modulus varies with temperature $T$ as $Y = Y_0[1 - \beta(T - T_0)]$ . Assuming the distance between the ceiling and the scale remains strictly $L_0$ , and $\alpha \Delta T \ll 1$ , $\beta \Delta T \ll 1$ , what is the apparent weight of the block (the normal force recorded by the scale)?

Options

A.
$M(g+a) - Y_0 A_0 \alpha \Delta T [1 + (\beta - \alpha)\Delta T]$
B.
$M(g+a) - Y_0 A_0 \alpha \Delta T [1 + (\beta - 2\alpha)\Delta T]$
C.
$M(g+a) - Y_0 A_0 \alpha \Delta T [1 + \beta \Delta T]$
D.
$M(g-a) - Y_0 A_0 \alpha \Delta T [1 + (\beta + \alpha)\Delta T]$

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

Apparent Weight in Accelerating Elevators|Thermal Stress Calculation|Young's Modulus Sensitivity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize non-inertial frame dynamics with advanced thermal stress calculations.

The primary trap lies in treating the spring constant of the wire as $k = Y'A_0/L_0$ or $k = Y_0A_0/L_0$ , ignoring that thermal expansion alters not just the unstretched length (creating the extension $\Delta x$ ) but also the geometric parameters ( $A$ and $L$ ) that define the stiffness $k$ .

Students often forget to recalculate $k' = Y'A'/L'$ using the temperature-modified area and unstretched length, leading to an incorrect binomial expansion coefficient in the thermal tension term.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Apparent Weight in Accelerating Elevators
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free