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Question #43532

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The Question

A uniform elastic rod of length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ is initially unstretched. One end is fixed to a rigid wall. At $t=0$ , a motorized mechanism attached to the free end begins to stretch the rod. The mechanism is programmed to deliver a transient mechanical power $P(t) = P_0 e^{-t/\tau}$ entirely into the rod, where $P_0$ and $\tau$ are positive constants. Assume the rod's mass is negligible, the process is isothermal, and all delivered work is stored as elastic potential energy. If the breaking stress of the rod's material is $\sigma_b$ , at what time $t_b$ will the rod break? (Assume $2 Y P_0 \tau > A L \sigma_b^2$ ).

Options

A.
$\tau \ln \left( \frac{2 Y P_0 \tau}{2 Y P_0 \tau - A L \sigma_b^2} \right)$
B.
$\tau \ln \left( \frac{Y P_0 \tau}{Y P_0 \tau - A L \sigma_b^2} \right)$
C.
$\tau \ln \left( \frac{P_0 \tau}{A L \sigma_b} \right)$
D.
$\tau \ln \left( \frac{2 Y P_0 \tau + A L \sigma_b^2}{2 Y P_0 \tau} \right)$

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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

Stress vs. Strain Definition|Breaking Stress Concept|Transient Current Analogy

Why This Question Is Tricky

Common Mistakes

This question tests the Energy-Equivalence Principle in the context of a time-varying transient load.

A common trap is for students to erroneously attempt to extract a time-varying force $F(t)$ from the power equation and directly equate the instantaneous stress $F(t)/A$ to the breaking stress $\sigma_b$ .

This ignores the fundamental principle that stress is a state variable dependent on the accumulated internal strain energy ( $dU = P(t)dt$ ), not the instantaneous external power.

By forcing students to integrate the transient power $P(t)$ to find the accumulated work, and equating this to the elastic strain energy density $U = \frac{1}{2} \frac{AL}{Y} \sigma^2$ , the problem evaluates their ability to link non-conservative transient inputs (analogous to an RL circuit discharge) to conservative elastic potential energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Stress vs. Strain Definition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free