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Question #42213

Difficulty: 8/10 | 🟢 Free

The Question

A viscoelastic wire of volume $V$ (in $m^3$ ) is subjected to a cyclic tensile stress $\sigma(t) = \sigma_0 \sin(\omega t)$ (in $N/m^2$ ). Due to the elastic after-effect, the strain lags the stress and is given by $\varepsilon(t) = \varepsilon_0 \sin(\omega t - \delta)$ , where $\delta$ is the phase angle. Assuming the process is isothermal and all dissipated mechanical energy appears as heat, what is the total heat generated in the wire during one complete cycle?

Options

A.
$\pi V \sigma_0 \varepsilon_0 \sin \delta$
B.
$2\pi V \sigma_0 \varepsilon_0 \cos \delta$
C.
$\frac{1}{2} V \sigma_0 \varepsilon_0 \sin \delta$
D.
$0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Elastic After-Effect|First Law Application|State Function vs Path Function

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that all elastic processes are perfectly conservative and path-independent.

Students often assume that because a material is 'elastic', the net work done over a closed cycle is zero (Option D), or they incorrectly apply the static elastic strain energy formula $\frac{1}{2}\sigma\varepsilon$ (Option C).

The trap lies in the 'elastic after-effect' (viscoelasticity), which introduces a phase lag $\delta$ between stress and strain, creating a hysteresis loop.

By the First Law of Thermodynamics, the area of this loop ( $\oint \sigma d\varepsilon$ ) represents non-conservative work that is entirely dissipated as internal heat, proving the cycle is thermodynamically irreversible despite being mechanically cyclic.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Elastic After-Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free