A compressive force, F is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by \Delta T. The net change in its length is zero. Let \ell be the length of the rod, A its area of cross-section,Y its Young’s modulus, and \alpha its coefficient of linear expansion. Then, F is equal to :

Heat and Thermodynamics PYQ — JEE Main 2017 · 8 Apr · Shift 1 · Q64 with Solution

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Correct answer: C

If the rod is heated by $\Delta T$ , then its free expansion would be

$\Delta \ell_{\text{thermal}} = \alpha \ell \, \Delta T$

where:

A compressive force $F$ produces a contraction in the rod. Using Young’s modulus,

$Y = \frac{\text{stress}}{\text{strain}} = \frac{F/A}{\Delta \ell_{\text{mech}}/\ell}$

So,

$\Delta \ell_{\text{mech}} = \frac{F\ell}{AY}$

This is the magnitude of contraction.

Given that the rod is heated and compressed simultaneously, and the net change in length is zero, we must have

$\Delta \ell_{\text{thermal}} = \Delta \ell_{\text{mech}}$

Hence,

$\alpha \ell \, \Delta T = \frac{F\ell}{AY}$

Cancel $\ell$ from both sides:

$\alpha \, \Delta T = \frac{F}{AY}$

Therefore,

$F = AY\alpha \Delta T$

This corresponds to:

Option C: $AY\alpha \Delta T$