MHT CET · Physics · Mechanical Properties of Solids

A metal rod of length $L$ and cross-sectional area $A$ is heated through $T^{\circ} C$. What is the force required to prevent the expansion of the rod lengthwise?
$[ Y =$ Young's modulus of the material of rod, $\alpha=$ coefficient of linear expansion $]$

A $\frac{Y A α T}{(1 - α T)}$
B $\frac{Y A α T}{(1 + α T)}$
C $\frac{(1 - α T)}{Y A α T}$
D $\frac{(1 + α T)}{Y A α T}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{Y A α T}{(1 + α T)}$

Step-by-step Solution

Detailed explanation

∆ L = α L T

...... (i)
Also

Y = \frac{F}{A} . \frac{L (1 + α t)}{∆ L}

∆ L = \frac{F . L (1 + α t)}{A . Y}

........ (ii)
Equating (i) and (ii)

\frac{F . L (1 + α t)}{A . Y} = α L T

∴ F = \frac{A Y α T}{(1 + α t)}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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A metal rod of length \(L\) and cross-sectional area \(A\) is heated through \(T^{\circ} C\). What is the force required to prevent the expansion of the rod lengthwise?
\([ Y =\) Young's modulus of the material of rod, \(\alpha=\) coefficient of linear expansion \(]\)

Answer & Solution

Step-by-step Solution

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A metal rod of length \(L\) and cross-sectional area \(A\) is heated through \(T^{\circ} C\). What is the force required to prevent the expansion of the rod lengthwise?\([ Y =\) Young's modulus of the material of rod, \(\alpha=\) coefficient of linear expansion \(]\)

Step-by-step Solution

A metal rod of length \(L\) and cross-sectional area \(A\) is heated through \(T^{\circ} C\). What is the force required to prevent the expansion of the rod lengthwise?
\([ Y =\) Young's modulus of the material of rod, \(\alpha=\) coefficient of linear expansion \(]\)