Mechanical Properties of Solids

JEE Main level, single correct, medium

72 of 118

Question 72 of 118

A rod, of length LL at room temperature and uniform area of cross section A,A, is made of a metal having coefficient of linear expansion α/∘C\alpha /^{\circ}C It is observed that an external compressive force FF is applied on each of its ends, prevents any change in the length of the rod when its temperature rises by ∆TK∆TK Young's modulus, YY for this metal is:

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Answer: (C) FAα∆T\dfrac{F}{A\alpha ∆T}

Since the length of the rod does not change, the increment due to increase in temperature will be balanced by the compression due to applied force.

Y=FAΔll⇒Δll=FAYY=\dfrac{\dfrac{F}{A}}{\dfrac{\Delta l}{l}}\Rightarrow \dfrac{\Delta l}{l}=\dfrac{F}{AY}

Also Δll=αΔT\dfrac{\Delta l}{l}=\alpha \Delta T

∴FAY=αΔT∴\dfrac{F}{AY}=\alpha \Delta T

Y=FAαΔTY=\dfrac{F}{A\alpha \Delta T}

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