Mechanical Properties of Solids

JEE Main level, single correct, medium

83 of 118

Question 83 of 118

Two wires are made of the same material and have the same volume. However wire 1 has crosssectional area AA and wire-2 has cross-sectional area 3A3 A. If the length of wire 1 increases by Δx\Delta x on applying force FF, how much force is needed to stretch wire 2 by the same amount?

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Answer: (D) 9 F9 \mathrm{~F}

A1ℓ1=A2ℓ2⇒ℓ2=A1ℓ1 A2=A×ℓ13 A=ℓ13⇒ℓ1ℓ2=3Δx1=F1 Aγ×ℓ1Δx2=F23 Aγℓ2 Here Δx1=Δx2F23 Aγℓ2=F1 Aγℓ1 F2=3 F1×ℓ1ℓ2=3 F1×3=i)\begin{aligned} & \mathrm{A}_1 \ell_1=\mathrm{A}_2 \ell_2 \Rightarrow \ell_2=\frac{\mathrm{A}_1 \ell_1}{\mathrm{~A}_2}=\frac{\mathrm{A} \times \ell_1}{3 \mathrm{~A}}=\frac{\ell_1}{3} \Rightarrow \frac{\ell_1}{\ell_2}=3 \\ & \Delta \mathrm{x}_1=\frac{\mathrm{F}_1}{\mathrm{~A} \gamma} \times \ell_1 \\ & \Delta \mathrm{x}_2=\frac{\mathrm{F}_2}{3 \mathrm{~A} \gamma} \ell_2 \\ & \text { Here } \Delta \mathrm{x}_1=\Delta \mathrm{x}_2 \\ & \frac{\mathrm{F}_2}{3 \mathrm{~A} \gamma} \ell_2=\frac{\mathrm{F}_1}{\mathrm{~A} \gamma} \ell_1 \\ & \left.\mathrm{~F}_2=3 \mathrm{~F}_1 \times \frac{\ell_1}{\ell_2}=3 \mathrm{~F}_1 \times 3=\mathrm{i}\right) \end{aligned}

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