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Thermal Properties of Matter

JEE Main level, numerical answer, medium

19 of 126

Question 19 of 126

A steel rod of length $1m$ and cross-sectional area ${10}^{-4}{m}^{2}$ is heated from $0^{\circ}C$ to $200^{\circ}C$ without being allowed to extend or bend. The compressive tension produced in the rod is _____ $\times {10}^{4}N$ . (Given Young's modulus of steel $=2\times {10}^{11}N{m}^{-2}$ , coefficient of linear expansion $={10}^{-5}{K}^{-1}$ )

Show answer and solution

Answer: 4

The formula to calculate the Young's modulus of the material of the wire is given by

Y=\dfrac{FL}{A∆L}...\left(1\right)

The increase in length of the wire due to increase in temperature is given by

∆L=L\alpha \left({T}_{2}-{T}_{1}\right)...\left(2\right)

Substitute the expression for the extension in length from equation (2) into equation (1) and simplify to obtain the required compressive tension.

\begin{matrix}Y & = & \dfrac{FL}{AL\alpha \left({T}_{2}-{T}_{1}\right)} \\ & \Rightarrow & F=A\alpha Y\left({T}_{2}-{T}_{1}\right)...\left(3\right)\end{matrix}

Substitute the values of the known parameters into equation (3) to calculate the required compressive tension in the wire.

\begin{matrix}F & = & {10}^{-4}{m}^{2}\times {10}^{-5}{K}^{-1}\times 2\times {10}^{11}{Nm}^{-2}\times \left(200^{\circ}C-0^{\circ}C\right) \\ & = & 4\times {10}^{4}N\end{matrix}

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