A steel rod of length 1 m and cross-sectional area 10 - 4 m 2 is heated from 0 ° C to 200 ° C without being allowed to extend or bend. The compressive tension produced in the rod is _____ × 10 4 N . (

Question

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

TestHub · Accepted Answer

The formula to calculate the Young's modulus of the material of the wire is given by
$Y = \frac{F L}{A ∆ L} . . . (1)$
The increase in length of the wire due to increase in temperature is given by
$∆ L = L α (T_{2} - T_{1}) . . . (2)$
Substitute the expression for the extension in length from equation (2) into equation (1) and simplify to obtain the required compressive tension.
$\begin{array}{rcl} Y & = & \frac{F L}{A L α (T_{2} - T_{1})} \\ \Rightarrow & F = A α Y (T_{2} - T_{1}) . . . (3) \end{array}$
Substitute the values of the known parameters into equation (3) to calculate the required compressive tension in the wire.
$\begin{array}{rcl} F & = & 10^{- 4} m^{2} \times 10^{- 5} K^{- 1} \times 2 \times 10^{11} {Nm}^{- 2} \times (200 ° C - 0 ° C) \\ = & 4 \times 10^{4} N \end{array}$

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