Question

A steel beam is \[5\,{\text{m}}\] long at a temperature of \[20^\circ {\text{C}}\]. On a hot day, the temperature rises to \[40^\circ {\text{C}}\]. What is the change in the length of the beam due to thermal expansion?
Coefficient of linear expansion of steel \[\alpha = 1.2 \times {10^{ - 5}}\,^\circ {{\text{C}}^{ - 1}}\]

Answer 1

Hint: Use the expression for the linear thermal expansion of a solid material. This equation gives the relation between the original length, change in length, coefficient of linear thermal expansion and change in temperature of the solid material.
Formula used:
The expression for the linear thermal expansion of a solid material is
\[\Delta L = \alpha {L_0}\Delta T\] …… (1)
Here, \[\Delta L\] is the change in the length of the solid material, \[\alpha \] is the linear thermal expansion coefficient, \[L\] is the original length of the material and \[\Delta T\] is the change in the temperature.

Complete step by step answer:
The length of the steel beam is \[5\,{\text{m}}\] at a temperature of \[20^\circ {\text{C}}\]. The temperature increases to \[40^\circ {\text{C}}\] on a hot day.
Calculate the change in the temperature \[\Delta T\] of the steel beam.
\[\Delta T = {T_f} - {T_i}\]
Here, \[{T_f}\] is the final increased temperature of the steel beam and \[{T_i}\] is the initial temperature of the steel beam.
Substitute for \[{T_f}\] and \[20^\circ {\text{C}}\] for \[{T_i}\] in the above equation.
\[\Delta T = 40^\circ {\text{C}} - 20^\circ {\text{C}}\]
\[ \Rightarrow \Delta T = 20^\circ {\text{C}}\]
Hence, the change in the temperature of the steel beam is \[20^\circ {\text{C}}\].
Calculate the change in the length of the steel beam at the increased temperature \[40^\circ {\text{C}}\].
Substitute \[1.2 \times {10^{ - 5}}\,^\circ {{\text{C}}^{ - 1}}\] for \[\alpha \], \[5\,{\text{m}}\] for \[L\] and \[20^\circ {\text{C}}\] for \[\Delta T\] in equation (1).
\[\Delta L = \left( {1.2 \times {{10}^{ - 5}}\,^\circ {{\text{C}}^{ - 1}}} \right)\left( {5\,{\text{m}}} \right)\left( {20^\circ {\text{C}}} \right)\]
\[ \Rightarrow \Delta L = \left( {1.2 \times {{10}^{ - 5}}\,^\circ {{\text{C}}^{ - 1}}} \right)\left( {5\,{\text{m}}} \right)\left( {20^\circ {\text{C}}} \right)\]
\[ \Rightarrow \Delta L = 1.2 \times {10^{ - 3}}\,{\text{m}}\]
\[ \Rightarrow \Delta L = 1.2\,{\text{mm}}\]

Hence, the change in the length of the steel beam is \[1.2\,{\text{mm}}\].

Note:
Since the unit of the coefficient of the linear thermal expansion of the steel beam is given in degree Celsius, the change in the temperature of the steel beam is taken in degree Celsius. Otherwise, one should convert the unit of the convert in temperature of the given material in Kelvin.