Chapter 21, Problem 65A

Steel expands 1 part in 100,000 for each Celsius degree increase in temperature. If the 1.5-km main span of a steel suspension bridge had no expansion joints, how much longer would it be for a temperature increase of $20°C$?

To determine

The linear expansion of steel for the given temperature change.

Answer to Problem 65A

The linear expansion of steel for $20°\text{C}$ increase in temperature is 0.3m.

Explanation of Solution

Given:

Linear expansion coefficient of steel for 1 part , $\alpha =\frac{1}{100,000}={10}^{-5}$ for each degree increase.

Original length , $L_o=1.5\text{ km}$

  $\Delta T=20°\text{C}$

Formula used:

The linear expansion is given by the formula,

  $L=L_o(1+\alpha \Delta T)$

Calculation:

On substituting the values in the above equation,

  $\begin{array}{l}\Rightarrow L=1.5{\text{ km[1+(10}}^{-5}/°\text{C}\times 20°\text{C})]\\ \Rightarrow L=1.5003\text{ km}\end{array}$

Now, $20°\text{C}$ increase means, the steel will expand by 20 parts in 100,000.

By multiplying the length of the bridge by that ratio,

  $\begin{array}{l}\Rightarrow L=\frac{1.5003\text{ km}\times \text{20 parts}}{100,000}\\ \Rightarrow L=0.0003\text{ km}\\ \Rightarrow L=0.3\text{ m}\end{array}$

Therefore, the steel is $0.3\text{ m}$ long for a temperature increase of $20°\text{C}$ .