Chapter 13, Problem 81A

To determine

Increase in length of a $\text{300 m}$ steel bridge on a $30°\text{C}$ day in August compared to $-10°\text{C}$ night in January.

Answer to Problem 81A

Steel bridge will increase in length by $0.144m$ on a $30°\text{C}$ day in August compared to $-10°\text{C}$ night in January.

Explanation of Solution

Given:

Length of bridge, $L=\text{300 m}$

Higher temperature, $T_1=\text{30°C}$

Lower temperature, $T_2=\text{ -10°C}$

Linear coefficient of expansion, $\alpha \text{ = 12}\times {\text{10}}^{-6}/°\text{C}$

Formula used:

Linear thermal expansion

  $\Delta L=a{L}_1\Delta T$

Where, $\Delta L$ is the change in length  $L_1$ , $\alpha$ is a coefficient of linear expansion, $\Delta T$ is the change in temperature and $L_1$ is the length at temperature $T$

Calculation:

Substituting the given values, in the formula,

  $\Delta L=a{L}_1\Delta T$

  $\text{Δ}L{\text{ =12 × 10}}^{\text{-6}}/\text{°C}\text{× 300 m × }\left(\text{30 - }\left(\text{-10}\right)\right)\text{°C}$

  ${\text{=12 × 10}}^{\text{-6}}\text{ × 300 m × 40}$

  ${\text{=12 × 12 × 10}}^{\text{-6}}{\text{ × 10}}^{\text{3}}\text{ m}$

  ${\text{=144 ×10}}^{\text{-3}}\text{ m}$

  $\text{Δ}L\text{ = 0}\text{.144 m}$

Conclusion:

Increase in a length of steel bridge is $0.144m$ .