Question

The ratio of linear expansivity of copper to that of iron Is approximately 1:5. A specimen of iron and specimen of copper expand by the same amount per unit rise in temperature. What is the ratio of the length of iron to that of copper ?

Answer 1

The expansion per unit rise in temperature is given as follows:

$$\dfrac{\Delta L}{\Delta T} = \alpha L$$

where $\Delta L$ is the increase in length, $\Delta T$ is the rise in temperature, $\alpha$ is the expansivity, and $L$ is the initial length.

According to the text of the problem, $\left( \dfrac{\Delta L}{\Delta T} \right)_{cooper } = \left( \dfrac{\Delta L}{\Delta T} \right)_{iron }$ and $\dfrac{\alpha_{cooper}}{\alpha_{iron}} = \dfrac15$. Then the ration of the lengths is:

$$L_{cooper} = \dfrac{1}{\alpha_{cooper}} \left( \dfrac{\Delta L}{\Delta T} \right)_{cooper } \\ L_{iron} = \dfrac{1}{\alpha_{iron}} \left( \dfrac{\Delta L}{\Delta T} \right)_{iron }\\ \dfrac{L_{iron} }{L_{cooper} } = \dfrac{\dfrac{1}{\alpha_{iron}} \left( \dfrac{\Delta L}{\Delta T} \right)_{iron }}{ \dfrac{1}{\alpha_{cooper}} \left( \dfrac{\Delta L}{\Delta T} \right)_{cooper }} = \dfrac{\alpha_{cooper}}{\alpha_{iron}} = \dfrac15$$

Answer. 1:5.