Question

A brass scale is graduated at ${{10}^{\circ }}C$. What is the true length of a zinc rod which measures $60.00cm$ on this scale at ${{30}^{\circ }}C$? Coefficient of linear expansion of brass $=18\times {{10}^{-6}}{{K}^{-1}}$

Answer 1

The coefficient of linear expansion is an intrinsic property of every material. It varies from one material to another. In order to find the solution of the given question we will apply the formula of true length in linear expansion.
Formula Used:
$L={{L}^{'}}\left( 1+\alpha \Delta T \right)$

Complete answer:
Since, the scale is designed to measure from ${{10}^{\circ }}C$. Thus, the expansion will occur at ${{30}^{\circ }}C$ which will result in a wrong reading. So, to find the correct value we will apply the formula of true length.
It is given in the question that,
Increased temperature $={{10}^{\circ }}C$ Measured length ${{L}^{'}}=60.00cm$ Initial temperature $={{30}^{\circ }}C$
We need to calculate the true length of a zinc rod, using the formula of true length,
$L={{L}^{'}}\left( 1+\alpha \Delta T \right)$ where ${{L}^{'}}$ is the given length, $\alpha$is the coefficient of linear expansion of brass and $\Delta T$ is the change in temperature.
Substituting the values in the formula we will get,
$\begin{aligned} & L=60\left( 1+18\times {{10}^{-6}}\times \left( 30-10 \right) \right) \\\ & \Rightarrow L=60\left( 1+0.00036 \right) \\\ & \Rightarrow L=60.02cm \\\ \end{aligned}$
Hence, the true length of the zinc rod is $60.02cm$.

Additional Information:
Thermal expansion is defined as the tendency of an object to change its dimension either in length, area or volume because of heat. Thermal expansion is of three types – linear expansion, areal expansion and volume expansion. As the name suggests expansions due to change in length, area and volume due to temperature are known as linear expansion, areal expansion and volume expansion respectively.

Note:
Expansion is defined as change or increase in length. If the expansion is along one dimension, then it is known as linear expansion. The coefficient of linear expansion of materials depends upon the temperature. Therefore, the change in temperature will reflect the rate of expansion.