Question

When the diameter of a copper coin is raised by, its diameter increases by 0.2%. Then which of the following is correct?
(This question has multiple correct options)
A: percentage rise in the area of a face is 0.4%
B: percentage rise in the thickness is 0.4%
C: percentage rise in the volume is 0.4%
D: coefficient of linear expansion of copper is $0.25\times {{10}^{-4}}{{/}^{\circ }}C$

Answer 1

Hint:We know that the diameter has increased by 0.2%. This implies that the change has occurred linearly, that is a change in length as occurred. From this we can find the coefficient of linear expansion with which we can associate the coefficient of areal and volume expansion.

Formulas used:
For a temperature difference $\Delta T$
Linear expansion: $\dfrac{\Delta L}{L}=\alpha \Delta T$, where $\Delta L$is the difference in length, L is the actual length and $\alpha $ is the coefficient of linear expansion.
Areal expansion: $\dfrac{\Delta A}{A}=\beta \Delta T$ where $\Delta A$is the difference in length, A is the actual length and $\beta $ is the coefficient of linear expansion.
Volume expansion: $\dfrac{\Delta V}{V}=\gamma \Delta T$ where $\Delta V$is the difference in length, V is the actual length and $\gamma $ is the coefficient of linear expansion.
Also, $\beta =2\alpha $ and $\gamma =3\alpha $

Complete step by step answer:
We know that $\Delta T={{80}^{\circ }}C$
The diameter increases by 0.2%.
$\dfrac{\Delta L}{L}=\alpha \Delta T=0.2\%$
$\Rightarrow\dfrac{\Delta A}{A}=\beta \Delta T=2\alpha \Delta T=2\times 0.2=0.4\%$ (since $\beta =2\alpha $)
Hence, the percentage rise in the area of a face is 0.4%.
Option A is correct.
$\dfrac{\Delta L}{L}=0.2\%$= the rise in thickness
Option B is incorrect.
Also, $\dfrac{\Delta V}{V}=\gamma \Delta T=3\alpha \Delta T=3\times 0.2=0.6\%$
Hence, the percentage rise in the volume of a s 0.6%.
Option C is correct.
$\begin{align}
& \alpha \Delta T=0.2\% \\
& \therefore \alpha =\dfrac{0.2}{100\times 80}=0.25\times {{10}^{-4}} \\
\end{align}$

Hence, coefficient of linear expansion of copper is $0.25\times {{10}^{-4}}{{/}^{\circ }}C$.We can conclude that options A, C and D are correct among the given options.

Note: Applications of thermal expansion include expansion joints in bridges, thermometers, bimetallic strips, electricity pylons and so on.