Question

A brass sheet is $25\,cm$ long and $8\,cm$ breadth at $0^\circ C$. Its area at $100^\circ C$ is $\left( {\alpha \, = \,18\, \times \,{{10}^{ - 6}}/^\circ C} \right)$ :
A. $207.2\,c{m^2}$
B. $200.72\,c{m^2}$
C. $272\,c{m^2}$
D. $2000.72\,c{m^2}$

Answer 1

This question is from the topic Thermal Expansion of solids. The basic knowledge that is needed to solve this question is about how to relate the value of $\beta$ with $\alpha$, considering only $\alpha$ is given to us, while to find the expansion in area, we need the value of $\beta$

Complete step-by-step solution:
As told in the hint section, we will be using the knowledge about the relation of values of $\beta$ and $\alpha$ since $\beta$ is used to find the value of area expansion.
If you are having troubles in recalling the relation between the values of $\beta$ and $\alpha$, let us help you.
The relation between $\beta$ and $\alpha$ is:
$\beta \, = \,2\alpha$
We have already been given the value of $\alpha$ in the question:
$\alpha \, = \,18\, \times \,{10^{ - 6}}/^\circ C$
So, we can find the value of $\beta$ as:
$\beta \, = \,2\alpha \\\ \beta \, = \,2 \times \left( {18\, \times \,{{10}^{ - 6}}/^\circ C} \right) \\\ \beta \, = \,36\, \times \,{10^{ - 6}}/^\circ C \\\$
Now, that we have found out the value of coefficient of Area expansion, all we need to do is find the original area and then using the value of coefficient of area expansion, we can find the value of the new area at the given temperature.
We will use the following formula of Area expansion:
${A_2}\, = \,{A_1}\left( {1\, + \,\beta \Delta T} \right)$
In this formula, ${A_2}$ is the new area
${A_1}$ is the old area
$\beta$ is the coefficient of area expansion and,
$\Delta T$ is the temperature difference between the old temperature and the new temperature
Using the values that are given to us in the question:
${A_1}\, = \,l \times b \\\ {A_1}\, = \,25 \times 8\,c{m^2} \\\$
Now that we have the value of old area, we can find the value of new area as:
${A_2}\, = \,{A_1}\left( {1\, + \,\beta \Delta T} \right)$
${A_2}\, = \,200\left( {1\, + \,36 \times {{10}^{ - 6}} \times 100} \right) \\\ {A_2}\, = \,200.72\,c{m^2} \\\$
So, the value of the area at $100^\circ C$ is $200.72\,c{m^2}$.
Hence, the correct option is option (B).

Note:- Many students do wrong step as first considering the expansion in length and breadth and then finding the new area using the new, altered length and breadth, which gives them wrong answer if they don’t use approximation and neglect the value of ${\alpha ^2}$. Since, this method is independent of such a thing, we highly recommend you to use this method to solve such questions.