Question

An aluminium measuring rod, which is correct at $5{}^\circ C$ measures the length of a line as $80cm$ at $45{}^\circ C$. If thermal coefficient of linear expansion of aluminium is $2.5\times {{10}^{-5}}{}^\circ {{C}^{-1}}$, the correct length of the line is
$\begin{aligned} & A)80.08 \\\ & B)79.92 \\\ & C)81.12 \\\ & D)79.62 \\\ \end{aligned}$

Answer 1

When an object undergoes change in temperature, the object tends to expand or contract. Coefficient of thermal expansion of an object refers to the ratio of strain of the object to the change in temperature experienced by the object. Strain of an object refers to the ratio of change in length of the object to the actual length of the object.

Formula used:
$1)\varepsilon =\dfrac{\Delta L}{L}$
$2)\alpha =\dfrac{\Delta L}{L\Delta T}$

Complete step-by-step answer:
When an object is heated or cooled, the object tends to deform in shape. Such an object is said to experience a strain. Strain of an object is defined as the ratio of change in length of the object to the actual length of the object. It is given by
$\varepsilon =\dfrac{\Delta L}{L}$
where
$\varepsilon$ is the strain of an object
$\Delta L$ is the change in length of the object
$L$ is the actual length of the object
Let this be equation 1.
Coefficient of thermal expansion of the material of an object is defined as the ratio of strain of the object to the change in temperature experienced by that object. It is given by
$\alpha =\dfrac{strain}{\Delta T}=\dfrac{\Delta L}{L\Delta T}$
where
$\alpha$ is the coefficient of thermal expansion of the material of an object
$\Delta L$ is the change in length of the object
$L$ is the actual length of the object
$\Delta T$ is the change in temperature experienced by the object
Let this be equation 2.
Coming to our question, we are provided with an aluminium rod, whose coefficient of thermal expansion is $2.5\times {{10}^{-5}}{}^\circ {{C}^{-1}}$. This aluminium rod is used to measure the length of a line. It is said that this aluminium rod is correct in measuring the length of the given line, when it has a temperature of $5{}^\circ C$. This means that the actual length of the line and the measured length of the line are equal, if measured using the given aluminium rod at $5{}^\circ C$. It is also given that this rod measures the distance of the line as $80cm$ at $45{}^\circ C$. This suggests that the measured length of the line using the aluminium rod at $45{}^\circ C$ is equal to $80cm$. We are required to determine the actual length of the line.
From the information given above, we have
$\begin{aligned} & \alpha =2.5\times {{10}^{-5}}{}^\circ {{C}^{-1}} \\\ & \Delta T=45{}^\circ C-5{}^\circ C=40{}^\circ C \\\ & \dfrac{\Delta L}{L}=\dfrac{L-80cm}{L}=1-\dfrac{80cm}{L} \\\ \end{aligned}$
where
$\alpha$ is the coefficient of thermal expansion of aluminium
$\Delta L=(L-80cm)$, is the change in length of the line
$L$ is the actual length of the line
$\Delta T=40{}^\circ C$, is the change in temperature experienced by the line
Substituting these values in equation 2, we have
$\alpha =\dfrac{\Delta L}{L\Delta T}\Rightarrow 2.5\times {{10}^{-5}}{}^\circ {{C}^{-1}}=\dfrac{1-\dfrac{80cm}{L}}{40{}^\circ C}\Rightarrow 1-\dfrac{80cm}{L}={{10}^{-3}}\Rightarrow L=\dfrac{80cm}{1-0.001}=\dfrac{80cm}{0.999}=80.08cm$
Therefore, the actual length of the line is $80.08cm$. The correct option to be marked is $A$.

So, the correct answer is “Option A”.

Note: For this question, students can also use the formula of linear expansion as follows:
${{L}_{t}}={{L}_{0}}(1+\alpha \Delta T)$
where
${{L}_{t}}$ is the actual length of the line
${{L}_{0}}$ is the measured length of the line
$\alpha$ is the coefficient of thermal expansion
$\Delta T$ is the change in temperature experienced by the line
This method would be a direct approach to the answer.