Question

A metal tape gives correct measurement at $15{}^\circ C$. It is used to measure a distance of $100m$ at $45{}^\circ C$. The error in the measurement, if $\alpha =12\times {{10}^{-6}}{{\left( {}^\circ C \right)}^{-1}}$ is
$\begin{align}
  & A)36cm \\
 & B)36m \\
 & C)42mm \\
 & D)36mm \\
\end{align}$

Answer 1

Hint: Whenever a metal tape is subjected to heat, the measurements recorded by the tape change. This change in measurement due to thermal expansion of the metal tape is termed as error in the measurement of tape. Mathematically, the ratio of change in length to the original length of a metal tape is proportional to the change in temperature. The constant of proportionality is known as coefficient of thermal expansion, $\alpha $. The correct distance measured by the tape can be related to the length of tape. At the same time, the error in measurement of tape can be related to the change in length of tape, when subjected to heat.
Formula used:
$\dfrac{\Delta L}{L}=\alpha \Delta T$

Complete answer:
Whenever a measuring device such as metal tape is subjected to heat or temperature change, the measurements recorded by the tape tend to change. The ratio of change in length to the original length of the tape is proportional to the temperature change. Mathematically, change in length is given by
$\dfrac{\Delta L}{L}=\alpha \Delta T$
where
$\Delta L$ is the change in length of a metal tape when subjected to heat
$L$ is the original length of the metal tape
$\Delta T$ is the change in temperature
$\alpha $ is the coefficient of thermal expansion
Let this be equation 1.
Coming to our question, we are provided with a metal tape which gives correct measurement at $15{}^\circ C$. This measuring tape is used to measure a distance of $100m$ at $45{}^\circ C$. We are supposed to determine the error in the measurement, if $\alpha =12\times {{10}^{-6}}{{\left( {}^\circ C \right)}^{-1}}$.
Clearly, if the metal tape was used to measure the given distance at $15{}^\circ C$, then, the value of distance would have been $100m$. Let us call this distance the length of the metal tape. Clearly, from the information provided in the question, we have
$L=100m$ is the length of the metal tape (or the actual distance measured by the tape)
$\alpha =12\times {{10}^{-6}}{{\left( {}^\circ C \right)}^{-1}}$ is the coefficient of thermal expansion, as provided
$\Delta T=45{}^\circ C-15{}^\circ C=30{}^\circ C$ is the change in temperature, as provided
Substituting these values in equation 1, we have
$\dfrac{\Delta L}{L}=\alpha \Delta T\Rightarrow \Delta L=L\alpha \Delta T=100m\times 12\times {{10}^{-6}}{{({}^\circ C)}^{-1}}\times 30{}^\circ C=0.036m=36mm$
Therefore, change in length of the metal tape (or the error in measurement of the distance) $\Delta L$ is equal to $36mm$.

So, the correct answer is “Option D”.

Note:
Students need not get confused with the length of the metal tape and the distance measured by the metal tape. Both of these can be related while solving this problem because we are dealing with ratios in the required formula. Also, students need to be thorough with conversion formulas. Conversion formula used in this solution is:
$\begin{align}
  & 1m=1000mm \\
 & 0.036m=0.036\times 1000mm=36mm \\
\end{align}$