Question

(A): A meter scale measures correct lengths at $0\,{\text{C}}$. Using this scale, the length of an object is measured as $15\,{\text{m}}$ at $40\,{\text{C}}$. Then the correct length of that object is $15.012\,{\text{m}}$ ($\alpha = 20 \times {10^{ - 6}}\,{\text{/}}^\circ {\text{C}}$).
(B): True length=scale reading
A. A and B are correct and B is correct explanation for A
B. A and B are correct and B is not correct explanation for A
C. A is true and B is false
D. A is wrong and B is true

Answer 1

Use the expression for the linear thermal expansion of an object to check whether the given correct length of the object is true or not. This expression gives the relation between the original length of the object, coefficient of linear expansion and the change in temperature. Check whether B gives the correct explanation for A.

Formula used:
The expression for linear thermal expansion is given by
$L = {L_0}\left( {1 + \alpha \Delta T} \right)$ …… (1)
Here, $L$ is the expanded length of the material, ${L_0}$ is the original length of the material, $\alpha$ is the linear thermal expansion coefficient and $\Delta T$ is the change in temperature.

Complete step by step answer:
We have given that the length at temperature $40\,^\circ {\text{C}}$ is $15\,{\text{m}}$.
${L_0} = 15\,{\text{m}}$
The expanded length of the object is $15.012\,{\text{m}}$.
$L = 15.012\,{\text{m}}$
The coefficient of linear expansion is $20 \times {10^{ - 6}}\,{\text{/}}^\circ {\text{C}}$.
$\alpha = 20 \times {10^{ - 6}}\,{\text{/}}^\circ {\text{C}}$
The change in temperature of the object is $\Delta T = 40\,^\circ {\text{C}} - 0\,^\circ {\text{C}} = 40\,^\circ {\text{C}}$.

We can determine the expanded length of the object using equation (1).
Substitute $15\,{\text{m}}$ for ${L_0}$, $20 \times {10^{ - 6}}\,{\text{/}}^\circ {\text{C}}$ for $\alpha$ and $40\,^\circ {\text{C}}$ for $\Delta T$ in equation (1).
$L = \left( {15\,{\text{m}}} \right)\left[ {1 + \left( {20 \times {{10}^{ - 6}}\,{\text{/}}^\circ {\text{C}}} \right)\left( {40\,^\circ {\text{C}}} \right)} \right]$
$\Rightarrow L = \left( {15\,{\text{m}}} \right)\left[ {1 + 8 \times {{10}^{ - 4}}} \right]$
$\Rightarrow L = \left( {15\,{\text{m}}} \right)\left[ {1.0008} \right]$
$\Rightarrow L = 15.012\,{\text{m}}$
The above result shows that the expanded length of the object is $15.012\,{\text{m}}$.
Hence, the statement A is correct.

In the statement B, it is given that the true reading of the length of the object is equal to the scale reading. But from the above calculations, we can conclude that the true reading for the length of the object is not equal to the scale reading. Hence, the statement B is incorrect.
Therefore, A is correct but B is false.

Hence, the correct option is C.

Note: One can also check that the given measured lengths of the object at two different temperatures are correct or not by substituting the value in equation (1) and check that the value of the length at other temperature comes $15\,{\text{m}}$ or not. One can also check if the change in the length comes $0.012\,{\text{m}}$ after substituting all the values in equation (1).