Question

A pendulum clock loses 12s a day if the temperature is \[40{}^\circ C\] and gains 4s a day if the temperature is \[20{}^\circ C\]. The temperature at which the clock will show correct time, and the coefficient of linear expansion (α) of the metal of the pendulum shaft are respectively.
A. \[60{}^\circ C,\alpha 1.85\times {{10}^{-4}}/{}^\circ C\]
B. \[30{}^\circ C,\alpha 1.85\times {{10}^{-4}}/{}^\circ C\]
C. \[55{}^\circ C,\alpha 1.85\times {{10}^{-4}}/{}^\circ C\]
D. None of these

Answer 1

Hint: We will be using the formula for time period of pendulum and linear expansion. As mentioned in the question there is a change in length and time so we will form an equation irrespective the change mentioned in the question.

Formula used:
\[\begin{align}
  & T=2\pi \sqrt{\dfrac{1}{g}} \\
 & \Delta L=\alpha L\Delta T \\
\end{align}\]

Complete answer:
Linear expansion- Whenever there is temperature change the object undergoes elongation due to temperature variation. This is referred to as linear expansion or linear thermal expansion.
According to the question, the time of the pendulum clock changes as the temperature changes and we have to calculate at what temperature and coefficient of linear expansion the time will be correct.
We know the time period of pendulum is \[T=2\pi \sqrt{\dfrac{1}{g}}\]
Where g is the acceleration due to gravity and l be the length of the pendulum.
As there is change in the time period of the pendulum, let us assume the change in time be \[\Delta T\] and change in length be \[\Delta L\].
\[\dfrac{\Delta T}{T}=\dfrac{1}{2}\dfrac{\Delta L}{L}\]
We know the formula for linear expansion \[\Delta L=\alpha L\Delta T\]
Here\[\alpha \] is the coefficient of linear expansion
According to the question, clock gains 12 seconds
\[\dfrac{12}{T}=\dfrac{1}{2}\alpha (40-\theta )\]…………..(1)
When clock loses 4 seconds, we get
\[\dfrac{4}{T}=\dfrac{1}{2}\alpha (\theta -20)\]……………(2)
From equation (1) and (2)
\[3=\dfrac{40-\theta }{\theta -20}\]
Rearranging the terms
\[\begin{align}
  & 30-60=40-\theta \\
 & \theta =25{}^\circ C \\
\end{align}\]
Thus the temperature should be \[25{}^\circ C\].
Now we can substitute the value of in equation (1)
\[\dfrac{12}{T}=\dfrac{1}{2}\alpha (40-25)\]
Rearranging the terms
\[\begin{align}
  & \dfrac{12}{24\times 3600}=\dfrac{1}{2}\alpha (15) \\
 & \alpha =1.85\times {{10}^{-5}}/{}^\circ C \\
\end{align}\]
Thus \[\alpha =1.85\times {{10}^{-5}}/{}^\circ C\] is the coefficient of linear expansion in the pendulum clock.

Therefore option D is correct.

Note:
Coefficient of linear expansion varies from material to material. It depends upon the force and separation between the atoms. If we know the value for coefficient of linear expansion, change in length at a certain temperature can also be calculated.