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A clock which keeps correct time at ${25^ \circ }C$ has a pendulum made of a metal. The temperature falls to ${0^ \circ }C$. If the coefficient of linear expansion of the metal is $1.9 \times {10^{ - 5}}pe{r^ \circ }C$, then find the number of seconds the clock gains per day.

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Last updated date: 13th Jun 2024
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Answer
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Hint:We can use the relation connecting the gain or loss of time and the change in temperature to solve this problem. The relation connecting gain or loss in time and the change in temperature is given as
$\Delta t = \dfrac{1}{2} \times \alpha \times \Delta \theta \times t$
Where, $\Delta t$ is the gain or loss in time.
 $\alpha $ is the coefficient of linear expansion.
$\Delta \theta $ is the change in temperature and t is the time.
Since there is an increase in temperature the value that we get will be the gain in time.

Complete step by step solution:
It is given that a clock keeps correct time at ${25^ \circ }C$. So initial temperature ${\theta _1} = {25^ \circ }C$ .
The pendulum is made of metal. We need to find the number of seconds gained per day by the clock when temperature falls to ${0^ \circ }C$ .
So final temperature ${\theta _2} = {0^ \circ }C$
So, ${\theta _1} - {\theta _2} = 25 - 0 = {25^ \circ }C$
The relation connecting gain or loss in time and the change in temperature is given as
$\Delta t = \dfrac{1}{2} \times \alpha \times \Delta \theta \times t$
Where $\Delta t$ is the gain or loss in time.
$\alpha $ is the coefficient of linear expansion.
$\Delta \theta $ is the change in temperature and t is the time.
$\alpha $ is given as $1.9 \times {10^{ - 5}}pe{r^ \circ }C$
We need to calculate the gain of time per day.
Thus $t = 24 \times 60 \times 60\,s$
 Change in temperature is $\Delta \theta = {25^ \circ }C$
Now let us substitute all the given values in the equation.
$\Delta t = \dfrac{1}{2} \times 1 \cdot 9 \times {10^{ - 5}} \times 25 \times 24 \times 60 \times 60$
$\therefore \Delta t = 20\cdot520\,s$

This is the gain in time when the temperature falls to ${0^ \circ }C$ . We know that the time period of a pendulum increases with increase in length. When the temperature falls the length of the pendulum is decreasing so the time period becomes less. That means the time goes faster so we get a gain in time. So this value that we got is the number of seconds gained.

Note: In cases where the temperature is increasing from a low value to a high value, we will find that there will be a loss in time. Because in that case the length of the pendulum increases due to expansion of metal with temperature increase and hence the time period increases. An increase in time period means time is running slower in that case. So there will be loss of time. In our case as temperature is decreasing there is gain of time.