Question

A clock with a metal pendulum beating seconds keeps correct time at $0^{\circ}C$. If it loses 12.5 seconds a day at $25^{\circ}C$, the coefficient of linear expansion of the metal pendulum is
(a). $\dfrac{1}{86400}^{\circ}C$
(b). $\dfrac{1}{43200}^{\circ}C$
(c). $\dfrac{1}{14400}^{\circ}C$
(d). $\dfrac{1}{28800}^{\circ}C$

Answer 1

Hint: Calculate the loss or gain in the time and then the change in temperature. Then from the formula that is used to determine the number of seconds lost in a day, we can find the coefficient of linear expansion of the pendulum.
Formulae used:
Loss or gain in time, $\Delta t=\dfrac{1}{2}\alpha \Delta \theta$, where $\alpha$ is the coefficient of linear expansion and $\Delta \theta$ is the change in temperature.

Complete step-by-step answer:
It has been given that the loss of 12.5 seconds per day takes place. That is, $\Delta t=12.5\;s$.
And loss of time takes place due to change in temperature from $0^{\circ}C$ to $25^{\circ}C$. That is $\Delta \theta =25^{\circ}C$
We have to find the coefficient of linear expansion, $\alpha$ of the metal pendulum.
So, we can use the formula $\Delta t=\dfrac{1}{2}\alpha \Delta \theta$, where $\Delta t$ is the loss or gain in time, $\alpha$ is the coefficient of linear expansion and $\Delta \theta$ is the change in temperature.
As the loss of time is happening over a day. Therefore, upon rearranging the formula for calculating coefficient of linear expansion $\alpha$, we get
$\alpha=\dfrac{2\Delta t}{\Delta \theta \times 24\times 60\times 60}=\dfrac{2\times 12.5}{25\times 86400}$
$\implies \alpha=\dfrac{1}{86400}^{\circ}C$
Hence, option a is the correct answer.

Note: As the direct formula is in terms of time lost per second, one may forget to convert it to time lost per day that is done by multiplying the formula by $24\times 60\times 60$ seconds. The formula determines the time lost in one second, so for one day or 86400 seconds we need to multiply it with the number 86400.