# Effect of Temperature on Pendulum Clocks | Formula | Solved Examples

Variation Of Time Period Of Pendulum Clocks :

$\displaystyle T_o = 2\pi\sqrt{\frac{l_o}{g}}$

If temperature is increased by  Δ t,

$\displaystyle T = 2\pi\sqrt{\frac{l}{g}}$

$\displaystyle T = 2\pi\sqrt{\frac{l_o(1+ \alpha \Delta t)}{g}}$

(by using Binomial expansion)

$\displaystyle T = T_o (1 + \frac{1}{2}\alpha \Delta t)$

$\displaystyle T – T_o = T_o \frac{1}{2}\alpha \Delta t$

$\displaystyle \frac{\Delta T}{T} = \frac{1}{2}\alpha \Delta t$

Where ΔT = increase in time period

Illustration : A pendulum clock with a pendulum made of Invar has a period of 0.5 s and is accurate at 25º C. If the clock is used in a country where the temperature averages 35° C, what correction is necessary at the end of a month (30 days) to the time given by the clock?

Solution: In time interval , the clock will become slow (or will lose time) by

$\displaystyle \Delta t = \frac{1}{2}\alpha t \Delta \theta$

$\displaystyle \Delta t = \frac{1}{2}(7\times 10^{-7} ) (30\times 86400 ) (35 – 25)$

= 9.1 sec