# Effect of Temperature on the Time Period of a Simple Pendulum

## Effect of Temperature on the Time Period of a Simple Pendulum

The time represented by the clock hands of a pendulum clock depends on the number of oscillations performed by pendulum. Every time it reaches to its extreme position. The second hand of the clock advances by one second that means second hand moves by two seconds when one oscillation is complete.

A pendulum clock keeps proper time at temperature θ₀. If temperature is increased to θ (> θ₀) then due to linear expansion, length of pendulum and hence its time period will increase. Now,

Let, $$T=2\pi \sqrt{\frac{{{L}_{0}}}{g}}$$ at temperature θ₀ and $$T’=2\pi \sqrt{\frac{L}{g}}$$ at temperature θ.

$$\frac{T’}{T}=\sqrt{\frac{L’}{L}}=\sqrt{\frac{L(1+\alpha \Delta \theta )}{L}}=1+\frac{1}{2}\alpha \Delta \theta$$.

Therefore, change in time per unit time lapsed is: $$\frac{T’-T}{T}=\frac{1}{2}\alpha \Delta \theta$$.

Gain or loss in time in duration of T is: ΔT = ½ αΔθT. If T is the correct time then,

θ < θ₀, T’ < T – clock becomes fast and gain time.

θ > θ₀, T’ > T – clock becomes slow and loses time.

Final change in time period, $$\frac{\Delta T}{T}=\frac{1}{2}\alpha \Delta \theta$$.

The clock will lose time i.e. will become slow if θ’ > θ and will gain time i.e. will become fast if θ’ < θ. The gain or loss in time is independent of time period T and depends on the time interval (t). Since, coefficient of linear expansion (α) is very small for invar; hence pendulums are made of invar to show the correct time in all seasons.