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Question
As the temperature is increased, the time period of a pendulum ______.
Options
increases as its effective length increases even though its centre of mass still remains at the centre of the bob.
decreases as its effective length increases even though its centre of mass still remains at the centre of the bob.
increases as its effective length increases due to shifting of centre of mass below the centre of the bob.
decreases as its effective length remains same but the centre of mass shifts above the centre of the bob.
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Solution
As the temperature is increased, the time period of a pendulum increases as its effective length increases even though its centre of mass still remains at the centre of the bob.
Explanation:
A pendulum clock keeps proper time at temperature θ0. If the temperature is increased to θ (> θ0), then due to linear expansion, the length of pendulum increases and hence its time period will increase
Let T = `2πsqrt(L_0/g)` at temperature θ0
And T' = `2πsqrt(L/g)` at temperature θ.
`T^'/T = sqrt(L^'/L)`
= `sqrt((L[1 + αΔθ])/L`
= `1 + 1/2 αΔθ`
Therefore change (loss or gain) in time per unit time lapsed is `(T^' - T)/T = 1/2 αΔθ`
Fractional change in time period `(ΔT)/T = 1/2 αΔθ`
So, as the temperature increases, the length of pendulum increases and hence time period of the pendulum increases. Due to an increment in its time period, a pendulum clock becomes slow in summer and will lose time.
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