Identify the correct statement.

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(D) The time period of a simple pendulum is given by $T = 2\pi \sqrt{\frac{\ell}{g}}$.When temperature changes,the length changes as $\ell = \ell_0(1

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The fractional change in the time period of a pendulum on changing the temperature is independent of the length of the pendulum.

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(D) The time period of a simple pendulum is given by $T = 2\pi \sqrt{\frac{\ell}{g}}$.When temperature changes,the length changes as $\ell = \ell_0(1 + \alpha \Delta 	heta)$.The time period becomes $T' = 2\pi \sqrt{\frac{\ell_0(1 + \alpha \Delta 	heta)}{g}} = T(1 + \alpha \Delta 	heta)^{1/2}$.Using binomial approximation for small $\alpha \Delta 	heta$,$T' \approx T(1 + \frac{1}{2} \alpha \Delta 	heta)$.The fractional change in time period is $\frac{\Delta T}{T} = \frac{T' - T}{T} = \frac{1}{2} \alpha \Delta 	heta$.This expression is independent of the length $\ell$ of the pendulum. Therefore,statement $D$ is correct.

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