Identify the correct statement among the foll | vedclass

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(D) The time period of a simple pendulum is given by $T = 2\pi \sqrt{\frac{L}{g}}$.For a given angular amplitude $	heta_0$,the maximum velocity $v_{m

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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 $T = 2\pi \sqrt{\frac{L}{g}}$.For a given angular amplitude $	heta_0$,the maximum velocity $v_{max}$ is given by $v_{max} = \omega A = \sqrt{\frac{g}{L}} \cdot (L 	heta_0) = 	heta_0 \sqrt{gL}$.As the length $L$ increases,the maximum velocity $v_{max}$ increases,not decreases. Thus,$(c)$ is incorrect.Option $(a)$ is incorrect because the frequency $f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$ is independent of mass.Option $(b)$ is incorrect because the tension at any angle $	heta$ is $T = Mg \cos 	heta + \frac{Mv^2}{L}$. Since $\frac{Mv^2}{L} > 0$,the tension is always greater than $Mg \cos 	heta$.Option $(d)$ is correct. The time period is $T = 2\pi \sqrt{\frac{L}{g}}$. Taking logs and differentiating,$\frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L} = \frac{1}{2} \alpha \Delta 	heta$. This is independent of the length $L$.

Identify the correct statement among the following.

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