1. Tardigrade
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  4. The time period of a physical pendulum about some pivot point is T . When we take another pivot point, opposite of the first one such that the centre of mass of the physical pendulum lies on the line joining these two pivot points, we obtain the same time period. If the two points are separated by a distance L , then the time period T is

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Q. The time period of a physical pendulum about some pivot point is $T$ . When we take another pivot point, opposite of the first one such that the centre of mass of the physical pendulum lies on the line joining these two pivot points, we obtain the same time period. If the two points are separated by a distance $L$ , then the time period $T$ is

NTA AbhyasNTA Abhyas 2022

Solution:

$T=2 \pi \sqrt{\frac{I_{C}+m x^{2}}{m g x}}=2 \pi \sqrt{\frac{I_{C}+m(L-x)^{2}}{m g(L-x)}}$
$\frac{I_{C}}{x}+mx=\frac{I_{C}}{L - x}+m\left(\right.L-x\left.\right)$
$I_{C}\left(\frac{L - 2 x}{x \left(\right. L - x \left.\right)}\right)=m\left(\right.L-2x\left.\right)$
$I_{C}=mx \, \left(\right.L-x\left.\right)$
$T=2\pi \sqrt{\frac{m x L - m x^{2} + m x^{2}}{m g x}}=2\pi \sqrt{\frac{L}{g}}$