A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is 4m, then the time period of small oscillations will be ___________ s

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Solution

Q.
A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is 4m, then the time period of small oscillations will be ___________ s. $[t a k e g = π^{2}$ $m s^{- 2}]$ [2024]

Ans.
(8)    $T = 2 π \sqrt{\frac{l}{g_{eff}}}$

   $g = \frac{G M}{R^{2}}, at a height, h, g = \frac{G M}{{(R + h)}^{2}}$

   $g_{eff} = g {(\frac{R}{R + h})}^{2}, g_{eff} = g {(\frac{R}{2 R})}^{2}$

   $g_{eff} = \frac{g}{4}$

   $∴ T = 2 π \sqrt{\frac{l}{g_{eff}}} = 2 π \sqrt{\frac{4 \times 4}{g}} = \frac{(2 π) (4)}{\sqrt{g}}$

   $Given, g = π^{2}$

   $T = 8 \sec$

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