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(B) When a particle is at a distance $x$ from the center of the Earth, the gravitational force acting on it is directed towards the center and is give

Vedclass · Accepted Answer

$1$ hour $24$ minutes

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(B) When a particle is at a distance $x$ from the center of the Earth, the gravitational force acting on it is directed towards the center and is given by $F = -\frac{GMm}{R^3}x$.Since $F = ma$, the acceleration $a$ is given by $a = -\frac{GM}{R^3}x$.Using the relation $g = \frac{GM}{R^2}$, we can write the acceleration as $a = -\frac{g}{R}x$.Comparing this with the standard equation for simple harmonic motion, $a = -\omega^2 x$, we get $\omega^2 = \frac{g}{R}$, or $\omega = \sqrt{\frac{g}{R}}$.The time period $T$ is given by $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{R}{g}}$.Substituting the given values: $R = 6400 \, km = 6400 	imes 10^3 \, m$ and $g = 10 \, m/s^2$.$T = 2 	imes 3.14 	imes \sqrt{\frac{6400 	imes 10^3}{10}} = 2 	imes 3.14 	imes \sqrt{640000} = 2 	imes 3.14 	imes 800 \, s$.$T = 5024 \, s$.Converting to minutes: $T = \frac{5024}{60} \approx 83.73 \, minutes$, which is approximately $1$ hour and $24$ minutes.

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