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(D) The motion of a particle in a tunnel through the Earth is simple harmonic motion $(SHM)$.For a particle at a distance $r$ from the centre of the E

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$T_1 = T_2$

Vedclass · Answer

(D) The motion of a particle in a tunnel through the Earth is simple harmonic motion $(SHM)$.For a particle at a distance $r$ from the centre of the Earth, the gravitational force is $F = -(\frac{GMm}{R_e^3})r$.This force acts as a restoring force $F = -kr$, where $k = \frac{GMm}{R_e^3}$.The angular frequency of oscillation is $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{GM}{R_e^3}} = \sqrt{\frac{g}{R_e}}$.The time period of oscillation is $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{R_e}{g}}$.Since the time period $T$ depends only on the radius of the Earth $R_e$ and the acceleration due to gravity $g$, it is independent of the amplitude of oscillation (the distance from which the particle is released).Therefore, $T_1 = T_2$.

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