A tunnel is dug along the diameter of the Ear | vedclass

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(D) The gravitational force acting on a mass $m$ at a distance $r$ from the center of the Earth is given by $F = -\frac{GMm}{R^3}r$.This force is of t

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$T$

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(D) The gravitational force acting on a mass $m$ at a distance $r$ from the center of the Earth is given by $F = -\frac{GMm}{R^3}r$.This force is of the form $F = -kr$,where $k = \frac{GMm}{R^3}$.The motion is simple harmonic with an angular frequency $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{GM}{R^3}}$.The time period is $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{R^3}{GM}}$.Since the time period $T$ depends only on the mass of the Earth $M$ and its radius $R$,it is independent of the oscillating mass $m$.Therefore,if the mass is changed to $4m$,the time period remains $T$.

$A$ tunnel is dug along the diameter of the Earth of mass $M$. $A$ mass $m$ is released at a distance $X$ from the center,and the time period of oscillation is $T$. If a mass $4m$ is released from the same point,what will be the time period of oscillation?

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