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(A) Consider a body of mass $m$ at a distance $y$ from the center of the Earth inside a tunnel dug along the diameter. The gravitational force acting

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$T = 2\pi \sqrt {\frac{{{R_e}}}{g}} $

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(A) Consider a body of mass $m$ at a distance $y$ from the center of the Earth inside a tunnel dug along the diameter. The gravitational force acting on the body is due only to the mass of the Earth contained within a sphere of radius $y$.The acceleration due to gravity at a distance $y$ from the center is given by $g' = \frac{g y}{R_e}$,where $g$ is the acceleration due to gravity at the surface and $R_e$ is the radius of the Earth.The restoring force on the body is $F = -m g' = -m \left( \frac{g}{R_e} \right) y$.Since $F = -k y$,where $k = \frac{m g}{R_e}$ is the effective spring constant,the motion is simple harmonic.The time period $T$ is given by $T = 2\pi \sqrt{\frac{m}{k}} = 2\pi \sqrt{\frac{m}{m g / R_e}} = 2\pi \sqrt{\frac{R_e}{g}}$.

If a body is released into a tunnel dug across the diameter of the Earth,it executes simple harmonic motion with a time period of:

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