Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance $( R / 2)$ from the earth's centre, where $'R'$ is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period
A$\frac{2 \pi R }{ g }$
B$\frac{ g }{2 \pi R }$
C$\frac{1}{2 \pi} \sqrt{\frac{g}{R}}$
D$2 \pi \sqrt{\frac{ R }{ g }}$
JEE MAIN 2021, Diffcult
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D$2 \pi \sqrt{\frac{ R }{ g }}$
d Force along the tunnel
$F =-\left(\frac{ GMmr }{ R ^{3}}\right) \cos \theta$
$F =-\frac{ gm }{ R } x \left(\frac{ GM }{ R ^{2}}= g , r \cos \theta= x \right)$
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