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
JEE MAIN 2021, Diffcult
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Force along the tunnel
$F =-\left(\frac{ GMmr }{ R ^{3}}\right) \cos \theta$
$F =-\frac{ gm }{ R } x \left(\frac{ GM }{ R ^{2}}= g , r \cos \theta= x \right)$
$a=-\frac{g}{R} x$
$\omega^{2}=\frac{g}{R} \quad T=2 \pi \sqrt{\frac{R}{g}}$
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