Question 168 of 194

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance $\dfrac{R}{2}$ 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$\dfrac{1}{2\pi }\sqrt{\dfrac{g}{R}}$B$2\pi \sqrt{\dfrac{R}{g}}$C$\dfrac{g}{2\pi R}$DImage is virtual, opposite side of convex mirror.

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Answer: (B) $2\pi \sqrt{\dfrac{R}{g}}$

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