Question

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. $2 \pi \sqrt{\frac{ R }{ g }}$

Answer 1

Answer: (D)

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