A straight tunnel is dug through the center of the Earth. If a particle is dropped into the tunnel,

Question

A straight tunnel is dug through the center of the Earth. If a particle is dropped into the tunnel, what will be its time period of oscillation? (Let $R$ be the Earth's radius)

Professor PC Thomas Classes · Answer

The restoring force inside the Earth is proportional to the distance from the center, $F = -(mg/R)x$. Comparing with $F = -kx$, $k = mg/R$. Time period $T = 2\pi\sqrt{m/k} = 2\pi\sqrt{R/g}$.