All Problems Mechanical Oscillations
Problem 4.29
Imagine a shaft going all the way through the Earth from pole to pole along its rotation axis. Assuming the Earth to be a homogeneous ball and neglecting the air drag, find: (a) the equation of motion of a body falling down into the shaft; (b) how long does it take the body to reach the other end of the shaft; (c) the velocity of the body at the Earth's centre.
4.29. (a) \(\ddot{x}+(g / R) x=0,\) where \(x\) is the displacement of the body relative to the centre of the Earth, \(R\) is its radius, \(g\) is the \(\begin{array}{ll}\text { standard } & \text { free-fall } & \text { acceleration; }\end{array}\) (b) \(\tau=\pi \sqrt{R / g}=42 \quad\) min, (c) \(v=\sqrt{g R}=7.9 \mathrm{~km} / \mathrm{s}\)