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.

Reveal Answer
4.29. (a) x¨+(g/R)x=0,\ddot{x}+(g / R) x=0, where xx is the displacement of the body relative to the centre of the Earth, RR is its radius, gg is the  standard  free-fall  acceleration; \begin{array}{ll}\text { standard } & \text { free-fall } & \text { acceleration; }\end{array} (b) τ=πR/g=42\tau=\pi \sqrt{R / g}=42 \quad min, (c) v=gR=7.9 km/sv=\sqrt{g R}=7.9 \mathrm{~km} / \mathrm{s}
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