GRAVITY TRAIN
Drop a stone through the Earth: it oscillates in 84 min — the exact period of a skimming satellite, and any straight tunnel keeps the same clock.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Bore a tunnel clean through a planet — pole to pole, or from any city to any other — and drop a stone in. No air, no engine, just gravity. How long is the ride, and what decides it? The answer, from 1660s mathematics, is startling.
Field inside
Inside a uniform planet only the mass below your feet pulls on you — the shell above cancels itself out — so gravity does not stay constant. It grows straight from zero at the core to full strength at the surface. A force proportional to displacement is a spring.
The period
So the stone does not merely fall, it oscillates — simple harmonic motion, period 2π√(R³/GM). Written the other way, 2π√(R/g), that is exactly the period of a satellite skimming the surface. Fall through the planet or orbit it: the same clock.
Train & twin
Here is the pair, side by side: the stone in the tunnel and a satellite on the rim, released together. They keep step and arrive home together. Straight through the centre, the satellite’s shadow rides exactly on the stone — an orbit is this oscillation seen edge-on.
Audit
Audited: every chord has the same SHM period. The central chord reaches the core, where its maximum speed equals the surface-orbit speed.