GRAVITY
Shell theorem, drawn: g climbs straight to the surface, then falls as 1/r². Zero at the core.
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
A planet of uniform density — same stuff all the way down. The question sounds simple and is famously not: how strong is gravity, not just at the surface, but everywhere from the airless deep space above to the dead center of the core?
Two rules
Newton’s shell theorem splits the world in two. Stand outside, and the entire planet pulls as if its whole mass were squeezed to a point at the center. Burrow inside, and every shell of rock above your head pulls in all directions at once and cancels perfectly — only the ball of mass beneath your feet still counts.
Solve
The two rules have to agree where they meet. At the surface the outside law g = GM/r² and the inside law g = GM·r/R³ give the identical number — the value is continuous. But the slopes do not match, and that kink is the whole point: gravity is strongest exactly at the surface, and weakens whichever way you travel from it.
The sweep
Walk a probe from the core to deep space. On the way out through the rock, gravity climbs in a dead-straight line — halfway to the surface is exactly half the surface gravity. It peaks the instant you break the surface, then bleeds away as one-over-r-squared, the familiar inverse-square tail reaching out into the dark.
Audit
The seam is checked to the last bit: the inside and outside formulas return the identical surface value. Outside, g times r-squared holds the constant GM at sixty radii; inside, g over r holds GM over R-cubed. And at the exact center, gravity is zero — stand there and you float, pulled equally in every direction by the planet all around you.