Newton's Law of Universal Gravitation

Newton's insight was that the force pulling an apple down and the force bending the Moon's path into a circle are the same force, obeying the same rule. Any two masses attract each other along the line joining their centers, with a strength given by F = G m1 m2 / r^2. G is a tiny constant, 6.67 x 10⁻¹¹ N m²/kg², which is why you never feel the pull of your coffee mug. The force grows with more mass and shrinks fast as the distance between centers grows, because r is squared in the denominator.

That squaring is the whole personality of the law. Double the separation and the force drops to a quarter. Triple it and the force drops to a ninth. This inverse-square falloff is why gravity near Earth's surface feels essentially constant: Earth's radius is about 6,371 km, so climbing a 400 m building changes r by a tiny fraction of a percent, not enough to notice. The force only weakens noticeably when you move distances comparable to Earth's radius itself, like sending a satellite far into orbit.

This law sits right next to Newton's second law in the unit. Once you know F from the gravitation equation, you can set it equal to ma to find an object's acceleration, or use it as the centripetal force keeping a satellite or the Moon moving in a circle. It also explains why g = 9.8 m/s² is not a fundamental constant but something you can derive: g = G M_Earth / r², which is just this law divided by the falling object's mass.