Newton's law of universal gravitation
An apple dropping from a branch and the Moon circling Earth obey one and the same rule. The pull between two bodies grows with the product of their masses and falls off with the square of the distance between them: F = GMm/r². Double the distance and the force drops to a quarter. So why doesn't the Moon crash into Earth? It is in fact falling the whole time, but it also moves sideways so fast that it keeps missing.
The gold disk at the center is the mass M, and the eight arrows on the dashed ring show the direction and strength of the pull a test body would feel at each spot. Every arrow points at the center, and its length is the size of the force. Slide r outward and the ring widens while the arrows shrink with the square of the distance. At twice the distance the force is 1/4 as strong; at three times, only 1/9. A small step outward costs a lot of pull.
Raise the mass m₂ of the gold body with the slider and the two bodies still pull on each other with exactly the same force; the F in the readout is that one shared value. What changes is the acceleration. Each body accelerates by the force divided by its own mass, so the green arrow a₁ on the small fixed body m₁ grows in step with m₂, while the orange arrow a₂ on the heavy body keeps its length. That is why the apple, not the Earth, does the visible falling.
A circular orbit is falling and sideways motion in exact balance. The slider sets how fast the body is launched sideways from its starting point on the dashed circle, and the colored curve traces where it goes next. At exactly v = √(GM/r) the curve lies on the dashed circle. Launch it slower and it drops into an ellipse that dips inside the circle; much slower and it strikes the central body. Launch it faster and the ellipse stretches outward, until past a threshold the body never comes back and escapes.
A scale reads your weight, which is your mass times the local g. Raise the altitude and g = GM/r² falls, so the gold bar sinks along with the dashed g line. A scale aboard the International Space Station reads 0, but not because gravity is gone: the station and the astronaut are in free fall together, so nothing presses down on the scale. Switch to free fall and the bar drops to the floor while the dashed g line stays where it was.
Seen from outside, a round body like Earth pulls exactly as if all its mass sat at a single point at its center. This is the shell theorem. The plot shows the field of a point mass and of a uniform sphere together; the one you select is a solid gold curve and the other a faint dashed one. Beyond the vertical dashed line at R the two curves merge into one, and they part only inside. Within the sphere the pulls of the outer layers cancel, so g falls in a straight line toward the center, whereas the point-mass field keeps climbing as you close in. That is why a planet can be treated as a point.