Gravity Falls Off as the Square of Distance
The gravity between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Double the distance and the pull becomes four times weaker. This single line, F = GMm/r², explains an apple falling and the Moon orbiting Earth with the very same rule. The Moon does not fall in because it moves sideways fast enough to keep missing the Earth as it falls.
The arrows around the central mass show the direction and strength of the gravity felt there. As you increase the distance r, the arrows shrink steeply with the square of the distance. Double r and the pull is one quarter; triple it and it is one ninth. Slide r and watch for yourself how fast the field weakens.
Gravity pulls the two bodies toward each other with forces of exactly equal size. Yet the lighter one accelerates far more. Its acceleration is proportional to the other body's mass and divided by its own mass. Raise the heavier mass and you will see the two force arrows stay equal while only the lighter body's acceleration arrow grows.
An orbit is a balance between falling and slipping sideways at the same time. Move too slowly sideways and you plunge inward; move too fast and you fly outward. Only at exactly v = √(GM/r) does the amount of falling match the amount of sideways slip, giving a circle. Drag the speed slider to find the boundary between plunging, circling, and escaping.
Your weight is the mass of your body times the g at that height. As you rise, g = GM/r² shrinks, so the scale reading drops. But the scale reading zero on the International Space Station is not because gravity is gone. The station and the astronaut are both in free fall together, so they never press on each other. Raise the altitude and switch to free fall to see how the weight changes.
The gravity made by a round body like Earth looks, from outside, exactly as if all its mass were gathered at a single point at the center. This is the shell theorem. Outside the surface the point mass and the sphere overlap completely; only once you go inside do they differ. Toggle between point mass and sphere and you will see the two curves lie exactly on top of each other beyond the radius R. That is why treating a planet as one point is fine.