Every pair of objects with mass attracts each other. Newton's law of universal gravitation says that attraction is directly proportional to each object's mass, and inversely proportional to the square of the distance between their centers of mass.
Explore the inverse-square relationship directly below.
Adjust two masses and the distance between their centers. Watch how sharply the force drops off — this is the "inverse square" in Newton's law.
Try doubling r — the force drops to a quarter of its original value, not half. That's the inverse-square relationship in action.
A field models the effect of a non-contact force at every point in space, independent of whether a second object is actually there to feel it. The gravitational field created by a mass M is the force per unit mass it would exert on any test object placed at a given point:
The gravitational force an astronomical body exerts on a small nearby object is called weight:
Near Earth's surface, the distance from an object to Earth's center barely changes even as the object moves — so the gravitational force can be treated as essentially constant, with g ≈ 10 N/kg (the same 10 m/s² you've used since Unit 1).
A bathroom scale doesn't measure gravitational force directly — it measures the normal force pushing back on you, which is what "apparent weight" actually means. When you're not accelerating, this happens to equal your true gravitational force. The moment you accelerate, it doesn't.
Explore this directly with the classic elevator scenario below.
A 70 kg person stands on a scale inside an elevator. Set the elevator's acceleration — the scale reading (apparent weight, the normal force) responds instantly, while true gravitational force never changes.
This connects to a deeper idea: the equivalence principle states that an observer in a non-inertial (accelerating) reference frame cannot distinguish between their apparent weight and the gravitational force from an actual gravitational field. Standing in an accelerating rocket in deep space feels indistinguishable from standing on a planet's surface.
Mass shows up in physics wearing two very different hats. Inertial mass is the "m" in F = ma — a measure of how strongly an object resists a change in its motion. Gravitational mass is the "m" in Fg = GmM/r² — a measure of how strongly an object participates in gravitational attraction. These are, in principle, completely separate concepts measuring completely different things.
A uniform sphere isn't a point mass — but a remarkable result called Newton's shell theorem lets you treat it like one in most cases. The net gravitational force from a spherical mass distribution is the sum of forces from every differential piece of mass that makes it up, but that sum simplifies beautifully.
Treat the entire shell as a point mass located at its center.
The net gravitational force is exactly zero — everywhere inside.
Since the enclosed mass grows as r³ while the inverse-square law divides by r², the two powers of r don't cancel evenly — working through the substitution shows that the gravitational force inside a uniform sphere is directly proportional to r:
Explore the full picture below — g(r) both inside and outside a uniform sphere, in one continuous graph.
A uniform sphere of radius R. Drag through r — inside the sphere, only the enclosed mass matters, and g grows linearly with r. Outside, the whole sphere acts like a point mass at its center, and g falls off as 1/r².
Right at the surface — this is where both formulas agree, and g is at its maximum.
Modeling Earth as a uniform sphere of radius R and surface gravitational field g₀, find the gravitational field strength at a distance R/2 from Earth's center.