2.4 Gravitational Force

Newton's Law of Gravitation

Every pair of masses attracts with a force F = Gm₁m₂/r², where G = 6.67×10⁻¹¹ N·m²/kg² and r is the distance between centers. This force is always attractive, acts along the line joining the masses, and decreases with the square of distance — doubling r cuts force to 1/4.

Gravity as a Central, Vector Force

Gravitational force is a vector pointing from one mass toward the other. When multiple masses act on an object, find the net force by vector addition of each pairwise gravitational force. Superposition applies just like with electric forces.

Weight vs. Gravitational Force

Near a planet's surface, weight = mg, but this is just the gravitational force evaluated at r = planet's radius: mg = GMm/R², so g = GM/R². This shows g depends on the planet's mass and radius, and explains why g varies with altitude or on different planets.

Key terms

Newton's Law of Universal Gravitation

F = Gm₁m₂/r², an attractive force between any two masses acting along the line connecting them.

Gravitational constant G

G = 6.67×10⁻¹¹ N·m²/kg², the proportionality constant in Newton's law of gravitation.

Inverse-square law

Gravitational force decreases with the square of the distance between masses; doubling r quarters the force.

g at a planet's surface

g = GM/R², derived by setting weight (mg) equal to gravitational force (GMm/R²).

Superposition of gravitational forces

The net gravitational force on a mass from multiple sources is the vector sum of each individual pairwise force.

Try an AP-style question

At a distance of 2R from Earth's center (R = Earth's radius), what is g?

1. g/2
2. 2g
3. g/4
4. g