Newton’s Law of Gravitation

You’ll be able to

State the law of universal gravitation and compute the force between two masses
Relate a planet’s surface gravity to its mass and radius through g = GM/R²
Predict how gravitational force changes with separation using the inverse-square law

Every mass attracts every other mass

Newton’s law of universal gravitation says any two point masses attract along the line joining them with a force proportional to each mass and inversely proportional to the square of their separation. The constant G = 6.67 × 10⁻¹¹ N·m²/kg² is tiny, which is why gravity is only noticeable when at least one mass is astronomical. A uniform sphere acts gravitationally as if all its mass were at its center, so r is measured center-to-center.

Universal gravitation

F = G m₁ m₂ / r²

Attractive, directed along the line between the masses. G = 6.67 × 10⁻¹¹ N·m²/kg². Doubling either mass doubles F; doubling r cuts F to one quarter.

Surface gravity ties g to a planet

Set the gravitational force on a mass m at a planet’s surface equal to mg: G M m / R² = m g. The mass m cancels — the reason all objects fall with the same g — leaving g = GM/R². This is why g depends on the planet, not the falling object. Above the surface, replace R with the distance r from the center; because of the inverse square, g falls off as 1/r² as you climb.

Surface gravity

g = GM / R²

The free-fall acceleration at a spherical planet of mass M and radius R. At a distance r > R from the center, the local acceleration is GM/r².

Worked example

A planet has the same mass as Earth but half its radius. How does the free-fall acceleration g at its surface compare with Earth’s?

1.Surface gravity is g = GM/R², so with M fixed, g scales as 1/R².
2.Halving the radius replaces R with R/2, giving a factor 1/(R/2)² = 1/(R²/4) = 4/R².
3.Therefore g_planet = GM/(R/2)² = 4 GM/R² = 4 g_Earth.
4.The smaller radius brings the surface much closer to the mass, so the inverse-square law makes gravity four times stronger.

Answer: g_planet = 4 g_Earth — same mass but half the radius makes surface gravity four times larger

Checkpoint

The gravitational force between two masses is F. If the distance between their centers is doubled (masses unchanged), the new force is:

Checkpoint

A planet has twice Earth’s mass and twice Earth’s radius. Its surface gravity compared with Earth’s is:

On the exam

Keep the two formulas straight: F = Gm₁m₂/r² is the force between two masses, while g = GM/R² is the field (acceleration) one mass produces at distance R. Both are inverse-square — whenever a distance changes, square the ratio before scaling.

Answer the 2 checkpoints as you read.