13.3 Gravitational field of a point mass

Equating gravitational force GMm/r² with weight mg gives g=GM/r² for a point mass or spherical source outside its surface.

The test mass cancels, so g depends on source mass M and centre distance r, not on the mass being tested.

Doubling distance from a planet’s centre quarters g; changing the test object’s mass does not change g.

g is a field strength, not a universal constant; G is constant while g varies with location.

For a spherical source outside its surface, g=GM/r² in N kg⁻¹ or m s⁻².

Use M for the source and r from its centre. The inverse-square relation is a scaling law, not a claim that g changes linearly with height.

At twice the centre distance, a satellite experiences one quarter of the original field strength.

Do not use altitude alone when the planet’s radius is significant; r is radius plus altitude.

Although g=GM/r² varies with distance from Earth’s centre, the change is negligible over small everyday height ranges, so g≈9.8 m s⁻² is a useful approximation.

Use the constant-g model for local motion and mg∆h calculations, but switch to inverse-square reasoning for satellites or large altitude changes.

A building’s roof and ground have almost the same g compared with a spacecraft thousands of kilometres away.

“Constant near Earth” is an approximation, not evidence that the gravitational field has no spatial dependence.