Deriving g = GM/r² for a point mass
Combining Newton's law of gravitation with the definition g = F/m gives the field strength of a point mass M at distance r as g = GM/r², directed towards M.
F = GMm/r² and g = F/m ⇒ g = GM/r²What this actually means
This is syllabus LO (d), a named derivation, and it is worth two marks for two lines. Write F = GMm/r² for the force on a small test mass m, then divide by m using the definition g = F/m. The m cancels and you are done.
The physics point that earns the second mark is that m CANCELS. Field strength is a property of the source M and the position r only. It does not depend on what you put there to test it, which is exactly what 'per unit mass' was promising.
State the direction. g is a vector pointing towards the centre of M. A derivation that produces only a magnitude is incomplete if the question says 'derive an expression for the gravitational field strength'.
The result holds outside a uniform sphere as well as for a genuine point mass, with r measured from the centre. Inside a uniform sphere it fails completely: there g rises linearly from zero at the centre to the surface value.
Once you have this, notice you can also run it backwards. At a planet's surface g_s = GM/R², so GM = g_s R². That substitution kills G and M in one move and is the trick behind v_esc = √(2gR) and v_orbit = √(gr).
Quoting g = GM/r² without showing the m cancelling, or omitting the direction.
Prove it — watch it be true
- Choose the Earth planet preset and open the g vs r graph.
- Place the test mass probe outside the surface, read F and g, and confirm g = F/m for whatever probe mass you set.
- Double r on the probe and check the g readout falls to a quarter, confirming the GM/r² form.
- Switch planet presets and confirm g changes with M while the 1/r² shape does not.