Gravitational potential φ
● VERBATIM — examined word-for-wordGravitational potential at a point is the work done per unit mass by an external force in bringing a small test mass from infinity to that point, without change in kinetic energy. Scalar, unit J kg⁻¹.
φ = −GM/r (always negative; φ = 0 at infinity)What this actually means
Four things must be in the sentence: work done, PER UNIT MASS, from INFINITY to that point, and by an external force with no change in kinetic energy. The last clause is what stops the test mass arriving with a pile of KE and ruining the bookkeeping.
Infinity is the reference point where we DEFINE φ = 0. It is a choice, not a discovery, and it is the reason every real potential comes out negative. Say 'from infinity' explicitly; 'from far away' is not the same thing.
Potential is a scalar. Two masses? Add their potentials algebraically, both negative, no components, no angles. Contrast field strength, which needs vector addition. Examiners love a question that gives you both bodies and checks which rule you reach for.
The unit is J kg⁻¹, not J. J is potential ENERGY, U. Keep the two symbols apart in your working: φ per unit mass, U for an actual mass m, with U = mφ.
Practical use: potential is the quantity that makes energy questions easy. Work done moving a mass between two points is m(φ_final − φ_initial), and you never have to integrate anything.
Omitting 'per unit mass' or 'from infinity', which turns a definition of φ into a definition of nothing.
Prove it — watch it be true
- Look at the 3D potential well and note the surface sits at zero far out and dips downwards near the planet.
- Read the φ–r graph and confirm φ → 0 as r → ∞, which is the infinity reference in the definition.
- Use the work-done two-marker tool with marker A at infinity and marker B at your chosen point, and check the work per unit mass equals φ at B.