Gravitational Potential — Quick Summary

Mr Toogood's Physics · Gravitational fields

AQA 3.7.2.3
V = W/m
Gravitational potential
V = −GM/r
Radial field
ΔW = mΔV
Work done moving m
g = −ΔV/Δr
Potential gradient

Gravitational potential

V is the work done per unit mass bringing a small test mass from infinity to a point: V=W/m. This removes dependence on the test mass, just as g=F/m did for field strength. Units: J kg⁻¹.

Not potential energy: Eₚ=mV depends on the mass placed in the field; V itself does not — it's a property of the location.

V=0 is defined at infinity — a convention (not a physical fact), chosen because it gives the same universal reference for every mass and field.

Why V is negative

Graph of gravitational potential against distance, negative and rising to zero at infinity

V rises towards zero (from below) as distance from the source increases.

As an object falls from infinity (V=0) toward a source mass, gravity does positive work, so it gains KE — meaning its PE must fall below zero. Closer to the source = more negative V = deeper "potential well." A bigger source mass means a deeper well, and more energy needed to escape it.

The sign is physical, not just mathematical: a more negative V means more energy is needed to reach infinity from that point.

Deriving V and the work equation

V = −GM/r    ΔW = mΔV

The same approach re-derives the familiar ΔEp=mgΔh for small height changes near a surface (where r_A≈r_B≈r), using g=GM/r².

Common slip: write V=−GM/r out in full for each point before substituting into ΔV=V_B−V_A — juggling the signs mentally is where marks are lost.

Equipotential surfaces

Joining points of equal potential creates a "map" of the field, directly analogous to contour lines on a hill. No work is done moving along an equipotential — just as walking along a contour line doesn't change your height. Since V∝1/r, equipotential spacing increases further from the source.

potential gradient = ΔV/Δr

Near Earth's surface, the magnitude of the potential gradient is 9.81 J kg⁻¹m⁻¹ for small height changes — numerically the same as g.

Comparing the g–r and V–r graphs

Graph of gravitational field strength against distance, always positive, following a 1/r squared curve

g–r graph: always positive, 1/r² shape — steeper than V–r, never quite reaching zero.

  • g–r graph: always positive (gravity attracts); 1/r² curve; steep close in.
  • V–r graph: always negative; approaches zero from below at infinity; 1/r curve, less steep than g.

Linking g and V: gradients & areas

Comparison showing the tangent of the V-r graph equals negative g, and the area under the g-r graph equals the change in V

Tangent on V–r graph = −g; area under g–r graph = |ΔV|.

The gradient of a V–r graph at any point equals −g there; the area under a g–r graph between r₁ and r₂ equals the magnitude of ΔV over that range. The two graphs contain exactly the same information, presented differently.

Avoiding sign errors: don't sign-juggle algebraically under exam pressure — read the magnitude off the graph (gradient or area), then state the direction separately from physics: g always points towards the source mass.

Exam essentials

Key equations

  • V=W/m   V=−GM/r
  • ΔW=mΔV
  • g=−ΔV/Δr

Graph shapes

  • V–r: always negative, −1/r shape, →0 at infinity.
  • g–r: always positive, 1/r² shape, →0 at infinity (never reaches).

Common slips

  • V is energy per unit mass, not potential energy itself (Eₚ=mV).
  • Write V=−GM/r out in full at each point before subtracting — don't sign-juggle mentally.
  • Read magnitude from a graph; state direction (towards the source) separately from physics.