Physics 9702 · Paper 4

Gravitational Potential

Field strength describes the force per kilogram at a point. This lesson's quantity describes the energy per kilogram.

The definition and the minus sign

\phi = -\dfrac{GM}{r}

Symbols

$\phi$ = gravitational potential (Greek phi) (J kg⁻¹)
$G$ = gravitational constant (N m² kg⁻²)
$M$ = mass creating the field (kg)
$r$ = distance from the centre of M (m)

The two-mark definition, in mark-scheme words: at a point is the work done per unit mass in moving a mass from infinity to the point (9702/41/M/J/25 Q1(a)).
Note the reference point: potential is zero at infinity, far from every mass.

The potential well, sampled from φ = −GM/r: deepest (−φ₀) at the surface, halved in size by 2R, quartered by 4R, approaching zero at infinity — always negative on the way.

Why always negative (a marked two-liner): potential is zero at infinity, and gravity is attractive, so work is done by the field as a mass comes closer (or: work must be done on a mass to move it away to infinity). Everywhere closer than infinity, the energy is below zero (9702/42/F/M/24 Q1(a)).
The well picture: every mass sits in an energy dip. Climbing out (to infinity) costs energy; falling in releases it.

Worked example

Exam version: the potential at a planet's surface

Mars has mass 6.4 × 10²³ kg and radius 3.4 × 10⁶ m. Find the gravitational potential at its surface, with a unit. (9702/41/M/J/25 Q1(b)(ii))

$\phi = -\dfrac{GM}{R} = -\dfrac{6.67 \times 10^{-11} \times 6.4 \times 10^{23}}{3.4 \times 10^{6}} = -1.3 \times 10^{7} \text{ J kg}^{-1}$ .

Answer

\phi = -1.3 \times 10^{7} \text{ J kg}^{-1}

The minus sign and the unit J kg⁻¹ each carried a mark. Dropping either loses it.

Field strength = force per unit mass. Potential = energy (work done from infinity) per unit mass. Keep the two apart, and half the definition marks in this topic become easy.

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