Gravitational Potential

Field strength is about force. Potential is about energy.
Potential tells you the energy per kilogram at a point in the field.

What potential means

Zero is chosen at infinity. Moving closer to a mass makes gravitational potential lower, so every point that is not at infinity has a negative potential.

We choose potential to be zero at infinity, far from every mass.
Moving a mass closer to a planet lowers its potential. Moving it back to infinity requires energy.
$\phi$ gives this energy change for each kilogram.

Definition

2 marks

What is meant by gravitational potential at a point?

Model answer: The work done per unit mass in bringing a small test mass from infinity to that point (9702/41/M/J/25 Q1(a)).

Unit: J kg⁻¹.
Potential is a value for the point in the field. It does not depend on the mass later placed at that point.

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

Symbols

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

Why potential is negative

Gravity pulls inward. When a mass moves inward, gravitational potential energy changes into kinetic energy. Therefore, potential falls below zero.
So potential is negative everywhere except at infinity. It is more negative close to the mass and rises toward zero as r grows.
Exam wording (why negative, two marks): potential is zero at infinity, and work must be done on the mass to move it out to infinity (9702/42/F/M/24 Q1(a)).

Compared with electric potential

Electric potential is taught in the Electric Fields chapter. For this comparison, remember that it is potential energy per unit charge.
Same shape: the size of each potential is proportional to $1/r$ . Both potentials are zero at infinity.
One difference: gravity only pulls. Therefore, gravitational potential is always negative (9702/42/F/M/23 Q1(c)).
Electric potential can be positive or negative.

Worked example

Calculate potential at a planet's surface

Find the gravitational potential at the Earth's surface. Mass 6.0 × 10²⁴ kg, radius 6.4 × 10⁶ m.

Start with the potential equation.
$\phi=-\dfrac{GM}{r}$
First calculate the product GM.
$GM=4.0\times10^{14}\ \text{m}^3\text{ s}^{-2}$
Substitute the radius.
$\phi=-\dfrac{4.0\times10^{14}}{6.4\times10^6}$
$\phi=-6.3\times10^7\ \text{J kg}^{-1}$

Answer

-6.3{\times}10^7\,\mathrm{J\,kg^{-1}}

Always give the unit and the minus sign; both are marked.

Gravitational potential is the work done per kilogram from infinity. It is negative at every finite distance and becomes zero at infinity.

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