13.4 Gravitational Potential

1. What is Gravitational Potential?

The Basic Idea

Imagine you are very, very far away from any planet — so far that the planet's gravity has absolutely no effect on you. Physicists call this distance infinity. At infinity, we say the gravitational potential is zero. This is our starting reference point.

Now, if you move a small mass from infinity towards a planet, the planet's gravity pulls it in. Because the gravitational force is always attractive (it always pulls, never pushes), the field itself does the work as the mass moves closer. The mass does not need an external push — gravity drags it in naturally.

Gravitational potential (ϕ) at a point is defined as:

The unit of gravitational potential is joules per kilogram (J kg⁻¹).

Why is Gravitational Potential Always Negative?

This is one of the most important — and most commonly misunderstood — ideas in this topic. Here is why it is always negative:

• We define potential to be zero at infinity.
• As a mass moves from infinity towards a planet, gravity does work on the mass. The mass loses potential energy as it falls inward.
• Because we start at zero and then lose energy, the value drops below zero — it becomes negative.

Think of it like a bank account starting at zero: gravity keeps withdrawing energy as the mass moves closer, so the balance goes negative.

Key rule: Gravitational potential is always negative. It becomes less negative (i.e., increases towards zero) as you move further away from the planet. It becomes more negative as you move closer.

2. The Formula for Gravitational Potential

For a point mass (a single, isolated mass — like a planet treated as if all its mass is at one point), the gravitational potential at a distance r from its centre is:

$\phi = -\frac{GM}{r}$

What Each Symbol Means