What Is Gravitational Potential (Vg) and How Is It Different From Gravitational Potential Energy? (HL)

Gravitational potential $V_g$ is gravitational potential energy per unit mass at a point in a field. Gravitational potential energy $E_p$ is the energy of a particular system of masses, so it depends on the mass placed in the field.

The Reasoning

For an isolated spherical mass $M$, gravitational potential at distance $r$ from its centre is

$$V_g=-\frac{GM}{r}.$$

It is defined as the work done per unit mass in bringing a small test mass from infinity to that point. The reference value is $V_g=0$ at infinity. Because gravity is attractive, bringing masses together releases energy, so $V_g$ is negative at every finite distance from an isolated mass.

If a mass $m$ is placed at that point, its gravitational potential energy is

$$E_p=mV_g=-\frac{GMm}{r}.$$

For example, if $V_g=-2.0\times10^7\ \text{J kg}^{-1}$ and , then

$$E_p=mV_g=(3.0)(-2.0\times10^7)=-6.0\times10^7\ \text{J}.$$

Potential differences are often more useful than absolute values. When a mass moves from point A to point B, $\Delta V_g=V_{g,B}-V_{g,A}$, and the corresponding change in potential energy is . If the mass moves closer to , both quantities become more negative; the gravitational field does positive work while the system's potential energy decreases. For several source masses, gravitational potential is found by scalar superposition: . This is simpler than adding gravitational field vectors because no directions are required.

A common misconception is that gravitational potential is another name for gravitational potential energy. It is not: potential describes the field at a location, whereas potential energy describes a mass interacting with that field.

Exam Technique

This is Additional Higher Level (AHL) content in D.1 Gravitational fields. In calculations, identify whether the question asks for $V_g$ or $E_p$, include the negative sign, and use $E_p=mV_g$. For movement between two points, use .