Gravitational potential is the gravitational potential energy per unit mass at a point.
It tells us how much potential energy a \(1\ \mathrm{kg}\) test mass would have at that location.
1. Definition
If \(U\) is gravitational potential energy and \(m\) is the test mass, then gravitational potential is
\[
\boxed{
V=\frac{U}{m}
}.
\]
The unit of gravitational potential is
\[
\mathrm{J/kg}.
\]
2. Potential due to one spherical body
For a spherical body of mass \(M\), the gravitational potential outside the body is
\[
\boxed{
V=-\frac{GM}{r}
}.
\]
Here \(r\) is measured from the center of the body. If a point is at altitude \(h\) above the surface of a body of radius \(R\), then
\[
r=R+h.
\]
3. Why the potential is negative
The standard convention is
\[
V(\infty)=0.
\]
Since gravity is attractive, a point closer to the mass is gravitationally bound. Therefore the potential at a finite radius is negative:
\[
V(r)<0.
\]
As \(r\) increases, \(V\) becomes less negative and approaches zero from below.
4. Potential difference
If a point moves from radius \(r_1\) to radius \(r_2\), then
\[
V_1=-\frac{GM}{r_1},
\qquad
V_2=-\frac{GM}{r_2}.
\]
The potential difference is
\[
\boxed{
\Delta V=V_2-V_1
}.
\]
Substituting the formula gives
\[
\boxed{
\Delta V
=
GM\left(\frac{1}{r_1}-\frac{1}{r_2}\right)
}.
\]
If \(r_2>r_1\), then \(\Delta V>0\). Moving outward requires energy per kilogram.
If \(r_2
5. Relation to potential energy
Once \(V\) is known, the gravitational potential energy of a mass \(m\) is
\[
\boxed{
U=mV
}.
\]
Similarly, the change in potential energy is
\[
\boxed{
\Delta U=m\Delta V
}.
\]
6. Superposition of potentials
Gravitational potential is a scalar. This makes superposition simpler than vector field addition.
If several masses contribute to the potential at a point, the total potential is the sum:
\[
\boxed{
V_{\mathrm{total}}
=
\sum_i V_i
=
\sum_i
\left(
-\frac{Gm_i}{r_i}
\right)
}.
\]
Each \(r_i\) is the distance from the evaluation point to source mass \(m_i\).
7. Field strength from potential
Gravitational field strength is related to the spatial slope of the potential:
\[
\boxed{
\vec g=-\nabla V
}.
\]
In one radial dimension outside a spherical mass,
\[
V=-\frac{GM}{r}.
\]
Therefore,
\[
\left|\vec g\right|
=
\frac{GM}{r^2}.
\]
The field points toward decreasing \(r\), which means toward the attracting mass.
8. Equipotential surfaces
An equipotential surface is a surface on which the potential has the same value everywhere:
\[
V=\mathrm{constant}.
\]
Around a single spherical mass, equipotential surfaces are spheres. In a two-dimensional diagram, they appear as circles.
Moving along an equipotential requires no change in gravitational potential energy.
9. Worked example: potential at Earth’s surface
Use
\[
M_{\oplus}=5.9722\times10^{24}\ \mathrm{kg},
\qquad
R_{\oplus}=6.371\times10^6\ \mathrm{m}.
\]
At Earth’s surface,
\[
r=R_{\oplus}.
\]
Apply the formula:
\[
V=-\frac{GM_{\oplus}}{R_{\oplus}}.
\]
\[
V
=
-\frac{(6.67430\times10^{-11})(5.9722\times10^{24})}
{6.371\times10^6}.
\]
\[
V\approx -6.26\times10^7\ \mathrm{J/kg}.
\]
Since \(1\ \mathrm{MJ}=10^6\ \mathrm{J}\),
\[
\boxed{
V\approx -62.6\ \mathrm{MJ/kg}
}.
\]
Rounded values are often quoted as about
\[
\boxed{
V\approx -62.5\ \mathrm{MJ/kg}
}.
\]
10. Escape-speed connection
Escape speed is connected to gravitational potential. For escape with zero final speed at infinity,
\[
\frac12v_{\mathrm{esc}}^2+V=0.
\]
Therefore,
\[
\boxed{
v_{\mathrm{esc}}=\sqrt{-2V}
}.
\]
This is valid when \(V<0\), as it is around a gravitationally attracting body with zero potential at infinity.
11. Summary table
| Quantity |
Formula |
Meaning |
| Gravitational potential |
\(V=-GM/r\) |
Potential energy per kilogram at radius \(r\) |
| Potential difference |
\(\Delta V=GM(1/r_1-1/r_2)\) |
Work per kilogram for a slow radial move |
| Potential energy |
\(U=mV\) |
Energy of a mass \(m\) at potential \(V\) |
| Energy change |
\(\Delta U=m\Delta V\) |
Energy change for a mass \(m\) |
| Superposition |
\(V_{\mathrm{total}}=\sum_i-Gm_i/r_i\) |
Total potential from several masses |
| Field from potential |
\(\vec g=-\nabla V\) |
Field points downhill in potential |
| Escape speed |
\(v_{\mathrm{esc}}=\sqrt{-2V}\) |
Speed needed to reach infinity with zero final speed |
Key idea: gravitational potential is a scalar map of gravitational binding energy per kilogram.
