15.5 Gravitational Potential

Definition

Gravitational potential ($V$) at a point in a gravitational field is defined as the work done per unit mass in bringing a small test mass from infinity to that point:

$V=\frac{W}{m}$

Gravitational potential is a scalar quantity with SI unit J kg⁻¹.

---

Formula for Gravitational Potential

For a point mass $M$, the gravitational potential at a distance $r$ from its centre is:

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

where $G=6.67\times 10^{-11}{\text{ N m}}^2{\text{ kg}}^{-2}$ is the Universal Gravitational Constant.

Why is $V$ always negative?

• The reference point is chosen at infinity, where $V=0$.
• As a mass moves towards $M$, the gravitational field does positive work on it, so the potential decreases (becomes more negative).
• Equivalently, external work must be done against the field to move a mass away to infinity, confirming that $V<0$ everywhere in the field.

---

Gravitational Potential at Earth's Surface

At Earth's surface ($r=R_E$):

$V_{\text{surface}}=-\frac{G{M}_E}{R_E}$

Using $M_E=5.97\times 10^{24}\text{ kg}$ and $R_E=6.37\times 10^6\text{ m}$:

$V_{\text{surface}}\approx -6.25\times 10^7{\text{ J kg}}^{-1}$

---

Relationship Between $V$ and Gravitational Field Strength $g$

Gravitational field strength is the negative gradient of gravitational potential:

$g=-\frac{dV}{dr}$

This means $g$ points in the direction of decreasing $V$ (i.e., towards the mass $M$), consistent with the attractive nature of gravity.

Verification: Differentiating $V=-\frac{GM}{r}$:

$g=-\frac{d}{dr}\left(-\frac{GM}{r}\right)=-\frac{GM}{r^2}$

which matches the standard formula for gravitational field strength.

---

Gravitational Potential Energy

Gravitational potential energy ($U$) of a mass $m$ at a point where the gravitational potential is $V$ is:

$U=mV=-\frac{GMm}{r}$

Distinction between $V$ and $U$

The relationship $U=mV$ shows that gravitational potential leads directly to gravitational potential energy — $V$ is the potential energy per unit mass.

---

Why $\Delta PE=mgh$ is Only Valid Near Earth's Surface

The familiar formula $\Delta PE=mgh$ assumes $g$ is constant. This is only valid for small heights $h\ll R_E$ near Earth's surface.

At large distances, $g$ decreases with $r^2$, so the general formula must be used:

$U=-\frac{GMm}{r}$

The change in gravitational potential energy between two distances $r_1$ and $r_2$ is:

$\Delta U=GMm\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$

---

Worked Example

Calculate the gravitational potential at a distance of $2R_E$ from Earth's centre.

Given: $G=6.67\times 10^{-11}{\text{ N m}}^2{\text{ kg}}^{-2}$, $M_E=5.97\times 10^{24}\text{ kg}$, $R_E=6.37\times 10^6\text{ m}$

$V=-\frac{G{M}_E}{2R_E}=-\frac{(6.67\times 10^{-11})(5.97\times 10^{24})}{2\times 6.37\times 10^6}$

$V\approx -3.13\times 10^7{\text{ J kg}}^{-1}$

Note this is half the magnitude of the surface potential, consistent with $V\propto \frac{1}{r}$.

---

Summary

• Gravitational potential $V=-\frac{GM}{r}$ is always negative and a scalar.
• $V=0$ at infinity (reference point).
• Gravitational potential energy $U=mV=-\frac{GMm}{r}$.
• Field strength $g=-\frac{dV}{dr}$ (negative potential gradient).
• $\Delta PE=mgh$ is only valid near Earth's surface where $g$ is approximately constant.