1. Introduction

Gravitational Potential Energy (GPE) is the energy associated with the position of an object in a gravitational field.

In introductory physics, gravitational potential energy near Earth’s surface is often written as:

$$U=mgh$$

However, AP Physics C uses a more general expression that applies to planets, satellites, stars, and any two masses separated by a distance.

The gravitational force is a conservative force, which means gravitational potential energy can be defined for any position.

2. Review of Potential Energy

Potential energy is stored energy associated with position or configuration.

Examples include:

• gravitational potential energy
• elastic potential energy
• electric potential energy

For gravity, potential energy depends on the distance between interacting masses.

3. Gravitational Potential Energy of Two Masses

Definition

For two masses separated by a distance $r$:

U=−GMmrU = -\frac{GMm}{r}U=−rGMm​

where:

• $U$ = gravitational potential energy
• $G$ = universal gravitational constant
• $M$ = source mass
• $m$ = object mass
• $r$ = distance between centers

This is the general gravitational potential energy equation used in AP Physics C.

4. Why Is the Potential Energy Negative?

The negative sign is extremely important.

Gravity is an attractive force.

Because the two masses naturally move toward one another, energy must be added to separate them.

Therefore:

$$U<0$$

for all finite distances.

Interpretation

As the distance increases:

$$U\to 0$$

As the distance decreases:

$$U\to \text{more negative}$$

The system becomes more tightly bound.

5. Reference Point for Gravitational Potential Energy

For gravitational interactions:

$$U=0$$

is defined at:

$$r=\infty$$

This differs from the near-Earth equation:

$$U=mgh$$

which chooses an arbitrary zero level.

6. Relationship Between Force and Potential Energy

Gravitational force is related to potential energy by:

FG=−dUdrF_G = -\frac{dU}{dr}FG​=−drdU​

Substituting:

U=−GMmrU = -\frac{GMm}{r}U=−rGMm​

produces:

FG=GMmr2F_G = \frac{GMm}{r^2}FG​=r2GMm​

which is Newton’s Law of Gravitation.

This demonstrates the connection between force and potential energy.

7. Change in Gravitational Potential Energy

The change in gravitational potential energy is:

ΔU=Uf−Ui\Delta U = U_f-U_iΔU=Uf​−Ui​

Substituting the gravitational potential energy formula:

ΔU=−GMmrf+GMmri\Delta U = -\frac{GMm}{r_f} + \frac{GMm}{r_i}ΔU=−rf​GMm​+ri​GMm​

or

ΔU=GMm(1ri−1rf)\Delta U = GMm \left( \frac{1}{r_i} – \frac{1}{r_f} \right)ΔU=GMm(ri​1​−rf​1​)

8. Moving Away from Earth

If an object moves farther from Earth:

rf>rir_f>r_irf​>ri​

then:

$$\Delta U>0$$

Its gravitational potential energy increases.

Work must be done against gravity.

9. Moving Toward Earth

If an object falls toward Earth:

rf<rir_f<r_irf​<ri​

then:

$$\Delta U<0$$

Gravitational potential energy decreases.

The lost potential energy becomes kinetic energy.

10. Relationship to Mechanical Energy

The total mechanical energy of a gravitational system is:

$$E=K+U$$

where:

• $K$ = kinetic energy
• $U$ = gravitational potential energy

If only gravity acts:

$$E=\text{constant}$$

This is the principle of conservation of mechanical energy.

11. Gravitational Potential Energy Near Earth’s Surface

When the height is small compared to Earth’s radius:

h≪REh \ll R_Eh≪RE​

the general equation simplifies to:

$$U=mgh$$

where:

• $m$ = mass
• $g$ = gravitational field strength
• $h$ = height

This approximation is commonly used near Earth’s surface.

12. Deriving $U=mgh$

Starting with:

U=−GMmrU = -\frac{GMm}{r}U=−rGMm​

For small heights:

r=RE+hr=R_E+hr=RE​+h

Using a small-change approximation leads to:

$$\Delta U=mgh$$

Thus the familiar equation is actually an approximation of the general gravitational potential energy formula.

13. Escape Energy

To completely remove an object from a planet’s gravitational influence:

Uf=0U_f=0Uf​=0

at infinity.

The required energy equals the magnitude of the initial gravitational potential energy.

Eescape=GMmrE_{\text{escape}} = \frac{GMm}{r}Eescape​=rGMm​

This concept leads directly to escape velocity.

14. Circular Orbits

For a satellite in circular orbit:

Potential energy is:

U=−GMmrU = -\frac{GMm}{r}U=−rGMm​

Orbital kinetic energy is:

K=GMm2rK = \frac{GMm}{2r}K=2rGMm​

The total mechanical energy is:

E=−GMm2rE = -\frac{GMm}{2r}E=−2rGMm​

Important Result

Bound circular orbits always have:

$$E<0$$

Negative total energy indicates the object is gravitationally bound.

15. Example Problem

A satellite of mass:

$$m=1000kg$$

orbits Earth at:

r=7.0×106 mr=7.0\times10^6\,mr=7.0×106m

Calculate its gravitational potential energy.

Use:

ME=5.97×1024 kgM_E=5.97\times10^{24}\,kgME​=5.97×1024kg

Solution

Using:

U=−GMEmrU = -\frac{GM_Em}{r}U=−rGME​m​

Substitute values:

U=−(6.67×10−11)(5.97×1024)(1000)7.0×106U = -\frac{ (6.67\times10^{-11}) (5.97\times10^{24}) (1000) } {7.0\times10^6}U=−7.0×106(6.67×10−11)(5.97×1024)(1000)​ U≈−5.69×1010 JU \approx -5.69\times10^{10}\,JU≈−5.69×1010J

16. Energy Diagram Interpretation

As distance increases:

$$U\to 0$$

As distance decreases:

$$U\to -\infty$$

The gravitational potential-energy curve is always negative and approaches zero at large distances.

17. Common AP Physics C Mistakes

Mistake 1

Forgetting the negative sign.

U=−GMmrU = -\frac{GMm}{r}U=−rGMm​

The negative sign is essential.

Mistake 2

Using $U=mgh$ for satellites or planets.

Use:

U=−GMmrU = -\frac{GMm}{r}U=−rGMm​

for large-scale gravitational systems.

Mistake 3

Confusing potential energy with gravitational force.

Potential energy is measured in joules, while force is measured in newtons.

18. Connections to Other Topics

Newton’s Law of Gravitation

FG=GMmr2F_G = \frac{GMm}{r^2}FG​=r2GMm​

Conservation of Energy

Ki+Ui=Kf+UfK_i+U_i = K_f+U_fKi​+Ui​=Kf​+Uf​

Circular Orbits

E=−GMm2rE = -\frac{GMm}{2r}E=−2rGMm​

Escape Velocity

Depends directly on gravitational potential energy.

Summary

General gravitational potential energy:

U=−GMmrU = -\frac{GMm}{r}U=−rGMm​

Change in potential energy:

ΔU=GMm(1ri−1rf)\Delta U = GMm \left( \frac{1}{r_i} – \frac{1}{r_f} \right)ΔU=GMm(ri​1​−rf​1​)

Near-Earth approximation:

$$U=mgh$$

Total mechanical energy:

$$E=K+U$$

Key ideas:

• gravitational potential energy arises from the positions of interacting masses
• gravitational potential energy is negative because gravity is attractive
• the reference point is $U=0$ at infinity
• $mgh$ is only a near-Earth approximation
• gravitational potential energy is essential for analyzing orbits and energy conservation

Gravitational Potential Energy provides the energy-based framework for understanding planetary motion, satellite orbits, escape velocity, and gravitational systems throughout AP Physics C Mechanics.