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# Gravitational Potential Energy

by Ron Kurtus (revised 17 May 2011)

The *gravitational potential energy* between two objects of mass is the potential of motion caused by their gravitational attraction. The attraction of the objects turns the potential energy into the kinetic energy of motion, such that the objects will move toward each other.

Potential energy of two objects at a given separation is defined as the work required to move the objects from a *zero reference point* to that given separation. The force required to move the objects equals the incremental change of the potential energy for a change in separation.

Combining the incremental change equation with the *Universal Gravitation Equation* and then integrating from infinity to the given separation results in the equation for the potential energy. The kinetic energy and total energy can lead to the gravitational escape velocity equation and other applications.

Questions you may have include:

- What is the relationship of work, potential energy and force?
- What is the derivation of the gravitational potential energy equation?
- What are the kinetic and total energies?

This lesson will answer those questions. Useful tool: Units Conversion

## Work, potential energy and force

At a given separation, the gravitational potential energy (**PE**) between two objects is defined as the work required to move those objects from a zero reference point to that given separation.

### Work

Work is often defined as the product of the force to overcome a resistance and the displacement of the objects being moved.

When you move two objects apart, you do work to overcome the gravitational resistance. However, when the objects are allowed to move toward each other, the work is "given back" and is considered negative work.

### Zero reference point

The zero reference point is where the potential energy is considered to be null. In the case of gravitation between two objects, that zero reference point is at a hypothetical infinite separation.

Note: In the case of the potential energy from gravity, the surface of the Earth is the zero reference point, and the potential energy is the work required to lift the object to some heightPE = mgh.(

See Potential Energy of Gravity for more information.)

The work done to move two objects apart against the gravitational force to an infinity separation is positive work, while the work in the direction of gravitation is negative work.

### Force used instead of work

Since the zero reference point for gravitation is at infinity, the displacement is infinite. This makes the calculation of potential energy as a function of work difficult.

Thus, it is more convenient to use the fact that the gravitational force between two objects is equivalent to the incremental change in potential energy with respect to a change in separation:

F = dU/dR

where

**F**is the gravitational force in newtons (N)**dU**is the derivation or incremental change in potential energy in joules (J)**dR**is the incremental change in separation in meters (m)**dU/dR**is the first derivation of**U**with respect to**R**

Note: BothPEandUare commonly used to denote potential energy. Right now, we are usingU, since it is more convenient to statedUas the derivative of the potential energy.

This equation is used to find the gravitational potential energy.

## Derivation of PE equation

The derivation of the gravitational potential energy equation starts with the *Universal Gravitation Equation*:

F = GMm/R^{2}

where

**G**is the Universal Gravitational Constant**M**and**m**are the masses of the objects**R**is the separation between the centers of mass of the objects

Note: Each object has a center of mass. However, there is also a center of mass of the system, which is considered when dealing with kinetic energy.

Combining this equation with the previous increment equation, you get:

dU/dR = GMm/R^{2}^{}

dU = (GMm/R^{2})dR

At **R = ∞**, the potential energy is zero. Thus, you integrate over the range of **
U = U** to

**U =**0 and

**R = R**to

**R = ∞**:

∫dU = ∫(GMm/R^{2})dR

The result is:

U_{}= −GMm/R

or

PE = −GMm/R

Since the gravitational potential energy between two objects is defined as the work required to move those objects from a zero reference point to that given separation, and since that work is negative, the potential energy is also negative.

## Energy and applications

The force of gravitation attracts objects toward each other, so both will also have kinetic energy at any separation unless there is some external force to hold them apart.

Assuming that the only force is that of gravitation between the objects, the system is considered closed and the total energy remains constant, according to the *Law of Conservation of Energy*.

This can lead to the escape velocity equation and has implications in orbiting objects.

### Kinetic energy

The kinetic energy of the two objects is stated in the equations:

KE_{M}= MV^{2}/2

KE_{m}= mv^{2}/2

where

**KE**and_{M}**KE**are the kinetic energies of the objects in joules (J)_{m}**M**and**m**are the masses of the objects in kg**V**and**v**are the velocities of the objects toward the center of mass between them in m/s

Note: The velocities of the objects are relative to the center of mass between them. The relationship of those velocities follows ifM > m(Mis larger thanm), thenV < v.(

See Gravitation and Center of Mass for more information.)

### Total energy

According to the *Law of Conservation of Energy,* the total energy (**TE**) of a closed system remains constant:

TE = (KE_{M}+ KE_{m}) + PE

Values of **R** will determine values of **V**, **v** and **PE** accordingly.

### Escape velocity application

In the case where the mass of one object, **M**, is much greater than the mass of the other object (**M >> m**), the velocity of the larger object can be considered negligible and its kinetic energy equal to zero:

KE0_{M}=

In such a case, the total energy equation is approximately:

TE = KE_{m}+ PE

TE = mv^{2}/2 − GMm/R

Since the total energy at infinity is zero:

mv^{2}/2 = GMm/R

This leads to the escape velocity equation:

v_{e}= − √(2GM/R_{i})

where

**v**is the escape velocity vector_{e}**R**is the initial separation between the two objects_{i}

Note:vis negative, since it is moving away from the center of mass between the objects_{e}(

See Gravitational Escape Velocity Derivation for more information.)

### Orbit application

In the case where the objects are rotating about the center of mass between them, they have a potential energy according to their separation, but their kinetic energy in the direction of the line between their centers is zero. This means that although the objects are moving in orbit, there is no work against gravitation.

(

See Circular Planetary Orbits for more information.)

## Summary

The gravitational force of attraction between objects at some separation creates their *gravitational potential energy*, which can be turned into the kinetic energy of motion. Potential energy is defined as the work required to move the objects from a zero reference point to a separation in space.

Gravitational force equals the incremental change of the potential energy for a change in separation and is used to derive the potential energy equation. Resulting kinetic energy and total energy can lead to applications such as gravitational escape velocity equation.

You have great potential

## Resources and references

The following resources can be used for further study on the subject.

### Web sites

**Gravitational Potential Energy** - HyperPhysics

### Books

**Top-rated books on Escape Velocity and Space Travel**

## Questions and comments

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## Gravitational Potential Energy