### Table of Contents

### Definition

**Gravitational potential energy** refers to celestial bodies (for example Earth and a satellite), which are not in contact but still interact with each other through a force, called gravitational force. Gravitational potential energy is defined as the potential energy that an object with a mass has in relation to another object with a mass, due to gravity.

When one object starts to move towards the centre of the second object, the gravitational potential energy is converted into kinetic energy.

Gravitational potential energy can be seen as a generalisation of the potential energy.

The general equation (formula) for gravitational potential energy is given by the law of gravity and is equal to the work done against gravity in order to bring a mass to a given point in space.

### Formula

Two bodies interacting with each other due to gravity, have the gravitational potential energy defined as [1]:

_{1}· m

_{2}) / r

where:

- U [J] – gravitational potential energy
- G [N·m
^{2}/kg^{2}] – gravitational constant - m
_{1}[kg] – mass of first body - m
_{2}[kg] – mass of first body - r [m] – distance between the center of the bodies

The expression of gravitational potential energy is valid when the distance between the two bodies r [m] is bigger than the maximum between the radius of the bodies (assuming spherical bodies).

If the gravitational potential energy is calculate for the particular case of the Earth and a satellite, the expression (1) becomes:

_{E}· m) / (r

_{E}+ h)

where:

- m
_{E}[kg] – Earth’s mass, equal to 5.97·10^{24}kg - m [kg] – satellite’s mass
- r
_{E}[m] – Earth’s radius, equal to 6.37·10^{6}m - h [m] – distance above Earth’s surface

The gravitational potential energy is negative because the force between the bodies is attractive and we have taken the potential energy as zero when the distance between bodies is infinite. Since the force between the two bodies is attractive, there must be an external component which has to do positive work in order to increase the separation between bodies.

The work done by the external component produces an increase (less negative) in the gravitational potential energy as the two bodies are separated. The bigger the distance between bodies, the less negative the gravitational potential energy, which means, a lesser force is necessary to separate them since the gravitational pull is smaller.

The gravitational constant G [N·m^{2}/kg^{2}] (also called universal gravitational constant), is an empirical physical constant, with the value of:

^{-11}N·m

^{2}/kg

^{2}

The unit of measurement of **gravitational potential energy** is **joule** [J].

### Example

Calculate the gravitational potential energy of a 9 kg mass on the surface of the Earth (a) and at a altitude of 325 km (b).

**Step 1 (a)**. Calculate the gravitational potential energy at the surface of the Earth using equation (2). This means that the altitude is (h = 0 m):

^{-11}· 5.97 · 10

^{24}· 9) / (6.37 · 10

^{6}+ 0) = – 5.63 · 10

^{8}J

**Step 2**. Convert the altitude from [km] to [m] by multiplying the [km] with 1000:

^{5}m

**Step 3 (b)**. Calculate the gravitational potential energy at 325 km altitude, using equation (2):

^{-11}· 5.97 · 10

^{24}· 9) / (6.37 · 10

^{6}+ 3.25 · 10

^{5}) = – 5.63 · 10

^{8}J

As you can see, at higher altitude, the gravitational potential energy is less negative, which means that the Earth’s gravitational pull is less and it requires less energy to increase further the altitude.

### Calculator

The gravitational potential energy calculator allows you to calculate the gravitational potential energy of two bodies. You can use this calculator for Earth’s system, defined by equation (2) or any other two bodies, defined by equation (1). If the Earth’s system is not used, m_{E} [kg] will become m_{1} [kg] and m [kg] will become m_{2} [kg]. Also h [m] will need to be set to 0.

The default unit of measurement for energy is **Joule**. If you want the result displayed in another unit, use the dropt down list to choose and click the CALCULATE button again.

### References

[1] David Halliday, Robert Resnick, Jearl Walker, Fundamentals of Physics, 7th edition, John Wiley & Sons, 2004.

[2] Benjamin Crowell, Light and Matter – Physics, 2007.

[3] Raymond A. Serway and John W. Jr. Jewett, Physics for Scientists and Engineers, 6th edition, Brooks/Cole Publishing Co.,2004

[4] Jiansong Li, Jiyun Zhao, and Xiaochun Zhang, A Novel Energy Recovery System Integrating Flywheel and Flow Regeneration for a Hydraulic Excavator Boom System, Energies 2020.

[5] Leo H. Holthuijsen, Waves in oceanic and coastal waters, Cambridge University Press, 2007.

[6] Kira Grogg, Harvesting the Wind: The Physics of Wind Turbines, Carleton College, 2005.