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High school physics

Course: High school physics>Unit 7

Lesson 8: Gravitational potential energy at large distances

Gravitational potential energy at large distances review

Review the equations and skills related to gravitational potential energy at large distances, including how to apply conservations laws to objects in orbit.

Equations

EquationSymbolsMeaning in words
U, start subscript, G, end subscript, equals, minus, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, r, end fractionU, start subscript, G, end subscript is gravitational potential energy, G is the gravitational constant, m, start subscript, 1, end subscript and m, start subscript, 2, end subscript are masses, and r is the distance between centers of mass of the two objectsGravitational potential energy at large distances is directly proportional to the masses and inversely proportional to the distance between them. The gravitational potential energy increases as r increases.

How to apply conservation laws to orbits

Although the Earth orbits the Sun, it does not go around in a perfect circle, but rather takes an elliptical path (Figure 1).
This means that the Earth’s distance from Sun r varies throughout the orbit. There is no net external force or torque acting on the Sun-planet system, and the only force is gravity between the Sun and planet. Therefore, angular momentum and energy remain constant. However, the gravitational potential energy does change, because it depends on distance. As a result, kinetic energy also changes throughout an orbit, resulting in a higher speed when a planet is closer to the Sun.
When dealing with gravitational potential energy over large distances, we typically make a choice for the location of our zero point of gravitational potential energy at a distance r of infinity. This makes all values of the gravitational potential energy negative.
If we make our zero of potential energy at infinity, then the gravitational potential energy as a function of r is:
U, start subscript, G, end subscript, equals, minus, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, r, end fraction
For example, imagine we are landing on a planet. As we come closer to the planet, the radial distance between us and the planet decreases. As r decreases, we lose gravitational potential energy - in other words, U, start subscript, G, end subscript becomes more negative. Because energy is conserved, the velocity must increase, resulting in an increase in kinetic energy.

Common mistakes and misconceptions

1. Students forget that there must be two separated objects considered as the system to have potential energy. A single object cannot have potential energy with itself, but only with respect to another object. For example, the Moon only has gravitational potential energy relative to the Earth (or another object).
2. Sometimes people forget that gravitational potential energy at large distances is negative. We typically make a choice for the location of our zero point of gravitational potential energy at a distance r of infinity. This makes all values of the gravitational potential energy negative.

For deeper explanations of these concepts, see our video about gravitational potential energy at large distances.
To check your understanding and work toward mastering these concepts, check out our exercises:

Want to join the conversation?

• Why does the gravitational potential energy increases as r increases if the equation states that they are inversely proportional?
• Think about it, the formula gmh applies for small distances. But for ridiculously large distances, it makes sense that the further away an object gets, the smaller the gravitational potential energy it has. The further away the object gets, the less influence it will feel gravitationally. Hope this helps!
• I saw practice questions about the change of maximum gravitational potential energy of the Sun-planet system as a planet is struck by a large asteroid. In this situation:

1. How minimum gravitational potential energy changes?
2. How elliptical orbit of planet changes?