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.

Równania

Equation	Symbols	Meaning in words
$U_{G} = - \frac{G m_{1} m_{2}}{r}$ ‍	$U_{G}$ ‍ is gravitational potential energy, $G$ ‍ is the gravitational constant, $m_{1}$ ‍ and $m_{2}$ ‍ are masses, and $r$ ‍ is the distance between centers of mass of the two objects	Gravitational 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.

It turns out that it makes sense to do this because as the distance

r

becomes large, the gravitational force tends rapidly towards zero. When you are close to a planet you are effectively bound to the planet by gravity and need a lot of energy to escape. Strictly you have escaped only when

r

is infinity, but because of the inverse-square relationship, we can reach a distance where gravitational potential energy becomes very close to zero. For a spacecraft leaving earth, this can be said to occur at a height of about

5 \cdot 10^{7} m

above the surface which is about four times the Earth's diameter. At that height, the acceleration due to gravity has decreased to about

1 %

of the surface value.

If we make our zero of potential energy at infinity, then the gravitational potential energy as a function of

r

is:

U_{G} = - \frac{G m_{1} m_{2}}{r}

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_{G}

becomes more negative. Because energy is conserved, the velocity must increase, resulting in an increase in kinetic energy.

Często spotykane błędy i nieporozumienia

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).
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.

Dowiedz się więcej

For deeper explanations of these concepts, see our video about gravitational potential energy at large distances.

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