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!

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Course: High school physics > Unit 7

Lesson 8: Gravitational potential energy at large distances

Gravitational potential energy at large distances review

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

Equations

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.

Common mistakes and misconceptions

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.

Learn more

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:

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Yes yes
Posted 4 years ago. Direct link to Yes yes's post “Why does the gravitationa...”
Why does the gravitational potential energy increases as r increases if the equation states that they are inversely proportional?
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- obiwan kenobi
  Posted 4 years ago. Direct link to obiwan kenobi's post “Think about it, the formu...”
  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!
  Comment on obiwan kenobi's post “Think about it, the formu...”
  (3 votes)
Muhammad Olim
Posted 4 years ago. Direct link to Muhammad Olim's post “I saw practice questions ...”
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?
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Vihaan Gargava
Posted 10 months ago. Direct link to Vihaan Gargava's post “I didn't understood how c...”
I didn't understood how can gravitational potential energy increase if distance between the centre of masses increases....it(potential energy) should particularly decrease as the distance is increasing?
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- samraddhisagar07
  Posted 9 months ago. Direct link to samraddhisagar07's post “the gravitational potenti...”
  the gravitational potential energy is inversely proportional to radius and yea it should decrease as the radius increases but what you are forgetting here is a -ve sign so like for example the radius increases and the potential energy becomes -5 and then radius is reduced by half so the potential energy becomes -10 and as we know -10<-5
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Bikash Adhikari
Posted 4 years ago. Direct link to Bikash Adhikari's post “Can anyone explain the co...”
Can anyone explain the conservation of energy in the following case? I suppose throw a body in air, which means I did work on the body to increase its potential energy by mgh. And it comes down with increasing velocity and as it reaches down its energy is 1/2 mv^2 which is equal to mgh. But when it reaches down, it doesn't have any velocity nor does it have potential energy. So, where does that energy go?
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이태경
Posted 4 years ago. Direct link to 이태경's post “I also saw the practice q...”
I also saw the practice questions about the change of maximum gravitational potential energy of the planet and the sun-planet system and i don't really get the idea why the gravitational energy decreases in the 1st question
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- Fiona T
  Posted 3 years ago. Direct link to Fiona T's post “The gravitational potenti...”
  The gravitational potential energy decreases as the distance between the two bodies decreases.
  Looking at the equation in this article, there is a negative sign in front of the expression (G*m1*m2) / r.
  Ug and r are inversely proportional, so Ug decreases as the absolute value of r increases (and -r decreases, getting farther away from 0).
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