# Gravitational Potential Energy Concept Issue

Hi,
I am confused about the concept of gravitational potential energy. In my book the formula for calculating gravitational potential energy is:
GPE = - (GMm)/r

Now Lets suppose I want to calculate the gravitational potential energy of moon with respect to earth. Then:

r = 384,400 km = 384400000 m (According to NASA)
G = 6.6743 × 10^-11 (N*m^2)/kg^2 (Universal Gravitational Const)
M = 5.972 × 10^24 kg (Earth's Mass)
m = 7.342×10^22 kg (Moon's Mass)

Therefore:
GPE= - (6.6743 × 10^-11)x(5.972 × 10^24)x(7.342×10^22)/(384400000)
GPE = - 7.613012167 x 10^28 J

So
GPE Of Moon w.r.t Earth = - 7.613012167 x 10^28 J

Is gravitational potential energy of earth with respect to moon also equal to this? I mean, Is the following statement TRUE OR FALSE?

GPE OF Earth w.r.t Moon = - 7.613012167 x 10^28 J

Also the definition in my book of Gravitational Potential Energy is:

For two isolated masses m1 and m2 situated a distance r apart in a vacuum, then the gravitational potential energy Ep of the two masses is given by:

Ep = - Gm1m2/r

So If we apply this formula to both Moon and Earth. Is their GPE with respect to each other equal?
(edited 1 year ago)
Original post by Infinitetimes
Hi,
I am confused about the concept of gravitational potential energy. In my book the formula for calculating gravitational potential energy is:
GPE = - (GMm)/r

Now Lets suppose I want to calculate the gravitational potential energy of moon with respect to earth. Then:

r = 384,400 km = 384400000 m (According to NASA)
G = 6.6743 × 10^-11 (N*m^2)/kg^2 (Universal Gravitational Const)
M = 5.972 × 10^24 kg (Earth's Mass)
m = 7.342×10^22 kg (Moon's Mass)

Therefore:
GPE= - (6.6743 × 10^-11)x(5.972 × 10^24)x(7.342×10^22)/(384400000)
GPE = - 7.613012167 x 10^28 J

So
GPE Of Moon w.r.t Earth = - 7.613012167 x 10^28 J

Is gravitational potential energy of earth with respect to moon also equal to this? I mean, Is the following statement TRUE OR FALSE?

GPE OF Earth w.r.t Moon = - 7.613012167 x 10^28 J

Also the definition in my book of Gravitational Potential Energy is:

For two isolated masses m1 and m2 situated a distance r apart in a vacuum, then the gravitational potential energy Ep of the two masses is given by:

Ep = - Gm1m2/r

So If we apply this formula to both Moon and Earth. Is their GPE with respect to each other equal?

It's the amount of energy stored in the system (The Earth-Moon system) as a result of their having been separated.
You would regain this energy if the 2 bodies were allowed to come back together again.
You had to input this energy to separate them against their mutual attraction.

So yes. It doesn't matter whether you look at it from the Earth's point of view or the Moon's.
It's about the system comprising Earth and Moon together.

Btw
The same thing happens with electrical potential energy when you separate two opposite charges.
(edited 1 year ago)
Original post by Stonebridge
It's the amount of energy stored in the system (The Earth-Moon system) as a result of their having been separated.
You would regain this energy if the 2 bodies were allowed to come back together again.
You had to input this energy to separate them against their mutual attraction.

So yes. It doesn't matter whether you look at it from the Earth's point of view or the Moon's.
It's about the system comprising Earth and Moon together.

What do you mean by this "You would regain this energy if the 2 bodies were allowed to come back together again."?

Do you mean to say that the Gravitational Potential Energy Lost will be Gained again if r = 0.

i.e. GPE Loss by Separation = GPE Gain by Removing the separation i.e. if r = 0

Also both Moon and Earth will have gravitational potential energy right?
Original post by Infinitetimes
What do you mean by this "You would regain this energy if the 2 bodies were allowed to come back together again."?

Do you mean to say that the Gravitational Potential Energy Lost will be Gained again if r = 0.

i.e. GPE Loss by Separation = GPE Gain by Removing the separation i.e. if r = 0

Also both Moon and Earth will have gravitational potential energy right?

You need energy to separate the two masses - you do work against the gravitational attraction.
This work you do is stored as gravitational potential energy- in exactly the same way as when you lift up an object on the earth's surface, it gains GPE.
If the Earth and Moon were sitting in space (at rest) after being separated, they would now have GPE. (Calculated as you have said with that formula)
If they are then allowed to come together again, under the influence of their mutual attraction, they will both gain kinetic energy.
That is to say, the GPE they lose (as they get closer) is turned in to Kinetic Energy as the accelerate towards each other.

This is conservation of energy.
Original post by Stonebridge
You need energy to separate the two masses - you do work against the gravitational attraction.
This work you do is stored as gravitational potential energy- in exactly the same way as when you lift up an object on the earth's surface, it gains GPE.
If the Earth and Moon were sitting in space (at rest) after being separated, they would now have GPE. (Calculated as you have said with that formula)
If they are then allowed to come together again, under the influence of their mutual attraction, they will both gain kinetic energy.
That is to say, the GPE they lose (as they get closer) is turned in to Kinetic Energy as the accelerate towards each other.

This is conservation of energy.

In my book, they explain gravitational potential energy between two point masses like this. They say lets suppose one of the object is the test mass while the other one is the field-producing mass. Also they tell us if we take the test mass to a point which is at an infinite distance from the field producing mass (i.e. r = infinity) then the gravitational potential energy of the test mass with respect to the field producing mass will be zero. Now if we bring the two masses closer when there is a finite distance r between them, the gravitational potential energy of the test mass with respect to the field producing mass is negative.

The book didn't present the concept in the context you are telling me. You are saying energy is required to separate the masses and when the objects come closer together, this energy is converted to kinetic energy.

Can you tell me how energy is generated which is required to separate the objects? and Vice Versa, how energy is generated which is required to bring the objects closer together?
Original post by Infinitetimes
In my book, they explain gravitational potential energy between two point masses like this. They say lets suppose one of the object is the test mass while the other one is the field-producing mass. Also they tell us if we take the test mass to a point which is at an infinite distance from the field producing mass (i.e. r = infinity) then the gravitational potential energy of the test mass with respect to the field producing mass will be zero. Now if we bring the two masses closer when there is a finite distance r between them, the gravitational potential energy of the test mass with respect to the field producing mass is negative.

The book didn't present the concept in the context you are telling me. You are saying energy is required to separate the masses and when the objects come closer together, this energy is converted to kinetic energy.

Can you tell me how energy is generated which is required to separate the objects? and Vice Versa, how energy is generated which is required to bring the objects closer together?

You are now talking about gravitational PE in general terms, that is, the energy required to bring a mass from infinity to a point in a field. This is correct. Your original question was specifically about the Earth and Moon system. That's what I was replying to.

Potential energy is a catch all term for energy stored. It can be in a spring when stretched, for example. In our case it is in a gravitational field.
The energy is stored (as GPE) when you move an object in the field and do work on it.
The work you do is stored as GPE.
When you lift an object up on the Earth's surface you have to do work on it. (Apply a force through a distance)
The work you do is stored as GPE in that object.
If you allow the object to fall, that GPH is converted into kinetic energy.

In the case of the earth and Moon you have to imagine them being pulled apart - this requires energy. This is purely in the imagination, but however you do it, you need a force and it acts through a distance. This work, to separate the the 2, becomes the GPE.
If the 2 objects then, under gravity, come back together, this GPE is converted into kinetic energy.

The general definition of GPE takes infinity as its starting point and says that the GPE an object has is the energy required (work done) bringing another mass from infinity to that point. The 2 definitions amount to the same thing.
The difficulty here was in starting from the Earth Moon system.

If you use the general definition you first calculate the energy required (work done) to bring the Moon from infinity to wherever it is near the earth, and compare that with the energy required to bring it from there to a point where it coincides with the earth. (The 2 centres are together. Impossible in reality.)

The difference between those 2 is the answer you got with your initial calculation.
Original post by Stonebridge
You are now talking about gravitational PE in general terms, that is, the energy required to bring a mass from infinity to a point in a field. This is correct. Your original question was specifically about the Earth and Moon system. That's what I was replying to.

Potential energy is a catch all term for energy stored. It can be in a spring when stretched, for example. In our case it is in a gravitational field.
The energy is stored (as GPE) when you move an object in the field and do work on it.
The work you do is stored as GPE.
When you lift an object up on the Earth's surface you have to do work on it. (Apply a force through a distance)
The work you do is stored as GPE in that object.
If you allow the object to fall, that GPH is converted into kinetic energy.

In the case of the earth and Moon you have to imagine them being pulled apart - this requires energy. This is purely in the imagination, but however you do it, you need a force and it acts through a distance. This work, to separate the the 2, becomes the GPE.
If the 2 objects then, under gravity, come back together, this GPE is converted into kinetic energy.

The general definition of GPE takes infinity as its starting point and says that the GPE an object has is the energy required (work done) bringing another mass from infinity to that point. The 2 definitions amount to the same thing.
The difficulty here was in starting from the Earth Moon system.

If you use the general definition you first calculate the energy required (work done) to bring the Moon from infinity to wherever it is near the earth, and compare that with the energy required to bring it from there to a point where it coincides with the earth. (The 2 centres are together. Impossible in reality.)

The difference between those 2 is the answer you got with your initial calculation.

Thanks a lot. I understand it now