# Thread: What is the difference GP Energy and Gravitational Potential?

1. What is the difference Gravitational Potential Energy and Gravitational Potential? The difference confused me over the weekend for they both seem to use the same formula to be calculated. Did I read the formulas wrong?

2.

3. I answer your question with a simple 'proximity of mass determines gravity force.'
ie; The great attractor is over 500 million L/y across yet the 'Virgo cluster' is being pulled into it..
Understanding that the math.. is just a tool..
~ Does this help ?

4. Originally Posted by astromark
I answer your question with a simple 'proximity of mass determines gravity force.'
ie; The great attractor is over 500 million L/y across yet the 'Virgo cluster' is being pulled into it..
Understanding that the math.. is just a tool..
~ Does this help ?
Newton worked out that the gravitational force was proportional to the inverse of the distance squared,
i.e. halve the distance produces 4 times the strength, or conversely 2 times the distance yields 1/4 the strength as before.

5. The formula to calculate GPE is given the symbol U = mgh
m = mass of the object
g - acceleration due to gravity of a nearby large body i.e. the Earth
h = the change in height from the original position.

This is an approximation that works while the distance moved is only an insignificant fraction of r (the distance to the center of mass).

6. Originally Posted by Robittybob1
The formula to calculate GPE is given the symbol U = mgh
m = mass of the object
g - acceleration due to gravity of a nearby large body i.e. the Earth
h = the change in height from the original position.

This is an approximation that works while the distance moved is only an insignificant fraction of r (the distance to the center of mass).
If the height factor starts to get significant the gravitational acceleration will also decline, so they the use the alternative formula
U = -GmM/r where:
G = Universal gravitational constant.
m = mass of body being lifted
M = mass of larger body
r = the distance between the centers of mass.
You would have to do the calculation twice and take the difference as the amount of work done from one position to the other.

The energy required to reach escape velocity is what it takes to take the object to infinity. It isn't infinite because the force due to gravity is declining at a rate proportional to 1/(r^2)

7. So how does gravitational potential differ? Firstly it relates to a unit mass so one of the "m" in the GPE equations drops off
The symbol for GP is V
V = -GM/r so it really is the same as saying the GPE /unit mass at a distance r from a mass M. So I finally understand it now. (I think!)

8. Originally Posted by Robittybob1
What is the difference Gravitational Potential Energy and Gravitational Potential? The difference confused me over the weekend for they both seem to use the same formula to be calculated. Did I read the formulas wrong?
Gravitational potential energy is obviously potential energy while gravitational potential is gravitational potential energy per unit mass. How are you with electrostatics? I thought that I'd use an analogy. The electric potential energy U between a charged particle at the origin of charge Q and charge q a distance q away is

U = kQq/r

The electric potential at a distance r from the origin is U/q or

V = U/q = kQ/r

Do you recognize this? It's the potential function for a charged particle at the origin. If you were to take the negative gradient of V then you'd get the electric field at r, i.e.

Let's try the same thing with gravity. Let there be an object of mass M at the origin and a particle of mass m a distance r away. Then

U = GMm/r

The gravitational potential is gravitational energy per unit mass so

V = U/m = GM/r

which is the potential function for an object of mass M at the origin. We take the gradient to get

Where g is the gravitational field. The gravitational field is gravitational force per unit mass. To see what this means notice that the force on a particle of mass m by an object at the origin whose mass is M is given by

F = -GMmer/r2

where er is a unit vector pointing from the object to the particle, i.e. e = r/r. So the gravitational field at r is the force per unit mass, i.e.

g = F/m = -GMer/r2

As you can see this is the acceleration of a particle in a gravitational field at point r. It can be shown that

I hope that helps.

9. [QUOTE=physicist;591399]
Originally Posted by Robittybob1

U = GMm/r

The gravitational potential is gravitational energy per unit mass so

V = U/m = GM/r

which is the potential function for an object of mass M at the origin. We take the gradient to get

Where g is the gravitational field. The gravitational field is gravitational force per unit mass. To see what this means notice that the force on a particle of mass m by an object at the origin whose mass is M is given by

F = -GMmer/r2

where er is a unit vector pointing from the object to the particle, i.e. e = r/r. So the gravitational field at r is the force per unit mass, i.e.

g = F/m = -GMer/r2

As you can see this is the acceleration of a particle in a gravitational field at point r. It can be shown that

I hope that helps.
g = - grad U is wrong

g = - grad V is the correct expression

10. [QUOTE=Howard Roark;591925]
Originally Posted by physicist
Originally Posted by Robittybob1

U = GMm/r

The gravitational potential is gravitational energy per unit mass so

V = U/m = GM/r

which is the potential function for an object of mass M at the origin. We take the gradient to get

Where g is the gravitational field. The gravitational field is gravitational force per unit mass. To see what this means notice that the force on a particle of mass m by an object at the origin whose mass is M is given by

F = -GMmer/r2

where er is a unit vector pointing from the object to the particle, i.e. e = r/r. So the gravitational field at r is the force per unit mass, i.e.

g = F/m = -GMer/r2

As you can see this is the acceleration of a particle in a gravitational field at point r. It can be shown that

I hope that helps.
g = - grad U is wrong

g = - grad V is the correct expression
Physicist and Howard, this is good, thanks.

Being lazy - and more preoccupied with atoms and electrons than astronomical objects - I'd never really thought about Gravitational Potential in the way that I do about an electrostatic potential. I shall be more careful in how I use these terms in future.

11. Originally Posted by physicist
Originally Posted by Robittybob1
What is the difference Gravitational Potential Energy and Gravitational Potential? The difference confused me over the weekend for they both seem to use the same formula to be calculated. Did I read the formulas wrong?
Gravitational potential energy is obviously potential energy while gravitational potential is gravitational potential energy per unit mass. How are you with electrostatics? I thought that I'd use an analogy. The electric potential energy U between a charged particle at the origin of charge Q and charge q a distance q away is

U = kQq/r

The electric potential at a distance r from the origin is U/q or

V = U/q = kQ/r

Do you recognize this? It's the potential function for a charged particle at the origin. If you were to take the negative gradient of V then you'd get the electric field at r, i.e.

Let's try the same thing with gravity. Let there be an object of mass M at the origin and a particle of mass m a distance r away. Then

U = GMm/r

The gravitational potential is gravitational energy per unit mass so

V = U/m = GM/r

which is the potential function for an object of mass M at the origin. We take the gradient to get

Where g is the gravitational field. The gravitational field is gravitational force per unit mass. To see what this means notice that the force on a particle of mass m by an object at the origin whose mass is M is given by

F = -GMmer/r2

where er is a unit vector pointing from the object to the particle, i.e. e = r/r. So the gravitational field at r is the force per unit mass, i.e.

g = F/m = -GMer/r2

As you can see this is the acceleration of a particle in a gravitational field at point r. It can be shown that

I hope that helps.
Sorry I didn't get back to you over this, as I've been down with a bug, but I must admit I never try and understand electrical forces yet it seems clear there is some obvious parallelism between gravitational and electrical forces. Thanks.

12. Well yes, they are both field theories that are describing work (energy) and much of the math is the same.

13. Originally Posted by dan hunter
Well yes, they are both field theories that are describing work (energy) and much of the math is the same.
But it did make me wonder about the space-time aspects. Do they also have a resemblance? The electrical space-time warp would be so much greater. Something like this: The electrical force is 10^36 times as strong as the gravitational force. A mere dimple in the space-time fabric!

14. Originally Posted by Robittybob1
Sorry I didn't get back to you over this, as I've been down with a bug, but Imust admit I never try and understand electrical forces yet it seems clearthere is some obvious parallelism between gravitational and electrical forces. Thanks.
You're quite welcome. It's easy to show the difference between gravitationalpotential energy and gravitational potential if you have a similar example togo by. Potential differs from potential energy as the former is the later perunit “charge” whether the charge be electric charge or “gravitational charge”aka mass.

15. Originally Posted by physicist
Originally Posted by Robittybob1
Sorry I didn't get back to you over this, as I've been down with a bug, but Imust admit I never try and understand electrical forces yet it seems clearthere is some obvious parallelism between gravitational and electrical forces. Thanks.
You're quite welcome. It's easy to show the difference between gravitationalpotential energy and gravitational potential if you have a similar example togo by. Potential differs from potential energy as the former is the later per unit “charge” whether the charge be electric charge or “gravitational charge”aka mass.
Thanks. I am more into the biological sciences but for some reason the theory around gravitation have always held a fascination for I wanted to know how gravitational potential energy is stored. In the other thread there was an hypothesis put that it was stored in the atoms, was it in the form of space-time or something but the hypothesis seemed to get abandoned. I thought it quite courageous for I have never seen an explanation that really struck me, and they are few and far between.

16. Originally Posted by Robittybob1
Thanks. I am more into the biological sciences but for some reason the theory around gravitation have always held a fascination for I wanted to know how gravitational potential energy is stored. In the other thread there was an hypothesis put that it was stored in the atoms, was it in the form of space-time or something but the hypothesis seemed to get abandoned. I thought it quite courageous for I have never seen an explanation that really struck me, and they are few and far between.
Once again you're very welcome. I've followed the derivation for the energy density of an electric and magnetic field. In the integrations used to get the total energy you have to integrate over a particular region. We can express the work done to create the electric field as

This can also be expressed as

In the first integral we only need to integrate over the region where there is charge present and in the second integral we need to integrate where the electric field is, which could be over all of space. We could therefore write energy density u in the first case as

or as in the second case as

So where the energy is located is very different depending on how you look at it. In GR we use the field only as being where energy is located. I hope that helps.

17. Originally Posted by physicist
Originally Posted by Robittybob1
Thanks. I am more into the biological sciences but for some reason the theory around gravitation have always held a fascination for I wanted to know how gravitational potential energy is stored. In the other thread there was an hypothesis put that it was stored in the atoms, was it in the form of space-time or something but the hypothesis seemed to get abandoned. I thought it quite courageous for I have never seen an explanation that really struck me, and they are few and far between.
Once again you're very welcome. I've followed the derivation for the energy density of an electric and magnetic field. In the integrations used to get the total energy you have to integrate over a particular region. We can express the work done to create the electric field as

This can also be expressed as

.
The above formulas are patently incorrect, the dimensions are wrong. See here for the correct formulas.

18. Originally Posted by Howard Roark
Originally Posted by physicist
Originally Posted by Robittybob1
Thanks. I am more into the biological sciences but for some reason the theory around gravitation have always held a fascination for I wanted to know how gravitational potential energy is stored. In the other thread there was an hypothesis put that it was stored in the atoms, was it in the form of space-time or something but the hypothesis seemed to get abandoned. I thought it quite courageous for I have never seen an explanation that really struck me, and they are few and far between.
Once again you're very welcome. I've followed the derivation for the energy density of an electric and magnetic field. In the integrations used to get the total energy you have to integrate over a particular region. We can express the work done to create the electric field as

This can also be expressed as

.
The above formulas are patently incorrect, the dimensions are wrong. See here for the correct formulas.
I wish I knew what the maths all means. To me that is a page of hieroglyphics sorry. Thanks all the same.

19. Originally Posted by Howard Roark
The above formulas are patently incorrect, the dimensions are wrong.
Nope. You're quite wrong my friend. The next time you run into this kind of situation it'd be better for you to ask the person what they meant in a formula rather than claiming that you know what they meant and claim they were wrong by what you assumed that they meant. That's obviously a bad way to communicate online.

Sorry, but my equations are exactly correct. In the first equation, i.e. , the terms under the integral is charge density times potential times the volume element . You see, I have a textbook that I copied it out of as I always do in order to make certain that I don't make mistakes such as this. In this case I used Introduction to Electrodynamics - Forth Edition by David Griffiths. The third edition is online at Introduction to Electrodynamics (3rd Edition) | David J. Griffiths; Reed College | digital library BookOS

Simply click on [Download (pdf, 23.58 MB)] and turn to page 93 and read Equation (2.43) and you'll see this equation

(2.43) [tex]W = \frac{1}{2}\int \rho V d\tau[/tau]

which is exactly the equation that I posted here. How do I know what means? I know what it means because it's implied by its usage in the integral. That's how things are done in writing physics textbooks and physics journals. Weren't you aware of this?

That's to be understood because it has to be in order for the dimensions to be correct!

As far as the second equation, it's identical to the one you linked to so what are you claiming is wrong there since you said that the "formulas" are wrong? If its the volume element then why couldn't you just say that the volume had the wrong dimension or something. Why refer me to an entirely different page when you could simply tell me where the dimensional problem lie? Thanks.

20. Originally Posted by physicist
Originally Posted by Howard Roark
The above formulas are patently incorrect, the dimensions are wrong.
Nope. You're quite wrong my friend. The next time you run into this kind of situation it'd be better for you to ask the person what they meant in a formula rather than claiming that you know what they meant and claim they were wrong by what you assumed that they meant. That's obviously a bad way to communicate online.

Sorry, but my equations are exactly correct. In the first equation, i.e. , the terms under the integral is charge density times potential times the volume element .

is the established symbol for differential proper time.
There are other errors in your posts that you seem to have "missed".

21. Originally Posted by HowardRoark
is the established symbol for differential proper time.
I'm sorry to say this but I'm finding it difficult to accept that you actually believe this is true for all areas in physics such as electrodynamics. Are you actually going to tell me that in your entire life you've never seen a symbol that has more than one meaning which depends on the field of study?

It's a widely known fact throughout the field of physics that many symbols have more than one meaning, even within the same text. If I used V for volume would you have criticized me for saying that it was really potential or potential energy? If we're talking about quantum mechanics and I asked you what the symbol m stood for or what its units were, what would be your answer? Would you tell me that it was mass and the units were in kilograms or would you say that it was the magnetic quantum number and as such had no units? What does the term momentum mean? If we were talking about quantum mechanics then we'd be talking about canonical momentum whereas in classical mechanics we'd be talking about linear mechanical momentum. Where does it end?

Let me give you another, yet clearer, example from a text which explains this subject; from Classical Electrodynamics in a Nutshell by Anupam Garg, Princeton University Press, (2012). You can download the text from this website - Classical Electromagnetism in a Nutshell | Anupam Garg | digital library BookOS On page xxi the author writes
List of Symbols
...
The following is a list of the principle symbols used in this book, along with they key sections where their meanings are discussed or where they are first introduced. Many symbols do double and even triple duty -- they mean more than one thing. It is usually clear which meaning is intended from the context. In such cases, the table lists all the different usages. etc.
Once yet again, what a term means isdetermined by the context in which it’s used. Every skilled physicist knows this fact like the back of their hand. In many instances, authors of EMtextbooks will use for volume element such as in this case, i.e. Griffiths EM text. In this case it's a perfectexample. If you claim that he made a mistake then why not do what I always do in such instances. I contact the author so they can get the correction into the next edition of printing. If Griffiths agrees with thinks then I'm sure he'd appreciate you catching an error in his text since he also teaches special relativity in that text too. Imagine such a wonderful text by such a great author whose texts are used to teach those subjects at MIT having such mistakes in all four editions! Wow! If you'd like I can PM his e-mail address to you or I can contact the author himself and explain how a person in a forum found an error in his text. Which way do you prefer?

You shouldn't go around making such comments such as this because you're going to confuse people by leading them to believe that all symbols have one and only one meaning. That's not true at all. Every single physicist and mathematician knows this fact solid. In fact there simply aren't enough symbols for that to be the case. Here's another example; in general relativity represents an affine parameter. In electrodynamics it represents linear charge density. Then there are quantities which are found to be represented by two or more symbols such as kinetic energy. In Lagrangian mechanics and sometimes in relativity kinetic energy is given either the symbol T or K. In gravity or EM potential and potential energy is given the symbols U, V or .

Originally Posted by Howard Roark
There are other errors in your posts that you seem to have"missed".
What are these errors that you claim that I made and why didn't you state what they are but only claim it to be true?

22. Originally Posted by physicist

Originally Posted by Howard Roark
There are other errors in your posts that you seem to have"missed".
What are these errors that you claim that I made and why didn't you state what they are but only claim it to be true?
Click on the link, it will take you to the post where I corrected your errors.

23. Originally Posted by physicist
Introduction to Electrodynamics - Forth Edition by David Griffiths. The third edition is online at Introduction to Electrodynamics (3rd Edition) | David J. Griffiths; Reed College | digital library BookOS

Simply click on [Download (pdf, 23.58 MB)] and turn to page 93 and read Equation (2.43) and you'll see this equation
Thank you for the free textbook Physicist.

24. Originally Posted by Howard Roark
Originally Posted by physicist

Originally Posted by Howard Roark
There are other errors in your posts that you seem to have"missed".
What are these errors that you claim that I made and why didn't you state what they are but only claim it to be true?
Click on the link, it will take you to the post where I corrected your errors.
If I ignored your comment then I did so because it's wrong. When I get back from the store I'll recheck my work to see if I made the mistake of accidently switching U with V. The difference has always been confusing for me but I do recall checking my work each time I do this to make sure I don't screw it up.
However so far you seem to have the tendency to make claims that you are right and I'm wrong and that's that - In God's handwriting. In fact that's not been the case so far. We'll see when I get back home.

I'll assume that you now understand the concept of overloading symbols in physics equations now.

25. Originally Posted by physicist
Originally Posted by Howard Roark
Originally Posted by physicist

Originally Posted by Howard Roark
There are other errors in your posts that you seem to have"missed".
What are these errors that you claim that I made and why didn't you state what they are but only claim it to be true?
Click on the link, it will take you to the post where I corrected your errors.
If I ignored your comment then I did so because it's wrong.
My comment is correct, it is your math that is wrong.

26. Originally Posted by physicist
Classical Electrodynamics in a Nutshell by Anupam Garg, Princeton University Press, (2012).
And another one, bonus!
Thanks again Physicist!

27. Originally Posted by Howard Roark
My comment is correct, it is your math that is wrong.
You're quite wrong again here too. I demonstrated exactly what I asserted in no uncertain terms. I shouldn't have had to because what I said about symbols and their meaning is very well-known. All that you've done is made a claim and that's it. In no way are you right and my math is wrong unless I accidently switched U and V by accident. Period!

You really should do more that make assertions. You should back then up too. Especially when there's a disagreement about them. You should demonstrate that what you said is in fact true. Otherwise all you're doing is going from post to post and making claims.

28. Originally Posted by physicist
Originally Posted by Howard Roark
My comment is correct, it is your math that is wrong.
You're quite wrong again here too. I demonstrated exactly what I asserted in no uncertain terms.
Well, the errors you made are pretty basic, the fact that you refuse to admit and accept the corrections is quite surprising.

You really should do more that make assertions. You should back then up too.
I did, I posted the corrected formulas. All you need to do is to accept the corrections to your errors.

29. Originally Posted by Howard Roark

Originally Posted by physicist
Originally Posted by Robittybob1

U = GMm/r

The gravitational potential is gravitational energy per unit mass so

V = U/m = GM/r

which is the potential function for an object of mass M at the origin. We take the gradient to get

Where g is the gravitational field. The gravitational field is gravitational force per unit mass. To see what this means notice that the force on a particle of mass m by an object at the origin whose mass is M is given by

F = -GMmer/r2

where er is a unit vector pointing from the object to the particle, i.e. e = r/r. So the gravitational field at r is the force per unit mass, i.e.

g = F/m = -GMer/r2

As you can see this is the acceleration of a particle in a gravitational field at point r. It can be shown that

I hope that helps.
g = - grad U is wrong

g = - grad V is the correct expression
It is very simple, really:

So, contrary to your claims, you cannot have

On the other hand, as I corrected your error:

And that produces the correct answer:

30. Is there no one around to arbitrate? I'm no use in this matter?

31. Originally Posted by Howard Roark
Well, the errors you made are pretty basic, the fact that you refuse to admit and accept the corrections is quite surprising.
That is utter nonsense. I just saw them a minute or so before I left the house and went shopping and I just got back home. Now I need to go back and see whether I'm right or wrong. There's no refusing of me to admit any mistakes. When I check it and post again that will be it.

I did, I posted the corrected formulas. All you need to do is to accept the corrections to your errors.
Only if you're right or not. As I said before. For some odd reason the difference between Grav pot energy and grav pot has always confused me. I don't know why. Let me check it out. You need to be more patient than you are now.

32. Originally Posted by physicist
Originally Posted by Howard Roark
Well, the errors you made are pretty basic, the fact that you refuse to admit and accept the corrections is quite surprising.
That is utter nonsense. I just saw them a minute or so before I left the house and went shopping and I just got back home. Now I need to go back and see whether I'm right or wrong. There's no refusing of me to admit any mistakes. When I check it and post again that will be it.

I did, I posted the corrected formulas. All you need to do is to accept the corrections to your errors.
Only if you're right or not. As I said before. For some odd reason the difference between Grav pot energy and grav pot has always confused me. I don't know why. Let me check it out. You need to be more patient than you are now.
I have been very patient, I explained your errors to you three times already.

33. Originally Posted by Howard Roark
Well, the errors you made are pretty basic, the fact that you refuse to admit and accept the corrections is quite surprising.
That is utter nonsense. I just saw them a minute or so before I left the house and went shopping and I just got back home. Now I need to go back and see whether I'm right or wrong. There's no refusing of me to admit any mistakes. When I check it and post again that will be it.

I did, I posted the corrected formulas. All you need to do is to accept the corrections to your errors.
Only if you're right or not. As I said before. For some odd reason the difference between Grav pot energy and grav pot has always confused me. I don't know why. Let me check it out. You need to be more patient than you are now. It's not as if you're the most important thing in my life right now. I'm always very careful when checking equations. I'm slow in this respect. For some physicists that's the way it is, i.e. we're slow. But we're very good in the end. So please be patient please. Sheesh!

34. Originally Posted by physicist
Originally Posted by Howard Roark
Well, the errors you made are pretty basic, the fact that you refuse to admit and accept the corrections is quite surprising.
For some physicists that's the way it is, i.e. we're slow. But we're very good in the end.
Yes, you have been very slow admitting your errors. I agree with this part of your statement.

35. Originally Posted by HowardRoark
This is trolling. Please don't do it. Also please don't claim I made such statements. I said that I'm slow studying, not admitting my mistakes. I'm very well known for admittingmy mistakes by my friends.

When I finally sat down to dinner I stared studying. It all cameback to me in a lovely rush! I love it when that happens.

Let there be a gravitational field created by a mass M at the origin of thecoordinate system in the inertial frame S. The potential energy of a point mass m at a distance r is U = GMm/r. If the point mass M is held fixed at the origin then the force exerted on by the body at the origin is found to be

The gravitational acceleration of the point mass is then given by

Then upon dividing my m gives

where V = U/m. Therefore

See? As soon as I saw my error I thoroughly made the correction to it. Period. I didn't hold back or wait. This is the soonest I was able to do it. I had a delay due to a bout of my chronic pain syndrome. That's indeed a horrible thing to have to deal with and extremely hard to focus with.

36. Originally Posted by physicist
Originally Posted by HowardRoark
This is trolling.

I simply pointed out that it took you very long to admit to your errors.

37. Originally Posted by Howard Roark
Originally Posted by physicist
Originally Posted by HowardRoark
This is trolling.

I simply pointed out that it took you very long to admit to your errors.

I keep telling you that after you made your assertion that I was not 100% certain of the actual result until I sat down to dinner and worked it out while I was having dinner. I know you gave a calculation but I trust nothing unless I do it myself. And the moment I came to that conclusion I posted my retraction. Yet you are once again claiming ... it took you very long to admit to your errors. Please stop doing this? I'm not a liar. Everything I post here is true.

Perhaps you think I'm a liar and that I knew you were right all along but refused to admit it earlier. If so then what evidence to you have to support such an accusation?