# Potential Gravitational Energy into Force

1. May 28, 2013

### AlexVGheo

How do you derive the equation for gravitational force from the eqauation for potential gravitational energy: e=-Gm/x? For me it seems as though you take the derivative and that would work, but i don't understand why it does?

2. May 29, 2013

### Simon Bridge

Welcome to PF;

Gravity is a conservative force - which is why the force is the gradient of a potential.
Its a definition.

3. May 29, 2013

### AlexVGheo

Aha, is that also true for electric potential then?

4. May 29, 2013

### sudu.ghonge

For static cases, yes.

5. May 29, 2013

### WannabeNewton

It is true for any conservative force. If you accept the rather intuitive definition that a conservative force is one whose associated work function is path independent, then you can construct a well defined potential energy function for a conservative force using the work-energy theorem: $W = -\Delta U = U(a) - U(x) = \int _{a}^{x}F(x')dx'$. Now differentiate both sides and apply the fundamental theorem of calculus to obtain $F(x) = -\frac{\mathrm{d} U}{\mathrm{d} x}$. This can be trivially generalized to higher dimensions.

As such, you can always define an electric potential energy for electrostatic forces (it won't work for electrodynamic systems because the path independence will fail). There is, however, a way to get around this that you will learn about at some point or another in your physics education.

6. May 29, 2013

### Simon Bridge

... and you have your answer for the most likely interpretation :)
But your question is not really clear: is what also true? - are you asking if the static electric force on a charge is proportional to the gradient of the electric potential, if this is a definition for electric potential, or something else? There are quite a few ways that question finishes from the context.

I didn't pick it up before but post #1 says "potential energy" but the equation given is for potential - these are different, though related, concepts.

A lot of confusion can stem from just not being careful enough to say what you mean - it risks accidentally conflating mismatched ideas.

7. May 30, 2013

### AlexVGheo

Ok, so what is the difference between potential and potential energy? and what is the difference between pe =mgh and pe=-GMm/r?

8. May 30, 2013

### Simon Bridge

9. May 30, 2013

### BruceW

also, potential energy is a form of energy. potential is not, but is often related. for example, gravitational potential is like an energy per mass. and analogously, in electromagnetism, the electric potential is like an energy per charge.

10. May 31, 2013

### AlexVGheo

Thank you for the ilink, it was somewhat enlightening I think I understand the relationship between the forces and potential energy a bit better now =) but i still don't understand the difference beween potential and potential energy? For example if I lift up an object of mass m to a hight h then the potential energy is E = mgh so what is the potential?
And I don't understand the approximation, isn't it the job of a formula to be representative of the true value?
Also i am finding this all very interesting, unfortunately I am used to dealing in forces and so I don't know much about energy, is there a book you know of which can explain how to use energy instead of force?

11. May 31, 2013

### Simon Bridge

The potential energy is the work needed to get a mass to that height ... thus W=Fd=mgh.
The potential is the work per unit mass, so that would be W/m = gh

The strength of the gravitational field at a position is the acceleration due to gravity at that position - which is the negative gradient of the potential ... which is just -g.
The force is the negative gradient of the potential energy ... F=-mg.
... and this is equal to the rate of change of momentum.

It can take a while to get used to dealing with energy ... pretty much any physics text book should have something on it under "conservation of energy".

12. May 22, 2015

### Sun E Man

Alex for some practical understanding of potential energy look up calculations of Hydro Power.