Potential Energy and Conservative Force Question

[e(ho0n3]

Hello everyone,

I'm a little confused on how potential energy is related to a conservative force. Say some system has potential energy U. There is a relation stating that
$$\vec{F} = \nabla U$$
I understand the F is some conservative force, but does it represent the net conservative force acting on the system or on an object within the system or what?

e(ho0n3

 

[Doc Al]

I'm a little confused on how potential energy is related to a conservative force. Say some system has potential energy U. There is a relation stating that
$$\vec{F} = \nabla U$$

That should be:
$$\vec{F} = - \nabla U$$

I understand the F is some conservative force, but does it represent the net conservative force acting on the system or on an object within the system or what?

That conservative force is the force associated with that potential energy. Given the potential energy function you can calculate the force as above. For example, gravitational PE between two masses is:
$$GPE = - G\frac{m_1m_2}{r}$$
thus the gravitational force associated with this potential energy is:
$$F = - G\frac{m_1m_2}{r^2}$$

Does this get at your question at all? If not, ask again.

 

[e(ho0n3]

Still Confused

That should be:
$$\vec{F} = - \nabla U$$

Right.

That conservative force is the force associated with that potential energy. Given the potential energy function you can calculate the force as above. For example, gravitational PE between two masses is:
$$GPE = - G\frac{m_1m_2}{r}$$
thus the gravitational force associated with this potential energy is:
$$F = - G\frac{m_1m_2}{r^2}$$

Does this get at your question at all? If not, ask again.

But where is F acting. I guess in this simple case, F can act on either one of the masses since it's all the same. Let me make a more concrete example. Suppose we have a system of three particles with potential energy

$$U = -G\Big(\frac{m_1m_2}{r_{12}} + \frac{m_1m_3}{r_{13}} + \frac{m_2m_3}{r_{23}}\Big)$$​

You can calculate the force associated with this potential energy, but what does the force represent?

e(ho0n3

 

[Doc Al]

To find the force on m3, for example, you start with the PE of m3. Don't include the m1m2 term.

Interesting question, though. There is probably a subtlety that I am missing.

 

[remcook]

But where is F acting.

THe potential is defined everywhere (except the singularities). The forces at any point that arise form a force field. So it depends on the point in which you want to know the energy and force (force changes with position).

does it represent the net conservative force acting on the system or on an object within the system or what?

the force at a certain point within the system (of an object)

 

[Doc Al]

If you start with the potential (not potential energy), then you can calculate the field at any point. In our 3 body gravity example, if we want to calculate the force on m3, we need to find the potential at m3. As remcook notes you must exclude m3 to avoid the singularity. Then you can calculate the field, then the force on m3.

Try it and you'll see that the methods are equivalent.

 

[e(ho0n3]

THe potential is defined everywhere (except the singularities). The forces at any point that arise form a force field. So it depends on the point in which you want to know the energy and force (force changes with position).

OK. So the force in the system I described is the net force that will act on an external object within the system given that this external object is not located where the masses are. In other words, F is a vector field correspoding to the force of gravity produced by the three masses.

Am I right?
e(ho0n3

 

[remcook]

looks about right. But the third mass (for which the force holds then) is attracted by only two other bodies.

So I would also like to say:

Don't include the m1m2 term.