# A few questions about Potential Energy

• Kaushik
In summary: Thanks for the link!I would say the phrase "conservative force" is just another word for the case that the force is only dependent on position variables and is given as the gradient of a potential,$$\vec{F}=-\vec{\nabla} U(\vec{x}).$$Of course for energy conservation to hold this is only a sufficient but not a necessary condition. An example is the force on a moving charge in a magnetic field, $$\vec{F}=\frac{q}{c} \vec{v} \times \vec{B}(t,\vec{x}).$$for which also energy conservation holds (here energy being

#### Kaushik

Is it possible to briefly explain the potential energy concept?
• Why is potential energy only associated with conservative forces?
• Does potential energy really exist? Or Is it just kinetic energy from different reference frame?

Kaushik said:
Summary: I want to know about potential energy.

Is it possible to briefly explain the potential energy concept?
• Why is potential energy only associated with conservative forces?
• Does potential energy really exist? Or Is it just kinetic energy from different reference frame?

This is too broad a question for PF. You need to find a good reference, textbook or online.

To answer part of your question: PE exists in the sense that it can always be converted back into KE; and, it isn't just KE from a different reference frame.

Kaushik
Kaushik said:
Why is potential energy only associated with conservative forces?
That is just a matter of definition. Any force which is associated with a potential energy is called a conservative force.
Kaushik said:
Does potential energy really exist? Or Is it just kinetic energy from different reference frame?
Potential energy is not just kinetic energy from a different frame.

Kaushik
Dale said:
That is just a matter of definition.
Can you please define it for me?

Kaushik said:
Can you please define it for me?
A conservative force is any force such that ##F=-\nabla \phi## where ##\phi## is a scalar field called the potential.

jbriggs444
Kaushik said:
Can you please define it for me?

There is lots of reference material online about conservative forces and their properties. Which property you choose to define "conservative" is a matter of taste.

For example:

https://en.wikipedia.org/wiki/Conservative_force

Kaushik
I have a small doubt.
'All conservative force must be a function of position only and not of velocity or time'
Is this true? If yes, why?

Kaushik said:
I have a small doubt.
'All conservative force must be a function of position only and not of velocity or time'
Is this true? If yes, why?

Maybe that's something you could work out for yourself.

Kaushik
Kaushik said:
I have a small doubt.
'All conservative force must be a function of position only and not of velocity or time'
Is this true? If yes, why?
This is a hard one to work out for yourself. For a force to be conservative it is necessary that ##\frac{\partial}{\partial t}\phi=0## but it is not necessary that ##\frac{d}{dt}\phi=0##.

Kaushik
Dale said:
This is a hard one to work out for yourself. For a force to be conservative it is necessary that ##\frac{\partial}{\partial t}\phi=0## but it is not necessary that ##\frac{d}{dt}\phi=0##.
Can I get any link so that I can get to know more about it? I searched but I couldn't find any. It would be nice if you can help me .

Kaushik said:
Can I get any link so that I can get to know more about it? I searched but I couldn't find any. It would be nice if you can help me .
See the posts by @vanhees71 in this thread: https://www.physicsforums.com/threa...-constraints-mean-energy-conservation.816591/

Also, this seemed good: https://www.chm.uri.edu/dfreeman/chm531_pfizer_2009/cm.pdf but most books that talk about Lagrangian and Hamiltonian mechanics should have at least a brief mention. Even wikipedia: https://en.wikipedia.org/wiki/Lagrangian_mechanics

Kaushik said:
Why is potential energy only associated with conservative forces?

For the potential energy to be defined at a point it's necessary that the work done by the force around a closed loop be zero.

Kaushik said:
Can I get any link so that I can get to know more about it? I searched but I couldn't find any. It would be nice if you can help me .

Are you in a calculus-based college-level physics course, or is it a non-calculus course? Any college-level introductory physics textbook will do. Here's a link to a few, free provided you download the PDF.

https://openstax.org/subjects/science

Kaushik
Mister T said:
For the potential energy to be defined at a point it's necessary that the work done by the force around a closed loop be zero.
Are you in a calculus-based college-level physics course, or is it a non-calculus course? Any college-level introductory physics textbook will do. Here's a link to a few, free provided you download the PDF.

https://openstax.org/subjects/science

I would say the phrase "conservative force" is just another word for the case that the force is only dependent on position variables and is given as the gradient of a potential,
$$\vec{F}=-\vec{\nabla} U(\vec{x}).$$

Of course for energy conservation to hold this is only a sufficient but not a necessary condition. An example is the force on a moving charge in a magnetic field,
$$\vec{F}=\frac{q}{c} \vec{v} \times \vec{B}(t,\vec{x}).$$
for which also energy conservation holds (here energy being the kinetic energy only). Though (kinetic) energy is conserved here, one doesn't call this force "conservative", because it's not of the type described by this phrase.

Kaushik and PeroK
Is ##\Delta K.E + \Delta G.P.E + \Delta E.P.E = W_{ncf}## ,where 'ncf' stands for non conservative force?

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Kaushik said:
Is ##\Delta K.E + \Delta G.P.E + \Delta E.P.E = W_{ncf}## ,where 'ncf' stands for non conservative force?
With minor caveats, it is correct yes. You've split up all the forces acting on an object into three categories.

1. Gravitational force. Associated with a potential named "G.P.E."
2. Other forces that have associated potentials. Associated with an aggregate potential named "E.P.E."
3. Other forces not associated with potentials.

The equation comes, of course, from the work-energy theorem: ##\Delta K.E. = W = \Sigma F\cdot d##

You've simply taken the work from gravity and from all the other conservative forces and moved the associated terms over to the energy side of the equation as potentials. Since the potentials are defined in terms of the work done over a path, this is a perfectly valid thing to do.

Minor caveats:

The gravitational field has to be static. No gravitational slingshots.

If the object upon which work is being done is extended and is either non-rigid or is rotating then we need to compute the work done on the object by considering all external forces as acting on its center of mass. We need to compute the resulting kinetic energy based on total mass and the motion of the center of mass only. (i.e. we need to use center-of-mass work and bulk kinetic energy).

Kaushik