General Definition of Potential Energy - Conceptual Help

[Irfan Nafi]

My textbook states that an alternative definition of the change in potential energy is the work required of an external force to move an object without acceleration between two points.

I am confused on why it says acceleration. Wouldn't that mean that the acceleration is 0 and therefore the work done is zero? Thanks for the help.

 

[jambaugh]

The point about the acceleration is that you don't want to change the kinetic energy. All the work done then goes into the potential energy.

Example: raising a weight against gravity at a constant speed. Extending a spring at a constant speed.

Work done will be force time distance moved in the direction of that force. This can be done at a constant rate (speed) so there's no acceleration.

The only problem with this definition is that you could misapply it to frictional force. Moving a block across a frictional surface without acceleration will also require an external force and you don't change potential energy. You are rather converting work to heat.

 

[kuruman]

My textbook states that an alternative definition of the change in potential energy is the work required of an external force to move an object without acceleration between two points.

I should add that your textbook definition is incomplete if not incorrect and should have mentioned conservative forces. The change in potential energy is the negative of the work done by an external (conservative) force as the object is moved without acceleration between two points. For example, if you raise a mass straight up at constant speed (zero acceleration), the book gains gravitational potential energy, i.e. the change in potential energy is positive, but the work done by (the conservative force of) gravity is negative. The work done by your hand is positive in this case, however the force exerted by your hand is not conservative and there is no change in potential energy associated with it. So, other than the negative sign issue, as @jambaugh pointed out, your textbook definition allows room for confusion by not excluding non-conservative forces such as friction.

Regarding your initial question, if the acceleration is zero, only the work done by the net force is zero. There could be two forces acting on the object (see example of hand raising book) at constant speed, one doing non-zero positive work and the other doing non-zero negative work of equal magnitude so that their sum is zero.

 

[jbriggs444]

The change in potential energy is the negative of the work done by an external (conservative) force as the object is moved without acceleration between two points.

As I read the textbook definition, the "external" force is the non-conservative force (e.g. a hand) that is moving an object against the resistance of whatever conservative force one is considering to have an associated potential. The potential is the work done by this external (not-necessarily-conservative) force as the test object is moved from the reference point to the point where potential is to be determined.

i.e. we are in complete agreement except for assignment of names to forces.

 

[jambaugh]

I started to add further thoughts earlier and will add them now. Here's a way of thinking about potential energy.

Start with only the kinetic energy and consider the class of systems where kinetic energy is conserved as the system evolves over time. You can classify such cases as "free systems". Examples are "Newtons' particle in uniform motion in the absence of forces" but also such examples as rotary motion or complex systems of massive gears and lever arms but always rigid bodies in motion without springs or such.

Now consider systems where this is possibly not the case and define a larger class of systems where kinetic energy is not conserved but in which you can define an extended energy as kinetic energy plus some function of the configuration parameters of the system and call it total energy = kinetic energy + "other energy". For those systems where this total energy is conserved as the system evolves over time we call such isolated systems.

If the dynamics of an isolated system is time reversible then we can call the "other energy", potential energy. If there are irreversible components to the dynamics then we say the system is "dissipative" and try to describe some sort of entropy function.

I'm not sure this is quite correct but lately I've been thinking that "potential energy" is mostly a "fudge factor" we add to make energy stay conservative where possible.

I sometimes think of quantum tunneling as being the fact that the described potential is an expectation value for particle interactions and the tunneling particle has a certain probability of sneaking through the "barrier". It's like dodging raindrops as one runs from car to house. You can quantify the expected amount of water you pick up per distance traveled and draw a nice smooth graph called say "wetness potential" but there's always the slim chance you can just happen to get missed by each raindrop and there's no mystery in that. Likewise with tunneling across a potential barrier if you view the potential energy as an almost always true convention rather than a hard reality.

Maybe a bit off the wall? Certainly a serious tangent to the OP topic. Shall I start a thread on it?