B Understanding the Concepts and Formulas for Energy: Potential and Kinetic

[Avalon_18]

While I was doing some practice questions it hit me "what is energy". I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

 

[BvU]

Energy is a measure for work: if you do some work to move a book from the bottom shelf to the top shelf, the difference in potential energy of the book is the work you have done. Establishing a zero point for energy is awkward because in the real world all we have is energy differences.

 

[Merlin3189]

An interesting question to which I do not know the answer! My recollection is that I first thought about energy when learning about Joule and work-heat equivalence. But maybe I'd met KE before and not thought much about it.

If you do have a concept of KE, then PE seems obvious. When an object falls, it gains speed and hence KE due to the acceleration caused by gravity. So when an object is high up, it has the potential to gain KE by falling. Or if an object has speed, KE, then it has the potential to rise up against gravity.

Quantitatively the change in KE with height when moving in a gravitational field works out at mgh, so that becomes the PE.

 

[PeroK]

While I was doing some practice questions it hit me "what is energy". I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

There are many videos online on the subject "what is energy?". Perhaps what they all have in common is that it is not, in general. easy to define. One simple approach is to leave it essentially undefined, but list the different forms it can take: kinetic, potential, chemical, thermal, nuclear, radiation.

 

[Dale]

While I was doing some practice questions it hit me "what is energy". I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

There are several equally valid definitions used in different branches of physics. In Newtonian mechanics energy is defined as the capacity to do work. In thermodynamics an approach like @PeroK is used, energy is defined as kinetic energy ($\frac{1}{2}mv^2$) and anything that can be converted to kinetic energy. In Lagrangian mechanics it is defined as the conserved quantity associated with time translation symmetry of the Lagrangian.

 

[BeedS]

Do all forms of energy manifest as motion?

 

[PeroK]

Do all forms of energy manifest as motion?

Can you think of any that don't? See the list in post #4.

 

[BeedS]

Can you think of any that don't? See the list in post #4.

No
kinetic = Motion
potential = potential to move
chemical = Motion caused by interactions of atoms / particles
thermal = Motion of heat energy
nuclear = motion of particles and subatomic
radiation = Motion of photons

 

[PeroK]

No
kinetic = Motion
potential = potential to move
chemical = Motion caused by interactions of atoms / particles
thermal = Motion of heat energy
nuclear = motion of particles and subatomic
radiation = Motion of photons

I might question whether potential, chemical and nuclear are directly "motion".

 

[Dale]

Do all forms of energy manifest as motion?

No. KE and thermal energy of an ideal gas are motion. Thermal energy of other materials would have some of its energy in internal degrees of freedom corresponding to motion, but could also have non motion internal degrees of freedom if available. Most other forms of energy are not necessarily manifested as motion, although of course they can all be converted into KE. For example, electromagnetic energy can move (radiation) but you can also have static electromagnetic fields which hold energy without moving.

 

[jtbell]

While I was doing some practice questions it hit me "what is energy". I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

Does your textbook say anything about this stuff? It would help if we can use it as a starting point for discussion. Is this a grade-school or university textbook? The university level textbooks that I've used all make at least some attempt at this.

 

[A.T.]

While I was doing some practice questions it hit me "what is energy".

A quantity that is conserved in an isolated system over time.

I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

All the formulas for energy where defined such that the total energy is conserved in an isolated system over time.

 

[jbriggs444]

Here is Feynman's explanation: http://www.feynmanlectures.caltech.edu/I_04.html

 

[BeedS]

although of course they can all be converted into KE

That is when work is being done, something is converted to KE. What work doesn't involve moving something?

For example, electromagnetic energy can move (radiation) but you can also have static electromagnetic fields which hold energy without moving.

No work is being done with the static field (PE), when work is done the field will move(KE) the charge or whatever is being moved/worked/energized?

 

[Dale]

What work doesn't involve moving something?

None. $dW=F\cdot dx$

No work is being done with the static field

Sure, but it is still energy. The question was “do all forms of energy manifest as motion” not “do all forms of work manifest as motion”. Work and energy are closely related, but they are not the same thing.

 

[BeedS]

Sure, but it is still energy.

Yes in a stable configuration.

The question was “do all forms of energy manifest as motion” not “do all forms of work manifest as motion”. Work and energy are closely related, but they are not the same thing.

So all forms of work manifest as motion? Judging by your "None. dW=F⋅dx" reply, then yes?
And I need to look into my understanding of work and energy.
Thanks

 

[Dale]

So all forms of work manifest as motion? Judging by your "None. dW=F⋅dx" reply, then yes?

Yes, the dx term is motion of the object on which work is done.

 

[Twodogs]

Good question. I think you should hang on to it for a while. We can measure energy with incredible precision. We can describe its permutations with strict accounting. I don't know that we can say 'what' it is for the time being.

 

[Dale]

I don't know that we can say 'what' it is for the time being.

Sure we can. I gave three definitions for what it is in post 5.

 

[Twodogs]

Sure we can. I gave three definitions for what it is in post 5.

truth -- the quality or state of being true.
• that which is true or in accordance with fact or reality.
• a fact or belief that is accepted as true.

Certainly, you give a workable definition. Regards.

 

[BeedS]

And I need to look into my understanding of work and energy.

Energy = current motion (Since everything is moving, energy is the current motion caused by previous work done)
Work = change of motion (result of changing a objects energy through KE)

Is this better?

 

[A.T.]

Energy = current motion

What about potential energy?

 

[lomidrevo]

What about potential energy?

And we should not forget about the rest energy $E = m c^2$. Quite a good example of a energy not associated with a particle's motion.

 

[BeedS]

What about potential energy?

The object with potential is still created by things moving internally(sum of the objects scaled energy(its mass)). PE is potential motion, the PE object is still moving (KE) and has a potential to change that motion through work done.

 

[A.T.]

the PE object is still moving (KE)

What about two opposite charges at rest? They have PE without moving.

 

[BeedS]

What about two opposite charges at rest? They have PE without moving.

At rest relative to what? Only to the object itself and something with the same motion as the object?. The charges are still moving with the charged objects. The charges are considered part of the objects mass (PE)? Charges do contain mass? Charges do move with the object they are bound to?

 

[jbriggs444]

At rest relative to what?

Yes, every object that is at rest in one frame is moving in another. It does not follow that energy is motion or that rest is a meaningless concept.

Edit: You do realize that energy is a frame-relative concept? If we are talking about energy being present in an object at rest, we are working in a frame of reference within which the energy is non-zero and and the object is at rest.

 

[A.T.]

At rest relative to what?

Relative to the reference frame in which you calculate the energy.

 

[folkethefat]

What is mass?
The dimension of time might be related to energy?

In E=MC2, what does it mean? The energy bound in mass?

 

[lomidrevo]

What is mass?

In relativity, it is a property of an object - a quantity which is invariant for all observers. But mass can be defined in different ways too: https://en.wikipedia.org/wiki/Mass

The dimension of time might be related to energy?

No idea what do you mean by your question.

In E=MC2, what does it mean? The energy bound in mass?

I think this short video provides a nice introductory explanation:

 

[Kavorka]

It really is just easiest to think about it as "the capacity to do work". Essentially, work is measured in terms of energy (work and energy have the same units). This is because energy by itself doesn't mean much, it is in reference to what is it able to physically do that matters. For example, it means little to say you have a certain amount of potential energy relative to 20 feet underground below you. You certainly could possesses this capacity to do work, but it is meaningless in the context of the physical situation - you are not able to fall through the ground. Because energy can only be transformed and not destroyed, you can think of any context of energy in terms of this definition.

 

[A.T.]

It really is just easiest to think about it as "the capacity to do work".

Having energy around doesn't mean you can do work. When you dissipate energy, there is still the same amount of energy in the from of heat, but it cannot by itself be used to do work.

 

[folkethefat]

Is this correct then?
Time is measured by the ordering of events. "Events" always involves a change in energy. Energy is driving the change. So energy seems to be driving that which we measure as time?

 

[jbriggs444]

Is this correct then?
Time is measured by the ordering of events. "Events" always involves a change in energy. Energy is driving the change. So energy seems to be driving that which we measure as time?

I do not agree. An "event" is simply a location in four dimensional space-time. It does not involve a change in energy. How could it, since "energy" is a conserved quantity?

In particular, energy is the conserved quantity associated with time translation symmetry by Noether's theorem.

 

[Herman Trivilino]

While I was doing some practice questions it hit me "what is energy".

There's no definition that will help when the concept is being refined and used in different contexts. After a while the different contexts provide you with enough experience that you feel comfortable with the concept of energy. Like how we feel comfortable with the concept of money.

I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

Any college-level introductory physics textbook would have good explanations of those things.

 

[BvU]

I also don't understand how the formulas for potential and kinetic energy were derived. Was it due to their definitions or is there some reason behind their respective formulas.

Perhaps it's nice to add that in real life the concept of 'force' is much more vivid. Force has something to do with to the change in energy

Some force fields are conservative, hence the introduction of a potential (energy) with somehing like $\Delta {\rm PE} = \vec F \cdot \Delta \vec x\$.
From $$ \vec F = {\Delta\vec p\over \Delta t} \Rightarrow \quad \vec F \cdot \Delta \vec x= m{\Delta\vec v\over \Delta t} \cdot \Delta\vec x = m\;\Delta\vec v \cdot {\Delta\vec x\over \Delta t} = m\;\Delta\vec v \cdot \vec v = \Delta \left ( {1\over 2 }m \vec v^2 \right ) $$ comes the formula for kinetic energy.

It's all differentiation and integration -- which makes math so useful !

[edit] hehe if you do it right

 

[folkethefat]

I do not agree. An "event" is simply a location in four dimensional space-time. It does not involve a change in energy. How could it, since "energy" is a conserved quantity?

In particular, energy is the conserved quantity associated with time translation symmetry by Noether's theorem.

Ok! i see, i was confused about the definition of "event" in a physics context. I thougt it implied a change in the energycontent in a location in space, a defined volume.

 

[folkethefat]

There's no definition that will help when the concept is being refined and used in different contexts. After a while the different contexts provide you with enough experience that you feel comfortable with the concept of energy. Like how we feel comfortable with the concept of money. /

But to ask what "money is" except that its nice to have lots of, , is a legitimate question and there are many theoretical answers for it.
The question about if energy is something more fundamental than the different types of energy, leads beyond the standard model and that's the problem. I Think?

 

[sophiecentaur]

I do not agree. An "event" is simply a location in four dimensional space-time. It does not involve a change in energy.

Strictly, yes perhaps but an 'event' is a word used colloquially to describe 'something happening' and that does imply change because that sort of event has to be observable - which involves some change of Energy state.
I used to use the form of words "Energy is needed for anything to happen" in lower school introductions to Science. It was interesting to note that 'they' would always try to argue with that and to bring up examples that didn't fit. I always managed to explain away their examples. 'Death' would always be one of them.

 

[folkethefat]

Can all sorts/types of energy translate into some other kind? Heat for example.
Is there a hierarchy in this "translation"? Is heat the lowest form or state of energy?

 

[sophiecentaur]

Can all sorts/types of energy translate into some other kind? Heat for example.
Is there a hierarchy in this "translation"? Is heat the lowest form or state of energy?

We could turn this thread into a very practical discussion about Energy - which is quite valid, I think.
Heat Engines (the general term for a device to obtain mechanical work from the flow of heat) work best (higher efficiency) between large temperature differences. Once the work has been 'extracted' from a mass of (say) steam and the steam has condensed, you can sill get some work out of it by using a cold sink at an even lower temperature than 100C BUT NOT MUCH. So you could say that the energy in the condensed steam is a 'lower grade' than that of the superheated steam that is supplied by the original boiler.
The hot water would still be 'useful' for heating but not so much for doing work.
It's basically down hill all the way, though.
Your suggested "hierarchy" is ok as an arm waving term and if you want to talk in terms of relative usefulness. Considerations like possibilities of storing for the various forms of energy are very relevant sometimes. You could perhaps put Mechanical Potential Energy at the top because you can store large amounts of water at height or use a wound spring to store useful energy.

 

[A.T.]

Can all sorts/types of energy translate into some other kind?

Transform spontaneously or be transformed by doing work? Look up:
https://en.wikipedia.org/wiki/Second_law_of_thermodynamics

 

[Merlin3189]

If you think of energy as an accounting term, then all types of energy are equivalent. The energy simply tells you the conversion value, not whether and how a conversion process happens.
You suggest heat may be a low form of energy, but in a power station we use it to generate electricity, a more noble form. We can use that to drive motors and raise objects to give them potential energy, to charge batteries and give them chemical potential energy, produce light and other electromagnetic energy. I don't know if we can put nuclear energy back in the bottle, but I guess if we could, it might involve using large amounts of electrical energy in particle accelerators.

The thing about heat is that it is the kinetic energy of random motions of molecules, lots of them. If two molecules collide, it is normally perfectly elastic, energy is conserved and energy can be transferred from the slower to the faster or vice versa. But it is more likely for energy to be transferred from the faster to the slower. Statistically, when there are large numbers of molecules, energy gets shared out. Energy could not be concentrated in a small number of very fast molecules, because they would very quickly collide with the large number of very slow ones and lose some energy. These ideas lead us down the road of entropy and thermodynamics. Heat doesn't travel from a colder to a hotter (by conduction.) That is probably what you have in mind.

 

[stevieboy]

energy = mass x the square of the fastest velocity in the universe
the square of something is 2 dimensional
times a mass which is described as the 3rd dimension pretruding down from a 2d platform
so is energy not basically a cube or volume?
do we live in the 2d world?

 

[Dale]

is energy not basically a cube or volume?

In geometrized units speed is dimensionless, and the square of speed is therefore also dimensionless. So energy has geometrized dimensions of length, not length cubed.

 

[danachiraq]

Derivation for the equation of Potenstial Energy:
Let the work done on the object against gravity = W
Work done, W = force × displacement
Work done, W = mg × h
Work done, W = mgh

Since workdone on the object is equal to mgh, an energy equal to mgh units is gained by the object . This is the potential energy (Ep) of the object.
Ep = mgh

Derivation for the equation of Kinetic Energy:
The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is
v2 - u2 = 2aS
This gives
S = v 2 - u 2 2a

We know F = ma. Thus using above equations, we can write the workdone by the force, F as

W = ma × v 2 - u 2 2a
or
W = 1 2 m( v 2 - u 2 )If object is starting from its stationary position, that is, u = 0, then
W = 1 2 m v 2

It is clear that the work done is equal to the change in the kinetic energy of an object.

If u = 0, the work done will be W = 1 2 m v 2 .

Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is Ek = ½ mv2