Conceptual derivation of (classical mechanical) energy?

[Cleonis]

I think to really appreciate what energy is, you have to go beyond the approximation that is Newtonian physics and turn to relativity. From that vantage point, one can see that matter and energy are on equal par, as you say. Under Newton, matter and energy are two very different things, and one must view energy as essentially a fallout of the math under that approximation to "reality".

I guess I never gave that much thought. I am accustomed to the relativistic view on energy, and I guess I just brought that to bear on the Newtonian framework.

It may be that strictly within the theoretical framework of Newtonian theory it's possible to view energy as only a bookkeeping device, not atttributing physical reality to it - I don't know. Still, there is the empirical finding that we see conservation of energy. I think that that in itself is suggestive that energy is part of the physical world, independent of any theory we formulate.

 

[RoyLB]

You might enjoy "The Feynman Lectures on Physics," V1, Sect. 4-1: "What is energy". Herewith a couple of quotes: "...there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes ... the energy has a large number of different forms, and there is a formula for each one ... we have no knowledge of what energy is ..."

GRDixon,

I do indeed enjoy the Lectures and Feynman's other books. I was very disappointed that the great Feynman gave up when it came to explaining energy He explained quantum electrodynamics to the layman, but he couldn't explain energy to Caltech freshmen! IIRC, he promised in the lectures to derive the formula T=1/2mv^2, but I don't think he ever did

- Roy

 

[Studiot]

Isn't all this esoteric discussion is rather OTT for control engineering?
Does relativity have any relevance? (smile)

Here is an excerpt from my reference to whet you appetite for practical applied science.

…..The meaning of the word particle in the laws must be studied first. A particle is often said to be a point mass with no spatial extent. Atomic nuclei and electrons might be thought of as particles of this type, but Newton’s laws are not intended to apply to such small scale phenomena; usually quantum mechanics must be used instead. In classical mechanics the smallest piece of matter we need to consider contains enormous numbers of atoms and on this scale we can ignore atomic structure and think of matter as continuous.
Accordingly, we define a particle to be a material body whose dimensions, though not zero, are sufficiently small for the internal structure of the particle to be unimportant. The actual size permissible depends upon the particular physical problem. Thus the Earth may be treated as a single particle for the discussion of its movement around the sun, but a grain of sand cannot be treated as one in the formation of a sand dune. For our purposes the essential feature of a particle is that its position is sufficiently described by asingle vector r, the position vector from some origin

Glauert develops a similar argument to that presented here by Cleonis from this one fact right up to a complete derivation of total mechanical energy.

 

[RoyLB]

Studiot

Isn't all this esoteric discussion is rather OTT for control engineering?
Does relativity have any relevance? (smile)

For control engineering, I'd agree. For my own knowledge, its all relevant

Here is an excerpt from my reference to whet you appetite for practical applied science.



Glauert develops a similar argument to that presented here by Cleonis from this one fact right up to a complete derivation of total mechanical energy.

So what does he say about what energy is?

- Roy

 

[Studiot]

There's quite a few pages of it, as he develops a complete theory of kinetic, potential and rotational energy and the consequential total mechanical energy.

You have to take something as given and he starts from Newtons second law expressed as

F = mr\limits^{..}

 

[RoyLB]

There's quite a few pages of it, as he develops a complete theory of kinetic, potential and rotational energy and the consequential total mechanical energy.

You have to take something as given and he starts from Newtons second law expressed as

F = mr\limits^{..}

Studiot,

Nevermind the math (unless its truly novel). Does he opine at all as to the nature of energy?

- Roy

 

[Benson]

Hi All,

I'm currently teaching an introductory thermo and fluids course to 2nd year college diploma students who come through vocational high schools (so their mechanics conceptual fundamentals and mathematical fundamentals aren't great). I've started by giving them an overview of all the different forms of energy and some calculations for converting between them, and had similar problems explaining just why *is* energy equal to force x distance (I drew upon the a = F/m concept too).

I've now reached the point where I'm about to teach them calculating temperature rise in a closed system (no mass flow in or out) - e.g. a tank of liquid with a heating element of X Watts, a stirrer with a shaft power of Y Watts and heat loss of Z Watts, how long does it take to raise to a certain temperature. And of course with this you have the concept that part of the temperature rise is due to the heating element, and part is due to the stirrer doing work on the fluid. And then I got to thinking - how do I explain to them *how* or *why* - as in, through what mechanism - a spinning stirrer causes the temperature of a fluid to rise?

They understand the concept of friction, and intuitively that it dissipates heat - so would it be right to say that the temperature change (increase in internal energy) is due to increased friction between the water molecules as they move past/over/rub against each other with greater velocity?

Thanks,
Benson

 

[Philip Wood]

The following is naive, but, I would argue, logical. Can be made more sophisticated as required.

Starting point: Define the body's KE as the work it can do because of its motion.

Think of something like a boulder moving on a frictionless surface. Lasso it and exert a force F on it, so retarding it. Then

KE = Work done on rope by boulder as it slows down = ∫F.dx

But F = - \frac{d\textbf{v}}{dt}

This leads swiftly to the familiar formula.

 

[Riad]

In an isolated system all the input Heating Energy will go into increasing the internal temperature after some time. This increase involves the heat capacity of the liquid as is well known.
The input KE also becomes eventually internal energy and causes an additional increase in temperature to be calculated exactly the same way above as your units are all in watts in the two cases.
How the KE is converted to internal energy (expressed as an increase in temperature) is not difficult to explain. This conversion happens only if the fluid has a viscosity. Viscosity works like friction. Molecules of high velocity(KE) rubbing against others of smaller velocity causing them to accelerate(it is still a KE but not visible as it is only within the substance ie internal). This rubbing action happens between the fluid and blade material and also between the fluid and fluid of smaller velocity in the small eddies.

 

[Riad]

[Benson:Mar5-12, 07:58 AM Re: Conceptual derivation of (classical mechanical) energy? #57]
Just to complete my answer above;
To find the dynamic response of the container with stirrer, heating and losses;
Increase in temp per second=(heating power+stirrer power -lost power)/sum(mass.specific heat of all masses involved: rotor,liquid and container).
If you take this to be a differential increase, you could integrate wrt time and find the dynamic response.
Lost power=ext surface area* surface temp diff (with outside)*coeff of thermal convection.
I have neglected any temp gradient in the walls, which should be included in a lagged container. In this case take it as a conduction problem to outside with equivalent resistance to replace convection- ie inverse of convection coeff.