Conceptual derivation of (classical mechanical) energy?by RoyLB Tags: derivation, kinetic enegry, mechanics, work 

#37
May1210, 06:47 PM

P: 226

@Riad
There's a counterexample. Just imagine 3 charged masses aligned (named 1, 2 and 3. 2 is the central one). Suppose charge of 1 = charge of 3, and distance 12 = distance 23. Than 1 will exert in 2 a force, but only 1 will be displaced. You don't need to impose that those point masses have the same mass nor that they are equal. Just use Newton's definition F=dp/dt, and use Newton's third law in order to find that, regardless of the properties of thoses masses, or the quantity of bodies under study, we will always have conservation of momentum. In fact, you should start backwars: conservation of momentum always holds (even in relativity or quantum mechanics), but there are situations in which Newton's third law doesn't apply. So you can't really use Newton's third law to find cons. of momentum, but use cons. of momentum from Newton's third law. 



#38
May1310, 01:01 AM

PF Gold
P: 693

In the case of a very large nucleus, such as a Uranium nucleus, there is so much repulsive force between the protons that the nucleus is teetering on the brink of falling apart. A neutron absorbtion event can trigger nuclear fission. The combined mass of the two fission products is smaller than the mass of the original Uranium nucleus. In a sense the Uranium nucleus is under very large internal stress. There is the strong nuclear force that acts to keep the nucleus together, and the Coulomb repulsion between the protons acts to make the nucleus fission. This internal stress is a form of potential energy, and it has a corresponding inertial mass. The magnitude of that inertial mass is given by the following expression: [tex] m = \frac{E}{c^2} [/tex] 



#39
May1310, 03:32 AM

P: 5,462

Roy,
I find the most successful learning method is recursive. I call it a spiral. Only (a coherent)part of a subject is presented on each circuit of the centre. As we move outwards the diameter expands and our knowledge expands and more is presented. This is why the sort of treatise prepared by a professor is often poor (=heavy going) teaching material but an excellent reference once the subject has been mastered. I commend you to the little monograph written by M B Glauert, entitled Principles of Dynamics. This monograph is pitched at about the level you seem to require, and contains many explanatory insights, in particular his explanation of what is meant by a particle when introducing Newton's laws. 



#40
May1310, 06:06 AM

P: 23

My initial objection was that this seemed like pure mathematics to me. Acceleration, distance, and velocity are kinematically related (or if you prefer, are derivatives of one other), so of course acceleration times distance will yield velocity squared terms. Now that I've embraced that energy is all just mathematics at this level, your derivation is not too bad. I don't like that you multiplied both sides by a in equation (3) (which is OK for a mathematical proof), but this step is unnecessary. Just solve (2) for t and substitute into (1) and you can get to (4) pretty quickly. Thanks Roy 



#41
May1310, 06:29 AM

P: 23

Cleonis
If we do have conservative forces  that is if there exists a gradient of a potential function in space  then particles can be considered to be simply following a geodesic in this space. For every meter traveled on a nonflat gradient surface, the particle will accelerate a certain amount. This looks like a "force", but its "really" just a particle following a path in space. The distance traveled times that acceleration will yield velocity squared terms, and we have conservation of energy. Of course, I'm heavily borrowing from general relativity here. In this view, energy is the more fundamental concept, and forces are just derived from it.  Roy 



#42
May1310, 06:34 AM

P: 23

I think what you're saying is that to really appreciate energy, we have to go beyond the simplification of our Newtonian worldview and embrace relativity. Forget about this ineffable "potential energy stored in a spring". The compressed spring has more mass (imperceptible as it is to us) than the uncompressed one. In "realty", energy, mass, momentum are all aspects of the same thing.  Roy 



#43
May1310, 06:36 AM

P: 23

Thanks for the advice and the recommendation. I will look for it.  Roy 



#44
May1310, 06:37 AM

PF Gold
P: 693

About the relation between F=ma and E=1/2mv^{2} : I like the example of an electric car designed for regenerative braking. For simplicity let's assume that the electric generator of the car is 100% efficient in converting kinetic energy to electric energy. (With the electric energy converted to chemical potential energy in the car battery.) The car is braking over a distance 'd' along the road, maintaining uniform deceleration. Then the level of energy in the battery increases linear with distance. Hence the level of kinetic energy of the car is decreasing linear with distance. What intrigues me is that the quadratic expression E=1/2mv^{2} expresses a linear relation when viewed as a function of distance travelled. 



#45
May1310, 07:00 AM

PF Gold
P: 693

I find it tempting to speculate that 'probability' is the underlying theme of potential energy on one hand, and entropy on the other hand. We know that increase of entropy can drive an endothermic process. A mix of ice and salt will decrease in temperature as the ice dissolves in the brine. Right now I can't think of a counterexample, and I venture to say: any form of storing potential energy in elastic deformation or in chemical form is in one way or another a process of deforming away from a ground state. The ground state is the quantummechanically most probable state. Needless to say, this is highly speculative 



#46
May1310, 07:07 AM

P: 15

I can not quite get you.. if you move 1 then 3 moves also otherwise you have the center of mass (of the three) moving without an ext. force involved. It seems to me that if it is possible to negate this ballanced displacement idea, then momentum is not conserved.. note that all events are taking the same time period dt so velocity would be the same as momentum. The most interesting thing about this idea is that it can be used to justify the second and third laws of Newton and it can even explain inertia.. since to move one mass forward you need to push another back.. so that the sum of sum(distance x mass)= zero. 



#47
May1310, 07:24 AM

PF Gold
P: 693

The question you raised was whether the concept of Energy can be seen as essentially a bookkeeping device, useful as a tool, but that we shouldn't attribute physical existence to it. Well, the lesson from relativistic physics is that to Matter and Energy we have to attribute the same level of existence; they're on equal par, given that both matter and energy have inertial mass. The nuclear binding energy example is, I guess, the only example where we can actually measure the mass difference that arises from difference in internal energy. Theoretically a compressed spring has additional inertial mass, corresponding to the stored energy, but that mass difference is far, far too small to be measurable. 



#48
May1310, 08:57 AM

P: 250





#49
May1310, 09:48 AM

P: 23

All,
What do you guys think of this? I just found it in, from all places, Wikipedia: (http://en.wikipedia.org/wiki/Energy#...cept_of_energy) In classical physics energy is considered a scalar quantity, the canonical conjugate to time. In special relativity energy is also a scalar (although not a Lorentz scalar but a time component of the energymomentum 4vector).[14] In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of spacetime (= boosts). So, of course I checked (http://en.wikipedia.org/wiki/Canonical_conjugate) In physics, conjugate variables are pair of variables mathematically defined in such a way that they become Fourier transform duals of oneanother, or more generally are related through Pontryagin duality. Time and energy  as energy and frequency in quantum mechanics are directly proportional to each other. So  energy is connected to time via fourier transforms! As a controls guy I understand FTs. I also know they are important to the study of quantum mechanics, although I'm not that far along yet in my study of QM. (I did check Pontryagin duality, but that entry quickly lost me)  Roy 



#50
May1310, 12:02 PM

P: 23

We may agree. I was lamenting the fact that energy appeared to be just a bookkeeping device in Newtonian physics, when I knew it was more than that. 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". Or am I still misunderstanding your point?  Roy 



#51
May1310, 01:57 PM

PF Gold
P: 693

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. 



#52
May1310, 03:37 PM

P: 23

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 



#53
May1310, 04:14 PM

P: 5,462

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. 



#54
May1310, 04:28 PM

P: 23

Studiot
 Roy 


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