View Full Version : Conceptual derivation of (classical mechanical) energy?
I have a pedagogical question, and a philosophical question, both involving energy:
1) All the derivations I’ve ever seen of kinetic energy in both elementary and advanced texts (or in Physics Forum searches) either simply define T=1/2mv^2 or start by taking the dot product of force and distance. Once either of these steps are done, it’s just a matter of mathematics to get to the work energy theorem. However, pedagogically speaking, both approaches seem forced (if you’ll pardon the pun). Do you know of a more natural way to introduce and motivate the idea of (mechanical) energy? I assume it is natural to start from F=ma, since the non-expert can grasp the concepts of forces, masses, and accelerations pretty easily, especially after some simple experiments.
2) What exactly is energy, anyway? Since work involves a dot product, can it be considered the component of something? If so, what? I suspect that one might need to resort to relativity and/or thermodynamics to get the true picture of what energy is, not unlike how Least Action in classical physics can be “explained” by resorting to
quantum physical considerations
(see http://www.eftaylor.com/pub/QMtoNewtonsLaws.pdf)
Direct answers or pointers to relevant references would be appreciated.
Thanks!
Roy
Energy (and work) is a scalar so can't be a component of anything.
They are both physical concepts.
The dot product is a mathematical concept that results in a scalar.
In traditional mechanical education energy is defined as the capacity to do work. The concept is introduced after work has been defined in terms of forces and distances. Both these entities are subject to relativistic transformations.
So where are you coming from?
Here is a way I've discovered. The outline is:
All of our assumptions are
conservation of momentum p(v)
all forces are inverse square forces (or more generally \oint \vec{F}\cdot d\vec{r}=0 or F=-\nabla V(\vec{r}))
I make absolutely no assumption about the velocity dependence of the momentum. Momentum is just "something" that is conserved as a total.
I also never use "mass". It is not needed, but whoever wishes can define m=p/v
In the end I derive purely mathematically from vector momentum conservation that a scalar quantity E=\int v\,dp exists, which is conserved provided you take care of another new quantity that we can call potential energy.
\frac{d\vec{p}_i}{dt}=\sum_{\stackrel{j}{j\neq i}}\vec{F}_{ij}
\frac{d\vec{p}_i}{dt}\cdot d\vec{r}_i&=\sum_{\stackrel{j}{j\neq i}}\vec{F}_{ij}\cdot d\vec{r}_i
d\vec{p}_i\cdot\vec{v}_i&=\sum_{\stackrel{j}{j\neq i}}\vec{F}_{ij}\cdot d\vec{r}_i
\sum_i \int \vec{v}_i\cdot d\vec{p}_i&=\sum_{\stackrel{i,j}{j\leq i}}\int\left(\vec{F}_{ij}\cdo d\vec{r}_i+\vec{F}_{ji}\cdot d\vec{r}_j\right)
If we assume
\vec{F}_{ij}&=-\nabla_i V(\vec{r}_i,\vec{r}_j)+\vec{A}\times\frac{d\vec{r} _i}{dt}
(conservative force plus any perpendicular part)
and define a new unassigned quantity
E_i= \int\vec{v}_i\cdot d\vec{p}_i
then the RHS is integrable
\sum_i E_i+\sum_{\stackrel{i,j}{j\leq i}}V(\vec{r}_i,\vec{r}_j)&=\sum_i E_i(0)+\sum_{\stackrel{i,j}{j\leq i}}V_{ij}(0)=\text{const}
You see that whenever we have only inverse square laws, then conservation of momentum (which is completely undefined yet), implys conservation of a scalar we will call kinetic energy. Note that we need to add a correction factor which we will call potential energy.
If I now combine E=\int v\, dp with E=m (natural units) i.e. E=p/v, then I get all of the expressions for relativistic mechanics.
If on the other hand I unnaturally assume p/v=const=m, then I get classical mechanics.
So you see in my view energy is just an artificial number that happens to be conserved in addition to momentum conservation, since we only have inverse square law forces.
Without inverse square laws, maybe our world wouldn't conserve energy.
(I haven't generalized my proof to non-instantaneous forces yet :( )
Conservation of energy holds even with forces that do not follow an inverse square law!
(as in springs, which follow Hooke's law)
@Acut: Think about it! Even springs are based in the inverse square law of electrostatics.
Thanks eveyone for responding
@Studiot - I am an Aeronautical engineer specializing in control systems, so I am familiar with the points you made. I am trying to write a treatise that takes a reader from some exposure to physics and perhaps calculus to where he/she could understand some elementary automatic control theory. I think it would be useful to bring some energy methods to this treatment. However, I realized that I was unable to bridge the gap in a satisfactory way between F=ma and W=Fd or T=1/2mv^2.
@ Gerenuk - I think you've just showed that if all the forces are derivable from a potential function that conservation of energy holds. But that is just the condition for the forces to be conservative (although I'm not sure how A x dr/dt comes into the discussion). I think that 1/r^2 forces are just a subset of conservative forces. To be even more general, Noether's theorem indicates that as long as there's no explicit dependence on time that energy will be conserved.
But, having said all that, your second step is to take the dot product between the forces and the displacement. What motivated this step - except of course that you knew how the answer needed to come out in the end! As a student, I always felt I understood concepts better when all the steps in a derivation made sense and followed naturally from the step before.
- Roy
jack action
May9-10, 10:02 AM
The best definition I've seen for energy is the capacity to create an effect.
As for teaching it, I can't see a simpler image to grasp than force times distance. If you imagine a person pulling a sliding block by a cord, anyone can grasp the concept that the bigger the force, the more energy required and the longer the distance traveled, the more energy required. In fact, I think it's more obvious (intuitively) than the bigger is an acceleration, the bigger is the force in F=ma.
@ Gerenuk - I think you've just showed that if all the forces are derivable from a potential function that conservation of energy holds. But that is just the condition for the forces to be conservative (although I'm not sure how A x dr/dt comes into the discussion).
Well, I actually derive that something like energy exists at all. Why do you think that F=-\nabla V means anything? Don't forget that you have to imagine to be the first scientist on earth and no-one has ever discovered something like energy.
I suppose if you were to start a proof, you would not only restrict yourself to static *external* potentials, but also you would define dE=Fds for no apparent reason? I just made this proof more general and logical.
The "A x" term is merely to add an admissible degree of freedom without breaking the proof. And this term is needed to represent magnetism.
I think that 1/r^2 forces are just a subset of conservative forces.
I vaguely recall, that inverse square forces and zero gradient is equivalent in 3D (I guess provided forces should vanish at infinity).
To be even more general, Noether's theorem indicates that as long as there's no explicit dependence on time that energy will be conserved.
That energy conservation follows from Noether, is a myth. The Lagrangian formalism guesses an formalism that is already tailored to yield energy results. Noether for energy in a way states that if energy is time independent, then energy is conserved.
But, having said all that, your second step is to take the dot product between the forces and the displacement. What motivated this step - except of course that you knew how the answer needed to come out in the end!
With mathematical experience, one sees that one can exploit \int \nabla V\cdot d\vec{s}=\Delta V. So I really could have been lead by mathematical simplification ideas only.
The best definition I've seen for energy is the capacity to create an effect.
As for teaching it, I can't see a simpler image to grasp than force times distance. If you imagine a person pulling a sliding block by a cord, anyone can grasp the concept that the bigger the force, the more energy required and the longer the distance traveled, the more energy required. In fact, I think it's more obvious (intuitively) than the bigger is an acceleration, the bigger is the force in F=ma.
Jack,
You may have a point in that I certainly don't recall having this difficulty during my own undergraduate or even high school physics experience. Perhaps I am making too much of it.
But, as for my other philosophical question, why should force times distance be meaningful? Why is energy a scalar at all? Is energy just a mathematical fiction / convenience, like imaginary numbers (OK - perhaps I opened a can of worms with that statement :-) or is there some physical "reality" to it? I think it is real, in that the energy concept exists in thermodynamics apart from mechanics. Also, I think relativistic physics shows that energy is as real as mass and momentum - in fact they are all aspects of the same mass energy momentum 4-vector. Unfortunately, my understanding of physics is not yet sufficiently sophisticated to make sense of all these ideas.
- Roy
fluidistic
May9-10, 12:12 PM
The Lagrangian formalism guesses an formalism that is already tailored to yield energy results.
I'm interested in knowing more about this.
Here is what I know or believe to be true: in some closed systems (where there's no dissipative force), there's a quantity that is invariant with respect to time translations, i.e. a constant. This number is called the total energy and we can as you know separate it as kinetic and potential energy which aren't necessarily constant. The derivation of the kinetic energy comes from the Lagrangian of the system. And if the Lagrangian doesn't depend explicitly on time, then the energy is conserved.
[...]The derivation of the kinetic energy comes from the Lagrangian of the system. And if the Lagrangian doesn't depend explicitly on time, then the energy is conserved.
The point is: How do you write down the Lagrangian for a system? You put in your knowledge about kinetic energy or anything equivalent! So you already set up the stage to make it yield energy conservation at some point.
If you just say that there is some vector quantity momentum which is conserved, then your system could be any crazy world.
But as soon as you say, OK, and now I take it as a *new* law that the Lagrangian formalism holds and I will put in mv^2/2 in there, then you are already presupposing that eventually there will be something like a conserved energy.
fluidistic
May9-10, 12:25 PM
The point is: How do you write down the Lagrangian for a system? You put in your knowledge about kinetic energy or anything equivalent! So you already set up the stage to make it yield energy conservation at some point.
I agree that I also found the expression of the Lagrangian of a system with 1 particle somehow artificial.
I don't know if you had a look at a thread of mine I started about a week ago: http://www.physicsforums.com/showthread.php?t=399993. It might be of interest for this part of the discussion.
If you just say that there is some vector quantity momentum which is conserved, then your system could be any crazy world.
But as soon as you say, OK, and now I take it as a *new* law that the Lagrangian formalism holds and I will put in mv^2/2 in there, then you are already presupposing that eventually there will be something like a conserved energy.
This requires a lot of hours for me to think in order to fully agree with this. :smile:
Well, I actually derive that something like energy exists at all. Why do you think that F=-\nabla V means anything? Don't forget that you have to imagine to be the first scientist on earth and no-one has ever discovered something like energy.
I suppose if you were to start a proof, you would not only restrict yourself to static *external* potentials, but also you would define dE=Fds for no apparent reason? I just made this proof more general and logical.
The "A x" term is merely to add an admissible degree of freedom without breaking the proof. And this term is needed to represent magnetism.
I vaguely recall, that inverse square forces and zero gradient is equivalent in 3D (I guess provided forces should vanish at infinity).
That energy conservation follows from Noether, is a myth. The Lagrangian formalism guesses an formalism that is already tailored to yield energy results. Noether for energy in a way states that if energy is time independent, then energy is conserved.
With mathematical experience, one sees that one can exploit \int \nabla V\cdot d\vec{s}=\Delta V. So I really could have been lead by mathematical simplification ideas only.
@ Gerenuk
I originally asked 2 questions. The first is how to show the derivation of energy from simpler concepts, the second is about what energy actually is. In the first case, I don't need a rigorous proof, I just want each step to be understandable, even if it is purely heuristic. In my mind, taking F dot ds comes out of nowhere, but perhaps that's just me. I feel there ought to be a more elegant way to approach this argument. As for my second question, I don't think your proof really tells me what energy actually is - anymore than any other textbook argument, unless you think that energy really is a sort of convenient mathematical fiction. As I've said earlier, I don't think that this is the case.
I think your proof is probably fine from a mathematics standpoint (although I think you'd need to show where A x dr comes from more explicitly and I don't think you've shown that energy is a constant in this case - i.e. that dE/dt = 0). I think from a physics standpoint you'd have to show that forces are conservative (I'd accept F = dp/dt from experimental reasoning, or even just as a definition). I believe that physicists think that this is indeed the case for the fundamental forces. However, the mathematics and the physics of the conservation of energy still work for dissipative forces, as long as you consider heat to be a form of energy.
In fact I think that the (mass-) energy momentum 4 vector from relativity is the probably the more fundamental physical entity, and that "forces" being the rate of change of momentum can be likely derived from that principle. Its just that we humans can appreciate forces, masses, and accelerations from our everyday experience while energy is more abstract.
- Roy
I agree that I also found the expression of the Lagrangian of a system with 1 particle somehow artificial.
I don't know if you had a look at a thread of mine I started about a week ago: http://www.physicsforums.com/showthread.php?t=399993. It might be of interest for this part of the discussion.
This requires a lot of hours for me to think in order to fully agree with this. :smile:
@fluidistic, Gerenuk
I tend to agree with you that the Lagrangian formulation seems contrived. If you check here: http://www.eftaylor.com/pub/QMtoNewtonsLaws.pdf you'll see that the "least action" formulation is better understood from quantum mechanical principles (although I think they've left out some key portions of the argument in that paper)
- Roy
Thank you for your response Roy, I should have perhaps said, where are you going to, rather than where are you coming from, but you anticipated even that!
From what you say I can't see the point of mentioning dot products at all. You may not be aware that physicists and mathematicians work to a slightly different definition of both vectors and scalars. The physicists' one being more restrictive.
If you want to introduce physical concepts in a staged manner I wonder how you introduce the definition of a force?
I was appalled to learn that modern teaching in the UK introduces friction as the first force discussed.
I have found in my experience of gaining understanding of a subject that following the historic route usually suits the best. Historically new concepts mostly arose when the necessary ground had been laid and could be seen in the concept of that work and understanding.
So some development of mechanics preceeded development of what we now call classical thermodynamics and some development of electrical phenomenon was also stirring. So it became obvious that some equivalence of the mechanical work, heat, fluid and electrical energies was called for.
I think those whose educational development followed this route have less trouble with what I might call engineering physics than with say particle or sub particle physics.
In my mind, taking F dot ds comes out of nowhere, but perhaps that's just me.
Yes, and that's why I explained to you why someone who has never heard of F dot ds would still perform this operation. It's purely mathematical logic and an obvious mathematical simplification step.
although I think you'd need to show where A x dr comes from more explicitly and I don't think you've shown that energy is a constant in this case - i.e. that dE/dt = 0.
I get the impression that you are not reading what I write. If you are really interested in answers please read carefully. With the A x dr terms I'm trying to express a more general function that still allows for my proof. I'm not assuming any knowledge about real world physics and the choice F=\nabla V+A x dr is motivated only by the mathematical form of the expression.
And dE/dt=0 follows if you look at the step carefully. The A x dr term just cancels.
I think from a physics standpoint you'd have to show that forces are conservative (I'd accept F = dp/dt from experimental reasoning, or even just as a definition).
Note that at no point I'm using force. Force is just flow of momentum. So it's not a question of experimental reasoning or definitions.
I believe that physicists think that this is indeed the case for the fundamental forces. However, the mathematics and the physics of the conservation of energy still work for dissipative forces, as long as you consider heat to be a form of energy.
What is heat? It's a macroscopical concept. On the microscopic level there are only bouncing billard balls, so you cannot speak about heat.
In fact I think that the (mass-) energy momentum 4 vector from relativity is the probably the more fundamental physical entity, and that "forces" being the rate of change of momentum can be likely derived from that principle.
The problem with the 4 vector philosophy is, that it is overdetermined and it's a miracle why it is consistent. Given that 3 components are conserved and our world has inverse square laws, my derivation shows that the 4th component is conserved.
It's much more ugly to postulate two laws which could potentially be conflicting. It's nicer if you show how to derivate one of the law from the other.
From what you say I can't see the point of mentioning dot products at all. You may not be aware that physicists and mathematicians work to a slightly different definition of both vectors and scalars. The physicists' one being more restrictive.
I need the dot product anyway, perhaps in both the mathematical and the physical senses. Since you can see I have trouble bringing in the F ds concept, you can imagine that I really have a problem with introductory Controls texts that simply introduce Laplace and Fourier transforms. I plan to use the idea of infinite dimensional "dot products" to motivate these transforms.
If you want to introduce physical concepts in a staged manner I wonder how you introduce the definition of a force?
I was appalled to learn that modern teaching in the UK introduces friction as the first force discussed.
I was planning just to use an archetypical spring-mass-damper system to introduce the "intuitive" concept of force. My main concern is not rigor, but elegance. Having said that, I don't want to sacrifice accuracy for simplicity.
I have found in my experience of gaining understanding of a subject that following the historic route usually suits the best. Historically new concepts mostly arose when the necessary ground had been laid and could be seen in the concept of that work and understanding.
So some development of mechanics preceeded development of what we now call classical thermodynamics and some development of electrical phenomenon was also stirring. So it became obvious that some equivalence of the mechanical work, heat, fluid and electrical energies was called for.
I think those whose educational development followed this route have less trouble with what I might call engineering physics than with say particle or sub particle physics.
I'm really focusing on control systems, so I'll only bring in as much physics and math as I need. I think the historical approach - while fascinating - will take too much time. I'll probably focus the treatise on the engineering physics path and bring in mathematics via hypertext links as appropriate. Perhaps if I am ambitious enough I will add historical sidebars to the text.
I probably don't even really need energy that all that much. After I explain the spring mass damper EOM via the force picture, I want to just mention that it can also be explained by energy transfer, from potential to kinetic to dissipative. This opens the door to more sophisticated methods (I prefer Kane's method to Lagrange, but that's what I studied in grad school)
- Roy
Well, I can think of a perhaps more natural way of introducing energy.
Begin with Torricelli:
v^2 = vo^2 + 2*a*delta s
Apply it to free fall
v^2 = vo^2 + 2g(ho - h)
Rearrange:
v^2 + 2gh = vo ^2 + 2gho
You can see that it's a beginning: you have isolated on each side of the equation terms that only relate to the beginning and terms that only relate to the end.
Dividing both sides by 2 is just a mathematical operation. Multiplying it by m and getting the "right" expressions for kinetic and gravitational potential energy is a bit more tricky: use the spring-mass system and show you can do something similar (one side having only information about the past and one side having only information about future) with it. Then it becomes quite a natural thing to multiply the one-half-v^2 term by the mass of the object.
Now, this derivation follows from F=ma (so that you know the object's acceleration, which Torricelli demands) and only some very basic knowledge of kinematics. I think it's a nice way of showing how the concept of energy arises if you will not show them much physics. However, such derivations would hardly convince most physicists.
About "what is energy", notice that there's no universal definition of it. My books usually define it as "the capacity to do work" or, as jack action said "to produce a effect". Those definitions are unacceptable, however. You may have a lot of energy, but if you are in thermodynamical equilibrium, you can't do any useful work with it. I prefer thinking of it as a number which -as far as we know- is conserved with time.
@Gerenuk: Although the internal mechanism of springs is based on electrostatics, it isn't a requirement for energy to conserve. All we need is curl F = 0, and quite a lot of forces and force fields satisfy it.
@Gerenuk: Although the internal mechanism of springs is based on electrostatics, it isn't a requirement for energy to conserve. All we need is curl F = 0, and quite a lot of forces and force fields satisfy it.
OK, tell me one example, but it should be one where the force vanishes at infinity. Our world is made up of particles and it's natural to assume that fundamental forces should not depend on particles an infinite distance away. So which type of force between point objects is possible, which isn't inverse square?
I have the feeling curl F=0 is equivalent to inverse square in 3D.
Btw: Springs are objects that change shape. This doesn't count. We are talking about forces on test particles, which do not change any of the enviroment.
russ_watters
May9-10, 11:52 PM
But, having said all that, your second step is to take the dot product between the forces and the displacement. What motivated this step - except of course that you knew how the answer needed to come out in the end! They did try to explain this already, but let me try another way of saying it: what motivated that step (the first time anyone did it) was the realization that it was a useful thing to do. That's it. There need not be any deeper/larger meaning than that. People realized the mathematical relation was useful and gave it a name. That's it.
Studiot
May10-10, 06:31 AM
I need the dot product anyway, perhaps in both the mathematical and the physical senses. Since you can see I have trouble bringing in the F ds concept, you can imagine that I really have a problem with introductory Controls texts that simply introduce Laplace and Fourier transforms. I plan to use the idea of infinite dimensional "dot products" to motivate these transforms.
What I don't understand is the level of knowledge of the course attendees. Presumably this course has prerequisites?
You express the need to 'introduce' force and energy to students in a course that will be using significant calculus.
Why would anyone attending this course and presumably having enough calculus to study Laplace transforms not have enough physics/ applied maths to understand force and energy?
Edgardo
May10-10, 07:04 AM
@ Gerenuk
I originally asked 2 questions. The first is how to show the derivation of energy from simpler concepts, the second is about what energy actually is. In the first case, I don't need a rigorous proof, I just want each step to be understandable, even if it is purely heuristic. In my mind, taking F dot ds comes out of nowhere, but perhaps that's just me.
- Roy
Intuitively, defining work as W=Fd makes sense. Imagine a giant spring that you have to compress. The work that you have to "invest" depends on the force F and the distance d.
F dot d (where F and d are vectors) is useful when the direction of force and path are different
Cleonis
May10-10, 04:36 PM
[...]
Do you know of a more natural way to introduce and motivate the idea of (mechanical) energy?
[...]
(I may duplicate something that is already posted; I tend to like writing more than reading.)
Is there a way to get from F=ma to mechanical energy, in intuitive steps?
I assume the following:
- Assumption: when an object falls down, gravity is doing work, increasing the mechanical energy of the object.
- I assume the existence of gravitational potential energy.
- Assumption: when I lift and object I am doing work, increasing the gravitational potential energy of the object. The potential energy is defined to be proportional to the height.
When acceleration is uniform then covered distance 's' as a function of elapsed time 't' is:
s = \frac{1}{2}at^2 (1)
And instantaneous velocity as a function of time:
v = at (2)
Adapting expression (1):
a*s = \frac{1}{2}a^2t^2 (3)
Substituting according to expression (2)
a*s = \frac{1}{2}v^2 (4)
Substituting:
\frac{F}{m}s = \frac{1}{2}v^2 (5)
Moving the factor 'm' for mass to the other side:
F*s = \frac{1}{2}mv^2 (6)
What does expression (6) mean?
I think it expresses something about velocity: I can isolate the velocity:
v = \sqrt{\frac{2Fs}{m} } (7)
Let's say you're dropping an object from several heights. If you double the height you don't get double the velocity. To get double the velocity you have to quadruple the height. That's because once the falling object is going fast gravity has less time to accelerate it even more.
@Geranuk:
I think pretty much any function wich obey 1/r^n , n being any real larger than one (in order to make F=0 at infinity) will have curl F = 0. But I haven't done the math. So curl F =0 does not imply an inverse square law.
I honestly can't see why you've restricted yourself to forces wich go to 0 as distance approaches infinity. If I'm missing something, please explain. Perhaps you only wanted to avoid that I mentioned the spring potential again (why on earth Physics shouldn't be concerned with springs? I really can't get your point). However, I'm happy to give you another example of curl F=0 force wich does not obey inverse square law and which vanishes at infinity: the covalent bond in H2.
If you still find problems with my example above, since it is not "fundamental" or "between two point particles" (which also seem quite useless restrictions to me, please explain them!) the strong force does not follow inverse square law. An it is a fundamental force.
I'm puzzled by your views that everything should be reduced to fundamental inverse square law forces between two point particles. May I give you a suggestion? Show us how could you build a model of spring force using only inverse square laws that are fundamental and act only on point objects.
Have fun!
@Edgardo
I disagree it's intuitive. It's very common to confuse "to do work" with "getting tired"
Gerenuk
May10-10, 06:35 PM
@Geranuk:
I think pretty much any function wich obey 1/r^n , n being any real larger than one (in order to make F=0 at infinity) will have curl F = 0. But I haven't done the math. So curl F =0 does not imply an inverse square law.
Yes, true. I forgot why inverse square law was special. Maybe I can find that reasoning again.
I honestly can't see why you've restricted yourself to forces wich go to 0 as distance approaches infinity. If I'm missing something, please explain.
Because I want to talk about real infinities, i.e. lightyears away. And of course we do not want a physics, where some object lightyears away influences what happens on earth.
Note that the spring potential does not belong to the forces we are considering here.
Here a force means, that a test particle experiences a force without the whole world changing depending on the particle position. However, if you are talking about spring forces, then you imply that you adjust the spring for whichever distance the test particle is. That's something completely difference and not related to fundamental forces between particles.
Show us how could you build a model of spring force using only inverse square laws that are fundamental and act only on point objects.
That should be doable. I mean in the end there is no other force, than electrostatics, right? I'll get back to that later.
@Gerenuk
I'm wondering how to describe spring forces with more elemental, inverse square laws.
For inverse square laws go weaker with distance, on the opposite trend of spring forces.
I'm really curious now. If you find a solution, post it!
Inverse square law is special, since it's the only kind of forces that keep flux constant (which implies they obey some sort of Gauss' Law). This, aside from some minor curiosities, is the only reason why inverse square laws have a special place in Physics.
I've never studied quantum stuff very deeply, but I believe that I've read about a mechanism to influence particles very very very far away. So objects lightyears away may influence earth. In fact, some stars you see in the sky are lightyears away, and they indeed influence our eyes!
Thanks everyone for the spirited discussion.
I think I am coming to the conclusion that, for "classical" or "engineering" physics, energy is just a convenient mathematical "fiction", in the same way that Hook's Law and Normal Friction "forces" are just ways to compartmentalize manifestations of the underlying more fundamental "real" forces. Which is, of course, what many of you were saying, either implicitly or explicitly.
However, I still don't think "trust me on this; it works; I don't have time to review centuries of experiments and theories" is a very good teaching strategy, if a better approach exists. Here's the solution I propose: The mass bob of a two-dimensional pendulum is of course constrained to follow a circular arc. The force of gravity can therefore only cause motion in a direction perpendicular to the rigid rod of the pendulum. This naturally suggests investigating the component of gravity in that direction - that is taking the dot product of the force with the displacement or the velocity. The dot product of the gravity force with displacement yields the work, and that with the velocity yields power. From either one, its just a matter of mathematics to get to energy and all the rest. To me, that seems a little more elegant and natural than just taking the dot product for "no good reason". I think this approach could lead very nicely to an analytical mechanics discussion (Lagrange's or Kane's method)
- Roy
What I don't understand is the level of knowledge of the course attendees. Presumably this course has prerequisites?
You express the need to 'introduce' force and energy to students in a course that will be using significant calculus.
Why would anyone attending this course and presumably having enough calculus to study Laplace transforms not have enough physics/ applied maths to understand force and energy?
@Studiot,
I should take a step further back. This is not intended to be any sort of course. It will be more of a tutorial or treatise that lives on a web page somewhere. I am designing my own quadrotor control system. There is at least one guy (http://www.aeroquad.com/) who was able to cobble together a quadrotor control system based on bits and pieces he picked up on the web. But, he and his followers have no real understanding of closed loop feedback control systems. My intent is to give them sufficient background in the theory to understand how their PID control loops work. I think I can do this with the prerequisite of some exposure to calculus and physics.
I won't need Laplace transforms until much later. However, I have some ideas on how to introduce the Laplace and Fourier transforms that I think are more elegant and understandable than the way they are typically introduced in the controls curriculum. So, even though this project is initially geared toward non-specialists, I do think that as it expands it could eventually be of value to struggling EE, aero, or mech E undergraduates (or perhaps even grad students) who are studying control systems. I envision it to become a mostly self contained feedback control tutorial, that brings in the needed mathematics via hyperlinks as appropriate.
- Roy
Gerenuk
May12-10, 03:42 AM
In the classical sense energy could be seen as an auxiliary invariant, but I haven't checked if energy is more fundamental in special relativity. Maybe someone can comment on this?
That I want to check soon.
And I've explained it a couple of times: a mathematician knows that if he sees a gradient, he wants to dot it and integrate. That's a very good reason to do that.
In the book "Mathematical Aspects of Classical and Celestial Mechanics", 3rd edition, by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt, this paragraph appears on page 19(!)
------------
Example 1.1. A natural mechanical system is a triplet (M, T, V ), where M is a smooth configuration manifold, T is a Riemannian metric on M (the kinetic energy of the system), V is a smooth function on M (the potential of the force field). A Riemannian metric is a smooth function on the tangent bundle that is a positive definite quadratic form on each tangent plane. The Lagrange function is L = T + V (the function V : M → R is lifted to a function from TM into R in the obvious way).
-----------
Now, to me, the idea of kinetic energy being a "metric" makes sense since it is (typically? always?) a quadratic function of the coordinates, just like the (square) of the distance between two points in Euclidean space. So, is that all energy is - a length in some space? The rest of the terms used here don't quite make sense to me yet, and I don't think they're defined in this book (at least not in the pages prior to page 19). Do you think this line of attack makes any sense? If so, do you know of any good references that will explain some of these terms (manifold, tangent bundle, etc.?)
- Roy
And I've explained it a couple of times: a mathematician knows that if he sees a gradient, he wants to dot it and integrate. That's a very good reason to do that.
Gerenuk,
I don't dispute that the dot product makes sense from a mathematical standpoint. I would have thought it was obvious from what I had posted several times now, but my intended audience is not mathematicians, physicists nor engineers.
- Roy
Cleonis
May12-10, 11:59 AM
[...]
Now, to me, the idea of kinetic energy being a "metric" makes sense since it is (typically? always?) a quadratic function of the coordinates, just like the (square) of the distance between two points in Euclidean space.
[...]
Well, angular momentum m r^2 \omega is a quadratic function of 'r', but there does not seem to be a way of interpreting that as akin to a metric.
I agree that interpreting kinetic energy as related to a metric is tempting, but I think it's a dead end.
The challenge is, I suppose, to capture in words the most general sense of the concept of Energy.
It appears that materials have a state of zero energy, and that energy is accumulated when the material is pushed away from this ground state.
When an elastic material is deformed it stores energy, which is released when the material relaxes again. In elastic deformation the molecules of the material do not slide along each other (when they do slide you have plastic deformation, which dissipates energy). In the case of elastic deformation you are deforming the very molecules away from their ground state.
In a molecular bond there is a distance between the atoms that is the state of least energy. I suppose the most fundamental level of description we currently have is the following: given the quantum physics of molecular orbitals there is a state (of the molecule) that is the most probable. When a molecule is deformed away from that most probable state then it tends to return to the most probable state.
Another aspect: is it possible to view gravitational potential energy as a deformation away from a state of lowest energy? I think so. According to GR gravitational interaction is mediated by deformation of spacetime. (In other words: spacetime curvature acts as mediator of gravitational interaction.) I infer that when two objects are pulled apart then there is a net increase in spacetime deformation.
Finally, kinetic energy.
When two objects have a velocity relative to each other then that two-object system has the potential to do work.
Example, a electric car that is designed for regenerative braking. When the car and the Earth have a velocity relative to each other then regenerative braking will recharge the batteries. The state of lowest energy is where the two objects have no relative velocity.
So in all I like to think of the relative velocity of the two objects in a two-object system as a form of potential energy.
I don't so much think of the quadratic form, I think of the potential to do work, expressed as force acting over a certain distance.
What is special of course is that in the case of kinetic energy this potential is realized only at the very instant in time that the two objects actually interact. When two objects interact with each other, and not with other surrounding objects (as in a collision) then they are in effect a two-object system.
Cleonis
May12-10, 01:40 PM
[...]
I realized that I was unable to bridge the gap in a satisfactory way between F=ma and W=Fd or T=1/2mv^2.
[...]
Hi Roy,
I believe I did the very thing you requested: I bridged from F=ma to W=Fd, using just d=1/2*a*t2 and v=a*t. That was in post #24 of this thread (http://www.physicsforums.com/showpost.php?p=2711586&postcount=24). I'm very curious what you think about that demonstration.
Cleonis
So you see in my view energy is just an artificial number that happens to be conserved in addition to momentum conservation, since we only have inverse square law forces.
Without inverse square laws, maybe our world wouldn't conserve energy.
(I haven't generalized my proof to non-instantaneous forces yet :( )
I like your derivation as it includes cross forces too. In my thinking about this subject I cam to the conclusion that one can postulate a modified Newtons law of action and reaction by saying that for equal point masses ' the displacement of one point mass must be offset by an opposite displacement of another' . as if the world is sitting on a knife edge and moving one particle must be offset by another. If we accept this (and I could not find a counter example yet) then differentiate once and you get conservation of momentum and differentiate a second time and you get acceleration or the sum of forces zero.. as Newton put it originally. Non equal masses can also be included.
@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 1-2 = distance 2-3. 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.
Cleonis
May13-10, 01:01 AM
[...]
Is energy just a mathematical fiction / convenience, like imaginary numbers (OK - perhaps I opened a can of worms with that statement :-) or is there some physical "reality" to it?
[...]
In the case of nuclear physics the nuclear binding energy makes a difference. The mass of a nucleus is not just the sum of the constituent protons and neutrons, the internal energy has inertial mass too.
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:
m = \frac{E}{c^2}
Studiot
May13-10, 03:32 AM
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.
Hi Roy,
I believe I did the very thing you requested: I bridged from F=ma to W=Fd, using just d=1/2*a*t2 and v=a*t. That was in post #24 of this thread (http://www.physicsforums.com/showpost.php?p=2711586&postcount=24). I'm very curious what you think about that demonstration.
Cleonis
Cleonis,
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
Cleonis
Well, angular momentum m r^2 \omega is a quadratic function of 'r', but there does not seem to be a way of interpreting that as akin to a metric.
I agree that interpreting kinetic energy as related to a metric is tempting, but I think it's a dead end.
Perhaps it only works for quadratic functions of the velocity coordinates, and not the position ones
The challenge is, I suppose, to capture in words the most general sense of the concept of Energy.
It appears that materials have a state of zero energy, and that energy is accumulated when the material is pushed away from this ground state.
When an elastic material is deformed it stores energy, which is released when the material relaxes again. In elastic deformation the molecules of the material do not slide along each other (when they do slide you have plastic deformation, which dissipates energy). In the case of elastic deformation you are deforming the very molecules away from their ground state.
In a molecular bond there is a distance between the atoms that is the state of least energy. I suppose the most fundamental level of description we currently have is the following: given the quantum physics of molecular orbitals there is a state (of the molecule) that is the most probable. When a molecule is deformed away from that most probable state then it tends to return to the most probable state.
Another aspect: is it possible to view gravitational potential energy as a deformation away from a state of lowest energy? I think so. According to GR gravitational interaction is mediated by deformation of spacetime. (In other words: spacetime curvature acts as mediator of gravitational interaction.) I infer that when two objects are pulled apart then there is a net increase in spacetime deformation.
Finally, kinetic energy.
When two objects have a velocity relative to each other then that two-object system has the potential to do work.
Example, a electric car that is designed for regenerative braking. When the car and the Earth have a velocity relative to each other then regenerative braking will recharge the batteries. The state of lowest energy is where the two objects have no relative velocity.
So in all I like to think of the relative velocity of the two objects in a two-object system as a form of potential energy.
I don't so much think of the quadratic form, I think of the potential to do work, expressed as force acting over a certain distance.
What is special of course is that in the case of kinetic energy this potential is realized only at the very instant in time that the two objects actually interact. When two objects interact with each other, and not with other surrounding objects (as in a collision) then they are in effect a two-object system.
I like this. As you indicate, potential energy is a function of the relative configuration in space of the components of the system. Kinetic energy is a function of the relative velocities of the components of the system.
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 non-flat 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
In the case of nuclear physics the nuclear binding energy makes a difference. The mass of a nucleus is not just the sum of the constituent protons and neutrons, the internal energy has inertial mass too.
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:
m = \frac{E}{c^2}
Cleonis
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
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.
Studiot,
Thanks for the advice and the recommendation. I will look for it.
- Roy
Cleonis
May13-10, 06:37 AM
[...] this step is unnecessary. Just solve (2) for t and substitute into (1) and you can get to (4) pretty quickly. [...]
Paraphrasing Goethe: apologies, my friend, for the unnecessarily long derivation, I didn't have time to write a shorter one.
About the relation between F=ma and E=1/2mv2 :
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=1/2mv2 expresses a linear relation when viewed as a function of distance travelled.
Cleonis
May13-10, 07:00 AM
[...] In this view, energy is the more fundamental concept, and forces are just derived from it. [...]
I do think of the concept of 'force' as physics shorthand. When thinking in terms of newtonian dynamics the concept of force is among the fundamental tools of the thinking process.
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
@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 1-2 = distance 2-3. 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.
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.
Cleonis
May13-10, 07:24 AM
Cleonis
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.
Actually no, that's kind of a step away from what I intended.
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.
GRDixon
May13-10, 08:57 AM
I have a pedagogical question, and a philosophical question, both involving energy:
What exactly is energy, anyway?
Direct answers or pointers to relevant references would be appreciated.
Thanks!
Roy
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 ..."
All,
What do you guys think of this? I just found it in, from all places, Wikipedia:
(http://en.wikipedia.org/wiki/Energy#Regarding_applications_of_the_concept_of_en ergy)
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 energy-momentum 4-vector).[14] In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of space-time (= 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 one-another, 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
Actually no, that's kind of a step away from what I intended.
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.
Cleonis,
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
Cleonis
May13-10, 01:57 PM
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.
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 :smile: 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
May13-10, 04:14 PM
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.
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 :smile:
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
May13-10, 04:51 PM
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^{..}
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
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
Mar5-12, 09:37 AM
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.
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.
[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.
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