Help me rationalize conservation of energy

[maximiliano]

Help me rationalize "conservation of energy"

So, I understand conservation of energy, but...I must be missing something, because in my mind, there are some problems with it. Let me explain by using an example. I'm sure you'll set me straight.

Okay, let's say that I have a bow and arrow. I draw back the bow to its full extent, and shoot the arrow into a concrete wall. What has happened?

1) the chemical energy in my MUSCLES (which came from the food I ate, which was derived from...ultimately back to solar fusion, etc) was transferred to the BOW
2) The BOW was then charged with potential energy
3) When I released the BOW, most of the energy was transferred to the arrow thus charging it, the arrow, with KINETIC energy.
4) As the arrow is flying along its path, it is transferring small amounts of kinetic energy to the air molecules, and some work is being done in the form of moving air, thus overcoming its inertia, and generating relatively small amounts of kinetic energy there.
5) When the arrow smashes into the concrete wall, ALL of its remaining kinetic energy is used up doing work, such as destroying the structure of the arrow (let's assume it is smashed nearly into pulp), the sound of impact and damaging some smaller part of the concrete wall (small dent).

So...at this point, it seems to me that ALL of the energy that was ultimately in the moving arrow...has now done work and is gone forever. How is this not so? Seems to me, that for practical purposes...once work is done, energy is in fact gone forever.

Part II:

I cut a branch from a tree, and allow it to dry out. I then set it on fire, thus breaking the hydrocarbon molecules (which were created by the energy ultimately from the fusion of H to He plus energy...in our sun) into the components, or at least small parts (H, C, O, CO2, O2, etc). The heat I feel is just the energy that was once the molecular bonds, which no longer exist. Again, in my mind, ...I see work has been done, and those molecular bonds have been broken. The energy that was created from solar fusion has now performed its final possible level of "work"...and is now gone. Set me straight please. At the end of the day, I do know that mass is energy, and that mass can not be created nor destroyed...but it seems there must be more to this. When the sun burns out...is the same amount of energy as 10 billion years prior present, even though work was done?? It seems there must be a difference between absolute energy (energy of mass) and something like practical energy, which can be used to do work. ??

Sorry for length. This has been bugging me for years, yet I know it to be a very basic concept.

 

[IsometricPion]

So...at this point, it seems to me that ALL of the energy that was ultimately in the moving arrow...has now done work and is gone forever. How is this not so? Seems to me, that for practical purposes...once work is done, energy is in fact gone forever.

Energy does not do work, it simply exists. The energy that was in the arrow has gone into the energy of the sound waves, the heat energy of air and wall, etc. Conservation of energy does not always hold in an obvious manner in everyday situations because there are so many ways for the energy to be transformed into forms of energy we do not easily recognize. It is only we one looks at things microscopically that one can see that the energy is simply being spread out into a small increase in the kinetic energy of a very large number of atoms (and perhaps increasing the electrical or gravitational potential energy of some atoms).

When the sun burns out...is the same amount of energy as 10 billion years prior present, even though work was done?? It seems there must be a difference between absolute energy (energy of mass) and something like practical energy, which can be used to do work. ??

Yes, the same amount of energy is present it is just more spread out than when the Sun first began emitting light. The large majority of the energy went into producing light that will still exist when the Sun burns out. The rest of the energy is spread into an increase in the average kinetic energy (i.e., the temperature) of a very large number of atoms, those in the Earth for example.

The decrease in the usefulness of a given amount of energy is related to the 2nd law of thermodynamics: the entropy of a closed system never decreases. Entropy is a measure of how spread out the energy of a system is, and it obviously increases in all of the situations you mentioned. So, while the energy that was in the Sun at its formation still exists, the entropy associated with that energy has increased dramatically such that one could not do as much "useful" work with the energy in the forms it is in today.

 

[maximiliano]

thank you for the reply...but I'm still questioning. So, ALL of the kinetic energy in that arrow was just spread out? What about the work done, where a wooden arrow was turned to pulp?

A better example maybe- I take a 1/4" thick piece of steel rod, and with my hands, bend it to a 90 degree angle. No sound...no heat...just WORK DONE. seems to me, the energy (or that portion anyway) that was in my muscles...is now GONE. Work was done. The energy is gone forever. It could not have gone into the steel bar, because the exact same number of atoms and molecules are there, as before. No heat was created. NO sound was created. No light was created. No potential energy hiding anywhere. Just a bent bar and energy missing from my biological system ??

 

[IsometricPion]

A better example maybe- I take a 1/4" thick piece of steel rod, and with my hands, bend it to a 90 degree angle. No sound...no heat...just WORK DONE. seems to me, the energy (or that portion anyway) that was in my muscles...is now GONE. Work was done. The energy is gone forever. It could not have gone into the steel bar, because the exact same number of atoms and molecules are there, as before. No heat was created. NO sound was created. No light was created. No potential energy hiding anywhere. Just a bent bar and energy missing from my biological system ??

Biological systems are complicated. In order for your muscles to contract potential energy is released by the breakdown of ATP into ADP, this occurs even if you aren't doing any physical work moving an object. The contraction of your muscles produces heat (which is where essentially all of the energy from the ATP goes if you aren't doing any work on an external object). Generally when one bends an object one adds heat to it, for example if you bend a clothes hanger back and forth repeatedly the bent area will heat up. Since the rod has changed shape the distribution of the potential energy in the bonds holding it together has changed (though there may or may not be a net increase in the potential energy stored in the rod). The sum of the heat energy radiated away and the heat energy due to amount the temperatures of you and the rod increase is equal to the energy you used to bend the bar (assuming no change in the potential energy stored in the rod).

 

[Delta Kilo]

What you observe is called 2nd law of thermodynamics. While the total amount of energy does not change, it gets converted irreversibly into less organized and useful form, usually ending up as heat dispersed in the environment.

 

[nonequilibrium]

Hello maximiliano :)

Good first post! Your reasoning is entirely correct and is the same thought that got the 19th century physicists like Carnot, Helmholtz, Joule, ... thinking.

As you say, there is a first law of thermodynamics: conservation of energy. But as your reasoning shows, applying this law to the world around us, you notice something: even though "absolute energy" is conserved, it seems there is an important (but perhaps vague) distinction between "useful energy" and "useless energy". The 19th century physicists noticed this also, and called the useful energy W and the useless energy Q, known as work and heat respectively. Okay actually this is wrong, you cannot label some energy as Q and some energy as W, it leads to contradictions, rather we use Q and W for energy transfers instead of energy content. This last note might seem strange and that's understandable, as the 19th century physicists also struggled with it, but it's rather cleared out now. Textbooks will give you more details. But let me continue with my general story: it seems that the useful energy can transfer into useless energy and practically lost forever. This can actually be quantized by using entropy, developped implicitly in the work of Carnot (although he didn't name it entropy, and he even used a wrong theory of energy, but apparently it worked out until a certain extent) and Clausius. The result was a new thermodynamic quantity, like energy, called S (as you know, of course), and defined in terms of temperature and heat transfer.

This new quantity gave shape to a second law of thermodynamics: the entropy of an isolated system never decreases. The meaning of entropy was completely unintelligible, and even the definition itself was not very pleasing, but it all worked out and entropy indeed was never seen to decrease. If you would calculate the energy for your stories, you see that energy is constant, but if you calculate the entropy, at the end the entropy is much higher than in the beginning. The second law of thermodynamics in a sense characterizes an arrow of time: it distinguishes tomorrow from yesterday by a quantity that can only increase, never decrease. Arrows go broken and smash into walls; but smashed bits never fly out of walls into a new arrow.

It were Boltzmann and Gibbs who elucidated the concept of entropy and more generally the arrow of time. You have to understand that in the 19th century the notion of atom was quite controversial and the 2nd law was used as evidence against atoms: if nature would be made of atoms moving according to Newton's laws, and Newton's laws were time symmetrical (which they are, i.e. they are invariant under a change $t \rightarrow -t$), then how could we possibly live in a time asymmetrical world (which we do)?

Not only did Boltzmann solve this puzzle (it took him many years, even a genius like him struggled with unifying Newton & atoms on the one hand with entropy & time asymmetry on the other hand), he actually showed that the second law of thermodynamics is a logical consequence of the idea of atoms moving under Newton's laws. In effect, he showed that the second law follows from the first law (assuming the existence of atoms). The idea is wonderfully simple once you see it: imagine having your hand full with marbles and releasing them. What do you see? They drop onto the floor, but they don't stay in place, rather they spread. It seems, naively, as though there is a force pushing these marbles apart, as they spread across the room, analogous to an "entropic force" that the 2nd law suggests (a force driving things toward higher entropy) but upon further reflection you see that there is no force: it's simply more probable for the marbles to spread out than to not do it... That's it! You see that there actually is a chance that the marbles come back together, randomly, but you also see that that chance is really really really small. Likewise, Boltzmann saw that the 2nd law isn't an absolute law, it's just really really really likely.

The second law is a consequence of atoms and probability: think of your arrow example. The arrow collides with the wall. What happens microscopically? The atoms, first moving in unison, start bouncing into the atoms of the wall, at least the front tip of the arrow does, the back end bumps into the middle of the arrow, and all the atoms start bumping into each other chaotically. You see what happens: the kinetic energy stays kinetic energy, but just much more chaotic kinetic energy, much more random. Such chaotic kinetic energy is perceived by us by a hot arrow: if you touch the arrow afterwards, it will feel warmer than before, and this is because the chaotic atoms are bouncing against your hand, making the atoms in your hand also bounce like crazy, which is registered by your nerve endings.

You also see that actually there is a chance that all the randomly bouncing atoms in the arrow by pure luck start moving in the same direction, which would result into your arrow taking off into the sky again (from its own accord!). It is possible, think about it from an atomic point of view: Newton does not forbid it can happen. You just realize it will never happen because there are many more ways to be disorderly than orderly.

 

[russ_watters]

A better example maybe- I take a 1/4" thick piece of steel rod, and with my hands, bend it to a 90 degree angle. No sound...no heat...just WORK DONE. seems to me, the energy (or that portion anyway) that was in my muscles...is now GONE. Work was done. The energy is gone forever. It could not have gone into the steel bar, because the exact same number of atoms and molecules are there, as before. No heat was created. NO sound was created. No light was created. No potential energy hiding anywhere. Just a bent bar and energy missing from my biological system ??

Yes, heat is generated: try it with a paperclip, bending it a dozen times or so.

Energy also goes into permanent re-arranging of the molecules of the metal rod.

 

[maximiliano]

Mr. Vodka, et al; THANK YOU! I will have to give your answer a longer read...and some thought. I have zero real background in physics, in an academic sense (think I took a physics class in high school maybe...but never paid attention) . What I know, is mostly just my own thinking about things. I do a LOT of mountain biking...and over the many hundreds of hours, over the years, I think and think about how things work (everything). The idea of conservation of energy always bugged me, as in my head, it didn't work, but was so central. This actually started as I would climb a mountain on my bike , and then coast down the other side. I thought to myself, okay, on the way down, I just took mass of X and moved it a distance of _____ in _____ time, against _____resistance. That is work done. But, I didn't break a sweat. Well, it was obvious that I was storing potential energy as I was climbing, overcoming the acceleration force of gravity...then using that stored potential energy on the other side. But...when I got to the bottom, I was told (via conservation of energy) that the energy was still ...well, somewhere? In my mind, for the energy to still be there (or anywhere), then I was looking at 2+2-2=4. It appeared that something just came from nothing...or that work was done for "free".

Anyway, I'll give it a read again. I'm glad to see that I'm not the only one dumb enough to stumble on this one:tongue: Thanks again so much for the detailed response.

 

[russ_watters]

No, this is a common question. Kinda disconcerting though, isn't it?: all of that blood and sweat put into your mountain bike ultimately amounts to nothing more than a little waste heat dumped into the atmosphere!

 

[Claude Bile]

Hi maximiliano,

Best to begin by considering objects as point masses. That way we can construct the concept of energy step-by-step.

1) Work done on point masses can only result in translational motion (i.e. translational kinetic energy).
2) Work done on perfectly rigid extended bodies (discs, rods, spheres etc.) can result in both translational and rotational motion (translational and rotational KE).
3) Work done on elastic bodies (bendy rods, ropes, elastic bands) can result in potential energy in addition to the above. (Potential energy = energy (or work) that can be recovered at a later time that is not related to motion).
4) Work done on bodies that consist of a multitude of tiny masses (atoms) that each possesses their own internal energy result in thermal energy in addition to the above.
5) Similarly, chemical energy is derived from the atom-to-atom interactions that exist within the solid.

In summary, the more complex our model of what "an object" is, the more ways work can be done on it.

It probably doesn't answer all your questions, but I hope you find it insightful.

Claude.