## Conservation laws.

Perhaps the first conservation law to be discovered was the conservation of mass. It simply tells us that in any chemical or physical process, the total mass mass of a closed system remains constant. [We learned later, in the early 20th century, that this law needed to be modified to accomodate conversions of mass to energy and vice versa.]

In the context of physics, conserved means "does not change" over time.

We have seen how the concepts of kinetic energy, work, and momentum arise naturally from Newton's laws. As we use these concepts, we realize that they are part of even more general laws—conservation laws:

• Conservation of energy.
• Conservation of linear momentum.
• Conservation of angular momentum.

All conservation laws are stated to apply to closed systems. A closed system is any collection of things (usually particles or objects with mass) for which all outside influences on the system are absent or negligible. Of course no system can be perfectly closed. But if we can monitor all outside influences precisely, we can "correct" for them in order to test the conservation laws.

Mass and energy are scalar quantities, so in the accounting of totals we simply find the algebraic sum the individual contributions. (But we must remember that scalar quantities can have algebraic signs.) Linear and angular momentum are vector quantities, and must be summed by vector addition.

Conservation of momentum is, perhaps, one of the simpler of these last three conservation laws. Momentum comes in only one kind: mv.

A body may have energy in either of two ways:

• Kinetic energy, (1/2)mv2
• Potential energy.

Kinetic energy is energy due to a body's motion. Potential energy is energy due to a body's position in space relative to other bodies that exert forces upon it. A body at rest in your chosen reference frame has zero kinetic energy, but may have potential energy. A moving body has kinetic energy associated with it, and may also have potential energy.

Is the energy "in" the body? Is it something the body "posesses"? It certainly is not a substance, for that question was settled by experiment in the 18th century. Clearly it is not something associated only with the body. An automobile moving at speed v with respect to the roadway is (1/2)mv2. But with respect to a train moving at that same speed parallel to the roadway, the auto's kinetic energy is zero, for its relative velocity is zero with respect to the train. So the kinetic energy depends upon the measurement frame of reference. But whatever inertial (non-accelerating) reference frame you use, changes of kinetic energy will be unaffected by this choice. [We will have to revisit this question when we get to the subject of Einstein's special theory of relativity.]

Likewise, potential energy has a value dependent on your choice of measurement reference frame. A body resting on a table has zero gravitational potential energy with respect to the tabletop, but larger potential energy with respect to the floor below. But whatever "zero" inertial reference frame you choose when doing a problem, changes of potential energy will be unaffected by this choice.

## Where is the energy hiding?

But on a deeper level, consider a mass on a spring. If we compress the spring and latch the mass in place, we say the system (mass and spring) has potential energy. If we release the latch, that potential energy can appear in another form: kinetic energy of the moving mass and spring. But if, instead, we carefully detach the spring and remove it without disturbing the mass, the mass now has no potential energy due to the spring. So where "was" the potential energy? In the spring? In the mass? In both? If so, in what proportion? All of these are the wrong questions. We don't need to say "where" the energy resides, we need only say that there's energy "associated with" a certain configuration of objects (mass and spring) that are exerting forces on each other and prevented from moving because of the latch.

In the mass-spring example, we could carefully detach the mass from the spring, and through some latching mechanism keep the spring commpressed. Now the energy certainly isn't in the mass, but in the latched spring itself, due to the forced reconfiguration of the atoms and molecules that make up the spring. But to say this is just passing the buck. Is the energy in the atoms and molecules themselves, or in the "springy" force fields that hold matter together? We know the energy is associated with the spring and not something else. Suppose we had two identical springs. One is relaxed. One is compressed and locked into position. Both have the locking mechanism, however. We drop each in identical vats of acid and let them completely dissolve. The acid bath with the compressed spring will reach a slightly higher temperature (indicating that it gained more thermal energy) than the other one.

A similar question arises with other energy storage devices. A pair of metal plates form a capacitor, and it is capable of storing energy when its metal plates are charged oppositely. We know the amount of stored energy, because we can calculate or measure the work required to assemble those charges. Now is the stored energy in the plates, or in the electric field between the plates? For many practical computational purposes we say the energy is in the field, and we can calculate the energy by doing an integration over the space of the entire field. But isn't the field itself just a "fiction", a mathematical device without physical substance? We can't reach into a region of space and extract "field lines" any more than we can probe the space with an "energy detector" to measure the energy content of the space.

This returns us to where we started. Energy is not a "stuff" or a "substance" or a "fluid" or a "gas" or anything material. It is only an accounting device. It is a way of labeling things to help us know how those things behave when they interact. These labels are tags, like "this body is of mass m moving at speed v in a specified inertial reference frame, so it's energy is (1/2)mv2" and "this system has been configured by having an amount of work W done on it, so it can exchange that much energy with other things, under the right conditions." It's tempting to think of energy as a substance, as was done before the 18th century, but we must realize that that's too simplistic. It is one of the dangers of analogical thinking.

Potential energy:

A familiar type of potential energy is that due to the position of a body in the earth's gravitational field. If we move a box of mass m from the floor to a table at height H above the floor we must do work on the box. Imagine moving the box from floor to table at constant speed. By Newton's third law, a body moving at constant speed has zero acceleration, and therefore the net force on the box is zero as we lift it. I exert a force upward to raise the box the distance H. The earth's field exerts a force of equal size downward on it, of size mg. Therefore the force I exert on the box is also of size mg, but upward. The work I do in moving it the distance H is mgH. The box began at rest and ended at rest, so its kinetic energy has no net change. We say that the box now has a potential energy of mgH, with respect to the floor.

There's a minor quibble possible here. In lifting the box, I must have accelerated it at the start, and decelerated it at the end. So I had to do a bit of additional work at the start, but at the end the box did the same amount back on me. A proper calculus treatment could resolve this point, but it doesn't affect our conclusions, and is hardly necessary in order to make the conceptual point about potential energy.

An alternate interpretation is possible. The work I did in lifting the box was positive work on the box, because the force and displacement were in the same direction. At the same time the earth was doing the same amount of negative work on the box. I.e., the box was doing positive work on the earth. So one could say that the lifting work I was doing was actually work done on the earth, through the intermediary object, the box. Or one could say that I was doing work on the box-earth system, forcing the two objects (mass and box) farther apart and "parking" them at that new separation. (The table plays the same role as the latch on the copressed spring.)

The bottom line is this. In such situations, for the purposes of calculation of energy changes we "pretend" that the energy due to its position with respect to the earth is associated with that body. So when books speak of "the potential energy of a body" they mean the amount of work required by an external force to place the body in that particular position. It is a "bookkeping" method for keeping track of energy changes associated with that body.

For the "practical" person who just wants answers, all of this philosopizing may be annoying, and not particularly helpful. One is reminded of the old tale of the centipede who over-intellectualized a simple process—walking.

The Centipede.

A centipede was happy quite,
Until a frog in fun
Said "Pray, which leg comes after which?"
This raised her mind to such a pitch
She lay distracted in the ditch
Considering how to run.

—Anon

Other manifestations of potential energy

The mass and latched spring example above is a situation where a system has potential energy. When molecules are bonded together they have molecular potential energy. Atoms have potential energy due to the binding forces of their component parts, and so do atomic nucleii. Potential energy within atoms and molecules is present in all matter, and can be released in other forms when the internal geometric arrangement of the atoms and molecules changes.

Manifestations of kinetic energy

The consitutent parts of matter, molecules and atoms, are continually in motion, and this motion represents kinetic energy. A hot cup of coffee on the table has no bulk motion due to this, for the directions of the individual molecular velocities are disordered. The vector sum of their momenta at any given time is very nearly zero, so we observe no acceleration of the cup as a whole.

If someone pushed the cup, propeling it across the table, the momenta of the molecules in the cup now have a non-zero sum equal to the total mass of the cup times its velocity. We can say that the molecular velocities are now partly disordered, but also have an ordered additional component in the direction of motion of the cup. The coffee is still thermally hot due to the energies of the disordered components of velocity, but the ordered components do not contribute to the thermal temperature of the cup's contents.

We often find it convenient to treat these two "kinds" of kinetic energy separately: ordered and disordered. The thermal energy content (which determines a body's temperature) is due to the disordered motions. The kinetic energy of the system as a whole is due to the ordered motions.

So, we recognize thermal energy as a "kind" or "manifestation" of kinetic energy.

Now, let's reconsider the conservation of energy law. In using it, we must take into account all of the manifestations of energy we discussed above to obtain the net energy of the system. Then, we will find that the net (total) energy of the system is conserved.

Energy exchange between systems

Systems may exchange energy in three ways:

1. By forces acting across the system boundary, doing work on that body.
2. By forces acting across the system boundary, changing the internal thermal energy of the system.
3. By material objects crossing the system boundary, carrying along their energy into the system.

The third one is so obvious that we need not spend much time on it.

The second one may need comment. We reserve the word "heat" to be a measure of the amount of thermal energy increase of one system due to an equal amount of decrease in another system. We do not speak of "the heat in a body". The energy in the body is "thermal energy" or "internal energy". The word "heat is used only to represent changes of internal energy due to interactions across the system boundary.

Unfortunately some introductory physics and chemistry textbooks still use the word "heat" when they should speak of "internal thermal energy". Old habits die hard.

It may seem trivial, but we'll say it anyway. If system A and B interact, and because of the interaction A gains energy, then B loses that same amount of energy. This follows from conservation of energy applied to the super-system consisting of the two systems taken together.

Perhaps one reason that conservation of energy took so long to become understood and appreciated is the ever-present process of friction. A box sliding down an inclined plane loses energy due to the work done by microscopic forces at the box/plane interface. These processes are called friction. The kinetic and potential energies are not conserved since some energy is lost from the system as thermal energy produced by friction. That amount of energy is quite difficult to measure, for it increases the thermal energy of both bodies near the surface. It is sufficiently troublesome that elementary physics textbooks avoid it in many problems, saying "ignore friction" or "consider that friction is a negligible effect in this problem." Even when textbooks do problems with friction included, they often do so in ways that do an injustice to the physics concepts. [See Arnold B. Arons, A Guide to Introductory Physics Teaching (Wiley, 1990) Chapter 5.]

Momentum.

Momentum conservation is in some ways far simpler to understand and apply than energy. Momentum comes in only one variety, mv. We must always treat it as a vector, but aside from that there are no "gotchas" in its application.

Perhaps the only pitfall students encounter is to incorrectly assume that if something doesn't seem to be moving, then it can't have momentum. Consider a bouncing ball. The ball impacts the floor with velocity v, and rebounds with velocity -v, losing no kinetic energy. This seems reasonable to the student, because the floor, not moving, gains no kinetic energy. But the momentum of the ball changes from mv to -mv, a net change of 2mv in the upward direction, So the floor must have gained momentum 2mv downward. How can the floor gain no energy, but gain considerable momentum? The floor has such large effective mass that it can have momentum with very small (unnoticeable and unmeasurable) velocity. Fair enough. But then how can the floor have zero energy if it has nonzero momentum? Alas, the answer to this intriguing question requires some calculus. But you can get the flavor of the argument from this discussion

We saw that the energy of a system can be changed by two kinds of outside influences: work and heat, both due to forces acting across the interface. There's one way that the momentum of a system can be changed, also by an external force, acting across the interface, exerting an impulse, FΔt, on the system.

Are conservation laws absolute?

Progress in physics sometimes requires us to reinterpret old laws. Newton's laws have been modified by Einstein's relativity. The law of conservation of mass is now incorporated into the conservation of energy law, now that we realize that mass, by its very existence, has energy, and that we can convert mass to energy and vice versa. Can we expect to see further refinement of conservation of energy in the future, perhaps in a way that would allow us to get more energy out of a machine than we put into it?

Is it possible that someday we will find a situation in which momentum or energy is not quite conserved, that in some cases these quantities in a closed system might increase, or decrease?

These conjectures appeal to those who think that "anything is possible, if you are clever enough". They point out that we haven't yet examined every conceivable machine, process or device. Just because we have never observed a case of energy or momentum non-conservation, we shouldn't dogmatically rule it out. Most scientists scoff at such a notion. Why?

When is it reasonable to hope for a violation of a known and well-tested law? Why do most scientists feel totally confident that conservation laws of energy and momentum will never be found to be wrong, not even in one isolated instance?

Part of the reason is the tight integration of these laws with so many other laws that are equally well tested and reliable, Newton's third law, for example. If momentum were not conserved in a particular case, Newton's third law would not be correct either. It has been said that the conservation laws are the most fundamental laws of physics, from which other laws can be derived. Historically it was the other way around, those "other laws" were discovered first, and then served as a basis for the conservation laws.

Scientists know that a thing is not true because you want it to be true, or because you believe it in your gut, or even because mankind might need it to be true. They generally don't go looking for exceptions to a well-established law unless there's some good reason to suspect that such an exception might be found and some indication of where to look for it. In the case of conservation of energy, momentum, and angular momentum, there's not a shred of credible evidence, not one suggestion from theory, that there's any exception to these laws in macroscopic phenomena. There are plenty of other areas where there's reason to suspect that laws might need modification, and those are the ones actively pursued by experimenters and brainstormed by theorists.

—Donald E. Simanek, Feb, 2005, edited Jan, 2011.

Go to the next chapter, Conservation of Mass.
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