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Thinking like a physicist....

PostPosted: October 23rd, 2009, 7:10 pm
by linford86
I decided to start a series of lectures/monologues/whatever you'd like to call them to describe how one develops physics. To begin, I'll start with how one develops Newtonian physics. In a future edition, I will discuss how one proceeds from Newtonian physics to a more formalized version of classical mechanics. And then from classical mechanics to quantum mechanics. As the person presenting this material, this will afford me the benefit of reviewing the full course of undergraduate physics.

To begin, let's transport ourselves to an alien planet where no one knows physics. Let's assume that this alien world contains beings that think like we do - in fact, we are going to pretend to be one of them. Furthermore, the planet is comparable in size to Earth and at a similar distance from it's own star. This will allow us to imagine the process by which one first develops certain physical principles without being distracted by the fact that, on this planet, we already know what I am going to present. We might have noticed that there are patterns in the night sky, that the progression of one day to the next always occurs, and that there are other regularities in nature. Whatever regularities we might have noticed, the world is certainly a strange and complicated place. If we are going to have any success in understanding the world, we are presumably going to have to start with something horrendously simple.

Therefore, we start with two balls and two flat pieces of wood. Further, we can prop one of the boards up against a wall. Using only these materials (and maybe some more things to prop the boards into various positions), we will develop several deep and highly profound principles. Indeed, with our board and our ball we can begin to plumb the depths of the universe's greatest mysteries. I know that you are probably thinking that this is an exaggeration; you don't have to believe me since I have yet to demonstrate anything of much importance. But momentarily, I will show you step by step how we will proceed.

And so it begins. I prop board A up against the wall. Board B is propped up infront of board A so that it forms a "U". Now, on the boards, I have marked out even increments with a marker. There are 5 increments on each board. I place the ball at the 1rst increment on board A and let it go. Since I made sure the boards were crazy flat by putting a special surface on them, the ball goes down the board and up the other, all the way to the 1rst increment on B. I catch the ball once it starts to turn around. Now I move board B down more so that the angle between B and the floor has decreased. Again, I place the ball at the 1rst increment. And down it goes, past the 1rst increment on board B and almost to the second increment. I repeat this experiment ad nauseum, and each time B's incline is decreased, the ball moves further along B. And so now I imagine what would happen if I were to make B flat (parallel with the floor.) The ball would presumably move forever. There is a reason that the ball stops when rolling on the floor, but, remember, we are aliens who don't know physics so we do not yet know such things.

Okay, so what did we discover? We found out two things:

1. Once placed into motion, the ball will move forever if there is nothing to stop it.
2. There is something about the motion that is symmetric; the height to which the ball reached when it rolled onto the other side has something to do with the height at which it started. We can imagine that we found this to be consistent no matter where we started the ball. So the ball, somehow, "remembered" where we started it.

Now, if we are sufficiently clever, we can notice another thing about (1). If we place a ball at rest on the floor, the ball does not move unless I do something to make it start moving; I can kick it, for example, or blow on it or do various other procedures to put it into motion. Once in motion, by (1), the ball will simply continue moving forever. Therefore, we notice a similarity between the notions of "rest" and "motion". In particular, we observe that an object will retain its state of motion unless acted upon by something. We don't know quite what that something is yet, however, though we can speculate that it might have to do with the "memory" the ball has in observation (2). We do note that it seems like motion is just as natural a state for an object as is rest. Therefore, we proceed by investigating how an object's motion changes.

To investigate the change in motion, let's place the two balls on the floor about a foot apart. Now suppose that we carefully hit ball A in its exact center (this is important, as pool players probably know) and watch it collide with ball B. We notice that ball A stops moving and ball B begins to move at the same speed as ball A was moving at before it stopped. Something funny is going on here; it seems like the "memory" of ball A's motion has somehow been transfered to ball B. Furthermore, if ball B is heavier than ball A, we notice that the speed at which ball B rolls away with is slower than the speed at which ball A hit it. We also notice that if ball B is lighter than ball A then the speed at which ball B rolls away is faster than the speed at which A hit it.

From here, it seems natural to propose that there was some stuff - we can call it the amount of motion - which was transfered from A to B in the collision. This stuff is presumably somehow related to the memory that the ball had in the first experiment. And, furthermore, we can imagine that every time the ball interacts with anything there is some amount of motion that is transfered. So the motion never really ceases; it only gets transfered from one thing to another. And since the rate of change in the amount of motion seems to be important, we can define a new quantity as its rate of change:



where is the new quantity (we call it "force"), is the amount of motion (which we will now call by the shorter name "momentum"), and is time. We can note that this definition is always true and trivially so; saying that it is universally true is just a statement about the consistent use of language. You can call it a law if you wish (Newton does this) or you can just call the definition of force. I chose the latter. But keep in mind that all we have really observed so far is the way that these objects move and the way they move in response to each other. We have not yet said why is conserved in the interactions, though there is a powerful argument for this. I will present that argument in a moment. In a future instalment, I will describe an even more powerful argument from a beautiful principle called "Noether's Theorem".

Now, one thing that's worth turning to next is how we should define . For this, we note what we have discovered about the "quantity of motion" so far:

1. is proportional to the mass . We know this because a light body hitting a heavier one causes the heavier one to move slower than the lighter one and because the inverse is also true.
2. is proportional the velocity (i.e. the speed along a straight line; we think of this a funny arrow called a "vector".)

We therefore find it conveniant just to define:



We didn't have to definite it that way. We could have put a proportional constant in front for example. However, we could not have made it a more complicated function of or of without altering either of their definitions; since we have defined as the thing which is transfered in collisions, we know that when all of the motion is transfered, the amount of motion carried by the second body is equal to the amount originally carried by the first. This forces us into choosing a function which consistently produces the correct ending velocity for the second body. Of course, also works (take the root of both sides) but this is more complicated than it needs to be. Therefore, is the simplest form for consistent with experiments.

Now, we can calculate . Since in collisions, the mass usually does not change, we can assume that . If and only if this condition is met, we have that:



It is now natural to introduce a new notion - that of acceleration. I define the acceleration, , as . I therefore see one of the most famous physics equations of all time:



But we warned - to get here, we needed to assume that the mass did not change. Provided that the mass changes, we do not get our beloved equation. Of course, this runs contrary to what every high school physics teacher tells his classes; nonetheless, the argument is unassailable. After all, we have done little more so far than to define a few things in convenient ways based an a couple of experiments that we ran. In addition, we have re-traced the footsteps of our great scientific forefathers (we started with Galileo and now we are standing with Newton.) It was much more difficult for those guys to find all this stuff because they were rather confused about mathematics and several other things. Groping around the darkness, however, these are essential the same steps they took.

We can go farther still. We have yet to derive conservation of momentum, for example, or to figure out how to calculate the trajectory of a projectile. All of these things lie ahead of us, and if you are sufficiently mathematically gifted, you might already see how we are going to get there. But, first, we need an extra principle.

Imagine a ball resting on the top shelf of a bookcase. If we removed the shelf the ball is resting on, the ball will simply fall until it hits the next shelf. Likewise, if we remove that shelf, the ball will fall once more only to hit the next remaining shelf. We can imagine performing this procedure an arbitrarily large number of times provided that we have a large enough bookshelf (of course, make the bookshelf too large and the ball will no longer noticeably feel the influence of gravity. But we are assuming that we don't make a bookshelf that large which seems reasonable.) So what influence is it that keeps the ball from falling through the shelf it rests on? Obviously, once the shelf is removed, the ball feels a force since it accelerates and it has a non-zero mass. But what is it that counteracts the force while the ball sits on the shelf? Clearly, there is a force pushing back up on the ball in such a way that the net acceleration is zero. These two influences cancel each other out and they do so absolutely perfectly; if they only nearly canceled, we'd see the ball slowing slide through the shelf or slowly move upward. But we observe neither of those two things; the ball just sits there. We therefore conclude that the forces cancel completely.

But everywhere we look, we find that a similar symmetry exists; whenever A and B have forces between them, the force on A is the same as the force on B. We can proceed to measure this phenomenon using force gauges on various different things. And so we have observed an additional principle:

For every force that acts on an object A from some object B, there will exist an equal and oppositely directed force on B.

From here, we are in a position to show the conservation of momentum. Imagine some arbitrary volume of space containing objects 1, 2, 3, etc. Now we calculate the total rate of change of momentum in that volume:



But notice that each term is a force. And so we have a set of forces, each equal to the sum of the forces exerted on it by the other bodies:




etc.

Now, we collect those terms together and insert them into the total rate of change equation:




Immediately, you should notice that all of the terms cancel. Therefore,



Implying that is a constant.

In fact, this is true for a volume of indefinite size. Since the volume can be any arbitrary size that you wish, you could consider the entire universe and the argument would still work. And, thus, we have arrived at our first cosmological result - momentum is conserved for the entire universe. There is a deeper way to show this which does not depend on the notion of force and, like I said before, I will show that method in a future post.