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Okay. Today we’re gonna try to finish up our discussion of Newton’s laws and I wanted to go back to Kepler for a minute. You may remember Kepler’s Third Law. The way I have it up on the screen is the way you originally saw it P square equals A cubed. The period of a planet’s orbit in years squared is equal to the average distance of the sun from the planet cubed. Or half of the major axis of the planet. That’s a fairly simple equation. When Kepler used this equation and calculated the periods and distances of the planets, they all worked out nicely. And so he thought he had discovered a law of nature. It is, in many respects, a law of nature. Unfortunately, it’s incorrect as written. Isaac Newton, after he came up with his laws of gravitation, his laws of motion, realized that he could go back and derive Kepler’s three laws from his own equations and laws of motion. Because his laws of motion and equation for gravity are actually more general than Kepler’s, he reasoned that he should be able to come up with Kepler’s laws and he did. He showed with the law of gravitation that all orbits had to be circles, ellipses, parabolas or hyperbolas. Now, Kepler said all the planet orbits were ellipses. Well, ellipses are among that list. But Newton realized you could have any of those kinds of orbits. They were all permitted by gravity, by the law of gravitation. And so he was able to show that, yes, planets do go in elliptical orbits, but they could also go in circular orbits. It just so happens that a circle is only one possible case of an ellipse, right? It’s an ellipse that’s round. And so most of the planets don’t happen to go in that particular ellipse. They mostly go in flattened ellipses.
Actually, if you look at the orbit of Venus it is extremely close to a circle. So close that if Kepler had used measurements of the p lanet Venus instead of the planet Mars when he was deriving his elliptical orbits, he would’ve decided the orbits were circular. Because Venus’s orbit is extremely close to being circular. So it was almost an accident — I won’t say it was an accident because there were more observations of Mars, but it was a good thing that Kepler looked at Mars and not Venus or he never would’ve discovered that all planets moved in elliptical orbits. So Newton derived this — Newton was able to derive easily the law of a reas. As a planet sweeps around the sun, it sweeps out equal areas and equal intervals of time. All that’s saying is that the momentum of the planet is constant as it goes around the sun. So it follows right from the momentum discussion having to do with Newton’s First Law and Second Law. Then he realized he could derive Kepler’s Third Law from the law of gravitation, but this is what he came up with. There’s an extra couple of terms in the equation now. M1 + M Now where did they come from? Well, you remember in the law of gravitation the masses of the objects show up. Well, when he derived Kepler’s Third Law, the masses were still in there. Now, you look at those two equations and you say, “How did Kepler come up with an equation without the masses? Newton derives the equation and comes up with the masses. How did Kepler’s original equation work?” It’s obviously not the right equation. There should be masses in there. Well, you have to look at the units. The units for period are years, earth years. The
objects going around each other. You can use it for the moon of Pluto going around Pluto. You can use it for two stars going around each other. There are lots of double stars in the sky, stars that are actually going in orbit around each other. You can use this equation for that. As we’ll learn toward the end of the book, there are even double galaxies and they go around each other and they follow this equation. So it’s a very general equation once you put the masses in. The way Kepler derived it, it was only useful for planets in the sun. Okay. One more thing I wanted to mention having to do with Newton’s law of gravitation and here it is again, just so you remember what it looks like. The masses are there. The author of your text uses A and B. On the previous equation I used 1 and 2. It really doesn’t matter what you call them. We’re just talking about two masses interacting gravitationally and this is the equation that describes the forces between them. The R square in the denominator makes the equation what we generally refer to as an inverse square law, inverse meaning whatever it is is in the denominator and square meaning the term is squared. So this is an inverse square law equation because you have distance squared in the denominator. Turns out we have many forces in nature, many things in nature, that follow inverse square laws. This turns out to be a pretty general kind of thing. If you’re talking about magnetic forces between objects, they follow an inverse square law. The farther apart the objects are, the less magnetic force they have on each other. Electrical forces, same thing. The farther apart they are, the less force they have on each other, the closer they are, the more force, and it’s again an inverse square relationship where the distances apart are in the denominator and they’re squared.
We also have light. The way light spreads out, it follows an inverse square law. And let me show you a drawing of that. Okay. Wha t you have on this drawing is a lightbulb in the center just to show you light coming out from somewhere. If you are at the first distance, distance 1, from the lightbulb you will see that lightbulb a certain brightness. A certain amount of light will go into your eyes. A certain amount of light will hit an area as you can see up there, a small area, with a certain amount of brightness, a certain total amount of light will hit that small area. As you can see, as the light moves farther away, the area that that same amount of light will hit increases. That’s just because the light is spreading out. If you have a small sphere around a lightbulb, every bit of that sphere receives a lot of light and so the sphere is lit up a lot in each small area of the sphere. But if you have a larger sphere around the same lightbulb, then that same amount of light coming out of the lightbulb now has to coat, if you want to look at it that way, coat a larger sphere and so the light is spread out more thinly. Well, as yo u increase the size of the sphere, how does the light spread out more thinly? Well, we can call the radius of the sphere R. And if you doubled the radius, as is in this drawing, you are making the area that has to be covered four times as large. This is just geometry. If you have a surface of a sphere, the area of that surface goes as the square of the radius of that sphere. And so you’re spreading the light over a larger area. It is getting thinner or weaker on each bit of area. And so the light drops off in brightness as one over the square of the distance, or one over R square. So again even with light leaving a lightbulb, it follows an inverse square law. Gravity follows an inverse square law, light follows an inverse square
a force back on me, and you might notice it. Are you ready? I put a force on it and stopped. Did you notice the force it put on me? It shoved my arm down. Take another look. I’m going to stop the wheel. You see it put a force back on me? Newton’s Third Law. All forces are mutual. I put force on the wheel, it puts a force on me. Now we’re going to see some transfer of momentum that is more easy to notice as a transfer. You saw a slight transfer there but it wasn’t very obvious. And we’re going to use our student assistant here. You’re going to begin moving around. The first thing I want her to do is take hold of this bicycle wheel by the handles. Don’t hit yourself in the breakfast. Grab each handle. Now hold it very still. Don’t do anything. That wheel has momentum. She’s gonna put a force on the wheel by tilting it. That’s putting a force on it. Tilt it. It puts a force back on her. Tilt it some more. Now go the other way. She’s putting a force on the wheel by tilting it, it’s putting a force back on her by making her go around. Go the other way. It works. A transfer of momentum. Tilt it a little more. Don’t be afraid of it. Tilt it up. Keep going. Come back around. Get to the class and then stop. She’s got control. She can actually move herself around. All she’s doing is putting a force on the wheel to change it’s momentum. Remember if the you change the direction, you change the momentum. And it’s putting a force back on her, but the only way it can force her is to make her go around. There’s not much else it can do. And so it makes her go around. Do you want me to take that from you? Okay. Do you want to stop this and feel the momentum exchange? Not on that? Okay. Step down. Just grab it. You felt it? Yeah. Okay. Please go sit down. Thank you. Everybody see the exchange of momentum here? She puts a force on the wheel
by tilting it, it makes her go around. Now, sometimes you try to put a force on something but there’s no place to transfer the momentum. If you can’t transfer the momentum, you can’t do it. The momentum won’t go away. Now, I’m gonna show you an instance where this works. You see this string? Now, some of you are betting types. I can tell. If I let go of this wheel and I’m holding the string, what will happen to the wheel? Don’t laugh. What will happen? It’s gonna fall down. Anybody want to bet against that? It did. It fell down. Now what’s it gonna do? Didn’t fall down. It can’t get rid of its momentum. It can’t exchange it anywhere. Because the only place it can exchange it is through the string and the string is too flimsy. It’s moving around. So all the wheel can do is try to twist the string up but it can’t fall down because it has to transfer its momentum. Now, it’s slowly doing it because there’s friction and it’s transferring some in the string, but it’s an arduous process for it. Ahhhh, straighten it back up. I have to put a force on it to bring it back up. Doesn’t fall down. It has trouble exchanging the momentum, moving it somewhere else, so it just keeps it. Disobeys the law of gravitation. Watch for the exchange of momentum. See it? Where did the momentum go? It went into the earth. It went through me and into the floor, so the entire earth moved. It did. It took up the momentum from this wheel. Now, I got it from the earth in the first place so I give it back. You can transfer momentum. You can’t get rid of it, but you can transfer it. Any questions on that? Anybody else want to try? Anybody want to stand up here? Go around in circles? No? Okay. Actually, I would like another volunteer. We have one more experiment. Oh, come on. She can’t be the only brave one in the class. All right. Come on down. He’s gonna
down. But this is Newton’s First Law here. Just wants to keep moving unless I put a force on it. Second Law, if a put a force on it, it goes in the direction of the force. If I put a force on it, it puts a force back on me. You get a transfer of momentum. Any questions about that? Yeah? Student: When he was using the weights, was that also the Second Law? Well, he did use a force to bring the weights in. So, yeah, there was a force involved, caused him to accelerate, go around, but he was transferring the momentum from a large radius to a small o ne. And also the First Law is there. They want to keep going. They have a tendency to keep going. So all the laws are there. It’s just you have to pick ‘em apart. Any other questions on that? Okay. When Newton came up with these three laws of motion and the law of gravitation, he explained why all the planets go around the sun. He explained the why. See, people before that were arguing on what, what was happening. Were the planets going around the sun? Were the planets going around the earth? Was the earth going around the sun? Was the sun going around the earth? What was happening? He showed you why one of them was happening and the other wasn’t. There was no magic involved. The earth goes around the sun strictly because the sun has more mass. And so in any interaction of gravity, the earth is gonna do more moving. It’s gonna be accelerated more because its mass is less. So you’re using the Second Law of Newton’s laws of motion and the law of gravitation, and all of a sudden you can understand why moons go around planets because they have less mass. Planets go around the sun because they have less mass. But the sun goes around the Milky Way
galaxy because it has less mass than the galaxy. So there’s a hierarchy there based on mass. It’s not magic. There’s no center of the universe that we’re worrying about. It’s just how do the forces play out. Keep in mind — and maybe it’s not obvious — everything is moving. When a moon is going around a planet, the moon is going around a big orbit around the planet, but the planet is also moving because the moon is putting a force on the planet causing an acceleration. It’s just a much smaller one. So as the moon does this, the earth is doing this. They’re both moving together. Same with the sun and the earth or the sun and Jupiter. Jupiter’s going around the sun and the sun is going around a small circle with it. It’s just that you don’t notice that small circle much because Jupiter is going in such a larger orbit. Everything is moving. And so really if you look at what is moving and how it’s all moving, the sun is moving around something, the earth is moving around something, same thing. But to you it looks as if it’s going around the sun. It’s actually going around a point that is near the sun but then the sun is going around that point, too. We call that point the center of gravity. In other words, you have two objects pulling on each other, making each other go in orbit. The one with the most mass, the sun, goes in the small orbit. The one with the less mass, the earth, goes in the big orbit. But they’re going around a point that’s between them that we call the center of gravity. Sometimes it’s also referred to as the center of mass, but it’s actually better to use the term center of gravity because we’re talking about gravity here. We’re talking about two gravitational objects pulling each other. So there is a place that everything’s going
discovered yet. And so only out to Saturn was the solar system known. But today we know all of those planets. We don’t think there are any more. There’s always a possibility there’s another large object out there, but we have not seen any evidence for it whatsoever. So as far as we’re concerned, Pluto is the edge of the solar system. In fact, some astronomers think Neptune is the edge of the solar system because some astronomers don’t consider Pluto a planet. They think it’s too small and so they don’t really want to count it as a planet. Officially it’s still a planet. Actually, astronomers have been arguing over it so much that the International Astronomical Union had to take a vote -- is Pluto a planet or not — and Pluto won. It got more votes. So it’s a planet officially. Even though it’s rather small and some people didn’t want it to be a planet. Just a minor historical note. I said that Uranus, Neptune and Pluto had not been discovered. It turns out that Galileo probably saw Uranus. He drew it in one of his drawings. He even noted in one of his notebooks that he thought that star that he’d drawn had moved. But he didn’t follow it up. So he could have discovered the planet Uranus back around 1612. But he didn’t follow it up, he didn’t realize he was on to something, and so he just let it go. But he actually did have it on his drawings and did comment that it appeared to have moved since the last time he drew it. But then he didn’t think much of it, so — he probably thought, well, maybe he hadn’t put it in the right spot. That ends our discussion of Newton’s laws. We now move on to a slightly different topic in the next chapter and that is matter. We’ve been talking about masses as sort of general matter, stuff. Now we’re going to talk about it in terms of individual materials. The author has a nice picture at the beginning of the chapter in which he shows elements and
compounds. He talks about the discovery of elements using Dalton’s Atomic Theory to discuss it. Around 1800, Dalton was wondering what things were made of. He knew that the Greeks thousands of years before had talked about the concept of tiny particles being the building blocks of matter, and those tiny particles had been called atoms. And so Dalton wondered whether they were real and if the y were not all the same thing but maybe different kinds of things. And after doing many experiments, he essentially discovered elements. He discovered oxygen, nitrogen, iron, various other elements that were actually separate materials that could not be divided up further. He also realized there were other materials such as the salt shown here on the drawing that are compounds. They are elements put together, different elements. So in the case of salt you would have sodium and chlorine put together, two different elements to make something new. And so Dalton figured out that there were the indivisible kinds of materials, the elements, and then there were the groups of elements called compounds. And he essentially worked out the basis of the atomic theory. We’ve been working on that for quite a while. We now refer to the individual materials, the elements, as atoms. You can have different kinds of atoms: an atom of oxygen, an atom of nitrogen, an atom of chlorine. If you combine them together — take the atoms and combine them in various ways — you get what we call molecules. A molecule is anything from two atoms up to millions or billions of atoms. Maybe not billions, but at least millions. You can have extremely large molecules. And they can b e very complicated. They can be made of many different elements.
helium. But we then found it on the earth and realized that, yes, it is around. It’s just not very common. We also have oxygen, nitrogen, carbon, the basic building blocks that you hear about a lot. Those are the most common elements in nature. Now, if you want to narrow it down to the most common one or two, it’s hydrogen by a long shot. Seventy percent of the entire universe is hydrogen. And about 27 or 28 percent i s helium. You add those two together, you’re already up to about 98 percent. Ninety-eight percent of the entire universe is those two simple light gases, hydrogen and helium. Everything else, all other 90-some elements, only add up to 2 percent of the universe. It’s almost like a little bit of pollution in the hydrogen and helium. In some ways it is. Because they’re made from hydrogen and helium and they just kind of show up as a little bit of extra stuff in the hydrogen and helium. So you should be able to get a 98 on any exam that talks about elements in the universe as long as you can remember hydrogen and helium, because that’s 98 percent of the universe. Now, we’ll continue this discussion of elements next time.
