Universal Gravitation (1961)

You probably weren't surprised that when I released this ball, it dropped. I think you would have been surprised if it had stayed up. Now you suspect that it's suspended from a thread. But it isn't. The point is that you believe very strongly in the existence of a force called the force of gravity, a force which acts on every object that you've ever seen. What do we know about this force? We know it exists, and we know how to describe it. We've known this 300 years, primarily due to the work of Sir Isaac Newton. Newton proposed a law of universal gravitation, and that's what this film is about. Newton proposed the law on the basis of what he and others like Kepler and Galileo had observed about our solar system. But I'm not going to tell you about this just now. One of the most important things about the law is that it is universal, that it should be applicable anywhere in the universe. Well, you know that our sun is a star, and that there are other stars much like our sun, and they may have planetary systems, and there may be life, like ours or like much like ours on these planets. Dr. Hume and I are going to pretend to be on a planet, planet X, which is not in our solar system at all. And we are going to tell you on Earth how the law of gravitation was discovered on our planet, planet X. Take it away, planet X. Thank you, Earth. You're on Planet X. We now know that our solar system has certain features which are similar to yours. Our planet, our Earth, has a sun just like yours. But there are no other planets and no moons in our system. At night, we can see the stars rotating about our pole star. And in the day, we can see the sun rotating about us. It was quite a while before we realized that we were the ones who were rotating. We had a spherical planet and it was spinning once a day about its axis. The stars were not moving. We confirmed this notion by swinging a pendulum at the North Pole. But you know all about that. But there was evidence of another motion. The sun gradually changed position relative to the stars. In a year's time, it moved around in a circle until it was back where it started. The sun was moving around us in a circle. Or so many people thought until someone pointed out that we really couldn't tell by observing the sun's motion relative to the stars whether the sun was moving or we were moving or both of us moving. Here you see the sun moving against the fixed background of stars. But the sun is stationary at the center of this turntable and we were looking at it here from our Earth which is rotating around it. What would it look like if the sun were rotating around us? I'm going to move our Earth to the center of this turntable and our Sun to the edge. Now, let's look at the Sun's motion. It looks the same as before. All motion is relative. So, when we just describe motion, it doesn't matter whether we think of a heliocentric system or a geocentric system. But, when we try to explain the motion in terms of forces, there is a difference. We have low friction pucks here on Planet X, too. I'm going to talk about the dynamics of our solar system and because there are only two bodies in our solar system, these two pucks will help me to talk about it. This puck sees this one moving in a circle around it and this puck sees this one moving in a circle around it. Now, I'll let you see the same thing from directly overhead. You can see from your external point of view that both pucks are moving in circles. Now, I'll do this again with two pucks of different mass. This puck has a thicker base and therefore more mass. Again, this time, each puck sees the other moving around it in a circle. But, you see that the puck which has more mass is moving in a smaller circle. From your external point of view. This idea of point of view is very important. We always view things from a particular frame of reference. We are viewing this system from a frame of reference in which the laws of dynamics are valid, an inertial frame of reference. A frame of reference fixed on either of these pucks would not be an inertial frame because each puck has an acceleration relative to our inertial frame. But there is a point in this system which is not accelerating relative to us, the point about which the two pucks are rotating. A frame of reference at this point would be an inertial frame. Suppose I use an even larger mass ratio. Now this puck is moving in a very small circle. The center of rotation of this system is very close to the center of the large puck. A frame of reference therefore that is attached to the large puck is almost an inertial frame of reference. And the more massive this puck is, the closer this is to being an inertial frame. Now you can see what we mean when we say that from a dynamical point of view it is meaningful to talk about one object as fixed and the other object as rotating around it. Now let's get back to thinking about our Earth and Sun. We on planet X had to believe that our laws of dynamics were valid anywhere in the universe. But of course, we had to find an inertial frame of reference in which to apply them. The best we could do was to think of a frame of reference in the stars. From the point of view of a frame of reference fixed in the stars, which is moving, our Earth or our Sun? Well, if we know that the Sun is more massive, then we know that it is nearly standing still. Like this, with the Earth going around it. The Earth could equally well be nearly at rest, provided it had the mass to swing it. However, until we discovered the law of gravitation, we had no way of knowing which was more massive, the Earth or the Sun. Just one more thing. What I have said about two pucks applies to our Earth and Sun only if there's a force between the Earth and Sun, which acts like the string. There must be a force of attraction between them and no other forces. If forces act from outside the system, then anything may happen. Dr. Ivy has explained that we couldn't understand our solar system until we found out about gravity. So, I'm going to go back to the beginning of the story. Everything we drop on planet X falls with the same acceleration. There is a mysterious force attracting everything in towards the center of our Earth. We invented the word gravity for this force, but of course, that doesn't make it any the less mysterious. We felt that gravity might extend into space, even as far as the Sun, and perhaps it, too, was a falling object. We went as far as working out the acceleration of the Sun relative to the Earth, and it was very different from the acceleration of objects at the Earth's surface. We didn't have any other object in space to make measurements on, so there the project ceased. We lost interest in gravity until recently when it became fashionable to think about space travel. Then it was clear that we must explore the space around our planet to see if there was any change in gravity as you we went away from the surface. We knew that at the surface the acceleration due to gravity was always directed towards the center of our earth and remain roughly constant in magnitude wherever we went. So we thought that as we moved away from the surface the magnitude might change, but the force would always be directed towards the center. It would be a central force. This would be just the kind of force necessary to hold a satellite in orbit. The satellites we planned to send up were to have a very small mass relative to our earth. So as far as the dynamics of these satellites is concerned, we could think of them moving in orbits about a fixed earth. This peg represents our earth and we could time the period of the satellite, measure the radius of its orbit, and calculate its acceleration. This acceleration would be due to gravity at this distance from the center of the earth. And if gravity behaves in the same way as it does at the surface, this acceleration would be the same for any mass of satellite at this distance. So in this way we hope to measure the strength of the gravitational field of our earth in space. By the time we were ready to send up a satellite, we were pretty advanced technically, even though we hadn't discovered the law of gravitation yet. We could track the satellite and record its position and velocity very accurately. This equipment is designed to demonstrate what the path of the satellite was like. The path will be displayed on this oscilloscope. The central spot represents the Earth. What we did was shoot the satellite straight up high enough so that we were sure our atmosphere wouldn't interfere with its motion. Then, we shot it sideways. We didn't know what the force of gravity would be on it at this height. We didn't know if there'd be any force on it for that matter. We just shot it sideways. If it had kept moving with constant velocity, we would have known that there was no force acting on it. If there was a gravitational force, and we happened to give the satellite just the right initial velocity, then we would see a circular orbit with the force of gravity the right size to be the centripetal force. Well, this is what we observed. The path was certainly not a circle. However, there must have been some force acting because the satellite did not keep moving with constant velocity. It went into orbit around us. We decided to find out what we could from this path. We noticed that although the speed of the satellite kept changing, that it swept out equal areas in equal times. Here's the Earth. Here's the satellite at some instant, some time later, and an equal amount of time later. This area is equal to this area. These distances are not the same, but these areas are. We deduced from our knowledge of dynamics that this meant that there was a force acting on the satellite which was always directed towards the Earth. What Dr. Hume called a central force. There is a central force acting up here just as there is at the Earth's surface. Well, then why didn't we get a circular orbit? Just because we didn't give the satellite the right initial velocity. It turned out that it was rather fortunate that we didn't have a circular orbit because this kind of a path explores space at different distances from the Earth's surface. When we looked more carefully at the orbit, we found that it was an ellipse with the Earth at one focus. What did this mean? We put our mathematicians to work on this, and it took some time, but finally one brilliant man came up with the solution. To have an orbit like this, the central force acting on the satellite must vary inversely as the square of the distance from the Earth. The distance R. F is proportional to 1 over R squared. It is rather difficult to show mathematically that an elliptic orbit with the Earth at one focus indicates an inverse square force law, but this model will help you to believe that this is so. The center of this model represents the Earth. Now, watch what happens when I release a ball at a certain distance from the Earth. It accelerates in towards the center. This represents the attraction of gravity. The shape of this model is carefully designed so that when you look at it from directly overhead, it seems to provide a force on the ball toward the center which varies inversely as the square of the distance. Now, I'm going to put the ball into orbit. The ball, because of friction, spirals in towards the Earth. This is what happens to a satellite if it is not high enough up in the atmosphere. Now, watch what happens again, only this time in slow motion from overhead. The orbit is an ellipse. One satellite gave us this information. But in science, we do confirming experiments whenever possible, so we kept on sending up satellites. The second satellite was in a different orbit, but it was an ellipse and again verified the inverse square force law. We kept on sending up satellites and we soon noticed an interesting fact. The easiest things to measure about a satellite are its average distance away and its period. We discovered that the ratio of the average distance cubed to the period squared is a constant. It's the same for all of the satellites that we send up. The actual number was 1 * 10 to the 13th m cubed per second squared. This number does not depend on the mass of the satellite. Now, we thought we were in a position to tell whether the sun was our satellite or not. For the sun, we knew the distance away and the period, so we could calculate this ratio. And it came out to be 3.3 * 10 to the 18th m cubed per second squared. It looks as though the sun is not our satellite. But we weren't really sure that this gravitational force law applied all the way out to the sun. All of the satellites that we had sent up were much closer to the earth than the sun is. So, we decided to send a satellite a way up to about the same distance that the sun is. We sent it up and began to track it. I'll change the scale so that you can see the sun in its circular orbit. There it is. On this scale, you can't see our earlier satellites at all. Now, here's the one we sent a way out. Its orbit really confused us for a while. It didn't move in a simple orbit at all. It actually turned around and went backwards sometimes, what we called a retrograde motion. We couldn't make sense out of the motion at all until somebody had an idea to look at the motion from the point of view of the sun rather than the earth. Here, the sun is fixed at the center. Now, the earth is going around the sun in a circular orbit. And look at the satellite. It is in an elliptic orbit round the sun. The motion is simple. We measured the average distance of this satellite from the Sun and its period around the Sun calculated the ratio R cubed T squared and it came out to be 3.3 * 10 to the 18th meters cubed per second squared. Now look at this. This is the number that we calculated thinking that the Sun went around the Earth. It didn't fit this but it does fit this. It looks as though the Earth goes around the Sun. We kept on sending more satellites up to great heights and they became Sun satellites. We calculated this ratio for the M and we always got this result. For Sun satellites, this is a constant. But it is a different constant than it is for Earth satellites. We had two systems of satellites, one around the Earth and one around the Sun. I'm going to concentrate on the Earth satellites and see what we can learn from this peculiar relationship. Now the orbits you have seen were ellipses but this would hold true for circular orbits as well. This is the force required to hold a satellite in a circular orbit. And it must be supplied by the gravitational attraction of the Earth on the satellite. Now we already know that the gravitational attraction would be inversely proportional to the radius of the orbit squared. And of course, gravitational force is always proportional to the mass of the body acted on. I'll put in a proportionality factor. This is the force needed. This is the force available. So, for a circular orbit to be stable, these two must be equal. Now, you'll notice that the mass of the satellite cancels from this equation. A satellite's orbit does not depend at all on its mass. Now, I'm going to rewrite this equation. R cubed over T squared is equal to the factor divided by 4 pi squared. This is the ratio that we found was a constant for all the Earth's satellites. So, this factor is a constant for all the Earth's satellites. But, it is not a universal constant. The factor would be different for the Sun's satellites. So, this factor must depend on some property of the Earth. But, what property? How could we ever find out? Well, we were fortunate here on Planet X in having a man of genius who pointed out that if the Earth exerted a certain size of force on the satellite, then the satellite must exert the same size of force on the Earth. Now, we already have an expression for the force of the Earth on the satellite. This same kind of expression must give the force of the satellite on the Earth. And these two must be equal. This factor depends on some property of the satellite. Now I'm going to rewrite this expression. The r squared of course cancel and the factor for the Earth divided by the mass of the Earth will be equal to the factor for the satellite divided by the mass of the satellite. This expression involves only the Earth. It doesn't depend whatsoever on the properties of the satellite. This expression involves only the satellite. It doesn't depend on the Earth. And what's more, it's the same for every Earth satellite. So perhaps it is reasonable to assume that this ratio of the factor divided by the mass is a universal constant. This means that the factor depends on the mass of the body, that the factor is proportional to the mass. Now if I used this expression, I would find that both of these expressions would equal the same expression. I have found one expression for the force between two bodies, a law of gravitation. Now I have used the Earth satellite system in talking about this. I could equally well have used the Sun satellites. And if I had, I would have found the same result except that the mass of the sun and its satellite would appear instead of the mass of the earth and its satellite. This is what led us to believe that this law held between any two masses in the universe. That the force on either mass is proportional to the product of the masses divided by the distance between them squared. This, we believe, is the law of universal gravitation. And G is a universal constant. Now we're in a position to calculate the relative mass of our earth and our sun. Dr. Ivy worked this out and you can show for yourself that it is true. The ratio of the masses is equal to the ratio of either the small K's or the large K's. Our sun is about a third of a million times more massive than our earth. This is why we couldn't think of our sun as a satellite of our earth. It has a much greater mass. So the in the inertial frame of reference of the stars, it is almost fixed with our earth traveling around it in orbit. Our satellite measurements gave us the values of the K's. These gave us the ratio of the masses. But in all this, we still did not know the actual mass of either our sun or our earth. To find these, we would need to know the value of the universal constant. We realized that we would have to do an experiment to measure the size of the force between two bodies in the laboratory where we knew both the masses. Then we could calculate G. In the experiment, the masses were as large as we could conveniently handle, but even so, the force was very small and required careful measurement. But we finally did it. And this ended our investigation of gravity. We have answered the age-old question of whether our sun goes around us or we around it. And we have found the masses of both of the bodies in our solar system. Perhaps our theory of gravity may need to be refined a bit, but at the moment, we are confident that our space travel program has a solid scientific basis. And that's how the law of gravitation was discovered on planet X. The way it was done was different than it was in our solar system because the system was different, but the result is the same because the law is universal. We pretended to be on planet X because we wanted to add experimental evidence, bit by bit, using the satellites we sent up so that you could see the steps involved in the discovery of the gravitational law. But this isn't how it happened in our solar system at all. It was discovered long before there were man-made satellites. And this was only possible because there were satellites. We call them the planets and moons. Most of the steps that we described on planet X were taken by one man, Sir Isaac Newton. You may have noticed that in the play we said things like a brilliant suggestion was made or we had a man of genius, so on. On Earth, this was always the same man, Newton. Newton could not do experiments with satellites. He had, in his work, to use the evidence of others about the motion of the planets and the moon and the sun. What's more, he didn't have the principles of mechanics and the tools of mathematics at his disposal as the planet X scientists did. He had to invent them. He formulated the laws of dynamics. He developed the necessary mathematics. For instance, he showed that the kind of elliptic orbits that we had could only result from an inverse square force law. Newton realized that the planets are not only attracted to the sun but to each other and that their orbits are slightly affected by this. In fact, if a planet has no moons, its mass can be determined from the way that it perturbs the orbits of the other planets. Newton realized that our moon is attracted not only to the Earth but also to the sun and its orbit is perturbed slightly. But in our discussion of satellites, we ignored the effect that the sun would have on our Earth satellites and the even smaller effect that the Earth would have on the sun satellites. And this was justifiable because really all these effects are very small. One thing to remember is that Newton only knew the relative masses of the bodies in the solar system. It wasn't till about 100 years later that Cavendish performed the experiment to measure the universal constant. Over the years, the evidence in favor of Newton's law has been so convincing that when it came time for us to think about space travel, the fact that the satellites we launched bore out predictions based on Newton's law wasn't even newsworthy. Mhm.