All right. The acceleration is a vector that’s made up out of a time derivatives of X, Y, and Z, or X1, X2, and X3. So, for each component — for each component, one, two, or three, the acceleration — which let me indicate, let’s just call it A. The acceleration is just equal — the components of it are equaled to the second derivatives of the coordinates with respect to time. That’s what acceleration is. The first derivative of position is called velocity. We can take this to be component by component X1, X2, and X3. The first derivative is velocity. The second derivative is acceleration. We can write this in vector notation. I won’t bother but we all know what we mean. I hope we all know what we mean by acceleration and velocity. And so, Newton’s equations are then summarized – not summarized but rewritten — as the force on an object, whatever it is, component by component, is equal to the mass times the second derivative of the component of position. So, that’s the summary of — I think it’s Newton’s first and second law. I can never remember which they are.
Newton’s first law, of course, is simply the statement that if there are no forces then there’s no acceleration. That’s Newton’s first law. Equal and opposite. Right. And so this summarizes both the first and second law. I never understood why there was a first and second law. It seemed to me that it was one law, F equals MA.
All right. Now, let’s begin even previous to Newton with Galilean gravity. Gravity as how Galileo understood it. Actually, I’m not sure how much of these mathematics Galileo did or didn’t understand. He certainly knew what acceleration was. He measured it. I don’t know that he had the — he certainly didn’t have calculus but he knew what acceleration was. So, what Galileo studied was the motion of objects in the gravitational field of the earth in the approximation that the earth is flat. Now, Galileo knew that the earth wasn’t flat but he studied gravity in the approximation where you never moved very far from the surface of the earth. And if you don’t move very far from the surface of the earth, you might as well take the surface of the earth to be flat and the significance of that is two-fold. First of all, the direction of gravitational forces is the same everywhere. This is not true, of course, if the earth is curved then gravity will point toward the center. But in the flat space approximation, gravity points down. Down everywhere is always in the same direction. And second of all, perhaps a little less obvious but nevertheless true, in the approximation where the earth is infinite and flat, goes on and on forever, infinite and flat, the gravitational force doesn’t depend on how high you are. Same gravitational force here as here. The implication of that is that the acceleration of gravity, the force apart from the mass of an object, the acceleration on an object is independent of where you put it. 2And so Galileo either did or didn’t realize — again, I don’t know exactly what Galileo did or didn’t know. But what he said was the equivalent of saying that the force of an object in the flat space approximation is very simple. It, first of all, has only one component, pointing downward. If we take the upward sense of things to be positive, then we would say that the force is — let’s just say that the component of the force in the X2 direction, the vertical direction, is equal to minus — the minus simply means that the force is downward — and it’s proportional to the mass of the object times a constant called the gravitational acceleration.
Now, the fact that it’s constant everywhere, in other words, mass times G doesn’t vary from place to place. That’s this fact that gravity doesn’t depend on where you are in the flat space approximation. But the fact that the force is proportional to the mass of an object, that is not obvious. In fact, for most forces, it is not true. For electric forces, the force is proportional to the electric charge, not to the mass. And so gravitational forces are at a special the strength of the gravitational force of an object is proportional to its mass. That characterizes gravity almost completely. That’s the special thing about gravity. The force is proportional itself to the mass.
Well, if we combine F equals MA with the force law — this is the law of force — then what we find is that mass times acceleration the second X, now this is the vertical component, by DT squared is equal to minus — that is the minus – MG period. That’s it. Now, the interesting thing that happens in gravity is that the mass cancels out from both sides. That is what’s special about gravity. The mass cancels out from both sides. And the consequence of that is that the motion of an object, its acceleration, doesn’t depend on the mass — doesn’t depend on anything about the particle. The particle, object– I’ll use the word particle. I don’t necessarily mean a point small particle, a baseball is a particle, an eraser is a particle, a piece of chalk is a particle. That the motion of an object doesn’t depend on the mass of the object or anything else. The result of that is that if you take two objects of quite different mass and you drop them, they fall exactly the same way. Galileo did that experiment. I don’t know whether he really threw something off the Leaning Tower of Pisa or not. It’s not important. He did balls down an inclined plane. I don’t know whether he actually did or didn’t. I know the myth is that he didn’t. I find it very difficult to believe that he didn’t. I’ve been in Pisa. Last week I was in Pisa and I took a look at the Leaning Tower of Pisa. Galileo was born and lived in Pisa. He was interested in gravity. How it would be possible that he wouldn’t think of dropping something off the Leaning Tower is beyond my comprehension. You look at that tower and say, “That tower is good for one thing: Dropping things off. ”