This program is brought to you by Stanford University. Please visit us at stanford.edu. Gravity.
Gravity is a rather special force. It's unusual. It has different than electrical forces, magnetic forces, and it's connected in some way with geometric properties of space, space and time. And that connection is, of course, the general theory of relativity. Before we start, tonight, for the most part, we will not be dealing with the general theory of relativity.
We will be dealing with gravity in its oldest and simplest mathematical form. form, well perhaps not the oldest and simplest, but Newtonian gravity, and going a little beyond what Newton, certainly nothing that Newton would not have recognized or couldn't have grasped, Newton could grasp anything, but some ways of thinking about it which will not be found in Newton's actual work, but still Newtonian gravity. Newtonian gravity set up in a way way that is useful for going on to the general theory.
OK, let's begin with Newton's equations. The first equation, of course, is F equals ma. Force is equal to mass times acceleration. Let's assume that we have a frame of reference, a frame of reference, and it means a set of coordinates and it's a collection of clocks, and that frame of reference is what is called an inertial frame of reference. An inertial frame of reference simply means one which, if there are no objects around to exert forces on a particular, let's call it a test object, a test object is just some object, a small particle or anything else, that we use to test out the very...
various fields, force fields that might be acting on it. An inertial frame is one which, when there are no objects around to exert forces, that object will move with uniform motion with no acceleration. That's the idea of an inertial frame of reference.
And so if you're an inertial frame of reference and you have a pin and you just let it go, it stays there. It doesn't move. If you give it a push...
push, it will move off with uniform velocity. That's the idea of an inertial frame of reference. And in an inertial frame of reference, the basic Newtonian equation number one, I always forget which law is which. There's Newton's first law, second law, and third law. I never can remember which is which, but they're all pretty much summarized by F equals mass.
Mass times acceleration. This is a vector equation. I expect people know what a vector is.
A three-vector equation. We'll come later to four vectors, where when space and time are united into space-time. But for the moment, space is space and time is time.
And a vector means a thing, which is a pointer in a direction in space. It has a magnitude and it has components. So component by component.
the x component of the force is equal to the mass of the object times the x component of acceleration, y component, z component, and so forth. And in order to indicate that something is a vector equation, I'll try to remember to put an arrow over vectors. The mass is not a vector.
The mass is simply a number. Every particle has a mass. Every object has a mass. And in Newtonian physics, the mass is a vector.
is conserved, it does not change. Now of course the mass of this cup of coffee here can change, it's lighter now, but it only changes because mass has been transported from one place to another, so you can change the mass of an object by whacking off a piece of it, but if you don't, change the number of particles, change the number of molecules, change the and so forth, then the mass is a conserved, unchanging quantity. So that's the first equation. Now let me write that in another form.
The other form, we imagine we have a coordinate system. and X a Y and a Z I don't have enough directions on the blackboard to draw Z I won't bother there's X Y and Z sometimes we just call them X 1 X 2 and X 3 I guess I can draw it in X 3 is over here someplace X Y and Z and A particle has a position, which means it has a set of three coordinates. Sometimes we will summarize the collection of the three coordinates, x1, x2, and x3. Incidentally, x1 and x2 and x3 are components of a vector. The components, they are components of...
the position vector of the particle. Position vector of the particle, I will often call either small r or large r, depending on the particular context. R stands for radius, but the radius simply means the distance between a point and the origin for example.
We're really talking now about a thing with three components x y & z and it's the radial vector. The radial vector, this is the same thing as the components of the vector R. All right.
The acceleration is a vector that's made up out of the time derivatives of x, y, and z, or x1, x2, and x3. So for each component, the for each component, one, two, or three, the acceleration, which let me indicate, well, let's just call it A, the acceleration is just equal, the components of it are equal 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. 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.
That's the second law. The third law is equal and opposite. Equal and opposite, right.
So this summarizes both the first and second law. I never understood why there was a first and second law. It seemed to me there was just one, F equals ma. All right. Now let's...
let's begin even previous to Newton with Galilean gravity gravity as Galileo understood it actually I'm not sure how much of this mathematics Galileo did or didn't understand he certainly knew what acceleration was, he measured it I don't know that he had the, well, 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 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. earth to be flat and the significance of that is twofold.
First of all the direction of gravitational forces is the same everywheres. 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 bit 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, since force, apart from the mass of an object, the acceleration on an object is independent of the way you put it. And so Galileo...
Galileo either did or didn't realize, well, he, again, I don't know exactly what Galileo did or didn't know, but what he said was equivalent to saying that the force on 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 the component of the force in the 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 is 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's not true. For electric forces, the force is proportional to the electric charge, not to the mass. And so gravitational forces are rather special.
The strength of the gravitational force on an object is proportional to its mass. That characterizes gravity almost completely. That's the special thing about gravity.
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 d second x, now this is the vertical component, by dt squared, is equal to minus, that's 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, it doesn't depend on anything about the particle. 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 the object doesn't depend on the mass of the object.
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 you say that tower is good for one thing, dropping things off.
Now I don't know maybe the Doge... whoever they called the guy at the time said, no, no, Galileo, you can't drop things from the tower. You'll kill somebody.
So maybe he didn't, but he must have surely thought of it. All right, so the result, had he done it, and had he not had to worry about such spurious effects as air resistance, would be that a cannonball and a feather would fall in exactly the same way, independent of the mass. And the equation. would just say the acceleration would first of all be downward that's the minus sign and equal to this constant G excuse me let me now G is a number it's 10 meters per second per second At the surface of the Earth, at the surface of the Moon, it's something smaller, on the surface of Jupiter it's something larger, so it does depend on the mass of the planet, but the acceleration doesn't depend on the mass of the object you're dropping. It depends on the mass of the object you're dropping it onto, but not the mass of the object that's dropping.
That fact, that gravitational motion is completely independent of mass, is called, or it's the simplest version of something that's called the equivalence principle. Why it's called the equivalence principle we'll come to later, what's equivalent to what, at this stage we could just say gravity is equivalent between all different objects independent of their mass, but that is not exactly what the equivalence, an equivalence principle was about. That has a consequence, an interesting consequence.
Supposing I take some object which is made up out of something which is very unrigid. Just a collection of point masses, maybe, let's even say they're not even exerting any forces on each other. It's a cloud, a very diffuse cloud of particles, and we watch it fall.
Let's suppose we start each particle from rest. not all at the same height, and we let them all fall. Some particles are heavy, some particles are light, some of them may be big, some of them may be small. How does the whole thing fall? The answer is all of the particles fall at exactly the same rate.
The consequence of it is that the shape of this object doesn't deform as it falls. It stays absolutely unchanged. The relationship between...
the neighboring parts are unchanged there are no stresses or strains which tend to deform the object so even if the object were held together by some sort of struts or whatever there would be no forces on those struts because everything falls together okay the consequence of that is a falling in a gravitational field is undetectable you can't tell that you're falling in a gravity gravitational field by, when I say you can't tell, certainly you can tell the difference between free fall and standing on the earth. Alright, that's not the point. The point is that you can't tell by looking at your neighbors or anything else that there's a force being exerted on you and that that force that's being exerted on you is pulling you downward.
You might as well for all practical purposes be infinitely far from the earth with no gravity at all and just sitting there, because as far as you can tell, there's no tendency for the gravitational field to deform this object or anything else. You cannot tell the difference between being in free space, infinitely far from anything with no forces, and falling freely in a gravitational field. That's another statement of the equivalence principle.
Well, in fact, not the technical period, but... But so far, not mechanically detectable. Or would it be optically detectable? No.
No. For example, these particles could be equipped with lasers. Lasers and optical detectors of some sort.
What's that? Oh, you could certainly tell if you were standing on the floor here, you could tell that something was falling toward you. But the question is, from within this object by itself, without looking at the floor, without knowing the floor was...
something that wasn't moving from moving well you can't tell whether you're falling in its... yeah, yeah if there was something that was not falling it would only be because there was some other force on it like a beam or a tower of some sort holding it up why? because this object, if there are no other forces on it, only the gravitational forces, it will fall at the same rate as this. Alright, so that's another expression of the equivalence principle, that you cannot tell the difference between being in free free space far from any gravitating object versus being in a gravitational field.
Now we're going to modify this. This of course is not quite true in a real gravitational field, but in this flat space approximation where everything moves together, you cannot tell that there's a gravitational field or at least you cannot tell the difference, not without seeing the floor in any case. The self-contained object here does not experience any gravitational field.
anything different than it would experience far from any gravitating object standing still, or in uniform motion. I have a question. What's that? Yeah.
Well, we can tell when we're accelerating. No, you can't tell when you're accelerating. Well, you can't feel that you're accelerating.
Isn't that because there's a connection between the different objects? Okay, here's what you can't tell. If you go up to the top of a high building, and you close your eyes, and you step off and go into free fall, you will feel exactly the same. You'll feel weird.
I mean, that's not the way you usually feel because your stomach will come up and, you know, do some funny things. You know, you might lose it. But the point is, you would feel exactly the same discomfort in outer space, far from any gravitating object, just standing still.
You'll feel exactly the same peculiar feelings. Thank you. What are those peculiar feelings due to? They're not due to falling, they're due to not fall.
Well, they're due to the fact that when you stand on the earth here, there are forces on the bottoms of your feet which keep you from falling, and if the earth suddenly disappeared from under my feet, sure enough, my feet would feel funny because they're used to having those forces exerted on their bottoms. You get it, I hope. So the fact that you feel funny in free fall is because you're not used to free fall. It doesn't matter whether you're infinitely far from any gravitating object standing still or freely falling in the presence of a gravitational field.
now as I said this will have to be modified in a little bit there are such things as tidal forces those tidal forces are due to the fact that the earth is curved and that the gravitational field is not the same in every, same direction in every point, and that it varies with height, that's due to the finiteness of the Earth. But in the flat space, in the flat Earth approximation, where the Earth is infinitely big, pulling uniformly, there is no other effect of gravity that is any different than being in free space. OK, again, that's known as the equivalent principle. Now let's go on beyond the flat space or the flat Earth approximation and move on to Newton's theory of gravity.
Newton's theory of gravity. It says every object in the universe exerts a gravitational force on every other object in the universe. Let's start with just two of them. Equal and opposite.
Attractive. Attractive means that the direction of the force on one object is toward the other one. equal and opposite forces and the magnitude of the force the magnitude of the force of one object on another Let's let's characterize them by a mass Let's call this one little m Think of it as a lighter mass and this one which we can imagine is a heavier object.
We'll call it big M Alright Newton's law of force Is that the force? As proportional to the product of the masses, making either mass heavier will increase the force. The product of the masses, big M times little m, inversely proportional to the square of the distance between them, let's call that R squared. Let's call the distance between them R. And there's a numerical constant.
This law by itself could not possibly be right. It's not dimensionally consistent. If you work out the dimensions of force, mass, mass, and R... it's not dimensionally consistent, there has to be a numerical constant in there. And that numerical constant is called capital G, Newton's constant, and it's very small.
It's a very small constant. I'll write down what it is. G is equal to 6, or 6.7 roughly, times 10 to the minus 11th, which is a small number.
So on the face of it, it seems that gravity is a very weak force. You might not think that gravity is such a weak force, but to convince yourself it's a weak force, there's a simple experiment that you can do. Weak by comparison with other forces. I've done this for classes and you can do it yourself.
Just take an object hanging by a string and two experiments. The first experiment, take a little object here and electrically charge it. You electrically charge it by rubbing it on your sweater.
That doesn't put very much pressure on it. much electric charge on it, but it charges it up enough to feel some electrostatic force. And then take another object of exactly the same kind, rub it on your shirt, and put it over here.
What happens? They repel. And the fact that they repel means that... This string will shift and you'll see it shift. Take another example.
Take your little ball there to be iron and put a magnet next to it. Again, you'll see quite an easily detectable deflection of the string holding it. Next. Take a 10,000 pound weight and put it over here.
Guess what happens? Undetectable. You cannot see anything happen. The gravitational force is much, much weaker than most other kinds of forces. And that's due to, or not due to, but the...
not due to that the fact that it's so weak is encapsulated in this small number here another way to say it is if you take two masses each of one kilo not one kilometer one kilogram good healthy mass, right, nice chunk of iron, m, m, and you separate them by one meter, then the force between them is just g, and it's 6.7 times 10 to the minus 11, the units being Newtons, so it's very weak force, but weak as it is, we feel it rather strenuously, we feel it strongly, because the earth is so darn heavy, so the heaviness of the earth makes up for the smallness of of g and um so we wake up in the morning feeling like we don't want to get out of bed because gravity is holding us down So that force is measuring the force between the large one and the small one? Both, both, they're equal and opposite, equal and opposite, that's the rule, that's Newton's third law the forces are equal and opposite So the force on the large one due to the small one is the same as the force of the small one on the large one. And, uh, but it is proportional to the product of the masses. So, the meaning of that is, I'm not, well, I'm heavier than I like to be, but, uh, but I'm not very heavy.
I'm certainly not heavy enough to deflect the hanging, uh, weight significantly. But I do exert a force on the Earth, which is exactly equal and opposite to the force that the very heavy Earth exerts on me. Okay.
Why does the earth accelerate... If I drop from a certain height, I accelerate down, the earth hardly accelerates at all, even though the forces are equal. Why is...
why is it that the earth if the forces are equal my force on the earth and the earth's force on me are equal why is it that the earth accelerates so little and I accelerate so much yeah, because the acceleration of the earth is so much acceleration involves two things, it involves the force and the mass the bigger the mass, the less the acceleration for a given force so the earth doesn't excel, yeah, question? how did Newton arrive at that equation for the gravitational force? I think it was largely a guess, but there was, so it was an educated guess and um I don't know, What was the key?
It was larger, ah, no, no, it was from Kepler's, it was from Kepler's laws. It was from Kepler's laws. He worked out, roughly speaking, I don't know exactly what he did. He was rather secretive and he didn't really know.
really tell people what he did. But the piece of knowledge that he had was Kepler's laws of motion, planetary motion. And my guess is that he just wrote down a general force and realized that that he would get Kepler's laws of motion for the inverse square law. I don't believe he had any underlying theoretical reason to believe in the inverse square law.
Edmund Halley actually asked him, what kind of force law do you need for conic section coordinates? And he had already performed the calculations years before. Yeah. So yeah.
Actually, I don't think that's correct. He asked a question. for inverse square laws, and I think that Newton already knew the solution was an ellipse. Well, no, no, it wasn't the ellipse which was the, the orbits might have been circular.
It was the fact that the period varies as the three-halves power of the radius. Alright? The period of motion is the circular motion has an acceleration toward the center.
Any motion in the circle is accelerated toward the center. If you know the period and the radius then you know the acceleration toward the center. Okay, we could write let's let's do it anybody know what if I know the angular frequency the angular frequency of going around in an orbit that's called Omega do you know what the and it's basically just the inverse period, okay?
Omega is roughly the inverse period, number of cycles per second. What's the, what is the acceleration of a thing moving in a circular orbit? Anybody remember? Omega squared R.
Omega squared R, that's the acceleration. Now supposing he sets that equal to some unknown force law, F of R, and then divides by r, then he finds omega as a function of the radius of the orbit. Okay, well let's do it for the real case.
For the real case inverse square law, f of r is 1 over r squared, so this would be 1 over r cubed. And in that form it is Kepler's second law? I don't remember which one it is. It's the law that says that the frequency or the period, the square of the period, is proportional to the cube of the radius. That was the law of Kepler so from Kepler's laws he easily could have that one law he could easily deduce that the force was proportional to one over r squared I think that's probably historically what what he did Then on top of that, he realized if you didn't have a perfectly circular orbit, then the inverse square law was the unique law which would give elliptical orbits.
So it was a two-step thing. Thank you. If two objects are touching, do you measure from the center?
Well, then of course there are other forces on them. If the two objects are actually touching each other, there are all sorts of forces between them that are not just gravitational. Electrostatic forces, atomic forces, forces, nuclear forces, so you have to modify the whole story. As the distance approaches zero, then the... Then it breaks down.
Then it breaks down, yeah. Then it breaks down when they get so close... that other important forces come into play.
The other important forces, for example, are the forces that are holding this object and preventing it from falling. These, we usually call them contact forces, but in fact what they really are is various kinds of electrostatic forces between the atoms and molecules in the table and the atoms and molecules in here. So other kinds of forces. All right. Incidentally, let me just point out our...
if we're talking about other kinds of force laws, for example, electrostatic force laws, then the force, we still have F equals MA, but the force law, the force law will not... be that the force is somehow proportional to the mass times something else, but it could be the electric charge. If it's the electric charge, then electrically uncharged objects will have no forces on them and they won't accelerate.
Electrically charged objects will accelerate in an electric field. So electrical forces don't have this universal property that everything falls or everything moves in the same way. Uncharged particles move differently than charged particles with respect to electrostatic forces.
They move the same way with respect to gravitational forces. n is repulsion and attraction, whereas gravitational forces are always attractive. Where's my gravitational force? I lost it. Yeah, here it is.
All right, so that's Newtonian gravity between two objects. For simplicity, let's just put one of them, the heavy one, at the origin of coordinates and study the motion of the light one. Then, oh incidentally, one usually puts, let me refine this a little bit. As I've written it here, I haven't really expressed it as a vector equation.
This is the magnitude of the force between two objects. Thought of as a vector equation. We have to provide a direction for the force.
Vectors have directions. What direction is the force on this particle? Well, the answer is, it's along the radial direction itself. So let's call...
The radial distance r, or the radial vector r, then the force on little m here is along the direction r, but it's also opposite to the direction of r. The radial vector relative to the origin over here points this way. On the other hand, the force points in the opposite direction.
If we want to make a real vector equation, equation out of this. We first of all have to put a minus sign. That indicates that the force is opposite to the direction of the radial distance here. But we have to also put something in which tells us what direction the force is in. It's along the radial direction.
But wait a minute. If I multiply it by r up here, I had better divide it by another factor of r downstairs to keep the magnitude unchanged. The magnitude of the force is 1 over r squared. If I were to just randomly come and multiply it by r, that would make the magnitude bigger by a factor of r, so I have to divide it by the magnitude of r. this is Newton's force law expressed in vector form.
Now, let's imagine that we have a whole assembly of particles, a whole bunch of them. They're all exerting forces on one another in pairs. They exert exactly the force that Newton wrote down.
But what's the total force on a particle? Let's label these particles. This is the first one, the second one, the third one, the fourth one, dot, dot, dot, dot, dot. This is the ith one over here.
So I is a running index which labels which particle we're talking about. The force on the ith particle, let's call it F sub I, And let's remember that it's a vector. It's equal to the sum.
Now, this is not an obvious fact that when you have two objects exerting a force on the third, that the force is... necessarily equal to the sum of the two forces of the two of the two objects, you know what I mean but it is a fact anyway obvious or not obvious, it is a fact, gravity does work that way at least in the Newtonian approximation with Einstein it breaks down a little bit but in Newtonian physics the force is the sum and so it's a sum of all the other particles Let's write that J not equal to I. That means it's a sum over all not equal to I.
So the force on the first particle doesn't come from the first particle. It comes from the second particle, third particle, fourth particle, and so forth. Each individual force involves M sub I, the force of the Ith particle, times the mass of the Jth particle.
Product of the masses divided by the square of the distance between them, let's call that Rij squared. The distance between the i-th particle is i and j, the distance between the i-th particle and the j-th particle is Rij, but then just as we did before we have to give it a direction. minus sign here that indicates that it's attractive another Rij upstairs but that's a vector Rij and make this cube downstairs all right so that says that the force on the i-th particle is the sum of all the forces due to all the other ones of the product of their masses inverse square in the denominator and the direction of each individual force on this particle is toward the other. Alright this is a vector sum The minus indicates that it's attractive. Oh, excellent.
Perfect. But you've got the vector going from i to j. Oh.
Let's see. It's a vector going from r to j. Yes, there is a question of the sign of this vector over here.
So, yeah, your absolute, let's see. Yeah, I actually think it's, yeah, you're right. You're absolutely right. The way I've written it, there should not be a minus sign here. All right?
But if I put RJI there, then there would be a minus sign, right? So you're right. But in any case, every one of the forces is attractive.
All right? And what we have to do is to add them up. We have to add them up as vectors.
And so there's some resulting vector, some resultant vector, which doesn't point toward any one of them in particular, but points in some direction which is determined by the... the vector sum of all the others. But the interesting fact is, if we combine this, this is the fourth something i-th particle. If we combine it with Newton's equations, let's combine it with Newton's f equals ma equations, then this is f. This is on the i-th particle.
This is equal to the mass of the i-th particle times the acceleration of the i-th particle. Again, vector equations. Now, the sum here is over all the other particles. We're focusing on number i.
I, the mass of the i-th particle, will cancel out of this equation. I don't want to throw it away, but let's just circle it and now put it over on the side. We notice that the acceleration of the i-th particle does not depend on its mass again.
Once again, because the mass occurs in both sides of the equation, it can be canceled out, and the motion of the i-th particle does not... depend on the mass of the ith particle. It depends on the masses of all the other ones. All the other ones come in, but the mass of the ith particle cancels out of the equation. So what that means is if we had a whole bunch of particles...
here, and we added one more over here, its motion would not depend on the mass of that particle. It depends on the mass of all the other ones, but it doesn't depend on the mass of the i-th particle here. That's, again, equivalence principle, that the motion of a particle doesn't depend on its mass.
And again, if we had a whole bunch of particles here, if they were close enough together, they were all all move in the same way. Before I discuss a little more mathematics, Let's just discuss tidal forces, what tidal forces are. Can I ask one question? Yeah. Once you set this whole thing into motion dynamically, we have all different masses, and each particle is going to be affected by each one differently.
Yes, yes, yes. I think I know what you're going to ask. Does this mean that every particle in there is going to experience a uniform acceleration? No, no, no.
No, no, no. The acceleration is not uniform. The acceleration will get larger when it gets closer to one of the particles. It won't be uniform anymore. It won't be uniform now because the force is not independent of where you are.
Now the force depends on where you are relative to the objects that are exerting the force. It was only in the flat Earth approximately. where the force didn't depend on where you were.
Now the force varies, so it's larger when you're far away. Sorry, it's smaller when you're far away. It's larger when you're in close.
But is it going to be changing in a... It changes in a vector form with each individual... Particles, each one of them is changing position. Yeah. And so, is the dynamics that every one of them is going towards the center of gravity of the entire...
Not necessarily. I mean, they could be flying apart from each other, but they will be accelerating toward each other. Okay, if I throw this eraser into the air with greater than the escape velocity, it's not going to turn around and fall back down, right?
No, the question is, is the acceleration a uniform acceleration, or is it a changing dynamic acceleration? Changing with what? With respect to what?
Time? Oh, it changes with respect to time because the object moves further and further away. In the two-mass system, I call that a uniform acceleration.
Uniform with respect to what? It's not uniform. The radius is changing and it's inverse cube of the radius.
Inverse square. Right. One of these can't be...
Inverse square. yeah let's take the earth here's the earth and we drop a small mass from far away as that mass moves in its acceleration increases why does its acceleration increase? its acceleration increases because the radial distance gets smaller so in that sense it's not the...
all right now once the gravitational force depends on distance then it's not really quite true that you don't feel any... anything in a gravitational field, you feel something which is to some extent different than you would feel in free space without any gravitational field. The reason is more or less obvious. Here you are, here's the Earth. Now you or me or whoever it is happens to be extremely tall.
A couple of thousand miles tall. Well, this person's feet are being pulled by the gravitational field more than his head, or another way of saying the same thing is if, let's imagine that the person is very loosely held together. He's just more less a gas of a... we are pretty loosely held together, at least I am.
The acceleration on the lower portions of his body are larger than the accelerations on the upper portions of his body. So it's quite clear what happens to him. He gets stretched.
He doesn't get a sense of falling as such. He gets a sense of stretching, being stretched. Feet being pulled away from his head.
At the same time, let's... All right, so... Let's change his shape a little bit. I just spent a week, two weeks in Italy and my shape changes whenever I go to Italy and it tends to get more horizontal.
My head is here, my feet are here, and now I'm this way. Still loosely put together. Now what?
Well, not only does the force depend on the distance, but it also depends on the direction. The force on my left end over here is this way. my right end over here is this way.
The force on the top of my head is down, but it's weaker than the force on my feet. So there are two effects. One effect is to stretch me vertically. It's because my head is not being pulled as hard as my feet.
But the other effect is to be squished horizontally by the fact that the forces on the left end of me are pointing slightly to the right, and the forces on the right end of me are pointing... slightly to the left. So a loosely knit person like this, falling in free fall near a real planet or a real gravitational object, which has a real Newtonian gravitational field around it, will experience a distortion, will experience a degree of distortion and a degree of being stretched vertically, being compressed horizontally.
But if If the object is small enough, what does small enough mean? Let's suppose the object that's falling is small enough. If it's small enough, then the gradient of the gravitational field across the size of the object will be negligible. And so all parts of it will experience the same gravitational acceleration. All right, so tidal forces, these are tidal forces, these forces which tend to tear things apart vertically and squish them this way, tidal forces.
Tidal forces are forces which are real you feel them I mean yeah do you recall if Newton calculated lunar tides? oh I think he did he certainly knew the cause of the tides yeah I don't know to what extent he calculated what do you mean calculated the as in this kind of a system with the moon and the sun well I doubt that he was capable I'm not sure whether he estimated the height of the deformation of the oceans or not but I think he did understand this much about tides. Okay, so that's the, that's what's called tidal force, and remember the tidal force has this effect of stretching, and In particular, if we take the Earth, just to tell you why it's called tidal forces, of course it's because it has to do with tides, I'm sure you all know the story, but if this is the moon down here, then the moon exerting forces on the Earth exerts tidal forces on the Earth, which means that the moon is exerting forces on the Earth. which means to some extent it tends to stretch it this way and squash it this way well the earth is pretty rigid so it doesn't deform very much due to the moon but what's not rigid is the layer of water around it and so the layer of water tends to get stretched and squeezed and so it gets deformed into a A deformed shell of water with a bump on this side and a bump on that side. All right, I'm not going to go any more deeply into that.
That I'm sure you've all seen. Okay, but let's define now what we mean by the gravitational field. The gravitational field is abstracted from this formula.
We have a bunch of particles. Question? Yeah. Don't you have, need some sort of a coordinate geometry so that when you have the poor guy in the middle is being pulled by all the other guys on the side?
I'm not explaining it right, but it's always negative, is that what you're saying? No, I'm saying it's always attractive. All right, so you have, but what about the other guys that are pulling upon him from different directions?
Well, let's suppose there's somebody over here, and we're talking about the force on this person over here. Yeah. Obviously there's one force pushing this way and another force pushing that way.
They're all negative. No, they're all opposite to the direction of the object which is pulling on them. That's what this minus sign says. Well, you kind of retracted the minus sign at the front and reversed the J-I.
Yeah. So it's the direction. We can get rid of the minus sign at the front there by making the sign J. Rij and rji are opposite to each other. One of them is the vector between i and j, i and j, and the other one is the vector from j to i, so they're equal and opposite to each other.
The minus sign... there look as far as the minus sign goes all it means is that every one of these particles is pulling on this particle toward it as opposed to pushing away from it it's just a convention which keeps track of attraction instead of repulsion. Yeah, for the, for the, for the eif mass, if that's the right word.
Yeah. That makes sense, but if you, if you look at it as a kind of an ensemble, wouldn't there be a nonlinear kind of component to it? Because the eif guy affects the j-th guy, and then when you compute the j-th guy, you know what I mean?
When you take into account the motion. Yeah. Now what this, what this formula is for is supposing you know the positions of all the others. You know that. Then what is the force on one additional one?
But you're perfectly right. Once you let the system evolve, then each one will cause a change in motion in the other one, and so it becomes a complicated, as you say, nonlinear mess. But this formula is a formula for if you knew the position and location of every particle, this would be the force. You need to solve some equations to know the force. know how the particles move.
But if you know where they are, then this is the force on the ith particle. All right, let's come to the idea of the gravitational field. The gravitational field is in some way similar to the electric field of an electric charge. It's the combined effect of all the masses Everywhere's.
And the way you define it is as follows. You imagine one more particle, one more particle. You can take it to be a very light particle so it doesn't influence the motion of the others. Add one more particle in your imagination. You don't really have to add it in your imagination.
And ask what the force on it is. The force is the sum of the force due to all the others. It is proportional.
Each term... ...is proportional to the mass of this extra particle. This extra particle, which may be imaginary, is called a test particle.
It's a thing that you're imagining testing out the gravitational field with. You take a light little particle and you put it here and you see how it accelerates. Knowing how it accelerates tells you how much force is on it.
In fact, it just tells you how it accelerates. And you can go around and imagine putting it in different places and mapping out the force field that's on that particle. Or the acceleration field.
Since we already know that the force is proportional to the mass, then we can just concentrate on the acceleration. The acceleration, all particles will have the same acceleration, independent of their mass. So we don't even have to know what the mass of the particle is.
We put something over there, a little bit of dust, and we see how it accelerates. Acceleration is a vector, and so we map out in space the acceleration of a particle at every point in space, either imaginary or real particle, and that gives us a vector field at every point in space. Every point in space, there is a...
gravitational field of acceleration, it can be thought of as the acceleration, you don't have to think of it as force, acceleration, the acceleration of a point mass located at that position. It's a vector, it has a direction, it has a magnitude, and it's a function of position. So we just give it a name.
The acceleration due to all the gravitating objects, it's a vector, and it depends on position. Here, x means location, it means all of the components of position, x, y, and z, and it depends on all the other masses in the problem. That is what's called a gravitational field. It's very similar to the electric field, except the electric field is force per unit charge. It's the force on an object divided by the charge on the object.
The gravitational field is the force on the object divided by the mass. on the object. Since the force is proportional to the mass, the acceleration field doesn't depend on which kind of particle we're talking about. So that's the idea of a gravitational field.
It's a vector field, and it varies from place to place. And of course, if the particles are moving, it also varies in time. If everything is in motion, the gravitational field will also depend on time. We can even work out what it is.
We know what the force on the i-th particle is. The force on a particle is the mass times the acceleration. So if we want to find the acceleration, let's take the i-th particle to be the test particle. Little i represents the test particle over here.
Let's erase. We erase the intermediate step over here and write that this is m i times a i, but let me call it now capital A. The acceleration of a particle at position x is given by the right hand side.
And we can cross out the m i because it cancels from both sides. So here's a formula for the gravitational field at an arbitrary point due to a whole bunch of massive objects. A whole bunch of massive objects. An arbitrary particle put over here will accelerate in some direction that's determined by all the others and that acceleration is the gravitation, the definition, is the definition of the gravitational field.
Okay let's um let's take a little break, usually take a break at about this time and I recover my breath. To go on, we need a little bit of fancy mathematics. We need a piece of mathematics called Gauss's theorem.
And Gauss's theorem involves integrals, derivatives, divergences, and we need to spell those things out. They're a central part of the theory of gravity and much of these things we've done in the context of electrical forces, in particular the concept of divergence, divergence of a vector field. So I'm not going to spend a lot of time on it. If you need to fill in, then I suggest you just find any little book on vector calculus and find out what a divergence and a gradient and a curl, we won't do curl today, what those concepts are.
are and look up Gauss's theorem and they're not terribly hard but we're going to go through them fairly quickly here since we've done them several times in the past. Alright, imagine that we have a vector field. Let's call that vector field A.
It could be the field of acceleration and that's the way I'm going to use it but for the moment it's just an arbitrary vector field, A. It depends on position. When I say it's a field, the implication is that it depends on position.
Now I probably made it completely unreadable. A of x varies from point to point. I want to define a concept called the divergence of the field.
Now, it's called the divergence because what it has to do is the way the field is spreading out away from a point. For example, a characteristic situation where we would have a strong divergence for a field is if the field was spreading out from a point like that. The field is diverging away from the point. Incidentally, if the field is pointing inward, then one might say the field has a convergence, but we simply say it has a negative divergence.
So divergence can be positive or negative. And there's a mathematical expression which represents the degree to which the field is spreading out like that. It is called the divergence. I'm going to write it down, and it's a good thing to get familiar with.
Certainly, if you're going to follow this course, it's a good thing to get familiar with. But if you're going to follow any kind of physics course past freshman physics, the idea of divergence is very important. Supposing the field A has a set of components.
One, two, and three component, or we could call them the x, y, and z component. Now I'll use x, y, and z. I have x, y, and z, which I previously called x1, x2, and x3.
It has components ax, ay, and az. Those are the three components of the field. Well, the divergence has to do, among other things, with the way the field varies in space. If the field is the same everywhere as in space, what does that mean? That would mean the field has both not only the same magnitude but the same direction everywhere in space, then it just points in the same direction everywhere with the same magnitude.
It certainly has no tendency to spread out. When does a field have a tendency to spread out? when the field varies, for example, it could be small over here, growing bigger, growing bigger, growing bigger, and we might even go in the opposite direction and discover that it's in the opposite direction and getting bigger in that direction. Then clearly there's a tendency for the field to spread out away from the center here. The same thing could be true if it were varying in the vertical direction or if it were varying in the other horizontal direction.
direction and so the divergence whatever it is has to do with derivatives of the components of the field I'll just tell you exactly what it is it is equal to the divergence of the field is written this way upside down triangle And the meaning of this symbol, the meaning of an upside-down triangle, is always that it has to do with the derivatives, the three derivatives, the three partial derivatives, derivative with respect to x, y, and z. And this is by definition... The derivative with respect to x of the x component of A plus the derivative with respect to y of the y component of A plus the derivative with respect to z of the z component of A. That's definition.
What's not a definition... There's a theorem, and it's called Gauss's theorem. I'm sorry, is that a vector or a scalar quantity? No, that's a scalar quantity. That's a scalar quantity.
Yeah, it's a scalar quantity. So it's, let me write it. It's the derivative of a sub x with respect to x, that's what this means, plus the derivative of a sub y with respect to y, plus the derivative of a sub z with respect to z.
Yeah, so the arrows you were drawn over there, those were just A on the other board. You drew some arrows on the other board that are now hidden. Yeah. Those were just A. Yeah.
Not the divergence. Right. Those were A. And A has a divergence when it's spreading out away from a point, but the divergence is itself a scalar quantity. Okay?
Let me try to give you some idea of what divergence means in a context where you can visualize it. Imagine that we have a flat lake. All right, just the water thin, a shallow lake. And water is coming up from underneath. It's being pumped in from somewhere underneath.
What happens if the water is being pumped in? Of course, it tends to spread out. Let's assume that the high, let's assume the depth can't change.
We put a lid over the whole thing so it can't change its depth. We pump some water in from underneath and it spreads out. We suck some water out from underneath and it spreads.
in it anti spreads it has so the spreading water has a divergence water coming in towards the towards the place where it's being sucked out it has a convergence or a negative divergence now we can be more precise about that we look down at the lake from above and we see all the water is moving of course it's moving if it's being pumped in the world it's moving and there is a velocity vector at every point there is a velocity vector so at every point in this lake there's a velocity vector and in particular if there's water being pumped in from the center here right underneath the bottom of the lake there's some water being pumped in the water will spread out away from that point okay and there'll be a divergence where the water is being pumped in okay if the water is being pumped out then exactly the opposite the the arrows point inward and there's a negative divergence. The, if there's no divergence, then for example a simple situation with no divergence, that doesn't mean the water is not moving, but a simple example with no divergence is the water is all moving together. You know the river is simultaneous, the lake is all simultaneously moving in the same direction with the same velocity. It can do that without any water being pumped in, but if you found that the water was moving to the right direction, right on this side and the left on that side, you'd be pretty sure that somewhere in between water had to be pumped in, right? If you found the water was spreading out away from a line this way here and this way here, then you'd be pretty sure that some water was being pumped in from underneath along this line here.
Well, you would see it another way, you would discover that the X component of the velocity has a derivative. It's different over here than it is over here. The x component of the velocity varies along the x direction. So the fact that the x component of the velocity is varying along the x direction is an indication that there's some water being pumped in here. Likewise, if you discovered that the water was flowing up over here, down over here, you would expect that in here somewhere some water was being pumped in.
So derivatives of the velocity are often an indication that there's some water being pumped in from underneath. That pumping in of the water of the water is the divergence of the velocity vector. Now, the water, of course, is being pumped in from underneath.
So there's a direction of flow, but it's coming from underneath. There's no sense of direction. Well.
OK. That's what divergence is. The diagrams you already have on the other board behind there with the arrows, I think, is.
Yeah. I think that's what I mean. OK. You take, say, the rightmost arrow, and you.
draw a circle between the head and tail in between, then you can see the in and the out. The in arrow and the out arrow of a circle right in between those two. And let's say that the bigger arrow is created by a steeper slope of the street. It's just faster.
It's going faster. It's going faster. And because of that, there is a divergence there that's basically, it's sort of the difference between the in and the out.
Right. That's right. That's right. If we draw a circle around here, we would see that more, since the water is moving faster over here than it is over here, more water is flowing out over here than is coming in over here.
Where's it coming from? It must be pumped in. The fact that there's more water flowing out on one side than is coming in from the other side must indicate that there's a net inflow from somewhere else, and the somewhere else would be from the pump-in water from underneath. So that's the idea of divergence. Could it also be because it's thinning out?
Would that be a crazy example, like the lake got shallower? Yeah, well, okay. I took a, all right, so let's be very specific now.
I kept the lake having an absolutely uniform height, and let's also suppose that the density of water, water is an incompressible fluid. It can't be squeezed. It can't be stretched. Then the velocity vector would be the right thing to think about then. Yeah, but you could have, no, you're right you could have a velocity vector having a divergence because the water is not because water is flowing in but because it's thinning out yeah that's that's also possible okay but let's keep it simple All right, and you can have the idea of a divergence makes sense in three dimensions just as well as two dimensions.
You simply have to imagine that all of space is filled with water, and there are some hidden pipes coming in, depositing water in different places so that it's spreading out away from points in three-dimensional space. In three-dimensional space, this is the expression for the divergence. If this were the velocity vector at every point, you would calculate this quantity, and that would tell you how much new water is coming in at each point of space. So that's the divergence.
Now, there's a theorem which the hint of the theorem was just given by Michael there. It's called Gauss's theorem. And it says something very intuitively obvious.
You take a surface, any surface. Take any surface or any curve in two dimensions, and now suppose there's a vector field. Vector field points.
Think of it as the flow of water. And now let's take the total amount of water that's flowing out of the surface. Obviously there's some water flowing out over here, and of course we want to subtract the water that's flowing in.
Let's calculate the total amount of water that's flowing out of the surface. That's an integral over the surface. Why is it an integral? Because we have to add up the flows of water outward.
Where the water is coming inward, that's just negative flow, negative outward flow. We add up the total outward flow by breaking up the surface into little pieces and asking how much flow is coming out from each little piece here, how much water is passing out through the surface. If the water is incompressible, incompressible means density is fixed, and furthermore, the depth of the water is being kept fixed. There's only one way that water can come out of the surface, and that's if it's being pumped in, if there's a divergence.
The divergence could be over here, could be over here, could be over here, could be over here. In fact, anywhere where there's a divergence will cause an effect in which water will flow out of this region here. So there's a connection. There's a connection between what's going on on the boundary of this region, how much water is flowing through the boundary on the one hand, and what the divergence is in the interior. There's a connection between the two, and that connection is called Gauss's theorem.
What it says is that the integral of the divergence in the interior, that's the total amount of flow coming in from outside, from underneath the bottom of the lake, the total integrated, and now by integrated I mean in the sense of an integral, the integrated amount of flow in, that's the integral of the divergence, The integral over the interior, in the three-dimensional case, it would be integral dxdy dz over the interior of this region of the divergence of A. if you like to think of A as the velocity field, that's fine, is equal to the total amount of flow that's going out through the boundary. And how do we write that?
The total amount of... the flow that's flowing outward through the boundary, we break up, let's take the three-dimensional case, we break up the boundary into little cells. Each little cell is a little area. Let's call each one of those little areas d sigma.
d sigma, sigma stands for surface area. Sigma is the Greek letter sigma, it stands for surface area. This three-dimensional integral Over the interior here is equal to a two-dimensional integral, d sigma over the surface, and it is just the component of A perpendicular to the surface.
Let's call it A perpendicular to the surface, d sigma. A perpendicular to the surface is the amount of flow that's coming out of each one of these little boxes. Notice, incidentally, that if there's a flow along the surface... it doesn't give rise to any fluid coming out.
It's only the flow perpendicular to the surface, the component of the flow perpendicular to the surface, which carries fluid from the inside to the outside. So we integrate the perpendicular component of the flow. over the surface, that's the sigma here, that gives us the total amount of fluid coming out per unit time, for example, and that has to be equal to the amount of fluid that's being generated in the interior by the divergence.
This is Gauss's theorem, the relationship between the integral of the divergence on the interior of some region and an integral over the boundary where... where it's measuring the flux, the amount of stuff that's coming out through the boundary. Fundamental theorem.
And let's see what it says now. Any questions about Gauss's theorem here? You'll see how it works. I'll show you how it works in a minute. Yeah.
Yeah. You mentioned that for water, as in compressible, does that mean that we're dealing with a compressible product? Well, yeah.
Yeah, you could have, sure, if you had a compressible fluid, you could discover that the fluid out in the boundary here is all moving inwards in every direction without any new fluid being formed. In fact, what's happening is just the fluid is getting squeezed. But if the fluid can't squeeze, if you cannot compress it, then the only way that fluid could be flowing in is if it's being removed somehow from the center.
If it's being removed by invisible pipes that are carrying it off. So that means the divergence in the case of water would be zero? Would you integrate it over a volume? If there was no water coming in, it wouldn't be zero.
If there was a source of the water, divergence is the same as source. Source of water is the source of new water coming in from elsewhere is right. So in the example with the two-dimensional lake, the source...
is water flowing in from underneath. The sink, which is the negative of a source, is the water flowing out. And in the two-dimensional example, this wouldn't be a two-dimensional surface integral. It would be the integral in here equal to a one-dimensional surface integral coming out.
All right, let me show you how you use this. Let me show you how you use this. and what it has to do with what we've said up till now about gravity. I hope we'll have time.
Let's imagine that we have a source. It could be water, but let's take a three-dimensional case. There's a divergence of a vector field.
Let's say A. There's a divergence of a vector field. dot a, and it's concentrated in some region of space that's a little sphere, in some region of space that has spherical symmetry.
In other words, it doesn't mean it doesn't mean that the divergence is uniform over here, but it means that it has the symmetry of a sphere. Everything is symmetrical with respect to rotations. Let's suppose that there's a divergence of the fluid.
And it's restricted completely to be within here. It could be strong near the center and weak near the outside, or it could be weak near the center and strong near the outside, but a certain total amount of fluid, or a certain total divergence, an integrated divergence, is occurring with nice spherical shape. Okay, let's see if we can use that to figure out what the field, what the A field is. There's a del dot A in here. And now let's see, can we figure out what the field is elsewhere, outside of here?
So what we do is we draw a surface around there. We draw a surface around there. And now we're going to use Gauss's theorem. First of all, let's look at the left side. The left side has the integral of the divergence of the vector field.
The vector field, or the divergence, is completely restricted to some finite sphere in here. What is, incidentally, for the flow case, for the fluid flow case, what would be the integral of the divergence? Does anybody know? If it really was a flow of a fluid. It would be the total amount of fluid that was flowing in per unit time.
It would be the flow per unit time that's coming into the system. But whatever it is, this integral doesn't depend on the radius of the sphere as long as the sphere... this outer sphere here is bigger than this region. Why? Because the integral over the divergence of A is entirely concentrated in this region here, and there's zero divergence on the outside.
So first of all, First of all, the left hand side is independent of the radius of this outer sphere, as long as the radius of the outer sphere is bigger than this concentration of divergence here. So it's a number. All together it's a number. Let's call that number M. No, not M.
Let's just, Q. That's the left-hand side, and it doesn't depend on the radius. On the other hand, what is the right-hand side? Well, there's a flow going out, and if everything is nice and spherically symmetric, then the flow is going to go... radially outward.
It's going to be a pure radially outward directed flow if the flow is spherically symmetric. Radially outward directed flow means that the flow is perpendicular to the surface of the sphere. So the perpendicular component of A is just the magnitude of A.
That's it. It's just the magnitude of A. And it's the same everywhere as on the sphere.
Why is it the same? Because everything has spherical symmetry. Spherical symmetry, the a that appears here, is constant over this whole sphere. So this integral is nothing but the magnitude of a times the area of the total sphere.
If I take an integral over a surface, a spherical surface like this, of something which doesn't depend on where I am on the sphere, then it's just, you can take this on the outside, the magnitude of the field, and the integral d sigma is just the total surface area of the sphere. What's the total surface area of the sphere? Four-thirds pi r minus four-pius.
No third, just four pi. 4 pi r squared. Oh, yeah.
4 pi r squared times the magnitude of the field is equal to q. So look what we have. We have that the magnitude of the field is equal to the total integrated divergence divided by 4 pi.
4 pi is just a number, times r squared. Does that look familiar? It's a vector field.
It's pointed radially outward. Well, it's pointed radially outward if the divergence is positive. If the divergence is positive, it's pointed radially outward, and its magnitude is 1 over r squared.
It's exactly the gravitational field of a point particle at the center here. That doesn't look like magnitude A. Hm? That's the part you've got to scale everything. The magnitude of A.
Yeah. That's why we have to put a direction in here. You know what this r hat, this r over r is? It's a unit vector pointing in the radial direction. It's a vector of unit length pointing in the radial direction.
So it's quite clear from the picture that the a field is pointing radially outward. That's what this says over here. In any case, the magnitude of the field, it points radially outward. It has magnitude Q, and it falls off like 1 over r squared, exactly like the Newtonian field of a point mass.
So a point mass can be thought of as a concentrated divergence Of the gravitational field right at the center. A point mass, a literal point mass, can be thought of as a concentrated... a concentrated divergence of the gravitational field.
concentrated in some very, very small little volume. Think of it, if you like, you can think of the gravitational field as the flow field or the velocity field of a fluid that's spreading out. Oh, incidentally, of course, I've got the sign wrong here. The real gravitational acceleration points inward.
Which is an indication that this divergence is negative. The divergence is more like a convergence sucking fluid in. So the Newtonian gravitational field is isolated. isomorphic is mathematically equivalent or mathematically similar to a flow field to a flow of water or whatever other fluid where it's all being sucked out from a single point and as you can see the velocity field itself, or in this case, the gravitational field, but the velocity field would go like 1 over r squared.
That's a useful analogy. That is not to say that space is a flow of anything. It's a mathematical analogy that's useful to understand the 1 over r squared force law, that it is mathematically similar to a field of velocity flow from a flow.
that's being generated right at the center at a point. Okay that's a useful observation but notice something else. Supposing now instead of having the flow concentrated at the center here, supposing the flow was concentrated over a sphere which was bigger but the same total amount of flow It would not change the answer. As long as the total amount of flow is fixed, the way that it flows out through here is also fixed.
This is Newton's theorem. Newton's theorem in the gravitational context says that the gravitational field of an object outside the object ...is independent of whether the object is a point mass at the center, or whether it's a spread-out mass, or whether it's a spread-out mass this big. As long as you're outside the object, and as long as the object is... symmetrically symmetric, in other words, as long as the object is shaped like a sphere, and you're outside of it, on the outside of it, outside of where the mass distribution is, then the gravitational field of it doesn't depend on what you do. whether it's a point, it's a spread out object, whether it's denser at the center and less dense at the outside, less dense in the inside, more dense in the outside all it depends on is the total amount of mass, the total amount of mass is like the total amount of flow through coming into the...
that theorem is very fundamental and important to thinking about gravity For example, supposing we are interested in the motion of an object near the surface of the Earth, but not so near that we can make the flat space approximation. Let's say at a distance 2 or 3 or 1 times the radius of the Earth. Well, that object is attracted by this point, it's attracted by this point, it's attracted by that point, it's close to this point, it's far from this point, it sounds like a hellish...
problem to figure out what the gravitational effect on this point is but no this tells you the gravitational field is exactly the same as if the same total mass was concentrated right at the center That's Newton's theorem. And it's a marvelous theorem. It's a great piece of luck for him, because without it, he couldn't have solved his equations. He knew.
He had an argument. it may have been essentially this argument, I'm not sure exactly what argument he made, but he knew that with the 1 over r squared force law, and only the 1 over r squared force law, wouldn't have been true if it was 1 over r cubed, 1 over r to the 4th, 1 over r to the 7th. With the 1 over r squared force law, a spherical distribution of mass behaves exactly as if all the mass was concentrated right at the center, as long as you're outside the mass. So that's what made it possible for Newton to easily solve his own equations, that every object, as long as it's spherical in shape, behaves as if it were a point mass. So if you're down in a mine shaft that doesn't hold?
That's right. If it's not in the mineshaft, it doesn't hold. But that doesn't mean you can't figure out what's going on. You can figure out what's going on.
I don't think we'll do it tonight. It's a little too late. But yes, we can work out what would happen in the mineshaft. But that's right. It doesn't hold in a mineshaft.
For example, supposing you dig a mine shaft right down through the center of the earth. Okay? And now you get very close to the center of the earth. How much force do you expect to be pulling you toward the center?
Not much. Certainly much less. than if all the mass were concentrated right at the center.
You've got... It's not even obvious which way the force is, but it is toward the center, but it's very small. You displace away from the center of the Earth a little bit, there's a tiny, tiny little force.
Much, much less than as if all the mass was squashed toward the center. So, right, it doesn't work... for that case.
Another interesting case is supposing you have a shell of material. To have a shell of material, think about a shell of source, fluid flowing in. Fluid is flowing in from the outside onto this blackboard, and all the little pipes are arranged on a circle like this.
What does the fluid flow look like in different places? Well, the answer is on the outside, it looks exactly the same as if everything were concentrated on a point. But what about in the interior?
What would you guess? Nothing. Nothing, everything is just flowing out away from here, and there's no flow in here at all.
How could there be? Which direction would it be in? So there's no flow in here.
Wouldn't you have the distance argument? Like, if you're closer to the surface of the inner shell, wouldn't that be more force towards that? No, you see, you use Gauss's theorem. Let's do Gauss's theorem.
Gauss's theorem says, okay, let's take a shell. the field, the integrated field coming out of that shell is equal to the integrated divergence in here but there is no divergence in here, so the net integrated field coming out is zero no field on the interior of the shell, field on the exterior of the shell so the consequence is that if you made a spherical shell of material like that the interior would be absolutely identical to what it would be if there was no gravitating material there at all on the other hand on the outside you would have a field which would be absolutely identical to what happens at the center. Now, there is an analog of this in the general theory of relativity. We'll get to it. Basically what it says is the field of anything, as long as it's spherically symmetric on the outside, looks identical to the field of a black hole.
But I think we'll finish for tonight. Go over divergence and all those Gauss's theorem. Gauss's theorem is central. There would be no gravity without Gauss's theorem.
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