Now we're going to talk more about gravity. So up till now, we said gravity is just a constant acceleration down. It's a force proportional to mass near the surface of the Earth, always points down. Everything accelerates the same due to gravity, and that's all we talked about. But of course, gravity is more than that, right? If you look, we usually have what we call universal gravitation. Meaning not just the surface of the Earth, but any two objects in the universe. And what gravity does, and the good part is, I like to say we are all attractive. This makes us happy, although some people don't want to be attractive. But that's the basic idea, every two objects in the universe attract each other. By a formula that looks like this, Fg = G, a new constant, times the mass of one times the mass of the other over their separation. Okay, so if we want to think about the graph, here's mass 1, here's mass 2, here is an axis between them. So the gravitational force between two masses always acts along the axis that connects them. If you're doing multiple masses and adding all the forces, then you treat them as vectors, you vectors sum them. But the individual gravitational force between two, you can always get the direction just by drawing a line between them, you can always draw line. So it's always attractive, so this one feels the gravitational force that way, and this one feels the gravitational force that way. And they both feel it by this formula, and you can see they make an action-reaction pair. Because they feel the same force, the on force this one, G m1 m2 over r squared, where r is the separation. The force on this one, G m1 m2 over r squared. They both depend on m1 and m2, so they both get the same magnitude. So let's look at a few of these, make sure we know when everything is. G is the gravitational constant, which I can never remember the value for some reason. I don't know why, I always have to look it up, it's 6.674 x 10 to the -11. And you can tell what the unit must be from this formula, it must be newtons meters squared per kilogram squared. Newton meters squared per kilogram squared, but of course, you're going to double check, yeah. So that's G In that formula, m1 and m2 are the masses, In kilograms, of course, if you're working in MKS units. And r, to be specific, r is the separation, Of their center of mass. It's not from the edge to the edge, for uniform spheres, it's from the center to the center. Now I've drawn it as a vector here, so let's think about the direction of the vector, the direction of each force. And I'll just put attractive. Okay, so for this one, it's that way, and for this one, it's that way, it's toward the other mass. And for the magnitude, I'll remind you that it's equal. Each one feels the same magnitude, that's why it makes an action-reaction pair. And that's why if two things start moving toward each other gravitationally, it doesn't break momentum conservation. This one builds up momentum this way, this one builds up momentum this way. Momentum is a vector sum, so the momentum would remain 0. They look like they're building up energy, you're gaining kinetic energy, but they have potential energy, you're losing potential energy. So energy is also conserved. Now if you're doing problems where you really need a more formal vector statement than that, let me show you how we do this with vectors. In this case, we would say mass 1, mass 2. So I'll say here's your vector notation. You would say the force 1-2, so that means the force that 1 causes 2 to feel. So F1-2 is a force on mass 2 due to mass 1. Okay, it has a negative sign, and you'll see why in a minute. Big G, mass 1, mass 2, r1-2. Writing it this way, that's just the distance between 1 and 2, it's always positive. It's really the magnitude of the vector from 1 to 2, but I'm going to write it in a way that that's just a positive number, it's a magnitude. Squared, of course. I did squared over here, yeah, squared. And now we need the unit vector 1-2, that gets you the direction, right? So what this means is, the force on 2 due to 1, the vector r1-2 goes from 1 to 2. The unit vector r1-2 is in that direction from 1 to 2 with a magnitude of 1. Right, so I could right here, there is the unit vector 1-2. So since it points in the direction 1 to 2, and since the gravitational force is attractive, the force will be the opposite direction. That's why there's a negative sign there, okay? So this is the vector way to write it. And now let's imagine we switched 1 and 2, what if it was F2-1? The force that 2 causes 1 to feel, okay? F2-1 would be -G m1 m2 over r2-1 squared, that's all the same. But it'll be the unit vector r2-1, unit vector r2-1. Well, the vector r 2 to 1 is this way, therefore the unit vector is also that way, r hat 2-1. So that expression would be this way, except it has a negative sign on it. So there you go, okay? So the unit vector points the wrong way for an attractive force. This is what makes it an attractive force.