Newton's law of universal gravitation
Viewing g as the value of Earth's gravitational field near the surface
What I want to do in this video is think about the two different ways of interpreting lowercase g. Which as we've talked about before, many textbooks will give you as either 9.81 meters per second squared downward or towards the Earth's center. Or sometimes it's given with a negative quantity that signifies the direction, which is essentially downwards, negative 9.81 meters per second squared. And probably the most typical way to interpret this value, as the acceleration due to gravity near Earth's surface for an object in free fall. And this is what we're going to focus on this video. And the reason why I'm stressing this last part is because we know of many objects that are near the surface of the Earth that are not in free fall. For example, I am near the surface of the Earth right now, and I am not in free fall. What's happening to me right now is I'm sitting in a chair. And so this is my chair-- draw a little stick drawing on my chair, and this is me. And let's just say that the chair is supporting all my weight. So I have-- my legs are flying in the air. So this is me. And so what's happening right now? If I were in free fall, I would be accelerating towards the center of the Earth at 9.81 meters per second squared. But what's happening is, all of the force due to gravity is being completely offset by the normal force from the surface of the chair onto my pants, and so this is normal force. And now I'll make them both as vectors. So the net force in my situation-- the net force is equal to 0, especially in this vertical direction. And because the net force is equal to 0, I am not accelerating towards the center of the Earth. I am not in free fall. And because this 9.81 meters per second squared still seems relevant to my situation-- I'll talk about that in a second. But I'm not an object in free fall. Another way to interpret this is not as the acceleration due to gravity near Earth's surface for an object in free fall, although it is that-- a maybe more general way to interpret this is the gravitational-- or Earth's gravitational field. Or it's really the average acceleration, or the average, because it actually changes slightly throughout the surface of the Earth. But another way to view this, as the average gravitational field at Earth's surface. Let me write it that way in pink. So the average gravitational field-- and we'll talk about what a field means in the physics context in a second-- the average gravitational field at Earth's surface. And this is a little bit more of an abstract thing-- we'll talk about that in a second-- but it does help us think about how g is related to this scenario where I am not an object in free fall. A field, when you think of it in the physics context-- slightly more abstract notion when you start thinking about it in the mathematics context-- but in the physics context, a field is just something that associates a quantity with every point in space. So this is just a quantity with every point in space. And it can actually be a scalar quantity, in which case we call it a scalar field, and in which case it would just be a value. Or it could be a vector quantity, which would be a magnitude and a direction associated with every point in space. In which case you are dealing with a vector field. And the reason why this is called a field is, because at near Earth's surface, if you give me a mass-- so for example-- actually, I don't know what my mass is in kilograms. But if you're near Earth's surface and you give me a mass-- so let's say that mass right over there is 10 kilograms-- you can use g to figure out the actual force of gravity on that object at that point in space. So for example, if this has a mass of 10 kilograms, then we know-- and this right over here is the surface of the Earth, so that's the center of the Earth. So it actually associates a vector quantity whose magnitude,-- so its direction is towards the center of the Earth, and the magnitude of this vector quantity is going to be the mass times g. And you could take-- since we're already specifying the direction, we could say 9.81 meters per second squared towards the center of the Earth. And so in this situation, it would be 10 kilograms times 9.81 meters per second squared. Which is 98.1. And even this I've rounded a little bit, so it's actually approximate number. 98.1 kilogram meters per second squared which is the unit of force, or 98.1 newtons. And this thing might not be in free fall, so this is why g is relevant even in a situation where the object isn't in free fall. g has given us the force per unit mass-- the force per mass of gravity on an object near the surface of the Earth. Another way to think about it-- so this is the average gravitational field, and what it's giving is force per mass. So you give me a mass near Earth's surface-- whether it's an object in free fall or not-- you multiply that mass times g, because it's giving you force per mass, and it will give you the force of gravity acting on that object near the surface of the Earth, whether or not it's in free fall. So I just want to make this little distinction, because although g tends to be referred to this way right over here. Sometimes you might encounter a stickler who says oh no, no, no, no, no but g is relevant even when an object is not in free fall. You obviously can't say that my acceleration when I'm sitting in my chair is 9.81 meters per second squared towards the center of the Earth. I am not accelerating towards the center of the Earth. And so they'll say, oh no, no, no, no, you can't just call this acceleration. It is true, it is the acceleration when an object is in free fall near the surface of the Earth-- if you don't have really air resistance, if the net force really is the force of gravity-- then this really would be the object's acceleration. But it becomes relevant, and we know most objects that we know of aren't in free fall. Obviously, an object in free fall doesn't stay in free fall for long. It eventually hits something. But we know that now g is actually relevant to all objects. It tells us the force per mass And it's tempting to call it always acceleration-- because the units are acceleration-- but even when you talk about in terms of the gravitational field, it's still the same quantity. It still has the exact same units, the same magnitude, and the same direction-- it's just a different way of viewing it. Here, acceleration for an object in free fall. Here, something to multiply by mass to figure out the force due to gravity.