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## AP®︎/College Physics 1

### Course: AP®︎/College Physics 1>Unit 3

Lesson 4: Gravitational fields and acceleration due to gravity on different planets

# Viewing g as the value of Earth's gravitational field near the surface

There are two ways to interpret the constant g = (-)9.81 m/s². The first interpretation is the acceleration due to gravity of an object in free fall near Earth's surface. The second interpretation is the average gravitational field at Earth's surface, which can be used to calculate the force of gravity on an object. Created by Sal Khan.

## Want to join the conversation?

• This video got me thinking. Suppose that my friend and I are tunneling downwards towards the Earth's center. As we proceed, I reason out that we should start to weigh less, since some of the Earth's mass that used to be beneath us is now "above" us (but not directly above us, of course) and is exerting a force that starts to contradict the force pulling us towards the Earth's center. Yet my friend reasons out that we should start to weigh more since according to Newton's law of universal gravitation, which states that F = G(m1)(m2)/r^2, the closer we get to Earth's center, the smaller the distance between the centers of our masses and of the Earth, which translates to a greater force.

We then think about what would happen to our weights when we reach the Earth's center. I say that we should become weightless, since there are equal amounts of mass pulling us in all directions, and therefore all forces should cancel out (we assumed that the Earth is a perfect sphere). My friend says, "No. Since the distance between the centers of our masses and of the Earth is zero, we would experience... infinite force?"

Which of us is correct? And what are your views on this?
• The law of gravitation treats the two objects as particles, meaning all the mass is located at one central point (this is Gauss's law, and only applies when the mass being influenced is outside of the sphere). However, in reality, the mass of the earth is distributed all around this blue sphere we call home. Although conceptually you are correct and your friend is wrong (forces of gravity influence you from all directions, ultimately canceling out), the pressure and heat at the center of the earth would compress/melt you to death before you could enjoy such a feeling of weightlessness.
• What is the cause of gravitation?

Is it even known yet?
Does the theory of general relativity explain it or is it a fundamental interaction carried by a hypothetical graviton particle?
• From a quantum field theory perspective all forces are carried by particles referred to as gauge bosons, for gravity it would be the Graviton but we do not really have a quantum description of gravity that is mathematically consistent.

The best description we have of gravity is General Relativity and it depicts gravity as curvature of space-time.
• I don't understand what (kg*m)/s^2 means. How can you have kilogram meters? And how do seconds matter for the force experienced by an object on the ground?
• lets say we have a point mass(something as small as a point with some mass) now lets say u tie a string to it and pull it, to pull it you have to apply a force, and how would you define the amount of force you have you put in to flex your strength? like sal says pause and think, now sumin u have had a go at it, we work this out together, you will say, i pulled a mass of 'x' kilograms, 'x' meters away, in 'x^2' amount of time(aka the measure of the relative changes in your environment while you were at work). and thats the description of force. mass, distance, time, they are the dimensions, the descriptions we made to describe things in our real life. atleast thats my understanding of it.
• How were scientists able to determine that g is equal to 9.81 m/s^2? And how are they able to determine the force of gravity on other planets?
• By Newton's law of universal gravitation F1 = F2 = G* (m1*m2)/r^2
we multiply the Gravitational constant G = 6.673X10^-11 by the earth's mass divided by the earth's radius which will give us F/m2 = acceleration derived from the formula F = ma.
• So, every time I trip over, I fall at an acceleration of 9.81 m/s^2 ?
• It actually depends upon exactly where on Earth you fall. g is not truly constant; it varies from location to location. 9.8 m/s^2 is just an average. The true value varies with your latitude and longitude (mainly due to surface variations in Earth's density). g also varies with altitude according to the formula: g/g' = (r'/r)^2, where g' equals g at the Earth's (or other celestial body's surface) and r' equals distance from the center of gravity of Earth (or other celestial body). For example, if you are at an altitude of 2.2x10^7 m (22000 km), the value of g drops to 0.512 m/s!
• At , it talks about the average gravitational field on the Earth's surface. I wanted to know how far does an object's gravity effect on matter? If it has no limit, even though it is negligible, does combing the distance and mass of objects near by give a combined for of gravity? (For example would the Earth's gravitational pull on the other planets be greater than 9.8 m/s^2, would it be the same, or based on the distance that other objects are in space it would decrease)

...and if theoretically there is no limit to how far matter's gravity is affecting other matter, than wouldn't that mean that the whole universe is pulling on itself?
• To answer your last question first, yes, there is no limit to the effect of gravity, and this does indeed mean that the whole universe is 'pulling' on itself. However, the strength of a gravitational pull gets weaker depending on the square of the distance between two objects, so at interstellar distances the gravitational effect of, say, our Sun would rapidly become very weak.

The Earth's pull at its surface is 9.8 m/s^2, but an object at its surface is only about 6400 km from the centre. The Moon is 384000 km away, which is 60 times as far, so the Earth's gravitational pull on the Moon is 60^2 (which is 3600) times as weak -- only 2.7 millimeters per second squared. Going out to other planets, Jupiter is 2000 times further away than the Moon, so the Earth's pull on Jupiter is a whopping 15 billion times smaller than the Earth's pull at its own surface. So the numbers do get very small very quickly.
• what exatly is normal force
• The acceleration in free fall is -9.81 m/s2 , but while skydiving you would be in equilibrium, but shouldn't it still be 9.8 m/s2 ?
• that's true because when you are sitting in a chair you are not free falling, and that chair is supporting your weight.
• I didn't really get the meaning of a field, at . Could someone please explain that?
How does a field associate EVERY point in space with something? Don't they also have an end? (This is only relevant if my understanding of a field so far is correct)
• Think of temperature, you can define a temperature at every point in the room you are in. This ability to define a temperature at a point doesn't stop at the walls of the room, you can continue to do this throughout all of space.
• For an object at the poles and the earth is rotating , why is there no centripetal acceleration for the object . Also what if the earth is stationary
• If you are at the pole, the radius from the axis of rotation is zero, so your centripetal acceleration from the Earth's rotation will also be zero.
a = v²/r = rω²
If r=0, then a=0.
If you are at the equator, you will experience the largest centripetal acceleration since the distance from you to the axis of rotation is at a maximum (the full radius of the Earth).
The distance from the axis of rotation ends up being R*cos(Θ) where Θ is your latitude on the Earth's surface and R is the Earth's radius. The centripetal acceleration from the Earth's rotation then becomes:
a = R*cos(Θ)*ω²

## Video transcript

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.