Gravitational forces and fields

0:00- [Instructor] Gravity holds solar systems together and galaxies together. But how do we model it? That's what we wanna find out in this video. So let's begin. In Newtonian mechanics, gravity is modeled using what we call Newton's universal law, which states that any two masses, whether it's a star and a planet, or two asteroids, or maybe two stones even for that matter, any two masses in the universe will always attract each other.

0:27Always. There will be an attractive force between them, and that is the force of gravity. So in Newtonian mechanics, we model gravity as an attractive force. But what does this force depend on? Well, if the masses of the objects where I say M1 and M2, and if the distance between their center of masses is r, then according to the universal law, the magnitude of the force, the gravitational force will be proportional to each mass, and it's inversely proportional to the square of the distance between the centers of masses.

1:03That's why this law is also called the inverse square law. And of course, if you replace the proportionality with an equal to, then we'll have a proportionality constant. This is called the universal gravitational constant. And we figure out its value experimentally and it happens to be a very tiny number.

1:23This is the reason why we don't feel the effects of gravity on everyday objects because the masses are too small. For the force to be big enough, we need at least one of the masses to be big enough. Anyways, what do you think is the unit of this universal constant?

1:36Can you pause and just think about it just from this equation? Okay, if we rearrange for G, we get F times r squared divided by M1, M2, right? So it'll be newton-metre square divided by kilogram square. But what about the direction of the force? Well, the force is always attractive.

1:55We always consider the force to be exerted at the center of mass, of planets or stars or any system for that matter, and that force is always along the line joining the two centers of masses. Remember that, okay? And when r over here is always calculated from, you know, it's calculated between the two centers of masses.

2:14Alright, let's apply the universal law to the Earth and the apple system. Well again, let's say the mass of the apple is just small m and the mass of the Earth to be capital M. And let's say the distance between the center of masses, that's important, you have to always consider the distance between the center of masses, okay?

2:29Let's call that to be r. Then, if we plug in, we get the force of gravity between the two of them would be G times mass of the Earth, times mass of the apple divided by r square. But we know from Newton's second law, force equals mass times acceleration. So if you apply Newton's second law to this apple, this force of gravity acting on it should equal mass of that apple times the acceleration of the apple.

2:55And we know the acceleration of the apple is just G, acceleration due to gravity. If there are no other forces acting on it, that acceleration is gonna be G. So we can plug that in over here, and we get m times g. And again, we are only considering the magnitude over here, okay?

3:09So look, the masses cancel and what do I end up with? I end up with an equation for gravitational acceleration. The equation tells me that the acceleration due to gravity at any point depends on two things. One is the mass of the planet, which makes sense. More massive the object, more gravity, more acceleration.

3:27And it also depends upon how far you are. The farther you go, the weaker gravity gets, right? And as a result, the acceleration would be smaller. So that's what this is saying. And if we were to calculate this acceleration for Earth close to it, then this r, which is the distance from the center to the apple, pretty much becomes the radius of the Earth, right?

3:46I mean, imagine you were like a couple of kilometers above the surface of the Earth. The Earth's radius is so big, it's about 6,400 kilometers that a couple of kilometers above the surface of the Earth is negligible, so you can pretty much assume this r, the distance between the centers, to be the radius of the Earth itself.

4:03So close to Earth's surface, we can approximate r to be just the radius of the Earth. And so if you now plug in the mass of the Earth, the radius of the Earth and the value of G, you will get the value of acceleration due to gravity close towards surface. And that is the famous 9.8 meters per second squared.

4:23Of course, historically, we knew the value of g experimentally, and we knew the value of r, the radius of the Earth was calculated. So we used the situation to actually estimate the mass of the Earth. That's amazing, if you're asking me like, we were able to use the universal law to figure out how massive our planet is.

4:42But here's an important question now. How exactly are objects able to attract each other without any contact? Remember, gravity is a non-contact force. So how is this Earth, for example, attracting the apple without being in contact with it? Well, here's how we like to think about it.

4:58Let's just focus on one of the objects. Let's consider the Earth. We like to think that the Earth produces an influence all around itself. We call that the gravitational field. We represent it using arrow marks and we call it the gravitational field. And the idea is that field exists everywhere, literally everywhere around the Earth.

5:17I mean, we can't draw the arrow marks everywhere, but it's there, okay? And by the way, what I've drawn is only a two dimensional version of it. Remember, we live in three dimensions. This is gonna be important later on. So the field actually exists in all the three dimensions, but let's just stick to the two dimensional version.

5:32And remember, the field does exist in all the three dimensions. And the idea is when another object, another mass comes in contact with that field, it experiences a force. That's how the force gets mediated. Why are the fields? So the Earth produces its own gravitational field, and then that field puts a force on the apple.

5:56And if you're wondering, yes, the apple will also produce its own gravitational field, which we've not shown over here. And when any other mass, like Earth for example, comes in contact with that field, it'll also experience a force. Every mass will produce its own gravitational field.

6:10And the length of the arrow mark is proportional to the strength of the field. So closer, the field is stronger. That's why we draw bigger arrow marks. Farther away, the field gets weaker. Farther you go, the weaker the field. So now the question is, can we quantify this gravitational field?

6:26Can we find an expression for it? Well, let's see, let's do that. So the magnitude of the gravitational field, let's find an expression for that. And before I do it, you might be wondering, well, wait a second, why are we using the symbol g? Isn't g acceleration due to gravity?

6:40Why are we using the same symbol for gravitational field? Yeah, it can be confusing, it'll make sense in a second, but let's now consider this to be the gravitational field, okay? So what is the magnitude of the gravitational field at any point? We like to define it, we like to quantify it as the strength of the gravitational force acting on some test mass, like an apple, divided by the mass of their object, divided by the test mass.

7:06So the idea is if you wanna calculate the gravitational field at any point, take a test mass, calculate the gravitational force divided by the mass. Why are we dividing it by the mass? Well, let's see, it'll make sense. Let's just substitute for the, you know, using the universal law.

7:20We know that the gravitational force is GMm divided by r squared. When you divide by the test mass, see what happens. The masses cancel out. And so the strength of the gravitational field no longer depends on the test mass. It only depends on the source, the mass that's producing the gravitational field.

7:42That makes sense, right? That's what you want. You want the gravitational field to depend on the source. It shouldn't depend on the test mass. That's the reason why we divided by the test mass, to cancel it out. So long story short, you can think of gravitational field at any point is the force, gravitational force per unit mass at that point.

7:57And what will be its units? Well, its units are going to be newtons per kilograms. But wait a second, you may be like, this equation expression looks identical to the expression that we just derived for gravitational acceleration, right? And the answer is yes, you get the same expression for gravitational acceleration as that for the gravitational field.

8:19That's why we use the same symbol. And so now we can see the full picture. You can think about g in two ways. You can either think of it as a gravitational acceleration. And when you do that, it's a measure of how, you know, the acceleration of an object purely under the force of gravity.

8:38And so in that case, we will, you know, think about its units in terms of meters per second squared. But you can now also think about g as the gravitational field strength. And when you do that, you think in terms of newtons per kilogram. So there are two different ways to think about g.

8:57But as long as there are no other forces acting on it, numerically, they'll be both equal to each other. So close to the Earth's surface, you can say that, hey, the gravitational acceleration is 9.8 meters per second squared. Or you can also say the gravitational field strength is 9.8 newtons per kilogram.

9:15That brings us to the last question. Is there a way we can get some intuition for why the gravitational force or the gravitational field goes down as one over r square? And the answer is yes, we can. See, right now we are looking at what we call the vector field model because we are using vectors to represent the gravitational field.

9:37But there's another way to do it. It's called the field line model because we're gonna use lines to represent gravitational fields, okay? How is this helpful you ask? Well, in the vector field model, the strength of the gravitational field is proportional to the length of the arrow mark.

9:56Closer to the Earth, the field is stronger, so we draw bigger arrow marks and farther away from the earth, the field gets weaker, we draw smaller arrow marks. But how do we think about the strength of the gravitational field over here? Well here, the strength is represented by how close these field lines are.

10:14Think about it. Close to the Earth, look, the field lines are very close to each other, so the field is strong. If we go farther away, look, the spacing between the field lines increases. That depicts that the gravitational field is becoming weaker. And just like before, this is in three dimensions.

10:33So we can say concretely the gravitational field strength is proportional to the density of the field lines, how many field lines we have per unit area. Okay, so with that, let's see how the gravitational field varies as we go farther away from this mass. Let's stick to the 2D representation, it's just easier to visualize.

10:58But remember it's in three dimensions, okay? So what we're gonna do is draw a couple of spheres. These are not circles, these are spheres. And think about the gravitational field strength on the surface of these two spheres, okay? So for that, let's think about the number of lines.

11:13What happens to the number of lines, the field lines as we go farther away from this sphere to this one? Well, the number of field lines stay the same because whatever field line is entering over here, the same field line enters through this sphere as well. So the number of field lines stays exactly the same.

11:27But what happens to the area, you know, as you go from this sphere to this one? Well, remember the area is proportional to the radius squared. So as we go farther away, the area increases as r squared. And so look, the number of field lines stay the same, but the area increases by r square, so the gravitational field reduces by r square.

11:52Does that make sense? The same field lines are spread over a much larger area. So this is the reason why gravitational field goes down as one over r square. At the end of the day, it's because we live in 3D space. That's pretty cool if you ask me.