Angular momentum of satellites | AP Physics | Khan Academy

0:00 [Instructor] Let's talk about angular momentum of orbiting stuff,

0:02 like planets and satellites.

0:04 To do that, let's start with a simple-sounding question.

0:07 I have a stone of mass m moving with some velocity v in a straight line.

0:13 Does this stone have angular momentum?

0:16 Well, we know angular momentum of any rigid body is

0:19 given as the product of its rotational inertia times angular velocity.

0:22 So our instincts might be, "Hey, this stone is not rotating.

0:26 It's just moving in a straight line.

0:27 So angular momentum is zero.

0:29 It doesn't have any angular momentum." (laughing) But that would be wrong.

0:33 Why?

0:34 Well, think of it this way.

0:35 Imagine there was a thin light rod, which was fixed at one end over here,

0:41 so that, you know, it can freely rotate like this.

0:44 Now, what do you think would happen if this stone were to go and hit

0:47 this rod so that its velocity is perfectly

0:50 perpendicular to the rod and it sticks to it?

0:52 Well, you can imagine it's going to start spinning with some speed, right?

0:56 And to figure out that speed,

0:57 we would need to know the rotational inertia of this rod.

1:00 But if you assume that the rod's mass

1:03 is negligible compared to that of the stone,

1:05 and if you also assume ideal conditions like say no friction and no

1:10 air resistance and all of that, then once it's sticks to this rod,

1:14 it will start spinning with the stone having that same linear speed forever,

1:20 which means now the system has angular momentum.

1:24 The system of rod and the stone together, it has some angular momentum, right?

1:29 But think about where did that angular momentum come from?

1:32 Remember, angular momentum of system cannot change unless

1:36 there is an external torque acting on the system.

1:39 Now, if we consider the rod and the stone as part of our system,

1:43 then as the stone hits that rod and sticks to it,

1:47 the forces and the torques are all internal to our system.

1:50 There are no external torques,

1:52 which means the angular momentum couldn't have changed.

1:56 So if the system has angular momentum here,

1:58 the system must also have angular momentum here as well.

2:01 And all of that angular momentum must be part of this stone,

2:04 because remember, this rod is not moving.

2:07 I mean, we have assumed it's mass to be negligible,

2:09 but even if it did have mass,

2:11 if it's at rest, it wouldn't contribute to the angular momentum,

2:14 which means the initial angular momentum of the system,

2:17 all of it must be part of just that stone,

2:20 which means even though this stone is moving in a straight line,

2:23 it's not spinning at all, it still has angular momentum.

2:28 Okay, but how much is that angular momentum?

2:31 Well, the angular movement of the stone here should be

2:33 the same as the angular momentum of the stone over here.

2:37 Remember, we are assuming that the rod has negligible mass,

2:40 so it has negligible rotational inertia.

2:42 So all the angular momentum is due to the stone.

2:46 Okay, so what's the angular momentum of the stone?

2:48 Well, let's just stick to its magnitude.

2:50 We can say the angular momentum is the momentum

2:52 of inertia of the stone times its angular speed.

2:55 I'm using the word speed because we are sticking to magnitude.

2:57 We'll talk about its direction later.

2:59 Okay, how much is the rotational inertia of the stone?

3:02 Well, if the stone is pretty tiny,

3:04 then we can say all of that mass is concentrated

3:06 at say the distance r from the axis of rotation.

3:10 So then the rotational inertia of the stone would be mr squared.

3:14 But what's the angular speed of the stone?

3:16 Well, angular speed equals just the linear speed v divided by r.

3:21 So over here, omega is v divide by r.

3:25 And so one r cancels and we get mvr.

3:28 And mv represents the magnitude of the momentum of that stone, right?

3:32 And so now we can write the angular momentum magnitude as just r times p.

3:37 And since the angular momentum is conserved because

3:40 there are no external torques acting on our system,

3:42 the magnitude of the stone's angular momentum even here must be r times p.

3:46 But here's a question.

3:48 Remember this thin rod that we imagined?

3:50 Well, that's just an imagination, that's not real, right?

3:54 So then what exactly is this r?

3:56 What does r truly represent in our angular momentum formula?

4:01 Well, think about r as the radius of the would-be circle

4:06 that the stone would be moving in if it was rotating.

4:10 The whole point is that, even though it's not rotating right now,

4:13 it has the ability to rotate sometime in the future.

4:16 That's why we say it has angular momentum

4:19 because it can rotate sometime in the future.

4:22 That's the whole idea.

4:23 So it can rotate about this point,

4:25 and that's why we say it has angular momentum about this point.

4:30 Okay, but saying that r is the radius of the would-be

4:34 circle along which that stone would rotate in the future,

4:38 (laughing) that's not very technical, right?

4:41 So how can we put it in more technical terms?

4:43 Well, we can say that this distance r represents the perpendicular distance

4:48 from this point on to the direction in which that stone is moving, look.

4:53 And so we can call this the r perpendicular times p.

4:59 Makes sense, right?

5:00 But remember, angular momentum is a vector quantity.

5:03 This is just its magnitude.

5:05 It also has direction.

5:06 How do we think about the direction

5:07 of the angular momentum of this stone right now?

5:10 Well, we can use our right-hand thumb rule.

5:12 You take your right hand and you curl

5:14 your four fingers along the direction of the rotation.

5:18 Then the thumb points in the direction of the angular momentum.

5:22 So if you curl your fingers over here

5:23 this way in the direction of rotation, look, the thumb points out of the screen,

5:27 so the angular momentum here is out of the screen.

5:30 But this means, and this is a very important thing about angular momentum,

5:34 that this angular momentum depends on our reference point.

5:38 Because remember, the value of our perpendicular depends on our reference point.

5:43 If I choose a different reference point,

5:45 the angular momentum of the stone would be different.

5:48 For example, consider the angular momentum

5:51 of the same stone with respect to this point.

5:55 Notice now the r perpendicular is much bigger.

5:59 So now the angular momentum magnitude is higher,

6:04 even though nothing has physically changed, it's the same stone,

6:07 it's the same situation, but if I consider it's angular momentum

6:09 with respect to this point, it's much higher.

6:12 Okay, what about angular momentum of that same stone with respect to this point?

6:16 Why don't you pause the video and think about its magnitude?

6:19 Would it be bigger, smaller?

6:21 And think about the direction as well.

6:23 Okay, so you can now see our perpendicular is much smaller.

6:27 So the angular momentum magnitude would be smaller.

6:29 But think about the direction of the rotation.

6:31 Now, if it were to rotate about this point, it would rotate this way, right?

6:36 Which means the angular momentum is not only smaller,

6:38 but it changes its direction.

6:40 So both its magnitude and direction depends on the point of reference.

6:44 Okay, finally, what do you think is

6:46 the angular momentum of this stone about this point?

6:49 Well, now our perpendicular is zero

6:51 because that point lies along this direction,

6:54 which means the angular momentum is zero.

6:57 Does that make sense?

6:58 Well, yeah, it's kind of like stone going

7:00 and hitting a hinge of a door, for example.

7:03 You can't make it rotate, right?

7:05 So now the angular momentum is zero, which means any point mass moving with some

7:11 velocity will have angular momentum given by this expression.

7:14 And that angular momentum depends on the choice of your reference point,

7:18 or you can think of that as your origin.

7:20 And now, since angular momentum depends on this reference point,

7:23 which we can think of it as origin,

7:25 it would make a lot of sense to define it in terms of a position

7:30 vector that we draw from that reference point

7:33 or from that origin to our point mass.

7:36 That way, we don't have to think about angular

7:38 momentum in terms of radius of some future imaginary circle.

7:41 Instead, we can talk about it in terms of its current position.

7:46 So our next question would be,

7:49 can we find an expression for r perpendicular in terms of its current position?

7:55 So we have a right-angle triangle.

7:57 So if I call this angle as theta, then I can write this r perpendicular in terms

8:02 of r and theta using some trigonometric ratio, right?

8:06 So which trigonometric ratio would we use?

8:08 Well, we can say sine theta is r

8:11 perpendicular divided by the hypotenuse, which is r.

8:14 And so from here, r perpendicular is just r times sine theta.

8:18 So I can say angular momentum equals r times sine theta times the momentum p.

8:25 And finally, we can now write this in its full glory,

8:30 the angular momentum as the vector product or the cross product of r and p.

8:37 Let's think about why do we write it this way.

8:39 First of all, whenever you take cross product of two vectors,

8:43 the magnitude of that cross product will be the magnitude of the first vector

8:48 times the magnitude of the second vector times sine of the angle between them.

8:52 And look, that's exactly what we have,

8:53 magnitude of r times magnitude of p times sine of the angle between them.

8:57 And so it makes sense to write this as a cross product.

9:00 And one question I used to usually have when I think about this angle is, "Wait,

9:03 is the angle this angle between these two vectors

9:06 or should we consider this angle between the two vectors?" Well,

9:10 technically it should be this angle because when we say,

9:13 "Angle between vectors," we have to consider them tail to tail.

9:17 However, here, it doesn't matter because

9:19 this angle is just 180 degrees minus theta,

9:22 and sine of 180 degrees minus theta is just sine theta.

9:27 So when you're dealing with cross products,

9:29 because you're dealing with sine theta,

9:31 either of them gives you the same result.

9:33 So for our purposes, we can just take this smaller angle.

9:36 But writing it this way also encodes the direction.

9:39 How do you think about the direction of the vector product?

9:42 Well, again, you use the right-hand rule.

9:44 Here, you start with your right hand,

9:46 and the four fingers should be in the direction of the first vector.

9:49 So your four fingers will be in the direction of the r vector,

9:53 and then you cross it from there to the second vector.

9:58 So we cross it from there to our p vector,

10:00 which is the same direction as v vector.

10:02 And when you do that, the thumb represents the direction of the cross product.

10:06 And notice that is in the same direction as the angular momentum.

10:10 And so look, this also gives us the right direction,

10:13 and that's why we write the angular momentum as a cross product.

10:17 It's basically writing the same way in a much more compact form.

10:20 It encodes both the magnitude and the direction in it.

10:23 And one final thing about the cross product is r

10:26 cross p is not the same as p cross r.

10:29 I mean, the magnitudes would be the same, but if you were to do p cross r,

10:33 then you would start with your forefingers along this vector,

10:36 the p vector, and then you would cross it towards the r vector.

10:39 So now your thumb would point inwards.

10:42 So r cross p is actually equal to the negative of p cross r,

10:45 so it's not commutative, okay?

10:47 So we need to be careful.

10:48 So angular momentum is not p cross r, it has to be r cross p.

10:53 Okay, now let's apply this to orbiting satellites and planets.

10:56 Let's consider the simple case of Earth going around the Sun,

10:58 and let's assume the orbit to be perfectly circular.

11:01 So the Earth has some mass m, it has some velocity v.

11:05 The question is, what happens to the angular momentum

11:08 of the Earth as it moves around the orbit?

11:11 The moment you hear the question, the first thing you should say is,

11:13 "About which point?" Because remember,

11:15 angular momentum depends on the reference point.

11:17 So let's consider the Sun's center to be the reference point.

11:21 What happens to the angular momentum of the Earth

11:23 about this point as it goes around the orbit?

11:26 Does it change?

11:26 Does it stay the same?

11:28 That's the question.

11:29 Well, one way to answer that question is,

11:30 we know the angular momentum of a system can only

11:32 change if there's an external torque acting on the system.

11:36 And how do we calculate torque?

11:37 In a similar way, we calculate torque as the cross product of the r vector,

11:42 the position vector, and the force.

11:45 So over here, our position vector from the origin

11:47 to the planet would look like this.

11:49 And is there a force acting on our planet?

11:51 Yes, force of gravity, right?

11:53 But does that force produce a torque?

11:55 That's the question.

11:56 Because if it does, then the angular momentum would change.

11:59 Well, what is the angle between r and f?

12:02 It's 180 degrees.

12:04 Sine of 180 degrees is zero, so the torque is zero about this point,

12:10 the force will be in the opposite direction of the position vector,

12:13 which means throughout the orbit,

12:14 the torque produced by the force of gravity about this point is always zero.

12:19 So its angular momentum stays conserved.

12:23 But at this point, you might say, "Well, isn't it obvious?

12:26 Like at every point the value of r, m, v, and theta, everywhere is the same,

12:31 so obviously the angular momentum must stay the same." Well,

12:35 in circular orbits, it is quite obvious.

12:37 But what about elliptical orbits?

12:40 Would the angular momentum stay the same?

12:42 Well, now look, the value of r changes,

12:45 and even the magnitude of velocity changes at every point.

12:48 The angle between r and v, the theta, that also changes.

12:52 So all of these are changing.

12:54 Now, it's not so straightforward.

12:56 Now, we can't tell just by looking at this whether the angular momentum,

13:00 you know, of this planet changes.

13:02 But if you think in terms of torque,

13:04 we can see it because, remember, the torque is still zero,

13:08 right, because the force of gravity is always

13:11 at the opposite direction of the position vector.

13:13 And therefore, we can immediately say, if the torque is zero everywhere,

13:17 then the angular momentum of the planet should stay conserved.

13:21 and this is a beautiful result.

13:23 We can use this to predict properties.

13:26 For example, if I knew the velocity of Earth at this point,

13:28 I can use that to predict what the velocity would be at this point.

13:31 It's a powerful principle.

13:33 And what's more important is that this is not just true for Earth and Sun,

13:37 this is true for any orbiting object,

13:40 whether you're a planet orbiting a star or a satellite orbiting a planet,

13:45 the force of gravity, because it is acting towards the center,

13:48 we call that as a central force, it cannot produce torque about the center.

13:53 And so the angular momentum of the orbiting planet

13:57 or satellite about the center of the parent star will always,

14:01 always stay conserved.

14:02 That's a powerful principle.

14:05 Now, having said that, remember, in our solar system,

14:06 there are other planets that are pulling on Earth as well,

14:09 and therefore they will produce a toque on Earth with respect to this point.

14:14 And so the Earth's angular momentum does change because of the other planets.

14:18 But if you consider all the planets as part of our system, then again,

14:23 the angular momentum of that new system would

14:26 stay conserved because gravity is a central force.

14:30 But of course, now if an interstellar asteroid came in, again,

14:33 that would change the angular momentum of our system.

14:36 And another thing is that remember that our planets are not just revolving,

14:40 they're also spinning about their own axis.

14:42 So that also adds to the angular momentum.

14:45 But of course, at this scale, the size of the planet is so minuscule,

14:49 we can just assume it to be a point object,

14:51 and we can consider its spin around its axis negligible at this scale.

14:54 So long story short, any mass having some velocity v will

14:58 have angular momentum given by this expression.

15:01 And that angular momentum depends on your point of reference.

15:06 And the cool thing is that the angular momentum of any

15:09 orbiting planet or a satellite about the center of the parent body,

15:15 that's important, will always be conserved, because gravity is a central force,

15:22 it cannot produce a torque about this point.