Conservation of angular momentum | AP Physics | Khan Academy

0:00 [Instructor] Did you know that a satellite or a space telescope can turn

0:03 by any amount about any axis in space without using any external force,

0:10 without any thrusters?

0:12 It can do that.

0:13 How?

0:14 By using a principle called conservation of angular momentum.

0:19 But what exactly is this?

0:20 Let's find out.

0:21 Let's start with something that we already know.

0:23 We know that if an object of mass m has some velocity v,

0:27 it has momentum, which is the product of mass and velocity.

0:31 Why do we care about this thing called as momentum?

0:33 Well, that's because if you have a bunch of particles,

0:36 not just one, but a bunch of them,

0:38 and if the total external force on that system of particles is 0,

0:44 then the momentum of the system,

0:46 which is basically the sum of individual momenta, that stays a constant.

0:51 It doesn't change, and we call

0:53 this the principle of conservation of linear momentum,

0:57 and this is a super powerful tool in making predictions.

1:00 For example, imagine a situation in which a block

1:02 is coming with some velocity and hits another block.

1:06 And let's say after the collision, they stick together and they move together.

1:09 We wanna find what that final velocity is.

1:12 Well, if you consider both m1, m2 as our system,

1:16 and if we ignore friction and air resistance,

1:18 then during the collision, the forces are all internal.

1:21 There are no external forces.

1:23 Then we can say the total momentum of the system before

1:28 the collision should equal the total momentum of the system after the collision.

1:33 And just from this, in a couple of steps,

1:36 we can figure out what that final velocity should be.

1:40 Isn't this awesome?

1:41 Why does it work out this way?

1:43 Well, that's because if there are no external forces acting on the system,

1:47 then during the collision,

1:48 when m1 puts a force on m2, m2 puts an equal but opposite force back on m1.

1:53 So, whatever momentum has gained by m2,

1:56 the same amount of momentum must be lost by m1,

1:58 and so the total momentum does not change.

2:01 Well, now, you might say, "In real--life cases, external forces are not 0.

2:05 You have friction, you have air resistance, right?" Yeah,

2:08 but if this was a situation just before the collision,

2:11 and this is the situation just after the collision, then during the collision,

2:16 the internal forces are much higher compared to the external ones,

2:18 and so we can ignore them.

2:20 We can pretty much say that, "Hey,

2:21 momentum changes mostly happen due to the internal forces." And so

2:25 we can pretty much use this principle and make pretty accurate predictions.

2:29 Okay, now, guess what?

2:31 If a rigid body is spinning, then we know it has angular momentum,

2:36 which is given as the product of its rotational inertia and angular velocity.

2:41 And many experiments show, for a bunch of rigid bodies,

2:45 if the net external torque on it is 0,

2:49 then the angular momentum of the entire system, which is, again,

2:53 basically the sum of individual angular momenta, that stays a constant.

3:00 So, now we have something called the conservation of angular momentum.

3:05 The idea is very, very similar to over here,

3:07 except that this will be used for rotation.

3:09 So, let's take a few examples.

3:12 Let's say we have a disc that is spinning on a frictionless table,

3:17 and let the angular velocity be omega1.

3:20 Now, let's say we drop another disc.

3:23 This is not initially rotating, okay?

3:26 We're gonna drop it on this one.

3:27 And there is friction between these two, so this is spinning, this is not.

3:31 What happens when you drop this on this one?

3:34 Well, because there is friction between these two,

3:36 the big disc is gonna put a torque on the smaller one,

3:40 trying to make it spin in this direction,

3:42 but of course, this small disc will put an equal

3:45 but an opposite torque on the bigger disc, slowing it down.

3:50 Eventually, they will have the same angular velocity, and when that happens,

3:55 the torque disappears,

3:56 and now they'll be both spinning together with the same angular velocity.

4:00 And notice, if I consider both the disc as part of my system,

4:04 there are no external torques, and therefore,

4:06 the total angular momentum of the system must stay conserved.

4:11 So, look, just with this principle,

4:12 I can now predict what that final angular velocity is going to be.

4:15 I can say that, "Hey, the total angular momentum of my system before,

4:20 which is just rotational inertia I1 of this disc

4:23 times its angular velocity omega1." And what's the direction?

4:27 Well, remember, for angular momentum, we can use our right hand thumb rule,

4:32 our four curling fingers curl in the direction of the angular velocity,

4:35 then the thumb points in the direction of the angular momentum.

4:38 So, over here, the angular momentum is just upwards, okay?

4:41 Plus the angular momentum of this one, which is just 0, I2 times 0.

4:46 That should equal the final angular momentum,

4:49 which is going to be the total rotational inertia,

4:51 which is just I1 plus I2 times the final angular velocity.

4:56 And look, just from this, I can calculate

4:59 what that final angular velocity is going to be.

5:02 Isn't it amazing?

5:03 Now, of course, one key thing over here is

5:05 that the rotational inertia not only depends on the mass,

5:07 but it also depends upon how that mass is distributed.

5:10 If the mass is distributed farther away from the axis of rotation,

5:13 the rotational inertia is higher.

5:15 So, the distribution of mass also matters.

5:18 For example, for a disc, the rotational inertia is given as 1/2 mR square,

5:23 where m is the mass of the disc and R is the radius,

5:25 so we could plug that in and simplify,

5:27 but you get the point, so we'll not do that.

5:30 Okay, let's take another example.

5:32 Consider spinning tires of an automobile.

5:34 Imagine the car is jacked up, so the tires are not touching the ground,

5:38 so the ground is not exerting any force or torque on these tires.

5:42 Right now, the clutch is not engaged,

5:45 so the two tires can spin independently because they're not connected.

5:49 So, let's say they're spinning with two different angular velocities,

5:53 omega1 and omega2, okay?

5:56 Now, what will happen if I engage the clutch?

6:00 Well, now these two axis are connected and they're

6:03 now forced to spin with one particular angular velocity,

6:07 so now they will have some new angular velocity omega.

6:11 If we ignore friction and if we ignore the mass of the clutch over

6:16 here and all of that, can you

6:18 predict what that final angular velocity should be?

6:22 Assume that the rotational inertia of this wheel

6:24 is I1 and that of this wheel is I2.

6:26 It will be a great idea to pause the video and see if you can

6:29 use the conservation of angular momentum to predict

6:30 what that final angular velocity should be.

6:34 All right, we're gonna consider the two wheels as the part of our system.

6:38 We're gonna ignore the mass of the axles and the clutch and everything, okay?

6:42 So, what's the total initial angular momentum?

6:44 Well, it's going to be the angular momentum of this wheel,

6:46 which is just I1 times omega1, and we have to be careful about the direction.

6:50 In this particular case,

6:51 the angular momentum of this wheel is going to be to the right,

6:54 and this wheel also has the angular momentum in the same direction,

6:58 so I can add it, so plus I2 omega2.

7:01 That should equal the total final angular momentum,

7:04 and the total final angular momentum would be just,

7:06 you know, I1 plus I2 times omega.

7:09 And boom!

7:10 From this, I can calculate what omega should be.

7:14 Of course, you could say that, "Hey, the two wheels are identical,

7:17 so I1 should be equal to I2." And you know,

7:20 we can further simplify this, but you get the point.

7:23 Okay, now consider a box floating in outer

7:26 space with a spinning disc inside of it.

7:29 The disc is connected to a motor and the motor is powered by a battery,

7:32 so everything, the disc, the motor, and the battery,

7:34 all of it is completely inside the box.

7:37 But because of the spinning disc, the entire,

7:39 you know, box disc system has some angular momentum, right?

7:44 But there are no external torques on this system,

7:47 so this angular momentum of this system must stay constant, right?

7:53 Okay, so now imagine we could remotely control

7:56 the motor and change the speed of this spinning disc.

7:59 What do you think would happen if we reduced its speed?

8:03 Well, the angular momentum of the disc would decrease,

8:06 but the total angular momentum must stay the same,

8:10 so how will nature ensure that?

8:12 The answer is that the box itself will begin to rotate in the same direction.

8:18 That way, the decrease in the disc's angular momentum is

8:21 compensated by an equal increase in the box's angular momentum,

8:25 keeping the total unchanged.

8:27 So, look, simply by slowing down the disc,

8:30 we can make the entire box start rotating.

8:33 No external force or torques needed.

8:35 Pretty cool, right?

8:37 Now, what do you think will happen if we

8:39 speed the disc back up to its original speed?

8:42 Well, now, the angular momentum of the disc returns to its original value,

8:46 so the box will stop rotating, again,

8:48 to keep the total angle of momentum the same as before.

8:51 Finally, what if we increase the disc speed?

8:56 Well, that would try to increase the total angular momentum, so to compensate,

9:00 the box would start rotating in the opposite direction,

9:03 again, ensuring the overall angular momentum remains the same.

9:07 Which means just by increasing or decreasing the speed of a spinning disc,

9:11 we can rotate this box by any amount in any direction.

9:16 This device is called a reaction wheel,

9:19 and this is how we control rotations of satellites.

9:23 So, here's an example of how these reaction wheels look like.

9:26 We have three reaction wheels to control rotations along three axes,

9:31 and we usually have another set as a backup

9:34 just in case some of the others stop working.

9:37 So, this is how, by using the reaction wheels,

9:40 we can rotate our satellites or our telescopes

9:44 by any amount along any particular axis.

9:47 Pretty awesome, right?

9:49 What's more awesome is that conservation of angular

9:51 momentum can also be applied to non-rigid bodies.

9:55 Take a look at this.

9:58 You have an ice skater who starts out spinning with low speed,

10:00 but then look, she starts speeding up!

10:04 Can you pause the video and think about what's going on over here?

10:08 See if you can explain this using

10:10 the idea of the conservation of angular momentum.

10:14 All right, let's see.

10:15 When she starts spinning, look at how her arms and legs are stretched out.

10:19 Because of this, a lot of particles are far away from the axis of rotation,

10:23 distributed far away from the axis of rotation.

10:25 As a result, her rotational inertia is pretty high.

10:27 She also has some angular velocity, and therefore,

10:30 she has some angular momentum, which is the product of these two.

10:34 Now, as she continues to spin,

10:35 we will see that she will pull her arms and legs closer.

10:39 As a result, a lot of particles now

10:40 are distributed close to the axis of rotation,

10:43 therefore, her rotational inertia goes down.

10:46 But what happens to her angular momentum?

10:49 It has to stay the same.

10:51 Why?

10:52 Well, because all the pooling forces were internal,

10:55 so all of those torques are internal,

10:57 and there are no external torques over here.

10:59 I mean, of course, there is force of friction, but that is very minimal,

11:03 so the torque produced by the friction is very minimal.

11:05 So, if I consider this ice skater as our system,

11:08 then there are pretty much no external torque acting on her, and so her total

11:12 angular momentum must stay the same as she pulls her arms and leg closer,

11:17 which means this product should stay the same,

11:19 which means as her rotational inertia decreases,

11:22 her angular velocity should increase to keep the product the same,

11:28 and that's why she speeds up.

11:31 Let's have a look now.

11:33 Here we go.

11:34 This is what's happening.

11:36 Isn't that amazing?

11:38 Wow!

11:39 And something very similar happens to the stars.

11:43 A star usually forms with huge swirling,

11:46 you know, clouds of gas gravitating together.

11:49 Now, if you consider this entire cloud as our system,

11:52 then particles are distributed very far away from the axis of rotation,

11:56 it's all spread out, so it has a very high rotational inertia.

11:59 And it also has some angular velocity,

12:01 and so it has some angular momentum, which is the product of these two.

12:06 But when eventually a star is formed, look.

12:10 Almost all these particles are distributed much closer than before.

12:14 So, the rotational inertia is much lower than before,

12:17 but the angular momentum must stay the same because

12:21 pretty much no external torque acts on our system,

12:25 and therefore, the product should stay the same,

12:27 and so the angular velocity of the star must be much higher than it was before.

12:32 Says that the product stays the same.

12:34 That's why the star will be spinning faster than before.

12:38 And then big stars run out of fuel to sustain nuclear fusion at the core,

12:43 gravity crushes the whole thing into a very tiny,

12:47 very compact object called neutron stars.

12:50 As a result, the rotational inertia becomes incredibly low, but again,

12:56 the angular momentum must stay pretty much the same as before,

13:00 and therefore, its angular velocity will be much,

13:04 much higher so that the product stays the same.

13:07 Now, this is not to scale.

13:08 Just to give you some feeling for numbers,

13:10 stars usually finish one rotation in, like,

13:13 a few days, but these neutron stars will

13:17 rotate tens or even hundreds of times per second.

13:21 So, now you can see why.

13:23 It's all because of conservation of angular momentum.

13:26 By the way, before we close, did you know how we actually discovered these?

13:30 Some of them actually shoot out beams

13:32 of electromagnetic radiation from their magnetic pulse.

13:34 And as the star spins, those beams sweep across the space like,

13:39 you know, a cosmic lighthouse, if you may.

13:41 And about 60 years ago,

13:43 astronomers noticed incredibly regular radio pulses arriving from deep space.

13:49 It was about 1.3 seconds apart.

13:51 And so the signal was so precise and it was so

13:54 regular that at first they thought it couldn't be natural at all.

13:57 Some even wondered if it might be intelligent alien life signaling us,

14:01 so the mysterious source was labeled LGM-1 for little

14:06 green men (chuckles) until we found more of them,

14:09 and then finally realized that they

14:11 were actually rapidly spinning neutron stars.