Angular momentum of rigid systems (with calculus)

0:00- [Instructor] Here's a ruler kept on a table. Suppose I flick it over here, what's gonna happen? Well, it's gonna go forward and it's gonna spin a bit, right? So when I flick it, the ruler gets both a translational motion and a rotational motion. So the question we wanna try to answer in this video is how do we predict how much linear and angular velocity it gains after I flick it?

0:23You may be wondering, why should we care about that? Well, over here, maybe it's not so high stakes, but what if you consider an orbiting satellite? Maybe we want to change its orientation a bit, or maybe it's spinning uncontrollably and we want to stop it remotely.

0:37Ooh, now we need to know how to compute all of this stuff, right? So over here, we're gonna learn how to do that by introducing a pretty cool idea called angular momentum. So let's begin. So how do we compute this? Well, we know that this ruler is undergoing both a translational and rotational motion, but the cool thing is we can analyze them separately.

0:58So let's first look at the translational motion and then try to analogously think about rotational motion, okay? So when it comes to translational motion, how do I figure out how much velocity this ruler has gained after I flicked it? Well, I can think in terms of linear momentum.

1:15Remember, linear momentum of any object equals the mass times the velocity of the object. So if I could somehow figure out what the linear momentum of this ruler is, after I flicked it, I'm done. Because then I can divide by the mass and I will find what its velocity is going to be.

1:33So I can use the idea of linear momentum to compute the linear velocity. Now, similarly, when it comes to rotation, we can define a quantity called angular momentum. It is analogous to linear momentum. Just like how linear momentum is a measure of how much motion, how much translational motion an object has, we can think of angular momentum as a measure of how much rotational motion an object has, okay?

2:06And how do we define it? How do we write an expression for it? We can write it analogously. So it'll be a great idea to pause the video and see if you can first make a guess of what this expression is going to be. Okay, so over here, linear momentum was mass times velocity, right?

2:24Well, instead of mass, what's analogous to mass when it comes to rotation? Well, that's rotational inertia, right? And what's analogous to velocity in rotation? Hey, that's angular velocity. So we define angular momentum as the product of rotational inertia and angular velocity.

2:43Let's see if this makes intuitive sense. If you increase angular velocity, the angular momentum increases. Does that make sense? Yeah, if an object starts spinning faster, well, it has more rotational motion, so it makes sense to say that it has more angular momentum.

2:59But the expression also tells us if you increase the rotational inertia, that too increases the angular momentum. Does that make sense? This is a little subtle. How do you increase the rotational inertia? Remember, rotational inertia is a measure of how a mass is distributed, right?

3:17For example, consider two spheres attached by a light rod. If the mass of the spheres are exactly the same, both of these objects have exactly the same mass, but if you're spinning them about their centers over here, they don't have the same rotational inertia.

3:31Over here, the masses are very close to the center, so it has lower rotational inertia, and over here the mass is just, you know, it's farther away from the center, so it has a higher rotational inertia. So if you were to spin both of them with the same angular velocity, then this one ends up having more angular momentum.

3:49Does that make sense? Well, yeah, I mean, look at this. If the mass is farther away from the center, it moves faster. Notice that these spheres are moving faster compared to this one, even though they're moving with the same angular speed, they have the same RPM, but because they're farther apart, those spheres are farther apart, they're moving faster.

4:08So it makes sense to think that this one has more rotational motion compared to this one, and therefore this one had more angular momentum compared to this one. Okay, next up, let's think about the direction. Just like how both linear momentum and velocity are vector quantities, angular momentum and angular velocities are also vector quantities.

4:30So how do we think about their direction? Let's start with the angular velocity. If a disc is spinning, let's say this particular way, then what is the direction of the angular velocity? Well, we use something called the right-hand rule. Over here we use our four fingers and we curl it in the direction of the spin, and then the thumb points to the direction of the angular velocity.

4:52So in this particular case, the direction of angular velocity is upwards. So you can imagine if it was spinning the other way around, then its angular velocity would be downwards. Okay, so that's angular velocity. What's the direction of angular momentum? Well, we might look at this and we say, "Hey, over here momentum has the same direction as velocity."

5:12So could we also say that angular momentum has the same direction as angular velocity? Ooh, this is subtle. It turns out not to be so in general. Now, in special cases, when we're dealing with rotation around something called the principal axis, which is basically an axis of symmetry, then the angular momentum does happen to be in the same direction as angular velocity, but that's a special case.

5:40So for example, in this particular case, this is our principal axis. So over here in this specific example, the angular momentum will also be in the same direction as angular velocity. Here is another principal axis. So if you were to spin the disc along this particular axis, kind of like when we flip a coin, right?

6:03Then it turns out this is also a principal axis of rotation, meaning along this axis, again, angular momentum and angular velocity, will be in the same direction. So with the direction over here, well again, we use our right-hand rule. We make sure that our four fingers are curling in the direction of rotations.

6:22Over here, we have to curl it this way, and so the angular velocity will be to the left. And so in this particular case, the angular momentum will also be to the left. But what's important is that the rotational inertia along this axis would be different compared to that along this axis, because the distribution of mass is different.

6:40In fact, along this axis, it turns out that the rotational inertia is actually lower compared to along this axis. And so even if I were to spin it with the same omega along this axis, the angular momentum would have a smaller magnitude. Okay, so these two were the principal axis, right?

6:58So the angular momentum and the angular velocity were having the same direction. But now consider this. What if this disc was spinning both along this and this axis simultaneously? Then its angular velocity would be a vector sum of these two, right? It'll be this way.

7:19And so the angular momentum would be the vector sum of these two, and it would be that way. And look, the angular momentum now is not pointing in the same direction as the angular velocity. So this is not a principal axis. Along this axis of rotation, the angular momentum and the angular velocity are not in the same direction.

7:43So long story short, whenever we're dealing with principal axes, these two are always in the same direction, and rotational inertia becomes just a simple number, a scalar. However, if you're dealing with non-principal axes, then they're not aligned in the same direction, then the rotational inertia is no longer a scalar, not a vector.

8:05It becomes something called rank-two tensor. But we don't have to worry about that because we're gonna restrict our analysis to principal axes only. So for our case, the angular momentum and angular velocity will point in the same direction, and rotational inertia is just a scalar.

8:21Okay, coming back to our original problem: If I could figure out how much angular momentum this ruler has gained after I flick it, then I'm done. Because then I can divide it by the momentum of inertia and calculate the angular velocity. But the big question now is how do I figure out how much angular momentum it gains after I flick it?

8:40Well, let's come back from translation of motion. How do we figure out how much linear momentum it gains after we flick it? Well, we can use Newton's second law. Newton's second law says the net force acting on an object equals mass times the acceleration of the object.

8:54What's the acceleration? The acceleration is the rate at which the velocity is changing. So it is m times dv over dt, and since m is a constant, I can put it inside the derivative and call this dt of d of mv. In other words, it's the rate of change of momentum.

9:11So the Newton's second law tells us that the net force acting on an object equals the rate at which its momentum is changing. So now I can rearrange this expression to get change in momentum, so let's do that. If I rearrange, then dp equals the net force times dt.

9:28Since this is a differential term, let's integrate it on both sides. When I do that on the left-hand side, I get the change in momentum. Now, that equals this term, which we call the impulse. So impulse is basically the product of the net force acting on an object, times the time of which that force is acting.

9:51Of course, if that force is changing over that time, we will have to integrate that, okay? So what I need to know is the impulse delivered by my flick. If I know that, that equals the change in momentum, and since I know the initial momentum is zero, I can use this to calculate the final momentum, and then from there I can calculate the linear velocity.

10:12Now, we can do the same thing for rotation. Let's write Newton's second law in the rotational form. It will be a great idea to pause the video and try this yourself, because again, everything is gonna be pretty analogous. All right, so how do we write it over here?

10:28Instead of net force when it comes to rotation, we think about net torque. That equals instead of mass, we'll have rotational inertia. Instead of acceleration, we have angular acceleration. What's angular acceleration you ask? Well, acceleration is a measure of how quickly the velocity is changing, right?

10:50Similarly, angular acceleration is a measure of how quickly the angular velocity is changing, okay? And just like over here, their directions are not always aligned, okay? And it comes to principal axis. If the rotation is along the principal axis, then the torque and the angular acceleration will be in the same direction.

11:09But if it's not, all bets are off. So again, we don't have to worry about that, because we're only gonna consider cases, you know, where the rotation is along the principal axis. Anyways, so I can write this as the rotational inertia times angular acceleration is rate of change of angular velocity.

11:26So instead of dv over dt, we'll get d omega over dt, and then I can pull this I into the derivative. And so look what we get. We get the net torque equals the rate of change of angular momentum, and look at how similar it is. Net force equals rate of change of linear momentum.

11:47Net torque equals rate of change of angular momentum. So just like before, we can rearrange this to write the expression for dl. Dl equals, you know, network times dt, and since this is the differential form we can integrate it on both sides. If you do that, we get on the left-hand side the change in angular momentum, that equals this quantity.

12:10Again, look how analogous it is to this quantity. So we will call this angular impulse. It's the product of the torque and the time of risk that torque was acting. Again, if that torque is changing over that time, then we have to integrate that. So if I know the impulse, I can calculate the change in momentum.

12:30If I know the angular impulse, I can calculate the change in angular momentum, and from that I can figure out what omega is going to be. So when we flicked over here, we imparted both an impulse and an angular impulse. That's why this object gained both linear and angular momentum.

12:52But what if I flick this ruler exactly at the center. That you can probably intuitively tell it wouldn't spin, it would just move forward. Only translation, no rotation. Why? Well, now notice, I still deliver an impulse and therefore there is a change in linear momentum, it gains some linear momentum, but now there is no longer an angular impulse.

13:11Why? Because the net torque is zero. There is no torque about the center, and so the angular impulse is zero. And so the angular momentum changes zero. And so no rotation is imparted. That makes sense, right? Finally, if you were to plot a graph of the net force acting on object versus time, then the area under that graph, the signed area under the graph, represents the change in momentum over that time interval, right?

13:39Because remember, interval is basically calculating area under the curve. Similarly, if you find the signed area under the net torque time graph, that gives you the change in angular momentum.