Angular momentum of rigid systems

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, 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.

4:05But because they're farther apart, those spheres are farther apart, they're moving faster. So 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.

4:22Just like how both linear momentum and velocity are vector quantities, angular momentum and angular velocities are also vector quantities. So 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?

4:39Well, 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. So in this particular case, the direction of angular velocity is upwards.

4:56So 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 are 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, then 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.

6:15So 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. Over here we have to call 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.

6:31But 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. In fact, along this axis, it turns out that the rotational inertia is actually lower compared to along this axis.

6:46And 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? So the angular momentum and the angular velocity were having the same direction, but now consider this.

7:04What 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. And so the angular momentum would be 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.

7:33So this is not a principal axis. Along this axis of rotation, the angular momentum and the angular velocity are not in the same direction. So, long story short, whenever we're dealing with principal axis, these two are always in the same direction, and rotational inertia becomes just a simple number, a scalar.

7:54However, 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. It becomes something called rank-two tensor, but we don't have to worry about that because we're gonna restrict our analysis to principal axis only.

8:14So for our case, the angular momentum and angular velocity will point in the same direction, and rotational inertia is just a scalar. Okay, 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 by the moment of inertia and calculate the angular velocity.

8:35But the big question now is, how do I figure out how much angular momentum it gains after I flick it? Well, let's come back from translational motion. How do we figure out how much linear momentum it gains after we flick it? Well, we can use Newton second law.

8:48According to Newton's second law, the net force acting on an object equals mass times its acceleration. But what exactly is acceleration? Well, it's a measure of how quickly the velocity is changing. It's the rate of change of velocity. So I can plug that in. So we can write this as m times delta v or delta T.

9:07Now, strictly speaking, we should call this the average rate of change of velocity because this value is averaged over that time interval delta t. So this is average acceleration over the time interval, and this is the average net force over that time interval.

9:23But I'm gonna stop using the word average over and over again because I wanna talk about this intuitively. Intuitively speaking, acceleration is a measure of how quickly the velocity is changing, right? Okay, now, if I pull this mass inside the delta, I'll get delta mv over delta t.

9:39What is mv? Hey, that's momentum. So this becomes delta p over delta t. In other words, this is rate of change of momentum. Again, average, okay? So this equation is saying that the net force acting on an object decides how quickly its momentum changes, and this is exactly what I wanted because I am interested in delta p.

10:03So now I can use this equation to rearrange and write it as delta p equals the net force, the average net force over the time delta t times delta t. And this product of the net force multiplied by the time over which that net force is acting, we call it the impulse.

10:25So this means that all I need to know is the impulse delivered by my finger on this ruler. if I know that, I'm done because that equals the change in momentum. And once I know how much the momentum has changed, I can figure out the final momentum and I can figure out the final velocity after I have flicked this ruler.

10:46All right, now, let's do the same thing for rotation. In fact, I want you to pause the video and see if you can try to write down Newton's second law for rotation yourself. It's gonna be analogous to this, so please first try it yourself. Okay, let's do that. Over here, we were dealing with net force.

11:03Over here, it's going to be net torque. And over here, that equals mass times acceleration. Here, instead of mass, we will have rotational inertia. And instead of acceleration, we will have angular acceleration. What's angular acceleration? Well, linear acceleration is a measure of how quickly linear velocity changes.

11:24Angular acceleration is a measure of how quickly angular velocity changes. So this equals rotational inertia times delta omega divided by delta T. And again, remember, because we are measuring this change over some time interval delta t, this is the average rate of change of angular velocity.

11:44So this is the average angular acceleration. So this is the average net torque. But anyways, just like before, I can now this rotational inertia inside the delta, and that gives me delta L. So this equation is saying the net torque equals rate of change of angular momentum.

12:06In other words, how quickly it's angular momentum is changing depends on the net torque acting on it. Again, average values, okay, but anyways, now I'm interested in how much the angular momentum has changed. So just like before, we can rearrange the situation to write delta L equals the net torque times delta t.

12:30And over here we call this impulse. Here we have an analogous version for rotation. We call this the angular impulse, which means over here I need to know how much angular impulse my finger delivered to this ruler. Once I know that, I am done, that equals how much the angular momentum has changed.

12:51And from that, I can calculate how much the final angular momentum is. And from there, I can calculate the final angular velocity once I have flicked this ruler. So putting it all together, when I flick this ruler, my force imparted both an impulse and an angular impulse on that ruler.

13:10As a result, it gained both linear and angular momentum. So it started moving and spinning together. But what if I had flicked the ruler over here? We can intuitively guess now. It wouldn't spin. It would just go forward. But why? Well, because this time, again, my force delivers an impulse, but it no longer delivers an angular impulse because this time the force acts through the center of mass, and as a result, there is no torque generated.

13:37So torque is zero. The net torque acting on this ruler is zero. Angular impulse is zero. Therefore, this time there's only a change in linear momentum, but no change in angular momentum. That's why this time the ruler does not spin at all. Finally, if I were to draw a graph of the net force acting on object versus the time, then the impulse is equal to the area, the signed area under that graph.

14:02So that total area would represents how much the momentum has changed over a given time interval. Similarly, if I were to plot a graph of the net torque versus time, then the signed area under that graph gives me angular impulse. So that would be a measure of how much the angular momentum has changed over a given time interval.