 Let's solve a couple of questions on conservation of angular momentum. For the first one we have a man holding onto two weights close to his body and he's standing on a platform that is freely rotating about a frictionless axle. Okay, so this person who is holding two weights is freely rotating, it's rotating about a frictionless axle. When he raises his arms away from his body, his angular velocity decreases. Which of the following best describes why the man's angular velocity decreased? We have to choose one answer out of these four options. As always pause the video and think through this question on your own first. Alright, hopefully you have given this a shot. Now over here let's see what the situation is. So this person who is holding these two weights, he is standing on a platform and the platform is rotating. It could be rotating anti-clockwise or clockwise, that doesn't really matter. Then initially his hands are like this, they are like this. Then he raises his hands and made them horizontal. And as a result of which angular velocity, the final angular velocity of the platform it decreased. So final angular velocity is less than the initial angular velocity. Okay, so the question is asking us why did this happen? Let's look at the options. The options are talking about moment of inertia, maybe the moment of inertia decreased while the angular momentum remained constant or maybe moment of inertia increased or angular momentum decreased. So there is a combination of options. But we can think about moment of inertia and angular momentum. So when we look at the situation, when we consider the platform, these weights and the person, the man as one system, we will realize that there is no actually there is no change in talk. When the person moves his hands and made them horizontal, there is no external talk which acts on the system really. Because the person moving around his hands and changing the location of the weights, it's still inside the system. There is no external, there is no friction as such, which is providing a talk in the opposite direction, some external talk. There is no external talk in this system. So even when the person moves his hands, even when the person moves his hands, they are really moving perpendicular to the axis of rotation, which will be like this. So his hands are really perpendicular to this plane. It's a perpendicular plane altogether. And more than that, the person and the weights, it's internal to the system. There is no external talk. So when tau external is 0, one quantity is conserved and that would be angular momentum, right? Just like how when external force was 0, there is no change in linear momentum. Similarly, when there is no external talk, the angular momentum is conserved. And we know that angular momentum, angular momentum is given by i omega, moment of inertia of the system, into the angular velocity. So that means if l i, if the initial angular momentum is equal to the final angular momentum, which is true because the angular momentum is conserved, since there is no external talk, this means that i initial, the initial moment of inertia into the initial angular velocity. This is equal to the final moment of inertia into the final angular velocity. And we know that the angular velocity is decreasing, omega f is less than omega i, which means for this relation to hold true, if omega i is increasing and omega f is decreasing, for this relation to hold true, the initial moment of inertia, it must be less, right? Compared to the final moment of inertia, so that the equality is maintained and both the sides are equal to each other. So the moment of inertia is really increased while the angular momentum, the angular momentum remains constant. And that is the last option. We can also try to understand it physically, why is the moment of inertia increasing? So now more of the weight, more of the weight is away, is away from the axis of rotation. And we knew that moment of inertia really is sigma m r square. So if you increase r, if you increase the distance from the axis, if there is more mass away from the axis, the moment of inertia will be more. And that is true for the case when the man makes his hands horizontal. Now more mass, more mass is away, away from the axis of rotation. So for this one, the right answer is option D. Let's look at one more question. Here we have a turtle placed at the edge of a platform with mass m and radius r. Okay, so the platform has a mass m and this radius, this radius is r. The platform spins freely about a frictionless axle with an initial angular velocity of omega i. The turtle then walks along a radial line toward the center of the platform. So this turtle starts moving to the center. When the turtle arrives at the center, the platform has a final angular velocity of omega f. And the question is to figure out the relationship between omega f and omega i. All right, now let's think about this. Here the turtle is at the edge of this platform to begin with. And at that point, there is an initial angular velocity. But now when the turtle starts moving towards the platform, turns out the angular velocity changed. Now there is some final angular velocity when the turtle arrives at the center. Now if you think about the platform and the turtle system, again there is no external torque to this system. Tau external is zero for this system as well. Turtle moving towards the center is still internal to the system. There is no external torque. And when there is no external torque, the angular momentum is conserved. Which means the initial angular momentum that is equal to the final angular momentum. That is the angular momentum when the turtle was at the edge is equal to the angular momentum of the system when the turtle is at the center. And if you write it in this manner, we know that angular momentum is, this is equal to i omega. So initial momentum of the system into the initial angular velocity of the system. This is equal to the final momentum of the system into the final angular velocity of this platform. Now we need to think about the relationship between omega f and omega i. But for that, we will really be able to do that only if we know the relationship between the momentum of the system and the initial momentum of the system. And the final momentum of the system. So initially when there is this platform and the turtle is at the edge, there is some moment of inertia, right? There is mass, there is an axis of rotation. There is an axis of rotation which goes through the center. But when now the turtle starts moving towards the center, there is some mass which is of the turtle that is moving towards the axis of rotation. And we know that moment of inertia was really given by sigma m r square. So if there is more mass that is moving towards the axis of the rotation, this r will decrease because this r was always a distance from the axis of rotation. And if more mass is concentrated around the axis, which will happen when the turtle is near the center, the moment of inertia of the system will decrease. So final moment of inertia, that's really less than the initial moment of inertia. This is less and this is more. So now for the equality to be maintained, for the angular momentum to be conserved, omega f has to be more, it has to be more than omega i. So that means the right answer is the first option. The final angular velocity has to be more than the initial angular velocity. So that the initial angular momentum of the system is equal to the final angular momentum of the system. Alright, you can try more questions from this exercise in the lesson. And if you're watching on YouTube, do check out the exercise link which is added in the description.