Opening/Closing Arms on Rotating Stool - Online Tutor, Practice Problems & Exam Prep

In a rotating system, the conservation of angular momentum is crucial. When a person on a rotating stool brings weights closer to their body, the moment of inertia decreases, causing an increase in rotational speed (RPM). The moment of inertia (I) is calculated using the formula $I^{=}(mr)²$. This principle illustrates how angular momentum (L) remains constant, as expressed by $L=I\omega$, where ω is angular velocity.

Holding weights on a spinning stool

Hey guys, let's check out this conservation of angular momentum example. So you stand on a stool that is free to rotate about a perpendicular axis to itself and through its center. So there's a stool. The stool, let's get a little stool going on here. And this thing can rotate around itself, about an axis perpendicular to it through its center. So imagine that the stool is this, and it rotates, and you're on top of it. Perpendicular means making a 90 degrees. Right? So this means that the stool rotates like this, with an axis through its center. So just like a normal stool if you are on it. So you stand on the stool, and it says that suppose that your combined moment of inertia, you and stool, is 8, and that it is the same whether you have your arms open or pressed about, around your body. Now that's not true, obviously, because if you reduce your if you close your arms, remember, the moment of inertia is some version of mrr2. If you close your arms, your r is going to drop which means your I drops. Okay. But in this problem, just to make things simpler, we're saying that this change is negligible. Okay. So when you close, it's going to stay the same. So initially, so that's not gonna happen. Initially, with arms open or closed, your I will be 8. So let me draw this here. You with your arms open, I=8, and then you with your arms closed over here, something like this, right? Close against your body. Your I will be 8 as well. And then it says here, that you stand on the stool with arms wide open, holding a 10-kilogram weight on each hand at a distance of 80 centimeters from the central axis. Now if you're on top of the stool, you're gonna rotate around yourself. So the central axis imagine is a line through your body. Okay. So the central axis looks like this. Okay. So what we're actually going to have is we're gonna have, we'll draw this one more time. We're gonna have one last time. You, holding weights with your arms wide open. So here are the weights you're holding. They have mass of 10. So the I of the system is not going to be 8. The I of the system is gonna be the I of you plus these two weights. Both weights have the same mass. They are the same distance from the axis, which is point 8 and point 8 meters. And these guys are 10 kilograms each. Okay. So the moment of inertia of the system is you plus 2 times the moment of inertia of the little mass because they're going to be the same moments of inertia. Okay. Your moment of inertia is 8 but, you'd have to calculate this piece here. Okay. So we want to first and that's what we want to find out first. We want to calculate the system's total moment of inertia about its central axis. So I already have that it's 8+2i. i is the moment of inertia of the little objects. The objects are point masses because they're shapeless. They have negligible size. The moment of inertia of a point mass, remember, is mr----

You stand on a stool that is free to rotate about an axis perpendicular to itself and through its center. The stool's moment of inertia around its central axis is 1.50 kg m² . Suppose you can model your body as a vertical solid cylinder (height = 1.80 m, radius = 20 cm, mass = 80 kg) with two horizontal thin rods as your arms (each:length = 80 cm, mass = 3 kg) that rotate at their ends, about the same axis, as shown. Suppose that your arms' contribution to the total moment of inertia is negligible if you have them pressed against your body, but significant if you have them wide open. If you initially spin at 5 rad/s with your arms against your body, how fast will you spin once you stretch them wide open? (Note:The system has 4 objects (stool + body + 2 arms), but initially only stool + body contribute to its moment of inertia)

• A. 3.1 rad/s
• B. 3.54 rad/s
• C. 4.38 rad/s
• D. 7.06 rad/s