Opening/Closing Arms on Rotating Stool: Study with Video Lessons, Practice Problems & Examples

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

Here’s what students ask on this topic:

What is the conservation of angular momentum in a rotating system?

The conservation of angular momentum in a rotating system states that the total angular momentum (L) of a closed system remains constant if no external torque acts on it. This principle is expressed by the equation L = Iω, where I is the moment of inertia and ω is the angular velocity. 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) to conserve angular momentum.

How does bringing weights closer to the body affect rotational speed on a rotating stool?

Bringing weights closer to the body on a rotating stool decreases the moment of inertia (I) because the distance (r) from the axis of rotation is reduced. According to the conservation of angular momentum (L = Iω), if the moment of inertia decreases, the angular velocity (ω) must increase to keep the angular momentum constant. This results in an increase in rotational speed (RPM).

How do you calculate the moment of inertia for a rotating system with weights?

The moment of inertia (I) for a rotating system with weights can be calculated using the formula I = I_person + 2(mr²), where I_person is the moment of inertia of the person, m is the mass of each weight, and r is the distance from the axis of rotation. For point masses, the moment of inertia is given by I = mr². Summing these values gives the total moment of inertia of the system.

What is the relationship between moment of inertia and angular velocity in a rotating system?

The relationship between moment of inertia (I) and angular velocity (ω) in a rotating system is governed by the conservation of angular momentum, expressed as L = Iω. If the moment of inertia decreases, the angular velocity must increase to maintain constant angular momentum, and vice versa. This inverse relationship ensures that the product of I and ω remains constant in the absence of external torques.

How do you convert RPM to angular velocity (ω) in radians per second?

To convert RPM (revolutions per minute) to angular velocity (ω) in radians per second, use the formula ω = (2π × RPM) / 60. This conversion accounts for the fact that one revolution is equal to 2π radians and there are 60 seconds in a minute. For example, if the RPM is 60, the angular velocity would be ω = (2π × 60) / 60 = 2π radians per second.