Opening/Closing Arms on Rotating Stool: Videos & Practice Problems

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

In this example of conservation of angular momentum, we explore how the moment of inertia affects the rotational motion of a system. Imagine standing on a stool that can rotate around a vertical axis through its center. The moment of inertia (I) is a measure of how mass is distributed relative to the axis of rotation, and it plays a crucial role in determining the angular momentum (L) of the system.

Initially, the combined moment of inertia of you and the stool is given as 8 kg·m², regardless of whether your arms are open or closed. However, in reality, closing your arms would reduce your moment of inertia, but for simplicity, we assume it remains constant in this scenario. When you hold a 10 kg weight in each hand at a distance of 0.8 meters from the central axis, the total moment of inertia of the system changes. The moment of inertia of a point mass is calculated using the formula:

$I^{=}mr{²}^{}$

For the two weights, the moment of inertia is:

$I=2mr{²}^{}$

Substituting the values, we find the total moment of inertia of the system to be 20.8 kg·m².

Next, we calculate the angular momentum using the relationship:

Where ω (angular velocity) can be derived from the given RPM (revolutions per minute) using the conversion:

Substituting the values, we find the angular momentum to be approximately 131 kg·m²/s.

When you bring the weights closer to your chest, effectively reducing their distance from the axis of rotation to zero, the moment of inertia of the weights becomes negligible. Thus, the new moment of inertia of the system is simply 8 kg·m². This change in moment of inertia leads to a change in angular velocity.

Using the conservation of angular momentum, we set the initial angular momentum equal to the final angular momentum:

$L\_i=L\_f$

Expanding this gives:

By substituting the known values and solving for the final RPM, we find that the new RPM is approximately 156. This increase in RPM illustrates the principle that as the moment of inertia decreases, the angular velocity must increase to conserve angular momentum.

This example highlights the relationship between moment of inertia, angular momentum, and rotational motion, demonstrating how changes in mass distribution can significantly affect the dynamics of a rotating system.

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

Answer: (B)

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.