b) On an old-fashioned rotating piano stool, a woman sits holding a pair of dumbbells at a distance of 0.60 m from the axis of rotation of the stool. She is given an angular velocity of 3.00 rad/s, after which she pulls the dumbbells in until they are only 0.20 m distant from the axis. The woman's moment of inertia about the axis of rotation is 5.00 kg m² and may be considered constant. Each dumbbell has a mass of 5.00 kg and may be considered a point mass. Ignore friction. (a) What is the initial angular momentum of the system? (b) What is the angular velocity of the system after the dumbbells are pulled in toward the axis? (c) Compute the kinetic energy of the system before and after the dumbbells are pulled in. Account for the difference, if any
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Physics Problem on Rotational Motion
#### Problem Statement:
**Scenario:**
On an old-fashioned rotating piano stool, a woman sits holding a pair of dumbbells at a distance of 0.60 meters from the axis of rotation of the stool. She is given an angular velocity of 3.00 radians per second (rad/s), after which she pulls the dumbbells in until they are only 0.20 meters distant from the axis.
**Given Data:**
- Woman's moment of inertia (I₁) about the axis of rotation: \(5.00 \, \text{kg} \cdot \text{m}^2\) (constant)
- Each dumbbell mass: \(5.00 \, \text{kg}\)
- Initial distance of dumbbells from axis: \(0.60 \, \text{m}\)
- Final distance of dumbbells from axis: \(0.20 \, \text{m}\)
- Initial angular velocity (ω₁): \(3.00 \, \text{rad/s}\)
#### Questions to be Answered:
1. **Initial Angular Momentum**
- What is the initial angular momentum of the system?
2. **Angular Velocity after Pulling in the Dumbbells**
- What is the angular velocity of the system after the dumbbells are pulled in toward the axis?
3. **Kinetic Energy Computation**
- Compute the kinetic energy of the system before and after the dumbbells are pulled in.
- Account for any difference in kinetic energy, if any.
##### Assumptions:
- Ignore friction.
- Dumbbells may be considered a point mass.
#### Calculation Steps:
1. **Initial Angular Momentum** (\(L_i\)):
The initial angular momentum can be calculated using the formula:
\[
L = I \cdot \omega
\]
where \(L\) is the angular momentum, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity.
2. **Final Angular Velocity** (\(\omega_f\)):
Using the law of conservation of angular momentum:
\[
L_i = L_f \implies I_i \cdot \omega_i = I_f \cdot \omega_f
\]
Solve for the final angular velocity (\(\omega_f\)).
3. **Kinetic Energy Computation** (K](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F701927fd-ceaf-4fad-8338-7d1e3c22653c%2Fa789bbce-bd91-4f02-a5b7-1512b1bef6e0%2Fjlmf4ha_processed.png&w=3840&q=75)
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