A playground merry-go-round of radius R = 3 m has two men each of mass m = 100 kg standing on the rim as it turns at 3.0 rad/s. The moment of inertia of the merry-go-round without the men is 12,000 kg · m2 (about its axle). If both men move to a radius r = R/2 what will be the new angular speed to three significant figures?
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.
A playground merry-go-round of radius R = 3 m has two men each of mass m = 100 kg standing on the rim as it turns at 3.0 rad/s. The moment of inertia of the merry-go-round without the men is 12,000 kg · m2 (about its axle). If both men move to a radius r = R/2 what will be the new angular speed to three significant figures?
![### Educational Resource: Conservation of Angular Momentum
#### Problem Statement:
A playground merry-go-round of radius \( R = 3 \, \text{m} \) has two men, each of mass \( m = 100 \, \text{kg} \), standing on the rim as it turns at \( 3.0 \, \text{rad/s} \). The moment of inertia of the merry-go-round without the men is \( 12{,}000 \, \text{kg} \cdot \text{m}^2 \) (about its axle). If both men move to a radius \( r = R/2 \), what will be the new angular speed to three significant figures?
#### Diagram Explanation:
The diagram illustrates a top view of the merry-go-round, with:
- A circle representing the merry-go-round.
- Two dots on the circle's rim representing the men.
- Both dots are labeled with \( m \), indicating their mass.
- The radius to the rim is labeled \( R \).
### Physics Concepts:
1. **Moment of Inertia (I):** A measure of an object's resistance to changes in its rotation rate. The total moment of inertia includes the merry-go-round and the men.
2. **Angular Speed (\(\omega\)):** The rate of rotation expressed in radians per second.
3. **Conservation of Angular Momentum (\(L\)):** The total angular momentum before and after an event remains constant, given by the equation \( L = I \cdot \omega \).
### Calculations:
1. **Initial Angular Momentum:**
- Total moment of inertia initially (\(I_{\text{initial}}\)) = \(I_{\text{merry-go-round}} + 2 \cdot m \cdot R^2\)
- \[ I_{\text{initial}} = 12{,}000 + 2 \cdot 100 \cdot (3)^2 = 12{,}000 + 1,800 = 13{,}800 \, \text{kg} \cdot \text{m}^2 \]
- Initial angular momentum (\(L_{\text{initial}}\)) = \(I_{\text{initial}} \cdot \omega\)
- \[ L_{\text{initial}} = 13{,}800 \cdot 3.0 = 41{,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ff5afa3-7a88-496b-9360-76d2f49b6ba3%2F4a77a1c6-65a0-459e-9bf4-31d73c5f0c79%2Fb538v6r_processed.png&w=3840&q=75)
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