A space station consists of a large, lightweight ring of radius R = 60 m and two small heavy control masses, each of mass M = 5.0 × 10ª kg, positioned at the outer edge. The space station rotates; workers standing on the inner surface of the ring experience an artificial gravity, by way of the normal force from their "floor". a. Determine the moment of inertia of the space station. (Treat the control weights as point masses; ignore the mass of the lightweight ring and the workers.) b. The station is initially turning counterclockwise at an angular speed of wi = 0.2 radians/s. What is the magnitude of the initial angular momentum of the station? C. What is the direction of the angular momentum vector?
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.
![**Problem Statement: Space Station Dynamics**
A space station consists of a large, lightweight ring with a radius \( R = 60 \, \text{m} \) and two small heavy control masses, each with a mass \( M = 5.0 \times 10^4 \, \text{kg} \), positioned at the outer edge. The space station rotates; workers standing on the inner surface of the ring experience an artificial gravity through the normal force from their “floor.”
**Problem Parts:**
a. **Moment of Inertia Calculation:**
Determine the moment of inertia of the space station. (Treat the control weights as point masses and ignore the mass of the lightweight ring and the workers.)
b. **Initial Angular Momentum:**
The station is initially turning counterclockwise at an angular speed of \( \omega_i = 0.2 \, \text{radians/s} \). What is the magnitude of the initial angular momentum of the station?
c. **Direction of Angular Momentum:**
What is the direction of the angular momentum vector?
**Solution Outline:**
1. **Moment of Inertia (I):**
- Consider the control masses as point masses located at the radius of the ring.
- Use the formula for moment of inertia for point masses:
\[
I = 2 \times M \times R^2
\]
- Substitute given values:
\[
I = 2 \times (5.0 \times 10^4 \, \text{kg}) \times (60 \, \text{m})^2
\]
2. **Angular Momentum (L):**
- Use the relationship \( L = I \times \omega_i \).
- Substitute the computed moment of inertia and given angular speed.
3. **Direction of Angular Momentum:**
- For a counterclockwise rotation, use the right-hand rule to determine the vector direction.
This structured breakdown is designed for educational purposes, aiding in the understanding of rotational dynamics in physics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F819d264c-a833-46aa-81dd-f353b8dcb1ab%2F601c74eb-0e8c-47b5-875f-769eec2d700c%2Fo0z0zu5_processed.png&w=3840&q=75)
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