6. A horizontal circular disk rotates freely about a vertical axis through its center. It has a mass of 100 kg and a radius of 2.0 m. A person of mass 60 kg stands at the edge of the disk while the disk is rotating at 2.0 rad/s. The person slowly walks from the edge toward the center of the disk. What is the angular speed of the system when the person is 0.50 m from the center?
6. A horizontal circular disk rotates freely about a vertical axis through its center. It has a mass of 100 kg and a radius of 2.0 m. A person of mass 60 kg stands at the edge of the disk while the disk is rotating at 2.0 rad/s. The person slowly walks from the edge toward the center of the disk. What is the angular speed of the system when the person is 0.50 m from the center?
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Please do the algebra to show how the answer is 4.1 rad/s
![**Physics Problem: Rotational Motion**
---
**Problem Statement:**
A horizontal circular disk rotates freely about a vertical axis through its center. It has a mass of 100 kg and a radius of 2.0 m. A person of mass 60 kg stands at the edge of the disk while the disk is rotating at 2.0 rad/s. The person slowly walks from the edge toward the center of the disk. What is the angular speed of the system when the person is 0.50 m from the center?
---
**Given:**
- Mass of the disk \( M = 100 \, \text{kg} \)
- Mass of the person \( m = 60 \, \text{kg} \)
- Radius of the disk \( R = 2.0 \, \text{m} \)
- Initial angular velocity \( \omega_0 = 2.0 \, \text{rad/s} \)
- Final distance of the person from the center \( r = 0.50 \, \text{m} \)
---
**Equations:**
1. **Conservation of Angular Momentum:**
\[ L = I \omega \]
2. **Moment of Inertia for Disk and Person:**
\[ I = I_{\text{disk}} + I_{\text{person}} \]
3. **Initial Angular Momentum:**
\[ I_{\text{disk}} = \frac{1}{2} MR^2 \]
\[ I_{\text{person}} = mr^2 \]
4. **Final Angular Momentum:**
\[ I_o \omega_o = I_f \omega_f \]
---
**Solution Steps:**
- **Initial Moment of Inertia:**
\[ I_o = \frac{1}{2} MR^2 + mR^2 \]
- **Equation Substitution:**
\[ \left(\frac{1}{2} \cdot 100 \cdot 2^2 + 60 \cdot 2^2\right) \cdot 2 = \left(\frac{1}{2} \cdot 100 \cdot 2^2 + 60 \cdot (0.5)^2\right) \cdot \omega_f \]
- **Inserting Numbers and Solving for Final Angular Speed \( \omega_f \):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2107c9d4-03cf-46c4-aa9a-f455311ac0bc%2F1461ba71-8b30-4932-925a-3263694de9dd%2Fljjstt.jpeg&w=3840&q=75)
Transcribed Image Text:**Physics Problem: Rotational Motion**
---
**Problem Statement:**
A horizontal circular disk rotates freely about a vertical axis through its center. It has a mass of 100 kg and a radius of 2.0 m. A person of mass 60 kg stands at the edge of the disk while the disk is rotating at 2.0 rad/s. The person slowly walks from the edge toward the center of the disk. What is the angular speed of the system when the person is 0.50 m from the center?
---
**Given:**
- Mass of the disk \( M = 100 \, \text{kg} \)
- Mass of the person \( m = 60 \, \text{kg} \)
- Radius of the disk \( R = 2.0 \, \text{m} \)
- Initial angular velocity \( \omega_0 = 2.0 \, \text{rad/s} \)
- Final distance of the person from the center \( r = 0.50 \, \text{m} \)
---
**Equations:**
1. **Conservation of Angular Momentum:**
\[ L = I \omega \]
2. **Moment of Inertia for Disk and Person:**
\[ I = I_{\text{disk}} + I_{\text{person}} \]
3. **Initial Angular Momentum:**
\[ I_{\text{disk}} = \frac{1}{2} MR^2 \]
\[ I_{\text{person}} = mr^2 \]
4. **Final Angular Momentum:**
\[ I_o \omega_o = I_f \omega_f \]
---
**Solution Steps:**
- **Initial Moment of Inertia:**
\[ I_o = \frac{1}{2} MR^2 + mR^2 \]
- **Equation Substitution:**
\[ \left(\frac{1}{2} \cdot 100 \cdot 2^2 + 60 \cdot 2^2\right) \cdot 2 = \left(\frac{1}{2} \cdot 100 \cdot 2^2 + 60 \cdot (0.5)^2\right) \cdot \omega_f \]
- **Inserting Numbers and Solving for Final Angular Speed \( \omega_f \):
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