College Physics
College Physics
11th Edition
ISBN: 9781305952300
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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### Chapter 08, Problem 29

**Interactive Solution 8.29** offers a model for this problem. The drive propeller of a ship starts from rest and accelerates at \( 2.49 \times 10^{-3} \, \text{rad/s}^2 \) for \( 2.14 \times 10^3 \) seconds. For the next \( 1.85 \times 10^3 \) seconds, the propeller rotates at a constant angular speed. Then it decelerates at \( 2.90 \times 10^{-3} \, \text{rad/s}^2 \) until it slows (without reversing direction) to an angular speed of \( 2.66 \, \text{rad/s} \). Find the total angular displacement of the propeller.

### Explanation:

This problem involves calculating the angular displacement of a ship's propeller as it undergoes three stages of motion: acceleration, constant speed, and deceleration. 

1. **Acceleration Phase:**
   - Initial angular speed (\(\omega_0\)) = 0 \( \text{rad/s} \) (starting from rest)
   - Acceleration (\(\alpha\)) = \( 2.49 \times 10^{-3} \, \text{rad/s}^2 \)
   - Time (\(t\)) = \( 2.14 \times 10^3 \) s

2. **Constant Speed Phase:**
   - Time (\(t\)) = \( 1.85 \times 10^3 \) s

3. **Deceleration Phase:**
   - Final angular speed (\(\omega_f\)) = \( 2.66 \, \text{rad/s} \)
   - Deceleration (\(\alpha\)) = \( 2.90 \times 10^{-3} \, \text{rad/s}^2 \)

By using the kinematic equations for rotational motion, you can calculate the angular displacement for each phase, then sum them to find the total angular displacement of the propeller. This type of problem helps in understanding concepts of rotational motion, angular acceleration, and displacement in a physical context.
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Transcribed Image Text:### Chapter 08, Problem 29 **Interactive Solution 8.29** offers a model for this problem. The drive propeller of a ship starts from rest and accelerates at \( 2.49 \times 10^{-3} \, \text{rad/s}^2 \) for \( 2.14 \times 10^3 \) seconds. For the next \( 1.85 \times 10^3 \) seconds, the propeller rotates at a constant angular speed. Then it decelerates at \( 2.90 \times 10^{-3} \, \text{rad/s}^2 \) until it slows (without reversing direction) to an angular speed of \( 2.66 \, \text{rad/s} \). Find the total angular displacement of the propeller. ### Explanation: This problem involves calculating the angular displacement of a ship's propeller as it undergoes three stages of motion: acceleration, constant speed, and deceleration. 1. **Acceleration Phase:** - Initial angular speed (\(\omega_0\)) = 0 \( \text{rad/s} \) (starting from rest) - Acceleration (\(\alpha\)) = \( 2.49 \times 10^{-3} \, \text{rad/s}^2 \) - Time (\(t\)) = \( 2.14 \times 10^3 \) s 2. **Constant Speed Phase:** - Time (\(t\)) = \( 1.85 \times 10^3 \) s 3. **Deceleration Phase:** - Final angular speed (\(\omega_f\)) = \( 2.66 \, \text{rad/s} \) - Deceleration (\(\alpha\)) = \( 2.90 \times 10^{-3} \, \text{rad/s}^2 \) By using the kinematic equations for rotational motion, you can calculate the angular displacement for each phase, then sum them to find the total angular displacement of the propeller. This type of problem helps in understanding concepts of rotational motion, angular acceleration, and displacement in a physical context.
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