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A hamster running in a wheel of radius 14 cm spins the wheel at 20 revolutions per minute. Round answers to 2 decimal places as needed. What is the angular velocity of the wheel in radians/sec? 02.09 0 At what linear velocity is the hamster running in cm/sec? 0 29.32 0 radians/sec cm/min

A hamster running in a wheel of radius 14 cm spins the wheel at 20 revolutions per minute. Round answers to 2 decimal places as needed. What is the angular velocity of the wheel in radians/sec? 02.09 0 At what linear velocity is the hamster running in cm/sec? 0 29.32 0 radians/sec cm/min

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.1: Angles
Problem 47E
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### Understanding Angular and Linear Velocity When a hamster runs inside a wheel of radius 14 cm, spinning the wheel at 20 revolutions per minute, we can determine both the angular velocity and the linear velocity of the hamster. Follow the steps below to understand how these calculations are made, rounding answers to 2 decimal places as needed. #### Problem Statement: 1. **Calculate the angular velocity of the wheel in radians/sec.** 2. **Determine the linear velocity at which the hamster is running in cm/sec.** ### Steps and Solutions: #### 1. Angular Velocity: **Given:** - Radius of the wheel (\( r \)) = 14 cm - Revolutions per minute (\( N \)) = 20 rev/min To find the angular velocity (\( \omega \)) in radians/sec: \[ \omega = \frac{2\pi N}{60} \] Detailed calculation: \[ \omega = \frac{2\pi \cdot 20}{60} = \frac{40\pi}{60} \approx 2.09 \text{ radians/sec} \] So, the angular velocity (\(\omega\)) is approximately **2.09 radians/sec**. **Input in box:** \[ 2.09 \] **Unit:** \[ radians/sec \] #### 2. Linear Velocity: **Given:** - Radius of the wheel (\( r \)) = 14 cm - Angular velocity (\( \omega \)) = 2.09 radians/sec To find the linear velocity (\( v \)), we use the relationship: \[ v = r \cdot \omega \] Detailed calculation: \[ v = 14 \text{ cm} \cdot 2.09 \text{ radians/sec} = 29.26 \text{ cm/sec} \approx 29.32 \text{ cm/sec} \] So, the linear velocity (\( v \)) is approximately **29.32 cm/sec**. **Input in box:** \[ 29.32 \] **Unit:** - Select \( cm/sec \) from the dropdown. ### Summary of Results: - **Angular Velocity:** 2.09 radians/sec - **Linear Velocity:** 29.32 cm/sec By understanding these calculations, students can better grasp the concepts of angular and linear velocities in rotational motion.
Transcribed Image Text:### Understanding Angular and Linear Velocity When a hamster runs inside a wheel of radius 14 cm, spinning the wheel at 20 revolutions per minute, we can determine both the angular velocity and the linear velocity of the hamster. Follow the steps below to understand how these calculations are made, rounding answers to 2 decimal places as needed. #### Problem Statement: 1. **Calculate the angular velocity of the wheel in radians/sec.** 2. **Determine the linear velocity at which the hamster is running in cm/sec.** ### Steps and Solutions: #### 1. Angular Velocity: **Given:** - Radius of the wheel (\( r \)) = 14 cm - Revolutions per minute (\( N \)) = 20 rev/min To find the angular velocity (\( \omega \)) in radians/sec: \[ \omega = \frac{2\pi N}{60} \] Detailed calculation: \[ \omega = \frac{2\pi \cdot 20}{60} = \frac{40\pi}{60} \approx 2.09 \text{ radians/sec} \] So, the angular velocity (\(\omega\)) is approximately **2.09 radians/sec**. **Input in box:** \[ 2.09 \] **Unit:** \[ radians/sec \] #### 2. Linear Velocity: **Given:** - Radius of the wheel (\( r \)) = 14 cm - Angular velocity (\( \omega \)) = 2.09 radians/sec To find the linear velocity (\( v \)), we use the relationship: \[ v = r \cdot \omega \] Detailed calculation: \[ v = 14 \text{ cm} \cdot 2.09 \text{ radians/sec} = 29.26 \text{ cm/sec} \approx 29.32 \text{ cm/sec} \] So, the linear velocity (\( v \)) is approximately **29.32 cm/sec**. **Input in box:** \[ 29.32 \] **Unit:** - Select \( cm/sec \) from the dropdown. ### Summary of Results: - **Angular Velocity:** 2.09 radians/sec - **Linear Velocity:** 29.32 cm/sec By understanding these calculations, students can better grasp the concepts of angular and linear velocities in rotational motion.
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