Understanding the relationship between angular velocity and linear speed is essential in many real-life scenarios, such as calculating how fast a car travels based on its wheel rotation. This article provides a step-by-step guide to finding speed (\( v \)) using angular velocity (\( \omega \)) and the radius (\( r \)) of a wheel, with relatable examples.
Formula to Find Speed
The linear speed (\( v \)) can be calculated from the angular velocity (\( \omega \)) and the radius of the wheel (\( r \)) using the formula:
\[ v = \omega \cdot r \]
where:
- \( v \) is the linear speed.
- \( \omega \) is the angular velocity (in radians per second).
- \( r \) is the radius of the wheel.
Example 1: Speed of a Car Based on Wheel Rotation
Scenario: You know that the wheels of a car are spinning at an angular velocity (\( \omega \)) of \( 10 \, \text{rad/s} \). The radius (\( r \)) of the car's wheels is \( 0.35 \, \text{meters} \). What is the car's speed?
Step-by-Step Calculation:
1. Given:
\[ \omega = 10 \, \text{rad/s} \]
\[ r = 0.35 \, \text{m} \]
2. Substitute Values into the Speed Formula:
\[ v = \omega \cdot r \]
\[ v = 10 \cdot 0.35 \]
3. Perform the Calculation:
\[ v = 3.5 \, \text{m/s} \]
Final Value
The car's speed is:
\[ v = 3.5 \, \text{m/s} \]
Example 2: Speed of a Bicycle Wheel
Scenario: A bicycle wheel rotates with an angular velocity (\( \omega \)) of \( 5 \, \text{rad/s} \) and has a radius (\( r \)) of \( 0.3 \, \text{meters} \). What is the bicycle's speed?
Step-by-Step Calculation:
1. Given:
\[ \omega = 5 \, \text{rad/s} \]
\[ r = 0.3 \, \text{m} \]
2. Substitute Values into the Speed Formula:
\[ v = \omega \cdot r \]
\[ v = 5 \cdot 0.3 \]
3. Perform the Calculation:
\[ v = 1.5 \, \text{m/s} \]
Final Value
The bicycle's speed is:
\[ v = 1.5 \, \text{m/s} \]
Example 3: Speed of a Rotating Platform
Scenario: A rotating platform at an amusement park has an angular velocity (\( \omega \)) of \( 2 \, \text{rad/s} \) and a radius (\( r \)) of \( 1.2 \, \text{meters} \). What is the speed at the edge of the platform?
Step-by-Step Calculation:
1. Given:
\[ \omega = 2 \, \text{rad/s} \]
\[ r = 1.2 \, \text{m} \]
2. Substitute Values into the Speed Formula:
\[ v = \omega \cdot r \]
\[ v = 2 \cdot 1.2 \]
3. Perform the Calculation:
\[ v = 2.4 \, \text{m/s} \]
Final Value
The speed at the edge of the rotating platform is:
\[ v = 2.4 \, \text{m/s} \]
Summary
To determine the linear speed (\( v \)) from angular velocity (\( \omega \)) and the radius of a wheel (\( r \)), use the formula:
\[ v = \omega \cdot r \]
In the examples provided:
1. A car with wheels spinning at \( 10 \, \text{rad/s} \) and a radius of \( 0.35 \, \text{m} \) has a speed of \( 3.5 \, \text{m/s} \).
2. A bicycle wheel rotating at \( 5 \, \text{rad/s} \) with a radius of \( 0.3 \, \text{m} \) results in a speed of \( 1.5 \, \text{m/s} \).
3. A rotating platform at \( 2 \, \text{rad/s} \) with a radius of \( 1.2 \, \text{m} \) has a speed of \( 2.4 \, \text{m/s} \).
These calculations are valuable for understanding how angular motion translates to linear speed in practical applications.
