To understand the relationship between linear speed and how fast a wheel is spinning, we use the concept of angular velocity. This article explains how to calculate the angular velocity (\( \omega \)) of a wheel given its linear speed (\( v \)) and radius (\( r \)) with real-life examples.
Formula to Determine Angular Velocity
The angular velocity (\( \omega \)) can be determined from the linear speed (\( v \)) and the radius of the wheel (\( r \)) using the formula:
\[ \omega = \dfrac{v}{r} \]
where:
- \( \omega \) is the angular velocity (in radians per second).
- \( v \) is the linear speed.
- \( r \) is the radius of the wheel.
Example 1: Angular Velocity of a Car Wheel
Scenario: A car is traveling at a linear speed (\( v \)) of \( 21 \, \text{m/s} \). The radius (\( r \)) of the car's wheels is \( 0.4 \, \text{meters} \). What is the angular velocity of the car's wheels?
Step-by-Step Calculation:
1. Given:
\[ v = 21 \, \text{m/s} \]
\[ r = 0.4 \, \text{m} \]
2. Substitute Values into the Angular Velocity Formula:
\[ \omega = \dfrac{v}{r} \]
\[ \omega = \dfrac{21}{0.4} \]
3. Perform the Calculation:
\[ \omega = 52.5 \, \text{rad/s} \]
Final Value
The car's wheels are spinning at:
\[ \omega = 52.5 \, \text{rad/s} \]
Example 2: Angular Velocity of a Bicycle Wheel
Scenario: A bicycle moves at a linear speed (\( v \)) of \( 6 \, \text{m/s} \). The radius (\( r \)) of the bicycle's wheels is \( 0.35 \, \text{meters} \). What is the angular velocity of the bicycle's wheels?
Step-by-Step Calculation:
1. Given:
\[ v = 6 \, \text{m/s} \]
\[ r = 0.35 \, \text{m} \]
2. Substitute Values into the Angular Velocity Formula:
\[ \omega = \dfrac{v}{r} \]
\[ \omega = \dfrac{6}{0.35} \]
3. Perform the Calculation:
\[ \omega = 17.14 \, \text{rad/s} \]
Final Value
The bicycle's wheels are spinning at:
\[ \omega = 17.14 \, \text{rad/s} \]
Example 3: Angular Velocity of a Rotating Playground Merry-Go-Round
Scenario: A playground merry-go-round rotates such that a point on the outer edge has a linear speed (\( v \)) of \( 3 \, \text{m/s} \). The radius (\( r \)) of the merry-go-round is \( 1.5 \, \text{meters} \). What is the angular velocity of the merry-go-round?
Step-by-Step Calculation:
1. Given:
\[ v = 3 \, \text{m/s} \]
\[ r = 1.5 \, \text{m} \]
2. Substitute Values into the Angular Velocity Formula:
\[ \omega = \dfrac{v}{r} \]
\[ \omega = \dfrac{3}{1.5} \]
3. Perform the Calculation:
\[ \omega = 2 \, \text{rad/s} \]
Final Value
The merry-go-round spins at:
\[ \omega = 2 \, \text{rad/s} \]
Summary
To determine how fast a wheel or any rotating object is spinning, given its linear speed (\( v \)) and radius (\( r \)), use the formula:
\[ \omega = \dfrac{v}{r} \]
In the examples provided:
1. A car moving at \( 21 \, \text{m/s} \) with a wheel radius of \( 0.4 \, \text{m} \) has wheels spinning at \( 52.5 \, \text{rad/s} \).
2. A bicycle traveling at \( 6 \, \text{m/s} \) with a wheel radius of \( 0.35 \, \text{m} \) has wheels spinning at \( 17.14 \, \text{rad/s} \).
3. A playground merry-go-round rotating at \( 3 \, \text{m/s} \) with a radius of \( 1.5 \, \text{m} \) spins at \( 2 \, \text{rad/s} \).
These calculations illustrate how to translate linear motion into angular velocity, which is crucial for understanding rotational dynamics in everyday contexts.
