How To Find Speed Using Angular Velocity

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.