Angular velocity measures how quickly an object rotates around a central point. It’s expressed in radians per second (rad/s). This article will demonstrate how to calculate angular velocity using the angle of rotation and time, with two detailed examples.
Formula for Angular Velocity
The formula for angular velocity (\( \omega \)) is:
\[ \omega = \dfrac{\theta}{t} \]
Where:
- \( \omega \) is the angular velocity in radians per second (\(\text{rad/s}\)).
- \( \theta \) is the angle of rotation in radians (\(\text{rad}\)).
- \( t \) is the time in seconds (\(\text{s}\)).
Converting Degrees to Radians
If the angle \(\theta\) is given in degrees, convert it to radians using:
\[ \theta_{\text{rad}} = \theta_{\text{deg}} \cdot \dfrac{\pi}{180} \]
Where:
- \(\theta_{\text{rad}}\) is the angle in radians.
- \(\theta_{\text{deg}}\) is the angle in degrees.
Example 1: Calculating Angular Velocity for an Initial Rotation
Let’s calculate the angular velocity for a specific angle of rotation over a given time.
Given:
- Angle of rotation \( \theta = 90^\circ \)
- Time \( t = 2 \, \text{s} \)
Step-by-Step Calculation:
Step 1: Convert the Angle from Degrees to Radians
\[ \theta_{\text{rad}} = 90 \cdot \dfrac{\pi}{180} = \dfrac{\pi}{2} \, \text{rad} \]
Since \(\pi \approx 3.1416\):
\[ \theta_{\text{rad}} = \dfrac{3.1416}{2} = 1.5708 \, \text{rad} \]
Step 2: Substitute the Values into the Angular Velocity Formula
\[ \omega = \dfrac{\theta}{t} = \dfrac{1.5708}{2} = 0.7854 \]
Final Value
The angular velocity is:
\[ \omega = 0.7854 \, \text{rad/s} \]
Example 2: Calculating Angular Velocity for a Larger Rotation and Different Time
Now, consider a different scenario with a larger angle and time.
Given:
- Angle of rotation \( \theta = 180^\circ \)
- Time \( t = 5 \, \text{s} \)
Step-by-Step Calculation:
Step 1: Convert the Angle from Degrees to Radians
\[ \theta_{\text{rad}} = 180 \cdot \dfrac{\pi}{180} = \pi \, \text{rad} \]
Since \(\pi \approx 3.1416\):
\[ \theta_{\text{rad}} = 3.1416 \, \text{rad} \]
Step 2: Substitute the Values into the Angular Velocity Formula
\[ \omega = \dfrac{\theta}{t} = \dfrac{3.1416}{5} = 0.6283 \]
Final Value
The angular velocity is:
\[ \omega = 0.6283 \, \text{rad/s} \]
Summary
To find the angular velocity (\( \omega \)), use the formula:
\[ \omega = \dfrac{\theta}{t} \]
If the angle is given in degrees, convert it to radians first:
\[ \theta_{\text{rad}} = \theta_{\text{deg}} \cdot \dfrac{\pi}{180} \]
Then apply the angular velocity formula. In the examples provided:
- For an angle of \(90^\circ\) over 2 seconds, the angular velocity is \(0.7854\) rad/s.
- For an angle of \(180^\circ\) over 5 seconds, the angular velocity is \(0.6283\) rad/s.
This approach provides a clear measure of rotational speed in radians per second, essential for various applications in physics and engineering.
