Angular rotation measures the angle through which an object rotates. To determine the change in angular rotation (\(\theta\)) when the initial angular velocity (\(\omega_1\)), final angular velocity (\(\omega_2\)), and time interval (\(t\)) are known, use the formula derived from the relationship between angular acceleration and angular velocity.
Formula for Change in Angular Rotation
The change in angular rotation (\(\theta\)) can be calculated using the following formula:
\[ \theta = \dfrac{\omega_1 + \omega_2}{2} \cdot t \]
Where:
- \(\theta\) is the change in angular rotation (in radians).
- \(\omega_1\) is the initial angular velocity (in radians per second).
- \(\omega_2\) is the final angular velocity (in radians per second).
- \(t\) is the time interval (in seconds).
This formula is derived from the kinematic equation for rotation:
\[ \theta = \omega_1 \cdot t + \dfrac{1}{2} \cdot \alpha \cdot t^2 \]
where \(\alpha\) is the angular acceleration. Since the average angular velocity over time is \(\dfrac{\omega_1 + \omega_2}{2}\), we simplify to:
\[ \theta = \left( \dfrac{\omega_1 + \omega_2}{2} \right) \cdot t \]
Example 1: Calculating Change in Angular Rotation with Initial Velocity Zero
Let's calculate the change in angular rotation when the initial angular velocity is zero.
Given:
- Initial angular velocity \( \omega_1 = 0 \, \text{rad/s} \)
- Final angular velocity \( \omega_2 = 2 \, \text{rad/s} \)
- Time interval \( t = 4 \, \text{s} \)
Step-by-Step Calculation:
Step 1: Substitute the Values into the Angular Rotation Formula
\[ \theta = \dfrac{\omega_1 + \omega_2}{2} \cdot t \]
\[ \theta = \dfrac{0 + 2}{2} \cdot 4 \]
\[ \theta = \dfrac{2}{2} \cdot 4 \]
\[ \theta = 1 \cdot 4 \]
Final Value
The change in angular rotation is:
\[ \theta = 4 \, \text{radians} \]
Example 2: Calculating Change in Angular Rotation with Initial and Final Non-Zero Velocities
Let's consider a scenario with non-zero initial and final angular velocities.
Given:
- Initial angular velocity \( \omega_1 = 3 \, \text{rad/s} \)
- Final angular velocity \( \omega_2 = 7 \, \text{rad/s} \)
- Time interval \( t = 5 \, \text{s} \)
Step-by-Step Calculation:
Step 1: Substitute the Values into the Angular Rotation Formula
\[ \theta = \dfrac{\omega_1 + \omega_2}{2} \cdot t \]
\[ \theta = \dfrac{3 + 7}{2} \cdot 5 \]
\[ \theta = \dfrac{10}{2} \cdot 5 \]
\[ \theta = 5 \cdot 5 \]
Final Value
The change in angular rotation is:
\[ \theta = 25 \, \text{radians} \]
Example 3: Calculating Change in Angular Rotation with Decreasing Angular Velocity
Let's consider a scenario where the angular velocity decreases over time.
Given:
- Initial angular velocity \( \omega_1 = 10 \, \text{rad/s} \)
- Final angular velocity \( \omega_2 = 4 \, \text{rad/s} \)
- Time interval \( t = 6 \, \text{s} \)
Step-by-Step Calculation:
Step 1: Substitute the Values into the Angular Rotation Formula
\[ \theta = \dfrac{\omega_1 + \omega_2}{2} \cdot t \]
\[ \theta = \dfrac{10 + 4}{2} \cdot 6 \]
\[ \theta = \dfrac{14}{2} \cdot 6 \]
\[ \theta = 7 \cdot 6 \]
Final Value
The change in angular rotation is:
\[ \theta = 42 \, \text{radians} \]
Summary
To find the change in angular rotation (\(\theta\)), use the formula:
\[ \theta = \dfrac{\omega_1 + \omega_2}{2} \cdot t \]
where:
- \(\omega_1\) is the initial angular velocity.
- \(\omega_2\) is the final angular velocity.
- \(t\) is the time interval.
In the provided examples:
- When initial angular velocity is \(0 \, \text{rad/s}\) and final angular velocity is \(2 \, \text{rad/s}\) over \(4 \, \text{s}\), \(\theta = 4 \, \text{radians}\).
- For initial \(3 \, \text{rad/s}\) and final \(7 \, \text{rad/s}\) over \(5 \, \text{s}\), \(\theta = 25 \, \text{radians}\).
- With decreasing angular velocity from \(10 \, \text{rad/s}\) to \(4 \, \text{rad/s}\) over \(6 \, \text{s}\), \(\theta = 42 \, \text{radians}\).
This method provides a straightforward approach to measure angular displacement over a given time interval, useful for applications in rotational motion analysis.
