How To Find The Change In Angular Rotation

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:

$$\theta ={\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}\cdot t$$

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:

$$\theta ={\omega }_1\cdot t+{\displaystyle \frac{1}{2}}\cdot \alpha \cdot t^2$$

where $\alpha$ is the angular acceleration. Since the average angular velocity over time is ${\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}$, we simplify to:

$$\theta =\left({\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}\right)\cdot t$$

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

$$\theta ={\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}\cdot t$$

$$\theta ={\displaystyle \frac{0+2}{2}}\cdot 4$$

$$\theta ={\displaystyle \frac{2}{2}}\cdot 4$$

$$\theta =1\cdot 4$$

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

$$\theta ={\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}\cdot t$$

$$\theta ={\displaystyle \frac{3+7}{2}}\cdot 5$$

$$\theta ={\displaystyle \frac{10}{2}}\cdot 5$$

$$\theta =5\cdot 5$$

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

$$\theta ={\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}\cdot t$$

$$\theta ={\displaystyle \frac{10+4}{2}}\cdot 6$$

$$\theta ={\displaystyle \frac{14}{2}}\cdot 6$$

$$\theta =7\cdot 6$$

Final Value

The change in angular rotation is:

$$\theta =42\text{radians}$$

Summary

To find the change in angular rotation ($\theta$), use the formula:

$$\theta ={\displaystyle \frac{{\omega }_1+{\omega }_2}{2}}\cdot t$$

where:

• ${\omega }_1$ is the initial angular velocity.
• ${\omega }_2$ is the final angular velocity.
• $t$ is the time interval.

In the provided examples:

1. When initial angular velocity is $0\text{rad/s}$ and final angular velocity is $2\text{rad/s}$ over $4\text{s}$, $\theta =4\text{radians}$.
2. For initial $3\text{rad/s}$ and final $7\text{rad/s}$ over $5\text{s}$, $\theta =25\text{radians}$.
3. 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.