Constant Angular Acceleration Calculator

What is Constant Angular Acceleration?

Constant angular acceleration is the rate at which an object's angular velocity changes over time, remaining constant throughout the motion. It occurs in rotational motion when the angular velocity of an object increases or decreases at a steady rate. This concept is fundamental in understanding the dynamics of rotating objects, such as wheels, gears, and celestial bodies.

Formulas

The key formulas for constant angular acceleration are:

1. \( \omega = \omega_0 + \alpha t \)
2. \( \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \)
3. \( \omega^2 = \omega_0^2 + 2\alpha \theta \)

Where:

• \( \omega_0 \) is the initial angular velocity (rad/s)
• \( \omega \) is the final angular velocity (rad/s)
• \( \alpha \) is the angular acceleration (rad/s²)
• \( t \) is the time (s)
• \( \theta \) is the angular displacement (rad)

Calculation Steps

Let's calculate the angular displacement using the second formula:

1. Given:
   • Initial angular velocity (\( \omega_0 \)) = 2 rad/s
   • Angular acceleration (\( \alpha \)) = 0.5 rad/s²
   • Time (\( t \)) = 4 s
2. Apply the formula: \[ \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \]
3. Substitute the known values: \[ \theta = (2 \text{ rad/s})(4 \text{ s}) + \frac{1}{2}(0.5 \text{ rad/s}^2)(4 \text{ s})^2 \]
4. Perform the calculation: \[ \theta = 8 \text{ rad} + 4 \text{ rad} = 12 \text{ rad} \]

Example and Visual Representation

Let's visualize constant angular acceleration:

This diagram illustrates:

• The circular path of the rotating object (blue circle)
• The initial angular velocity (\( \omega_0 \)) as a green arrow
• The constant angular acceleration (\( \alpha \)) as a red dashed curved arrow
• The angular displacement (\( \theta \)) as the yellow shaded sector