Angular acceleration is the rate at which angular velocity changes over time. This Angular Acceleration Calculator computes angular acceleration from the initial angular velocity, final angular velocity, and elapsed time, returning results in your chosen unit system.
You’ll also learn the core formula, how to avoid unit mistakes, and how to apply the result to real motion problems like rotating wheels, motors, and drills.
What Is Angular Acceleration?
Angular acceleration (usually written α) describes how quickly an object’s rotation speeds up or slows down. If angular velocity increases, α is positive. If angular velocity decreases, α is negative.
In everyday terms, angular acceleration tells you how aggressively a rotating system changes its spin rate over time.
Core Formula (The One You Use Every Time)
The relationship between angular acceleration, angular velocity, and time is:
α = (ωf − ωi) / t
- α = angular acceleration
- ωi = initial angular velocity
- ωf = final angular velocity
- t = time interval
Make sure you use consistent units for ω (angular velocity) and t (time). The calculator handles conversions for you.
Understanding Units (Radians vs. Degrees)
Angular velocity is commonly expressed in:
- rad/s (radians per second)
- deg/s (degrees per second)
Angular acceleration is then expressed in:
- rad/s² (radians per second squared)
- deg/s² (degrees per second squared)
The conversion between degrees and radians is:
1 rad = 57.2958 deg
So, when you switch between deg/s and rad/s, the acceleration result must also switch accordingly.
How to Use the Angular Acceleration Calculator
Enter the three required inputs and select the units you want to use. The calculator returns angular acceleration and verifies that time is valid.
- Initial angular velocity (ωi)
- Final angular velocity (ωf)
- Time interval (t)
Then choose output units (rad/s² or deg/s²). The result is computed using the same physical formula.
Practical Example 1: Motor Spin-Up
A small motor speeds up from 120 rad/s to 180 rad/s in 0.50 s. Find the angular acceleration.
- ωi = 120 rad/s
- ωf = 180 rad/s
- t = 0.50 s
Compute:
α = (180 − 120) / 0.50 = 60 / 0.50 = 120 rad/s²
This means the motor’s rotational speed increases by 120 radians per second every second.
Practical Example 2: Slowing a Rotating Disc
A rotating disc slows down from 300 deg/s to 60 deg/s over 2.0 s. Determine the angular acceleration and its sign.
- ωi = 300 deg/s
- ωf = 60 deg/s
- t = 2.0 s
Compute:
α = (60 − 300) / 2.0 = (−240) / 2.0 = −120 deg/s²
The negative sign shows the disc is decelerating (angular velocity is dropping).
Common Mistakes (And How to Avoid Them)
- Mixing degrees and radians: If ω is in deg/s but you interpret it as rad/s, your result will be off by a factor of about 57.3.
- Using the wrong time unit: 500 ms is not 500 s. Convert to seconds before using the formula.
- Forgetting the sign: If ωf < ωi, the acceleration must be negative.
The calculator prevents most of these errors by letting you select units and by rejecting invalid time values.
Frequently Asked Questions
What is the formula for angular acceleration?
Angular acceleration is the change in angular velocity divided by time: α = (ωf − ωi) / t. ωi is the starting angular velocity, ωf is the ending angular velocity, and t is the elapsed time. The sign shows speeding up or slowing down.
What units does angular acceleration use?
Angular acceleration uses squared time units paired with angular velocity units. If angular velocity is in rad/s, angular acceleration is in rad/s². If angular velocity is in deg/s, angular acceleration is in deg/s². Consistent units are required for correct results.
Can angular acceleration be negative?
Yes. Negative angular acceleration occurs when the angular velocity decreases over time, meaning ωf is smaller than ωi. A negative value indicates deceleration. The magnitude tells you how strong the slowing effect is, regardless of direction.
How do I convert between degrees and radians?
Use 1 rad = 57.2958 deg, or equivalently 1 deg = 0.0174533 rad. When converting angular velocity, apply the same conversion to get rad/s or deg/s, then compute acceleration in the matching unit system. Mixing degrees and radians causes errors.
Is angular acceleration the same as linear acceleration?
No. Linear acceleration describes changes in straight-line speed, while angular acceleration describes changes in rotational speed. They are linked through radius: a = αr for motion with constant radius r. Angular acceleration is measured in rad/s² or deg/s².
Bottom Line
Angular acceleration is computed directly from angular velocity change over time using α = (ωf − ωi) / t. With correct unit choices, you can confidently calculate how fast a rotating system speeds up or slows down.