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Angular Acceleration Calculator

Created by Dominik Czernia, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Jan 18, 2024


The angular acceleration calculator helps you find the angular acceleration of an object that rotates or moves around a circle. As you will soon see, the angular acceleration formula differs from the acceleration in linear motion, which you probably know very well.

Read on if you want to learn what are the angular acceleration units and what is the angular acceleration equation. You will find out that you can compute it with our angular acceleration calculator in two different ways.

You can also check our centrifugal force calculator, which is dedicated to circular motion too.

Angular acceleration definition

The rotational movement of an object is usually described by a physical quantity called angular velocity. It measures the angle by which an object rotates in a specific time. For example, imagine that a carousel in an amusement park performs full rotation within ten seconds. Its angular velocity is one rotation (360°) per ten seconds or 36° per second. See our angular velocity calculator to learn more about angular velocity.

Let's assume that our carousel starts to rotate faster and faster, not 36° but 50°, then 64° per second. Angular acceleration describes this rate of change of angular velocity and is caused by torque.

In the next section, we will see how to find the rate of change of angular velocity, i.e., the angular acceleration.

💡 Not sure what torque is? Our torque calculator can help explain.

Angular acceleration formula

Angular acceleration can be computed with our angular acceleration calculator in two different ways. We are using the below angular acceleration equations:

α=ω2ω1t,or:α=aR,\alpha = \frac{\omega_2 - \omega_1}{t},\quad \text{or:}\quad \alpha = \frac{a}{R},

where:

  • α\alpha – Angular acceleration;
  • ω1\omega_1 – Initial angular velocity;
  • ω1\omega_1 – Final angular velocity;
  • tt – Time of change of angular velocity;
  • aa – Tangential acceleration; and
  • RR – Radius of the circle (or the distance from an axis of rotation).

Tangential acceleration acts as a linear acceleration, which is perpendicular to the radius of the circle.

🔎 When an angular velocity is a scalar (not a vector), we should call it angular speed or angular frequency. Check out our angular frequency calculator to learn more about it!

Angular acceleration units

We can use several different units to express angular acceleration:

  • The most common are units of angle per second squared (e.g., rad/s², °/s²). This unit nicely illustrates the meaning of angular acceleration since the linear acceleration is expressed in m/s² or ft/s².

  • Sometimes, we omit the numerator leaving only the 1/s².

  • Since angular velocity can be expressed in hertz Hz = 1/s, we can also use this in angular acceleration receiving Hz/s. We have used this convention in our angular acceleration calculator.

The conversion between the above angular acceleration units is as follows rad/s² = 1/s² = Hz/s.

FAQ

What's the angular acceleration formula?

The angular acceleration formula is either:

α = (ω2 - ω1) / t

Where ω2 and ω1 are the angular velocities at the final and initial times, respectively, and t is the time interval. You can use this formula when you know the initial and final angular velocities and time.

Alternatively, you can use the following:

α = a / R

when you know the tangential acceleration a and radius R.

How do I find angular acceleration from torque?

To calculate the angular acceleration from torque:

  1. Use the formula:

    α = 𝜏/I

    Where 𝜏 is the torque and I is the moment of inertia.

  2. Substitute the given values into the formula and solve for α.

This will provide you with the angular acceleration of the object.

What's the angular velocity of a wheel with a constant 3.0 rad/s^2 acceleration?

After 8 seconds, the angular velocity of the wheel is 24 rad/s, starting from an initial angular velocity of zero. Here's how to calculate it:

  1. Use the formula for angular acceleration in terms of angular velocity:

    α = (ω2 - ω1) / t

    Where ω2 and ω1 are the final and initial angular velocities, respectively, and t is the time interval.

  2. Solve for ω2:

    ω2 = αt + ω1

  3. Substitute the known values:

    ω2 = 3.0 rad/s2 × 8 s + 0 rad/s
    ω2 = 24 rad/s

What's the difference between angular acceleration and linear acceleration?

Angular acceleration refers to the rate at which an object's rotational speed changes over time, and it's usually measured in units of radians per second squared (rad/s2). It is a measure of rotational motion.

In contrast, linear acceleration refers to the rate of change in an object's speed over time in a straight-line motion and is measured in units of distance per time squared (i.e., m/s2, ft/s2).

Dominik Czernia, PhD
How would you like to calculate angular acceleration?
Method
Angular velocity difference
Angular velocity difference
Initial angular velocity
rad/s
Final angular velocity
rad/s
Time
sec
Angular acceleration
rad
/s²
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