Angular Acceleration Calculator: Formula & Examples

This angular acceleration calculator works out the rate of change of angular velocity over time. You enter the initial angular velocity, final angular velocity, and the time interval, and it returns the average angular acceleration in radians per second squared (rad/s²). It also gives the equivalent in revolutions per second squared (rev/s²) for easier interpretation.

Formula

The average angular acceleration is defined as the change in angular velocity divided by the time over which that change occurs. In symbols:

α = (ω_final − ω_initial) / t

• α = average angular acceleration, in radians per second squared (rad/s²)
• ω_final = final angular velocity, in radians per second (rad/s)
• ω_initial = initial angular velocity, in radians per second (rad/s)
• t = time interval, in seconds (s)

If you prefer revolutions, divide the result by 2Ï€ because one full revolution equals 2Ï€ radians.

Worked Example

Using the default values from the calculator: initial angular velocity ω_initial = 0 rad/s, final angular velocity ω_final = 10 rad/s, and time t = 5 s.

1. Subtract the initial from the final angular velocity: 10 rad/s − 0 rad/s = 10 rad/s.
2. Divide by the time: 10 rad/s ÷ 5 s = 2 rad/s².
3. For revolutions: 2 rad/s² ÷ (2π) ≈ 0.32 rev/s².

The calculator shows 2.00 rad/s², matching the step-by-step result.

When to Use This Calculator

Use this tool whenever you need the average angular acceleration of a rotating object—a wheel, a turbine, a CD, a merry-go-round—given its starting and ending spin rates and the elapsed time. It is the rotational analog of linear acceleration (a = (v − u)/t). Typical scenarios include:

• Checking the acceleration of a car’s wheels from rest to a known speed.
• Estimating the angular acceleration of a flywheel in a machine.
• Solving physics homework problems that give initial and final angular velocities and time.

If the acceleration is not constant, this calculator gives the average over the interval, not the instantaneous value at any moment.

How to Read the Result

The output is the average rate at which angular velocity changes. A positive value means the object is spinning faster (or slowing down if it was already spinning in the opposite direction). A negative value means it is decelerating. The unit rad/s² tells you how many radians per second of angular velocity are gained (or lost) each second. For example, 2 rad/s² means the angular velocity increases by 2 rad/s every second.

To relate angular acceleration to linear (tangential) acceleration of a point at radius r from the axis, multiply: a_t = α × r. So a point 0.5 m from the center would have a tangential acceleration of 2 × 0.5 = 1 m/s².

Common Mistakes and Unit Mix-Ups

One frequent error is mixing degrees or revolutions with radians. The formula works only when angular velocities are in rad/s. If you have revolutions per minute (rpm), convert to rad/s first by multiplying by 2π/60 (since 1 rev = 2π rad and 1 min = 60 s). For example, 300 rpm = 300 × 2π/60 ≈ 31.42 rad/s.

Another mistake is forgetting that time must be in seconds. If your time is in minutes, convert to seconds before using the formula. Also, be careful with signs: if the object is slowing down, the angular acceleration will be negative when the angular velocity is positive. The calculator handles that automatically, but you must enter the correct sign for each velocity.

Finally, this is average acceleration. If the angular velocity changes non-uniformly, the average may differ from the instantaneous acceleration at any given moment. For instantaneous acceleration, you would need a derivative or a series of measurements.

Limitations of the Formula

This formula assumes the angular acceleration is constant over the time interval. If the acceleration varies, the average value is still useful, but it does not describe the motion at each instant. For non-constant acceleration, you would need calculus or a more detailed model. Also, the formula only considers the magnitude of the angular velocity change; it does not account for torque, moment of inertia, or forces that cause the acceleration—those are separate calculations.

Frequently Asked Questions

What is the difference between angular acceleration and centripetal acceleration?

Angular acceleration measures how fast the spin rate changes (rad/s²). Centripetal acceleration is the acceleration toward the center that keeps an object moving in a circle, equal to ω²r or v²/r. They are different: an object can have centripetal acceleration even if its angular velocity is constant (uniform circular motion), while angular acceleration is zero in that case.

How do I convert angular acceleration from rad/s² to rev/s²?

Divide by 2π. Since one revolution is 2π radians, 1 rad/s² = 1/(2π) rev/s² ≈ 0.159 rev/s². The calculator includes this conversion as a second output.

Can angular acceleration be negative?

Yes. A negative angular acceleration means the angular velocity is decreasing (if the initial velocity is positive) or increasing in the negative direction (if the object is spinning the other way). The sign depends on the chosen positive direction, which is usually counterclockwise.

What if the object starts from rest?

Then the initial angular velocity is 0 rad/s. You simply divide the final angular velocity by the time. For example, a wheel reaching 20 rad/s in 4 s has an angular acceleration of 20/4 = 5 rad/s².

How does angular acceleration relate to torque?

Newton’s second law for rotation is τ = Iα, where τ is torque, I is moment of inertia, and α is angular acceleration. So to find the torque needed to achieve a certain angular acceleration, multiply the moment of inertia by the angular acceleration. This calculator gives you α; you would need the object’s moment of inertia to compute torque.

Related

• Buoyant Force in Water: Formula and Calculation
• Half-Life Calculations: Formula and GCSE Examples
• Force Calculation Methods and Newton’s Second Law

in w0 | Initial angular velocity | rad/s | 0 | 0
in w1 | Final angular velocity | rad/s | 10 | 0
in t | Time interval | s | 5 | 0
out alpha | Angular acceleration | rad/s² | (w1 - w0) / t | 2
out alpha_rev | Angular acceleration | rev/s² | (w1 - w0) / (t * 2 * pi) | 2