Angular Acceleration Calculator
Compute angular acceleration α from velocity change, torque, or linear acceleration using three physics methods.
Select a calculation method, enter the required values, and instantly obtain the angular acceleration in rad/s².
Angular Acceleration Calculator
Compute angular acceleration α from velocity change, torque, or linear acceleration using three physics methods.
About the Angular Acceleration Calculator
Angular acceleration is the rate at which an object's angular velocity changes with time. It plays the same role in rotational motion that linear acceleration plays in translational motion. Denoted by the Greek letter α (alpha), it is measured in radians per second squared (rad/s²).
This calculator offers three methods to determine angular acceleration, each suited to different scenarios. The first method uses the kinematic relationship α = (ω − ω₀) / t, where ω₀ is the initial angular velocity, ω is the final angular velocity, and t is the elapsed time. This is the most direct approach when you have measured or specified the angular velocities at two moments in time and know how long the change took.
The second method applies Newton's second law for rotation: α = τ / I, where τ (tau) is the net torque applied to the rotating object and I is its moment of inertia. This is the rotational analogue of F = ma. The moment of inertia depends on both the mass distribution and the axis of rotation — it can be calculated from geometry for simple shapes such as solid discs, hollow cylinders, rods, and spheres, or measured experimentally for complex assemblies.
The third method converts linear acceleration to angular acceleration using the relationship α = a / r, where a is the tangential linear acceleration of a point on the rotating body and r is the perpendicular distance from the rotation axis to that point. This is useful when you can measure or compute the linear acceleration of a specific point on the rotating system, for example a point on the rim of a wheel.
Angular acceleration appears in many engineering and physics contexts: the spin-up of electric motors, the braking of flywheels, the manoeuvring of spacecraft attitude-control systems, the dynamics of gyroscopes, and the analysis of gear trains. Understanding and controlling angular acceleration is essential wherever rotational motion must start, stop, or change speed in a predictable, controlled way.
All three formulae assume a rigid body rotating about a fixed axis, and ignore relativistic effects and aerodynamic drag unless those are already incorporated in the torque value you supply. For variable torque or time-varying inertia, calculus-based integration is required.
Angular Acceleration Examples
Three worked examples illustrating each calculation method.
| Input | Result | Notes |
|---|---|---|
| Carousel: ω₀ = 0 rad/s, ω = 2.0 rad/s, t = 5 s | α = 0.4 rad/s² | Method: From Angular Velocities. α = (2.0 − 0) / 5 = 0.4 rad/s². |
| Flywheel: τ = 100 N·m, I = 25 kg·m² | α = 4 rad/s² | Method: From Torque & Inertia. α = 100 / 25 = 4 rad/s². |
| Wheel point: a = 3.0 m/s², r = 0.5 m | α = 6 rad/s² | Method: From Linear Acceleration. α = 3.0 / 0.5 = 6 rad/s². |
How to use the Angular Acceleration Calculator
- Select the calculation method from the dropdown: 'From Angular Velocities', 'From Torque & Inertia', or 'From Linear Acceleration'.
- For the velocity method, enter the initial angular velocity ω₀ (rad/s), final angular velocity ω (rad/s), and elapsed time t (s).
- For the torque method, enter the net torque τ (N·m) and the moment of inertia I (kg·m²).
- For the linear method, enter the tangential linear acceleration a (m/s²) and the radius r (m) from the rotation axis.
- Click Calculate to see the angular acceleration α in rad/s². Click Reset to clear all inputs.
Angular Acceleration FAQ
What is angular acceleration?
Angular acceleration α is the rate of change of angular velocity with time, measured in rad/s². It is the rotational equivalent of linear acceleration and follows Newton's second law for rotation: α = τ / I.
What is the difference between angular velocity and angular acceleration?
Angular velocity ω (rad/s) describes how fast an object is rotating. Angular acceleration α (rad/s²) describes how quickly that rotation rate is changing. A constant ω means zero α; a changing ω means non-zero α.
How is angular acceleration related to linear acceleration?
For a point at radius r from the rotation axis, tangential linear acceleration a = α × r. Centripetal acceleration also exists (directed inward) and equals ω² × r, but that is not caused by angular acceleration.
What units is angular acceleration expressed in?
Angular acceleration is expressed in radians per second squared (rad/s²). Since radians are dimensionless, this is equivalent to s⁻². In some engineering contexts you may see rev/min² (RPM/s), which can be converted: 1 RPM/s = π/30 rad/s².
How do I find the moment of inertia I?
For a solid disc: I = ½mr². For a solid sphere: I = ⅖mr². For a thin ring: I = mr². For complex assemblies, use the parallel-axis theorem or measure experimentally with a torsion pendulum setup.
Can angular acceleration be negative?
Yes. Negative angular acceleration (also called angular deceleration) means the object is slowing its rotation. The sign depends on the chosen positive direction of rotation — typically counterclockwise is positive by convention in 2D problems.