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Angular Acceleration Calculator

Find how quickly a rotating object's angular velocity changes over time.

How it is calculated

The tool computes α = (ωf − ωi) ÷ t. Here ωi and ωf are the starting and ending angular velocities in radians per second, and t is the time in seconds. The result α is in rad/s², and the tool also shows the change Δω = ωf − ωi in rad/s. Both are rounded to two decimals. The value is signed: a positive α means the spin is speeding up in the chosen direction, and a negative α means it is slowing down. Time must be above zero; zero or negative time gives no result. The tool does not convert units and it treats the change in speed as spread evenly over the interval, so the result is an average, not an instantaneous value.

Worked example

A shaft turns at 4 rad/s and reaches 31.4 rad/s after 6 s. Change in angular velocity: 31.4 − 4 = 27.40 rad/s. Divide by the time: 27.40 ÷ 6 = 4.5667, shown as 4.57 rad/s². The tool displays an angular acceleration of 4.57 rad/s² and a change of 27.40 rad/s. Run the other way, from 30 to 0 rad/s in 5 s, the result is −6.00 rad/s².

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Common mistakes

  • Entering speeds in RPM or degrees per second. The tool expects rad/s, so a motor going from 0 to 3000 RPM must first be converted: 3000 RPM is about 314.16 rad/s.
  • Entering time in minutes or milliseconds. The result is in rad/s², so the time has to be in seconds.
  • Dropping the sign. A spin-down from 30 to 0 rad/s gives −6.00 rad/s², and that is deceleration, not an error.
  • Swapping initial and final velocities. The sign of α flips, so a speeding-up wheel appears to slow down.
  • Treating the answer as the acceleration at every moment. It is the average over the interval, and real motors rarely accelerate evenly.

About this calculator

Angular acceleration (α) measures how quickly a rotating object's angular velocity (ω) changes over time, expressed as α = (ω_f − ω_i) / t and measured in radians per second squared. It is the rotational counterpart of linear acceleration in the same way angular velocity is the counterpart of linear velocity, and it appears throughout rotational kinematics wherever something spins up, slows down, or changes its rate of rotation — a motor shaft ramping up to operating speed, a centrifuge accelerating to its target RPM, a wheel skidding to a stop, or a figure skater's spin speeding up as they pull their arms in.

Mechanical and electrical engineers use angular acceleration to size motors and drivetrains (since torque relates to angular acceleration through a body's moment of inertia via τ = Iα), physics students use it as the rotational analogue of the standard SUVAT kinematics equations, and robotics and animation software use it to model realistic joint and rotor motion.

The change in angular velocity divided by the time it took: enter the starting and final angular velocity and the interval to get the angular acceleration.

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