Angular Acceleration Calculator
Calculate angular acceleration from angular velocity, torque, or tangential acceleration. Supports rad/s², deg/s², RPM/s with real-time unit conversion.
About
Angular acceleration α quantifies the rate at which an object's angular velocity changes over time. A miscalculated α in rotating machinery design leads to bearing failure, shaft fatigue, or resonance-induced catastrophic breakdown. This calculator computes α through four independent methods: direct angular velocity change (Δω ÷ Δt), torque-inertia ratio (τ ÷ I), initial/final angular velocity differencing, and tangential acceleration decomposition (at ÷ r). Results are output in rad/s2, deg/s2, rev/s2, and RPM/s simultaneously.
The tool assumes rigid-body rotation about a fixed axis with constant angular acceleration over the specified interval. It does not model variable-torque profiles, elastic deformation, or multi-axis precession. For geared systems, input the effective moment of inertia reflected to the shaft of interest. Pro tip: when working with electric motors, remember that rotor inertia published in datasheets is typically the rotor alone. Add the reflected load inertia before computing required torque.
Formulas
Angular acceleration is the time derivative of angular velocity. Four calculation methods are implemented.
Method 1 - Angular Velocity Change:
α = ΔωΔtMethod 2 - Newton's Second Law for Rotation:
α = τIMethod 3 - Initial & Final Angular Velocity:
α = ω2 − ω1ΔtMethod 4 - Tangential Acceleration:
α = atrWhere α = angular acceleration (rad/s2), Δω = change in angular velocity (rad/s), Δt = time interval (s), τ = net torque (N⋅m), I = moment of inertia (kg⋅m2), ω1 = initial angular velocity (rad/s), ω2 = final angular velocity (rad/s), at = tangential linear acceleration (m/s2), r = radius of rotation (m).
Unit conversion factors: 1 rev = 2π rad = 360°. Therefore 1 RPM/s = 2π60 rad/s2 ≈ 0.10472 rad/s2.
Reference Data
| Object / System | Typical Moment of Inertia I | Typical α | Application Context |
|---|---|---|---|
| Bicycle wheel | 0.1 kg⋅m2 | 2 - 5 rad/s2 | Pedal start from rest |
| Car engine crankshaft | 0.15 - 0.25 kg⋅m2 | 50 - 200 rad/s2 | Throttle response |
| Industrial flywheel | 50 - 500 kg⋅m2 | 0.5 - 5 rad/s2 | Energy storage spin-up |
| Ceiling fan | 0.3 - 0.8 kg⋅m2 | 1 - 3 rad/s2 | Speed change between settings |
| Hard disk platter (3.5") | 6.5 × 10−4 kg⋅m2 | 300 - 600 rad/s2 | Spin-up to 7200 RPM |
| Wind turbine rotor | 1 × 107 kg⋅m2 | 0.01 - 0.05 rad/s2 | Start-up in rated wind |
| Figure skater (spin) | 0.5 - 3.0 kg⋅m2 | 5 - 15 rad/s2 | Arms pull-in acceleration |
| Washing machine drum | 0.3 - 1.5 kg⋅m2 | 10 - 30 rad/s2 | Spin cycle ramp-up |
| Gyroscope (navigation grade) | 1 × 10−5 kg⋅m2 | 1000 - 5000 rad/s2 | Rapid spin-up |
| Earth (rotation) | 8.04 × 1037 kg⋅m2 | −5.4 × 10−22 rad/s2 | Tidal deceleration |
| Centrifuge (lab) | 0.005 - 0.05 kg⋅m2 | 500 - 2000 rad/s2 | Sample separation spin-up |
| Propeller (small aircraft) | 1 - 5 kg⋅m2 | 10 - 50 rad/s2 | Engine start to idle |
| CD/DVD disc | 2 × 10−4 kg⋅m2 | 100 - 400 rad/s2 | Variable-speed read |
| Turbine rotor (gas) | 5 - 50 kg⋅m2 | 20 - 100 rad/s2 | Power plant start-up |
| Pottery wheel | 2 - 8 kg⋅m2 | 0.5 - 3 rad/s2 | Kick-start spin |