Angular Acceleration Calculator

How to use this calculator

Five inputs, one click.

Initial Angular Velocity (ω₀): The rotational speed at the start of the time interval. Enter 0 if the object begins from rest.

Final Angular Velocity (ω): The rotational speed at the end of the interval. Use the same unit as ω₀.

Time (t): The duration over which the change in angular velocity occurs. Must be greater than zero.

Radius (r): Optional. The distance from the rotation axis to the point of interest. Required only if you want to calculate tangential acceleration.

Input unit: Choose rad/s or RPM. The calculator converts RPM to rad/s automatically before computing.

Click Calculate to see angular acceleration (α in rad/s²), tangential acceleration, angular displacement, and a chart of angular velocity over the time interval.

Rotational vs linear kinematics: the parallel equations

One of the most elegant features of classical mechanics is the structural symmetry between linear and rotational motion. Every linear kinematic equation has a direct rotational counterpart. The table below shows them side by side.

Linear quantity	Symbol	Rotational counterpart	Symbol
Displacement	x (m)	Angular displacement	θ (rad)
Velocity	v (m/s)	Angular velocity	ω (rad/s)
Acceleration	a (m/s²)	Angular acceleration	α (rad/s²)
Mass	m (kg)	Moment of inertia	I (kg·m²)
Force	F (N)	Torque	τ (N·m)

The kinematic equations transform the same way:

Linear	Rotational
v = v₀ + at	ω = ω₀ + αt
x = v₀t + ½at²	θ = ω₀t + ½αt²
v² = v₀² + 2ax	ω² = ω₀² + 2αθ

This symmetry means that any problem-solving technique that works for linear kinematics also works for rotational kinematics, with variable substitutions. Students who master one automatically understand the other.

Defining angular acceleration

Angular acceleration α is the rate of change of angular velocity with respect to time:

The SI unit is radians per second squared (rad/s²). The radian is a dimensionless unit (it is the ratio of arc length to radius), so rad/s² is equivalently s⁻².

Angular acceleration is a vector quantity. Its direction is along the rotation axis, given by the right-hand rule: curl the fingers of your right hand in the direction of rotation and your thumb points along the angular velocity vector. Angular acceleration either points the same way (speeding up) or opposite (slowing down).

For constant α, the exact relationship between angle and time is the rotational kinematic equation:

where θ is measured in radians. One full revolution equals 2π radians, so if θ = 6.28 rad, the object rotated once.

Tangential and centripetal acceleration in rotating systems

When a point is located at radius r from the rotation axis, its motion has two acceleration components:

Tangential acceleration results from angular acceleration and acts along the direction of motion:

This changes the speed of the point (faster or slower rotation).

Centripetal acceleration results from the circular path and points toward the center:

This changes the direction of motion (keeps the point on its circular path) without affecting speed.

The total acceleration of a point on a rotating object is the vector sum of these two components. For a rim point at radius r on a wheel with angular acceleration α and angular velocity ω:

Worked examples

Spinning bicycle wheel slowing to a stop

A bicycle wheel spinning at 120 RPM decelerates uniformly to rest in 8 seconds. Find α.

Motor startup with known torque

An electric motor has a moment of inertia I = 0.08 kg·m² and applies a net torque of 4 N·m. Find α and the time to reach 1800 RPM.

Earth’s rotation: negligible angular acceleration

Earth’s rotation is very slightly slowing due to tidal friction. The day lengthens by about 1.7 milliseconds per century.

Angular acceleration in engineering

Electric motors and generators

Every time a motor starts, stops, or changes speed it undergoes angular acceleration. Engineers specify the acceleration torque: τ = I × α, where I is the moment of inertia of the rotor and all connected rotating parts (load, coupling, gearbox input shaft). High-performance servo motors in CNC machines and robots need large angular acceleration to follow rapidly changing position commands.

For a motor driving a load through a gearbox with ratio n, the equivalent inertia seen at the motor shaft is:

Choosing the right gear ratio optimally matches the load inertia to the motor inertia, maximizing angular acceleration for a given torque.

Gas turbines and flywheels

Turbine rotors operating at 10,000-30,000 RPM have very large angular momentum. Their angular acceleration during spool-up and spool-down is carefully controlled to manage thermal stresses from differential expansion of the rotor and casing. Rapid angular acceleration would cause the rotor to grow faster than the casing, reducing tip clearances and potentially causing rubs.

Flywheels store energy as rotational kinetic energy E = ½Iω². During charging (spinning up), the electric motor provides positive α. During discharge (braking), the flywheel provides torque to the load and α is negative. The power delivered is:

Gyroscopes and angular momentum

A gyroscope spinning at high angular velocity ω has angular momentum L = Iω directed along the spin axis. When a torque τ is applied perpendicular to L, instead of tilting the axis, the gyroscope precesses: the spin axis rotates around the direction of the applied torque.

The precession angular velocity is:

This is why a spinning top doesn’t fall over immediately: gravity provides a horizontal torque, but instead of tipping, the top precesses around the vertical. The faster the spin, the slower the precession.

In aircraft attitude indicators, the gyroscope spin axis stays aligned with a fixed direction in space (because any disturbance torque causes slow precession rather than immediate tilting). This provides a stable reference for pitch and roll.

MEMS gyroscopes in phones work differently: they use a vibrating mass and the Coriolis effect to measure angular velocity, not angular acceleration directly. To get angular acceleration from a MEMS gyro, you differentiate the ω signal electronically.

Converting between linear and rotational quantities

For any point at radius r from the rotation axis:

Quantity	Formula	Units
Arc length	s = rθ	m
Tangential velocity	v = rω	m/s
Tangential acceleration	aₜ = rα	m/s²
Centripetal acceleration	a_c = rω²	m/s²

Converting RPM to rad/s: multiply by 2π/60 ≈ 0.10472.

Converting rev/s (Hz) to rad/s: multiply by 2π ≈ 6.2832.

Converting degrees per second to rad/s: multiply by π/180 ≈ 0.017453.

Practical measurement methods

Optical encoders attach a slotted disc to the rotating shaft. An LED and photodetector count pulses as the shaft turns. Dividing the pulse rate by the pulses-per-revolution gives angular velocity; differentiating this signal gives angular acceleration.

Resolver sensors produce sine and cosine outputs whose ratio tracks shaft angle continuously. They are more rugged than optical encoders and used in motors operating in dusty or hot environments. Differentiation of the angle signal gives ω and α.

Rate gyroscopes directly measure ω. The angular acceleration must be derived by numerical differentiation, which amplifies noise. In practice, engineers apply filters (Kalman or low-pass) to the differentiated signal.

Strain gauge torque transducers measure torque on a shaft. If the moment of inertia is known, α = τ_net / I. This approach measures average α over a rotation without needing fast angle data.

Accelerometers mounted off-axis at a known radius r can measure tangential acceleration directly (aₜ = αr), giving α = aₜ / r. This is a common approach in automotive wheel-speed sensors.

Angular acceleration in everyday machines

Most rotating machines start from rest and accelerate to an operating speed, then decelerate to stop. The angular acceleration during these phases determines how quickly the machine reaches operating conditions and how smoothly it handles load changes.

Electric motors: When an electric motor starts, it draws high current to generate the torque needed to accelerate the rotor from rest to operating speed. A motor spinning up to 1800 RPM (188.5 rad/s) in 2 seconds has an angular acceleration of approximately 94 rad/s². The motor’s moment of inertia determines how much torque is required: τ = I × α.

Hard disk drives: A hard drive spinning up from rest to 7200 RPM (754 rad/s) in about 8 seconds has an average angular acceleration of 94 rad/s² during spin-up. Drive controllers manage this acceleration to avoid mechanical stress on the bearing and platters.

Automotive engines: Engine angular acceleration when you press the accelerator is measured in RPM/second. A high-performance engine might rev from 1000 to 7000 RPM (6000 RPM change) in under 2 seconds: approximately 314 rad/s² average angular acceleration.

Understanding angular acceleration is essential for sizing motors, calculating starting torques, and designing control systems that prevent mechanical damage during rapid speed changes.