What is angular acceleration?

Angular acceleration is the rate at which an object's angular velocity changes over time. It is denoted by the Greek letter alpha (α) and measured in radians per second squared (rad/s²). A positive value means the object is spinning faster; a negative value (deceleration) means it is slowing down. The formula is α = Δω / t, where Δω is the change in angular velocity in rad/s and t is the elapsed time in seconds.

How do you calculate angular acceleration from torque?

Angular acceleration can also be found using Newton's second law for rotation: α = τ / I, where τ (tau) is the net torque applied in newton-metres (N·m) and I is the moment of inertia of the object in kilogram-metres squared (kg·m²). This is the rotational equivalent of the linear formula a = F / m. For example, a net torque of 50 N·m applied to a flywheel with a moment of inertia of 10 kg·m² produces an angular acceleration of 5 rad/s².

What is the difference between angular velocity and angular acceleration?

Angular velocity (ω) describes how fast an object is rotating at a given instant, measured in radians per second (rad/s) or revolutions per minute (rpm). Angular acceleration (α) describes how quickly that rotation rate is changing, measured in rad/s². A constant angular velocity means zero angular acceleration, just as a constant linear speed means zero linear acceleration. When a motor spins up from rest to full speed, angular acceleration is positive during that spin-up period and returns to zero once the target speed is reached.

Angular Acceleration Calculator

This calculator works out angular acceleration, the rate at which a rotating object speeds up or slows down, using whichever method suits the numbers you have. Choose Method 1 if you know how a rotation rate changed over time: enter the initial angular velocity, the final angular velocity, and the time elapsed, and the tool divides the change in angular velocity by time. Choose Method 2 if you know the forces involved: enter the net torque applied and the object's moment of inertia, and it divides torque by inertia using Newton's second law for rotation. Not sure your speed is already in radians per second? A built-in unit helper converts from rpm, degrees per second or revolutions per second first. The result is shown as angular acceleration in rad/s², plus the same figure converted to degrees per second squared and rpm per second, alongside a full calculation breakdown with the formula used, your inputs, the change in velocity, revolutions per second squared, and whether the object is accelerating or decelerating. It is useful for motor spin-up, braking, robotics, turbine startup and any situation where you need to know how quickly a spin rate is changing. Figures update instantly as you type, and results assume constant angular acceleration over the interval; for varying torque or inertia, calculus-based methods are needed instead.

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Reviewed June 2026 Standard rotational kinematics and Newton's second law for rotation.

Calculation Method

Method

Initial angular velocity (omega_i) rad/s

rad/s

Final angular velocity (omega_f) rad/s

rad/s

Time elapsed (t) seconds

Net torque (tau) N·m

N·m

Moment of inertia (I) kg·m²

kg·m²

Angular Velocity Unit Helper

Need to convert your speed to rad/s before entering above? Use this quick converter.

Input unit

Speed value

rad/s

Equivalent: 100.000 rad/s

Angular Acceleration Result

Angular Acceleration (alpha)
2.0000
rad/s²

In Degrees/s²

114.592

deg/s²

In rpm/s

19.099

rpm per second

Calculation Breakdown

Method usedKinematics

Formulaalpha = (omega_f - omega_i) / t

Initial omega (omega_i)0 rad/s

Final omega (omega_f)10 rad/s

Time (t)5 s

Change in omega (delta-omega)10 rad/s

Angular acceleration (alpha)2.0000 rad/s²

Unit Conversions

rad/s²2.0000

deg/s²114.592

rpm/s19.099

rev/s²0.31831

DirectionPositive (accelerating)

Result: An angular acceleration of 2.0000 rad/s² means the rotational speed increases by 2.0000 rad/s every second (114.592 deg/s² or 19.099 rpm/s).

How Angular Acceleration Is Calculated

Angular acceleration (alpha, α) measures how quickly an object's rate of rotation changes. There are two standard ways to calculate it.

Method 1: Kinematics (change in angular velocity / time)

The most common form is derived directly from the definition of acceleration:

alpha = (omega_f - omega_i) / t

Where omega_f is the final angular velocity in rad/s, omega_i is the initial angular velocity in rad/s, and t is the elapsed time in seconds. This is the rotational counterpart to linear acceleration a = (v_f - v_i) / t.

Method 2: Newton's Second Law for Rotation (torque / moment of inertia)

When a net torque acts on a rigid body, the resulting angular acceleration is:

alpha = tau / I

Where tau (τ) is the net torque in newton-metres (N·m) and I is the moment of inertia in kilogram-metres squared (kg·m²). This mirrors the linear form F = ma, rearranged to a = F / m.

Unit Conversions

From rad/s²	Multiply by	Result
rad/s²	1	rad/s²
rad/s²	180 / pi ≈ 57.2958	degrees/s²
rad/s²	60 / (2 × pi) ≈ 9.5493	rpm/s
rad/s²	1 / (2 × pi) ≈ 0.15915	revolutions/s²

Worked Example (matches default inputs)

A motor shaft accelerates from rest (0 rad/s) to 10 rad/s over 5 seconds.

delta-omega = 10 - 0 = 10 rad/s
alpha = 10 / 5 = 2.0000 rad/s²
In degrees/s²: 2.0000 x 57.2958 = 114.592 deg/s²
In rpm/s: 2.0000 x 9.5493 = 19.099 rpm/s

Common Applications

Motor spin-up: How quickly an electric motor reaches its rated speed from standstill.
Braking: The deceleration (negative alpha) of a rotating wheel when a brake is applied.
Robotics: Joint actuators accelerate their output shafts; alpha determines how fast a position change can be executed.
Turbines: Startup and shutdown sequences require controlled angular acceleration to avoid mechanical stress.
Gyroscopes and flywheels: Stored angular momentum and the rate at which it can be changed.

Sign Convention

Angular acceleration follows the right-hand rule by convention. If the fingers of your right hand curl in the direction of rotation, your thumb points along the axis of the angular velocity vector. A positive alpha means the object is speeding up in the positive rotational direction; a negative alpha means it is slowing down or reversing.

Related Calculators

Sources and method: Halliday, Resnick, and Krane, Physics (5th ed.), Chapter 11 (Rotation). Serway and Jewett, Physics for Scientists and Engineers, Chapter 10 (Rotation of a Rigid Object About a Fixed Axis). Standard SI definitions, International Bureau of Weights and Measures (BIPM).

This calculator computes angular acceleration from exact inputs using standard rotational kinematics and Newton's second law for rotation. Results assume constant angular acceleration over the time interval. For non-constant torque or varying inertia, calculus-based methods are required.