What is the formula for angular acceleration?

There are two formulas for angular acceleration: alpha = (omega2 - omega1) / t, where omega2 and omega1 are the final and initial angular velocities respectively and t is the time interval; or alpha = torque / I, where torque is the resulting rotational motion and I is the moment of inertia. Both formulas give the angular acceleration in radians per second squared. This calculator will select the appropriate formula based on the values entered.

How do you calculate angular acceleration using torque?

Divide the resulting torque by the moment of inertia: alpha = torque / I. This is a restatement of Newton's second law for rotational motion, which says that torque = I * alpha. For example, a wheel with 10 newton-meters of torque and a moment of inertia of 2 kilogram meters squared has an angular acceleration of 5 radians per second squared. Open the "Torque" section of the calculator, enter these two values, and the calculator will perform the division for you.

What is the difference between angular and linear acceleration?

Angular acceleration is the rate of change in angular velocity with respect to time and is measured in radians per second squared. It describes rotational motion. Linear acceleration is the rate of change in linear velocity along a straight line and is measured in meters per second squared. The two quantities are related: The tangential acceleration of a point on a rotating object is equal to the product of angular acceleration and radius, i.e., a_t = alpha * r.

What is the angular velocity of a wheel with constant angular acceleration?

Use the formula omega2 = omega1 + alpha * t. For a wheel starting from rest, omega1 is 0. So after constant angular acceleration of 3.0 radians per second squared for 8 seconds, the angular velocity will be 3.0 times 8, which is 24 radians per second. Enter the initial angular velocity, the angular acceleration and time, leave the final angular velocity field blank, and the calculator will give you your answer.

What is the unit of angular acceleration?

The SI unit for angular acceleration is radians per second squared. It can also be expressed as degrees per second squared; to convert, multiply the value in radians by 180 and divide it by pi (roughly 57.3). Since a radian is a dimensionless ratio, this unit is sometimes written as the reciprocal of seconds squared. This calculator displays results both in radians per second squared and degrees per second squared.

Angular Acceleration Calculator

Change in angular velocity over time

Initial angular velocity (ω₁)

Final angular velocity (ω₂)

Time interval (t)

Angular acceleration (α)

rad/s²

Enter any two of the initial angular velocity, the final angular velocity and the time, and the calculator finds the angular acceleration from α = (ω₂ − ω₁) / t. Leave any one of the boxes blank to solve for it instead. You can also find α from a torque and a moment of inertia, or from a tangential acceleration and a radius, in the sections below.

From torque and moment of inertia

From tangential (linear) acceleration

Please input values into the fields

When a rotating object is speeding up or slowing down, the angular acceleration tells you how quickly its rotational speed is changing. It's the rotational equivalent of linear acceleration: just as linear acceleration measures how quickly a velocity along a straight line changes, angular acceleration measures how quickly a rotation around an axis changes.

This calculator provides three methods to calculate angular acceleration and is applicable for any direction. If you leave a value blank, the calculator will compute that value.

What is angular acceleration?

Angular acceleration is denoted by the Greek letter alpha and is the rate of change in angular velocity with respect to time. Angular velocity itself is denoted by omega and measures how fast an object rotates, with units of radians per second. When a torque acts on a wheel, gear or disc, its angular velocity changes, and angular acceleration measures how quickly this change happens.

It is the first derivative of angular velocity with respect to time and the second derivative of angle with respect to time. The SI unit is radians per second squared. Since a radian is a pure ratio, this unit is sometimes expressed simply as the inverse square of seconds.

Formula for angular acceleration.

There are two core equations that are built into this calculator. The first equation is derived directly from the definition of angular acceleration, which is the change in angular velocity divided by the time it took for that change to occur.

\alpha = \dfrac{\omega_2 - \omega_1}{t}

In this formula, ω-two is the final angular velocity, ω-one is the initial angular velocity and t is the time interval. After rearranging, you get the equation of motion for a rotating object with constant angular acceleration.

\omega_2 = \omega_1 + \alpha\, t

The second formula is the statement of Newton's second law for rotational motion. Torque is the rotational equivalent of force, and moment of inertia is the rotational equivalent of mass.

\alpha = \dfrac{\tau}{I} \qquad \tau = I\,\alpha

The torque (Greek letter tau) is the resulting rotational force acting on an object while the moment of inertia I indicates how strongly an object resists a change in its rotation. The heavier an object is or the further the mass distribution is from the axis of rotation, the larger its moment of inertia and the smaller the angular acceleration for a given torque.

Relationship to linear acceleration

Angular acceleration is also related to the normal linear acceleration at a point on a disc. Tangential acceleration is the component of linear acceleration that lies in the direction of motion and it equals angular acceleration multiplied by the radius.

a_t = \alpha\, r \qquad \alpha = \dfrac{a_t}{r}

Imagine a yo-yo hanging from a string. If you halve the length of the string while keeping the same tangential acceleration, then the yo-yo will spin twice as fast since a smaller radius means a larger angular acceleration. This is the third method used in the calculator.

A complete example:

A wheel starts from rest and is given a constant angular acceleration of 3.0 radians per second squared. What is its angular speed after 8 seconds?

We'll start with the formula omega-two equals omega-one plus alpha times t. The initial angular velocity is zero, the angular acceleration is 3.0 and time is 8 seconds.

\omega_2 = 0 + 3.0 \times 8 = 24 \ \text{rad/s}

So the angular velocity of the wheel is 24 radians per second, which is the result after 8 seconds. If you enter 0 as initial angular velocity, 3 as angular acceleration and 8 as time and leave the field for final angular velocity blank, the calculator will output 24. It also shows that the angle swept out is about 96 radians, which is a little more than 15 rotations.

How to use this calculator.

Pick the method that best fits what information you already have. For the rate of change method, enter two out of three values: initial angular velocity, final angular velocity, and time. If you know the angular acceleration, also enter it and leave blank the value you want to calculate.

In the Torque method, activate the "Torque" section and enter torque and moment of inertia. In the Tangential method, activate the "Tangential" section and enter tangential acceleration and radius. You can even combine different methods: Enter torque and moment of inertia to calculate angular acceleration, then add initial angular velocity and time, and the calculator will predict final angular velocity for you.

Each field has a separate toggle for units. Enter the angular velocity in revolutions per minute (rpm) and read off the angular acceleration in radians per second squared, or convert the result to degrees per second squared. The calculator does all necessary conversions for you.

Angular acceleration units:

The SI unit of angular acceleration is the radian per second squared. The most commonly used units are radians per second squared and degrees per second squared. Since a radian is a ratio of two lengths, it is dimensionless, so the unit "radian" is sometimes abbreviated as the reciprocal of seconds squared.

Quantity	Symbol	SI unit
Angular acceleration	α	radian per second squared (rad/s²)
Angular velocity	ω	radian per second (rad/s); also rpm, rev/s
Time	t	second (s)
Torque	τ	newton metre (N·m)
Moment of inertia	I	kilogram metre squared (kg·m²)
Tangential acceleration	a_t	metre per second squared (m/s²)
Radius	r	metre (m)

This calculator always internally uses radians per second to represent angular velocity. So if you enter revolutions per minute (RPM), it will first convert that. One revolution per minute is equal to 2π divided by 60, which is approximately 0.105 radians per second.

Where does angular acceleration occur?

Angular acceleration occurs everywhere where a rotation takes place and its speed changes. Examples of this are the increase in the rotational speed of an automobile engine from idle to the limit, the acceleration of a hard disk drive, the beginning of the rotation of a carousel or the movement of an ice skater who draws his arms in.

Gears and pulleys are typical examples. When you pull on a rope wrapped around a pulley, the rotational motion of the pulley accelerates, and that rate of acceleration is angular acceleration. Also when you turn a screw with a wrench, the applied torque produces an instantaneous angular acceleration in the screw.

Common Mistakes

The most common mistake is to confuse angular and linear quantities. The unit of angular acceleration is radians per second squared, while the tangential acceleration of a point on a disk is measured in meters per second squared. Both are related by the radius but they are not equal.

Also pay attention to the unit of angular velocity. The conversion from revolutions per minute to radians per second is by a factor 2π divided by 60. So an engine turning at 3000 rpm has an angular velocity of about 314 radians per second, not 3000. Also remember that negative angular acceleration is not an error: it just means the rotation of the object is slowing down.

This tool is for general education and solving everyday problems. If used for research, engineering or safety critical applications please verify all values against the required standards and data for your project.

Frequently asked questions

What is the formula for angular acceleration?: There are two formulas for angular acceleration: alpha = (omega2 - omega1) / t, where omega2 and omega1 are the final and initial angular velocities respectively and t is the time interval; or alpha = torque / I, where torque is the resulting rotational motion and I is the moment of inertia. Both formulas give the angular acceleration in radians per second squared. This calculator will select the appropriate formula based on the values entered.
How do you calculate angular acceleration using torque?: Divide the resulting torque by the moment of inertia: alpha = torque / I. This is a restatement of Newton's second law for rotational motion, which says that torque = I * alpha. For example, a wheel with 10 newton-meters of torque and a moment of inertia of 2 kilogram meters squared has an angular acceleration of 5 radians per second squared. Open the "Torque" section of the calculator, enter these two values, and the calculator will perform the division for you.
What is the difference between angular and linear acceleration?: Angular acceleration is the rate of change in angular velocity with respect to time and is measured in radians per second squared. It describes rotational motion. Linear acceleration is the rate of change in linear velocity along a straight line and is measured in meters per second squared. The two quantities are related: The tangential acceleration of a point on a rotating object is equal to the product of angular acceleration and radius, i.e., a_t = alpha * r.
What is the angular velocity of a wheel with constant angular acceleration?: Use the formula omega2 = omega1 + alpha * t. For a wheel starting from rest, omega1 is 0. So after constant angular acceleration of 3.0 radians per second squared for 8 seconds, the angular velocity will be 3.0 times 8, which is 24 radians per second. Enter the initial angular velocity, the angular acceleration and time, leave the final angular velocity field blank, and the calculator will give you your answer.
What is the unit of angular acceleration?: The SI unit for angular acceleration is radians per second squared. It can also be expressed as degrees per second squared; to convert, multiply the value in radians by 180 and divide it by pi (roughly 57.3). Since a radian is a dimensionless ratio, this unit is sometimes written as the reciprocal of seconds squared. This calculator displays results both in radians per second squared and degrees per second squared.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Angular acceleration
Definition of angular acceleration, its relation to angular velocity and torque, and the distinction between spin and orbital angular acceleration.
OpenStax University Physics, Rotation with Constant Angular Acceleration
The rotational kinematics equations, including omega = omega0 + alpha t and the angle swept under a constant angular acceleration.
Wikipedia: Torque
Torque as the rotational analogue of force and the relation torque = I times alpha used to find angular acceleration from a torque and a moment of inertia.