Angular Acceleration Calculator finds alpha in rad/s² from angular velocity and time, or solves for speed or time. It also works from torque and inertia or acceleration and radius.
Angular acceleration is the change in angular velocity divided by the time it takes.
Radius and Result Unit
Speed Over Time
Calculation Steps
- Change in angular velocity: ω₂ − ω₁ = 40 − 10 = 30 rad/s
- Angular acceleration: α = 30 rad/s ÷ 5 s = 6 rad/s²
Acceleration by Time
| Time | Acceleration | In rpm/s |
|---|---|---|
| 1.25 s | 24 rad/s² | 229.2 rpm/s |
| 2.5 s | 12 rad/s² | 114.6 rpm/s |
| 5 s | 6 rad/s² | 57.3 rpm/s |
| 10 s | 3 rad/s² | 28.65 rpm/s |
| 20 s | 1.5 rad/s² | 14.32 rpm/s |
Each row makes the same change in angular velocity over a different time. The highlighted row is your case.
What the Angular Acceleration Calculator Solves
The Angular Acceleration Calculator finds how fast the spin of a rotating body changes. With the loaded values, a body going from 10 rad/s to 40 rad/s in 5 s has an angular acceleration of 6 rad/s². The same result is shown as 57.3 rpm per second, and the body turns 19.89 times on the way.
The Calculate From list has four starting points. They are angular velocity and time, tangential acceleration and radius, torque and moment of inertia, and angular velocity with the angle turned. The first three have a Solve For list, so any one value in the formula can be the unknown. The angle option has a Known Values choice in its place.
A Speed Over Time chart sits under the results and draws the angular velocity as a straight line between the two speeds. It works as a simple angular acceleration visualizer, since the slope of that line is the acceleration itself.
Angular Acceleration Formula
Angular acceleration is the rate at which angular velocity changes with time. OpenStax University Physics, section 10.1 defines it as the time rate of change of ω. Its symbol is the Greek letter alpha, α, while omega, ω, stands for angular velocity.
$$\alpha = \frac{\omega_2 – \omega_1}{t}$$
Here ω₁ is the initial angular velocity, ω₂ is the final angular velocity and t is the time taken. To calculate angular acceleration, subtract the first velocity from the second and divide by the time. For 10 to 40 rad/s in 5 s that is 30 ÷ 5, which gives 6 rad/s².
The formula is derived from the definition of angular velocity. Angular velocity is the rate of change of the angle, so angular acceleration is the rate of change of that rate. In calculus terms α is dω/dt, the second derivative of the angle. Rotational acceleration is another name for the same quantity.
| Time for a 30 rad/s change | Angular acceleration | In rpm per second |
|---|---|---|
| 1.25 s | 24 rad/s² | 229.2 rpm/s |
| 2.5 s | 12 rad/s² | 114.6 rpm/s |
| 5 s | 6 rad/s² | 57.3 rpm/s |
| 10 s | 3 rad/s² | 28.65 rpm/s |
| 20 s | 1.5 rad/s² | 14.32 rpm/s |
Angular acceleration to angular velocity
If the angular velocity is given along with α, the same formula returns the missing speed or the time. Starting at 10 rad/s with 5 rad/s² for 5 s, the final velocity is 35 rad/s. Working back from 40 rad/s, the initial velocity is 15 rad/s. Going from 10 to 40 rad/s at 5 rad/s² takes 6 s.
$$\omega_2 = \omega_1 + \alpha \times t$$
What are the units of angular acceleration?
The SI unit of angular acceleration is the radian per second squared, rad/s². Since the radian is a ratio of two lengths, the dimensional formula is simply T⁻², or M⁰L⁰T⁻² in full. The Angular Acceleration Calculator also reports the answer in degrees per second squared, revolutions per second squared and rpm per second.
One rad/s² equals 57.3 °/s², 0.1592 rev/s² and 9.549 rpm/s. The loaded 6 rad/s² is therefore 343.8 °/s², 0.9549 rev/s² and 57.3 rpm/s. The Result Unit bar switches the main answer between the four.
How to find angular acceleration from rpm
Enter both speeds in rpm and the tool converts them, since 1 rpm is 2π ÷ 60 rad/s. OpenStax works a bicycle wheel that goes from rest to 250 rpm in 5.00 s. The Angular Acceleration Calculator gives 26.18 rad/s for the final speed and 5.236 rad/s² for α, matching the book’s 5.24 rad/s². In rpm terms that is 50 rpm every second.
Angular to Linear Acceleration
Every point on a spinning body shares the same angular acceleration, but points farther from the axis speed up faster along their path. The link is the angular acceleration formula with radius, which OpenStax gives as equation 10.8.
$$a = \alpha \times r$$
Here a is the tangential acceleration in m/s² and r is the radius in meters. To convert angular acceleration to linear acceleration, multiply by the radius. An α of 5 rad/s² at 0.5 m gives 2.5 m/s². To go from acceleration to angular acceleration, divide instead, so 3 m/s² at 0.5 m is 6 rad/s².
| Radius | Angular acceleration for 3 m/s² | In rpm per second |
|---|---|---|
| 0.125 m | 24 rad/s² | 229.2 rpm/s |
| 0.25 m | 12 rad/s² | 114.6 rpm/s |
| 0.5 m | 6 rad/s² | 57.3 rpm/s |
| 1 m | 3 rad/s² | 28.65 rpm/s |
| 2 m | 1.5 rad/s² | 14.32 rpm/s |
In the Angular Acceleration Calculator this option doubles as a tangential acceleration calculator, and it can solve for the radius as well. With 3 m/s² and 5 rad/s² the radius is 0.6 m. The difference between angular and linear acceleration is what they measure. One is a change in spin rate in rad/s², the other a change in speed along a path in m/s².
What are the two types of acceleration?
A point on a rotating body has two kinds of acceleration at once. Tangential acceleration, α × r, changes its speed. Centripetal acceleration, ω² × r, points at the axis and only bends its path. In the loaded case a point 0.35 m out has 2.1 m/s² of tangential acceleration and 560 m/s² of centripetal acceleration at the final speed.
The two are at right angles, so the total comes from the Magnitude of Acceleration Calculator. For motion in a straight line, where acceleration is the change in velocity over time, the Acceleration Calculator is the one to use.
Angular Acceleration from Torque
A net torque is what causes angular acceleration. Newton’s second law for rotation says the sum of the torques equals the moment of inertia times the angular acceleration, which is equation 10.25 in OpenStax section 10.7.
$$\alpha = \frac{\tau}{I}$$
τ is the net torque in newton-meters and I is the moment of inertia in kg·m². To calculate angular acceleration using torque, divide the torque by the inertia. A torque of 12 N·m on 4 kg·m² gives 3 rad/s². Doubling the torque doubles α, and doubling the inertia halves it.
Angular acceleration is not equal to torque. The two are proportional, with the moment of inertia as the link between them. To calculate torque from inertia and angular acceleration, multiply them, so 4 kg·m² at 5 rad/s² needs 20 N·m. The power that torque delivers at a given rpm is worked out in the Work and Power Calculator.
When the torque comes from a push, set Torque From to Force × lever arm. Torque is then the force times its distance from the axis, with the force taken at a right angle to the arm. A push of 250 N at 1.5 m is 375 N·m.
OpenStax tests the law with that push on a merry-go-round, a 50.0 kg disk of radius 1.50 m. Pick Solid disk or cylinder, enter the mass and radius, and the Angular Acceleration Calculator finds an inertia of 56.25 kg·m² and an α of 6.667 rad/s². The book gives 6.67 rad/s².
The book then seats an 18.0 kg child 1.25 m from the center. The Added Point Mass field under the options bar adds m × r² for the child. The inertia rises to 84.38 kg·m² and α falls to 4.444 rad/s², against the book’s 4.44 rad/s².
How to find the angular acceleration of a rod?
A rod needs its moment of inertia first, and that depends on where the axis is. The Moment of Inertia list has both rod cases built in, along with a disk, a hoop and a sphere. The table uses a mass of 2 kg, a length or radius of 1.2 m and a torque of 12 N·m.
| Shape and axis | Moment of inertia | Value | α from 12 N·m |
|---|---|---|---|
| Thin rod, about its center | 1/12 × m × L² | 0.24 kg·m² | 50 rad/s² |
| Thin rod, about one end | 1/3 × m × L² | 0.96 kg·m² | 12.5 rad/s² |
| Solid sphere | 2/5 × m × R² | 1.152 kg·m² | 10.42 rad/s² |
| Solid disk or cylinder | 1/2 × m × R² | 1.44 kg·m² | 8.333 rad/s² |
| Hoop or thin ring | m × R² | 2.88 kg·m² | 4.167 rad/s² |
The same rod is four times harder to spin about its end than about its center. That is why the same 12 N·m gives 50 rad/s² in one case and 12.5 rad/s² in the other. Mass placed far from the axis always lowers the angular acceleration.
Angular Displacement and Acceleration
The relation between angular displacement and angular acceleration needs no time at all. For steady acceleration, OpenStax equation 10.13 in section 10.2 ties the two speeds to the angle turned.
$$\alpha = \frac{\omega_2^2 – \omega_1^2}{2 \times \theta}$$
θ is the angle turned in radians. A body that goes from 10 to 40 rad/s while turning through 150 rad has an α of 5 rad/s². The Angular Acceleration Calculator then recovers the time from the average angular velocity, which is 6 s here. The angle can also be entered in revolutions or degrees.
When the time is known but the final speed is not, switch Known Values to initial velocity, angle and time. This uses OpenStax equation 10.12, which gives the angle as ω₁ × t plus half of α × t².
$$\alpha = \frac{2 \times (\theta – \omega_1 \times t)}{t^2}$$
A body that starts at 10 rad/s and turns through 150 rad in 6 s would cover only 60 rad at a steady spin. The extra 90 rad gives an α of 5 rad/s², and the tool adds the final velocity of 40 rad/s.
These are the rotational twins of the straight-line equations in the SUVAT Calculator. Angle takes the place of distance, angular velocity the place of velocity, and α the place of acceleration.
Direction, Sign and Angular Speed
Is angular acceleration a vector?
Yes. Angular acceleration is a vector that lies along the axis of rotation. To find its direction, compare it with the angular velocity. It points the same way as ω when the body is speeding up and the opposite way when it is slowing down.
In the Angular Acceleration Calculator that shows up as a sign. A body that drops from 40 rad/s to rest in 5 s has an α of −8 rad/s², and the result is labeled Slowing Down. It still turns 15.92 times before it stops.
Angular velocities can be entered as negative numbers for spin in the opposite direction. A body that goes from 20 rad/s to −10 rad/s in 5 s has an α of −6 rad/s². The tool reports that it stops after 3.333 s and then turns the other way, so the 25 rad shown is the net angle.
Is angular speed the same as angular velocity?
Angular speed is the size of the angular velocity, with no direction attached. It is also called rotational speed. Angular speed cannot be negative, since it is only a magnitude. Angular velocity can be, because its sign marks clockwise against counterclockwise. The formula for either one is the angle turned divided by the time.
An example of angular acceleration in real life
A bicycle wheel is the everyday case. In the OpenStax example the wheel spins up to 250 rpm at 5.24 rad/s². When the brakes are applied with an α of −87.3 rad/s², it stops in 0.300 s. Solving for time in the tool with those values returns 0.2999 s, which rounds to the same figure. More rotation tools are listed under Rotational Motion Calculators.