Angular Acceleration Calculator
Find how quickly a spinning object speeds up or slows down, from angular speeds and time, from torque and inertia, or from tangential acceleration.
🌀 What is Angular Acceleration?
Angular acceleration is the rate at which the angular velocity of a rotating object changes with time. It is the rotational counterpart of ordinary acceleration. Where acceleration tells you how fast a car gains speed along a road, angular acceleration, written α and measured in radians per second squared, tells you how fast a wheel, a turbine, a hard drive platter or a spinning skater gains or loses rotation speed. A positive value means the angular velocity is growing in the positive direction, and a negative value means it is shrinking or growing the other way.
There are three common routes to the same quantity, and this calculator covers all of them. If you know two angular speeds and the time between them, α = (ω2 − ω1) / t. If you know the net torque and the moment of inertia, Newton's second law for rotation gives α = τ / I. If you know the acceleration of a point along its circular path and its distance from the axis, then α = at / r. Engineers use these relations to size motors that must spin up a load in a set time, to estimate braking of a flywheel, and to understand how a door, a pulley or a washing machine drum responds to a push.
Several points often cause confusion. Angular acceleration is not the same as centripetal acceleration. The first changes the spin rate, the second keeps a point on a circular path and exists even at constant spin. A negative α does not always mean slowing down, because slowing depends on whether α and ω have opposite signs. And the numbers you type in rpm or degrees per second must be converted to radians per second before they are used in the formulas, which this page does automatically while leaving your typed numbers untouched.
The moment of inertia I decides how hard it is to change the rotation of an object. It depends on the mass and on how far that mass sits from the axis. A solid disk has I = ½ M R2, a hoop has M R2 and a rod pivoted at one end has M L2 / 3. Pick a shape to have the calculator do this step, or enter a known value. For the speed itself, see the angular velocity calculator, and for the quantity that rotation carries, see the angular momentum calculator. For straight-line motion the acceleration calculator applies the same ideas.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Spinning Up a Wheel
A wheel goes from rest to 1200 rpm in 8 seconds
Example 2 — A Slowing Flywheel
A flywheel slows from 3000 rpm to 600 rpm in 12 seconds
Example 3 — Reversing Direction
A turntable goes from +90 °/s to −90 °/s in 2 seconds
Example 4 — Torque on a Disk
A 25 N m torque acts on a solid disk of mass 10 kg and radius 0.4 m
Example 5 — A Rod Pivoted at One End
A 6 N m torque acts on a 2 kg rod of length 1.5 m pivoted at one end
Example 6 — From Tangential Acceleration
A point 0.25 m from the axis speeds up along its path at 3 m/s²
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for angular acceleration?
Angular acceleration is the change in angular velocity divided by the time taken, alpha = (omega2 - omega1) / t, measured in rad/s squared. A wheel that goes from rest to 125.66 rad/s (1200 rpm) in 8 s has alpha = 15.708 rad/s squared.
How do you find angular acceleration from torque?
Use the rotational form of Newton's second law, alpha = torque / moment of inertia. A 25 N m torque on a solid disk of 10 kg and radius 0.4 m (I = 0.8 kg m squared) gives alpha = 31.25 rad/s squared.
How do you find angular acceleration from tangential acceleration?
Divide the tangential acceleration by the radius: alpha = a_t / r. A point 0.25 m from the axis accelerating along its path at 3 m/s squared corresponds to alpha = 12 rad/s squared.
What are the units of angular acceleration?
The SI unit is radians per second squared (rad/s squared). Other units are degrees per second squared, revolutions per second squared and rpm per second. 1 rad/s squared equals 57.2958 degrees/s squared and 9.5493 rpm per second.
Can angular acceleration be negative?
Yes. The sign shows the direction of the change relative to the positive rotation direction. A wheel slowing from 3000 rpm to 600 rpm in 12 s has alpha = -20.944 rad/s squared, which means it is decelerating.
How do I convert rpm to rad/s for angular acceleration?
Multiply rpm by 2 pi / 60, about 0.10472. So 1200 rpm is 125.66 rad/s. To convert an angular acceleration in rpm per second to rad/s squared use the same factor.
What is the moment of inertia of a disk, hoop and rod?
A solid disk about its axis is one half M R squared, a hoop is M R squared, a solid sphere is two fifths M R squared, a rod about its centre is M L squared over 12, and a rod about one end is M L squared over 3.
Is angular acceleration the same as centripetal acceleration?
No. Angular acceleration alpha describes how the spin rate changes. Centripetal acceleration omega squared times r points toward the axis and exists even when alpha is zero, whenever a point moves in a circle.
How do I find the angle turned during the acceleration?
For constant alpha, the angle is the average angular velocity times the time, theta = (omega1 + omega2) t / 2. The wheel in the first example turns 502.65 rad, which is exactly 80 revolutions.
What does zero angular acceleration mean?
It means the angular velocity is constant, either zero or a steady spin. Constant spin can continue with no net torque, just as constant linear velocity needs no net force.
Does the calculator work for a changing alpha?
It reports the average angular acceleration over the interval you enter. If the torque or the moment of inertia changes during the motion, the instantaneous value varies and you should use short intervals.