Angular Acceleration Calculator
Enter an initial and final angular velocity plus a time interval to find angular acceleration, or switch modes to solve from torque and moment of inertia, or from tangential acceleration and radius. The calculator works in rad/s, RPM, or deg/s and shows a step-by-step solution, the velocity-over-time chart, and derived values including angular displacement and final speed in RPM.
Formula
Worked example
A motor accelerates from rest (0 rad/s) to 10 rad/s in 5 s: alpha = (10 - 0) / 5 = 2 rad/s2. Angular displacement = 0*5 + 0.5*2*25 = 25 rad = 3.98 revolutions. In RPM/s: 2 / 0.10472 = 19.1 RPM/s.
What is angular acceleration?
Angular acceleration (symbol alpha, unit rad/s2) is the rate at which the angular velocity of a rotating body changes over time. Just as linear acceleration describes how quickly a straight-line speed changes, angular acceleration describes how quickly rotational speed changes. A positive alpha means the object is spinning faster; a negative alpha means it is slowing down. The concept is fundamental in mechanical engineering, robotics, and physics because almost every machine that spins (motors, wheels, turbines, gyroscopes) must accelerate or decelerate from one speed to another.
Three ways to calculate angular acceleration
There are three standard formulas, each suited to different known quantities. The first and most direct is alpha = (omega2 - omega1) / t, where omega1 and omega2 are the initial and final angular velocities and t is the elapsed time. This is used when speed-sensor or encoder data is available. The second is alpha = tau / I, the rotational equivalent of Newton's second law (F = ma): torque tau divided by moment of inertia I. This is the right formula for motor-sizing and structural dynamics problems. The third is alpha = a_t / r, which connects angular acceleration to the tangential (linear) acceleration a_t at a point on the rim at radius r from the axis. All three are implemented in the calculator above.
Units and conversions
The SI unit of angular acceleration is radians per second squared (rad/s2). Because a radian is dimensionless, rad/s2 is equivalent to 1/s2 or s(-2). Engineers also use RPM per second (RPM/s) and degrees per second squared (deg/s2). To convert: 1 rad/s2 = 9.5493 RPM/s = 57.296 deg/s2. Angular velocity is often given in RPM in industry (1 RPM = 2*pi/60 rad/s = 0.10472 rad/s), so when calculating alpha from a speed change in RPM, first convert both velocities to rad/s before dividing by time, or use the calculator's RPM mode directly.
Angular displacement and the kinematic equations
Under constant angular acceleration, three kinematic equations mirror the familiar linear-motion equations. The angular displacement swept during the interval is theta = omega0*t + 0.5*alpha*t2. The final velocity satisfies omega2 = omega0 + alpha*t. A velocity-squared relation also holds: omega22 = omega02 + 2*alpha*theta. The calculator computes theta and converts it to revolutions (1 revolution = 2*pi rad) so you can see how many complete turns the object makes during the acceleration phase. These results are shown in the step-by-step panel and plotted on the chart.
Angular acceleration in everyday systems
| System | Typical alpha (rad/s2) | Notes |
|---|---|---|
| Hard disk drive spindle (startup) | 100-300 | Reaches 7200 RPM in about 2-3 s |
| Electric motor (direct-on-line start) | 20-100 | Depends on rotor inertia and torque |
| Car wheel (emergency braking) | 50-200 | ABS limits slip; peak torque from brakes |
| Industrial fan (start-up) | 0.5-5 | Large inertia keeps alpha low |
| Servo motor (high-performance) | 1000-10000 | Very low inertia rotor, high torque density |
| Centrifuge (lab, startup) | 5-50 | Gradual ramp to avoid sample disruption |
| Gyroscope precession (small) | 0.001-0.1 | Very slow angular acceleration by design |
| Human forearm rotation | 10-30 | Approximate during rapid pronation/supination |
Approximate angular acceleration values for common rotating machines and devices.
Frequently asked questions
What is the formula for angular acceleration?
The most common formula is alpha = (omega2 - omega1) / t, where omega1 and omega2 are the initial and final angular velocities in rad/s and t is time in seconds. If you know torque and moment of inertia, use alpha = tau / I. If you know the tangential (linear) acceleration at a rim point at radius r, use alpha = a_t / r.
What units does angular acceleration use?
The SI unit is radians per second squared (rad/s2). Because the radian is dimensionless, this equals 1/s2. Other common units are RPM per second (RPM/s) and degrees per second squared (deg/s2). The conversion factors are 1 rad/s2 = 9.5493 RPM/s = 57.296 deg/s2.
How do I convert RPM to rad/s for this calculation?
Multiply RPM by 2*pi/60, which equals approximately 0.10472. For example, 100 RPM = 100 * 0.10472 = 10.472 rad/s. If both omega1 and omega2 are in RPM and you divide their difference by time in seconds, the result is already in RPM/s. To get rad/s2, multiply by 0.10472. The calculator handles this conversion automatically when you select RPM as the angular velocity unit.
Can angular acceleration be negative?
Yes. A negative alpha means the angular velocity is decreasing (deceleration or braking). The direction convention follows the right-hand rule for a given axis: counterclockwise rotation is positive, clockwise is negative, so a body slowing its counterclockwise spin has a negative angular acceleration for that axis.
What is the difference between angular acceleration and centripetal acceleration?
Angular acceleration (alpha) describes how fast the spin rate changes. Centripetal (radial) acceleration (a_c = omega2 * r) points toward the axis and exists whenever the body rotates, even at constant speed. Tangential acceleration (a_t = alpha * r) arises only when angular velocity is changing. Total linear acceleration at a rim point combines both centripetal and tangential components.
How is angular acceleration related to torque?
By Newton's second law for rotation: torque = moment of inertia times angular acceleration (tau = I * alpha). Doubling the torque doubles alpha; doubling the moment of inertia (more mass farther from the axis) halves alpha. This relationship is central to motor and drivetrain design.
What is moment of inertia and how does it affect angular acceleration?
Moment of inertia (I, in kg*m2) measures how hard it is to change a body's rotational speed. It depends on both the mass and how that mass is distributed around the axis. A hollow cylinder has a higher moment of inertia than a solid cylinder of the same mass, so it requires more torque to achieve the same alpha. For a point mass m at radius r, I = m*r2.