Angular Acceleration Calculator - omega, torque, and tangential

An angular acceleration calculator solves for the rate at which a rotating body's angular velocity changes, in rad/s^2, rev/s^2, or deg/s^2. The result comes from one of three equivalent branches: the kinematic form alpha equals (omega_f minus omega_i) divided by t, the dynamic form alpha equals net torque divided by I, and the tangential form alpha equals a_t divided by r.

• • Motor spin-up checks: Estimate how fast a motor-driven shaft reaches its target RPM given a starting speed and a fixed time, setting the torque request to the controller.
• • Flywheel and disk dynamics: Compute alpha for a flywheel, solid disk, or pulley when a known torque is applied, using the rigid-body I.
• • Physics homework and exam problems: Solve a rotational kinematics question for a wheel, pendulum, or turntable in rad/s^2 and deg/s^2.
• • Wheel and tire slip analysis: Compare a vehicle's tangential acceleration to the wheel's alpha to spot tire slip or miscalibration.

Alpha is a vector quantity, so a positive alpha spins the body up in the chosen positive sense, while a negative alpha means deceleration or reversal.

For the linear version of the same spin-up problem with displacement, velocity, and time, Kinematics Motion Calculator solves the SUVAT equations on a straight track.

The calculator evaluates the same physical quantity, alpha, in three different ways depending on the inputs available. Pick the branch that matches your data and the same alpha value is returned in rad/s^2, rev/s^2, and deg/s^2.

• alpha: Resulting alpha, in rad/s^2 by default.
• omega_i: Initial angular velocity at the start of the interval, in rad/s.
• omega_f: Final angular velocity at the end of the interval, in rad/s.
• t: Time interval over which omega changes, in seconds.
• tau: Net torque about the rotation axis, in N*m.
• I: Moment of inertia of the body about the rotation axis, in kg*m^2.
• a_t: Tangential linear acceleration at radius r, in m/s^2.
• r: Radius from the rotation axis to the point where a_t is measured, in metres.

All three branches reduce to the same alpha, so the best branch is the one that matches the data: a tachometer makes the kinematic branch ideal, a torque sensor and known geometry make the dynamic branch ideal, and an accelerometer at a known radius makes the tangential branch ideal.

According to OpenStax University Physics Volume 1, Section 10.2 (Rotation with Constant Angular Acceleration) and Section 10.7 (Newton's Second Law for Rotation), the kinematic form alpha equals omega_f minus omega_0 divided by t and the dynamic form tau_net equals I times alpha are the closed-form equations for constant alpha that the first two branches solve.

Once you have alpha, multiply it by the moment of inertia to get the net torque that the same drivetrain has to deliver, and Torque, Power & Speed Calculator covers the power and rotational speed side of that calculation.

Four core ideas drive the result: the kinematic form that ties alpha to a velocity change, the dynamic form that ties it to torque, the tangential form that ties it to linear motion at a radius, and the I reference values the dynamic branch uses.

All three forms use the same alpha, so switching branches is just a matter of which data you trust most. The result is always a signed quantity: positive alpha spins the body up in the chosen convention, and negative alpha represents braking or reversal.

The dynamic form alpha = tau / I is the rotational twin of F = m a, so when you want the linear force analogue of the same problem, Forces and Newton's Laws Calculator covers the translational case.

Pick the branch that matches your data, enter the inputs, and read alpha in your preferred unit. The same alpha is shown in rad/s^2, rev/s^2, and deg/s^2.

1. 1 Choose the branch: Select Kinematic for omega_i, omega_f, and t; Dynamic for net torque and I; or Tangential for a_t and a radius.
2. 2 Enter omega_i and omega_f (kinematic): Type omega_i and omega_f in rad/s. Convert RPM with omega_rad = RPM times 2*pi/60 (about 0.10472 rad/s per RPM).
3. 3 Enter the time interval (kinematic): Type the time interval in seconds over which omega changes from omega_i to omega_f.
4. 4 Enter torque and I (dynamic): Type the net torque in N*m and I in kg*m^2. For a solid disk use 1/2 m r^2; for a hoop use m r^2; for a solid sphere use 2/5 m r^2.
5. 5 Enter a_t and radius (tangential): Type the tangential acceleration in m/s^2 measured at a known radius r in metres.
6. 6 Read alpha in three units: Check the primary result in rad/s^2, then read the same value in rev/s^2 and deg/s^2.

Try a wheel spinning up from 60 RPM to 180 RPM in 4 s. Alpha should be about 3.1416 rad/s^2, or 0.5 rev/s^2 or 180 deg/s^2.

To turn alpha into the rotational work and power needed over the same spin-up interval, feed the omega and tau into Work, Energy & Power Calculator after you have your result.

A focused alpha tool gives all three branches in one view, returns the result in three useful units, and avoids dividing by a zero radius or a zero I.

• • Three branches, one answer: Switch between kinematic, dynamic, and tangential inputs without leaving the page; the same alpha is returned in rad/s^2, rev/s^2, and deg/s^2.
• • Unit-flexible inputs: Enter angular velocities, torques, and accelerations in SI units while seeing the kinematic and dynamic relationships in familiar form.
• • Three output units: Read alpha in rad/s^2 for physics, in rev/s^2 for drivetrain reporting, and in deg/s^2 for graphical or robotic applications.
• • Guards against invalid inputs: Zero time interval, zero I, and zero radius return zero alpha and surface a validation error, so a bad input never silently produces nonsense.
• • Matches textbook formulas: All three branches use the closed-form expressions from standard mechanics references, matching a homework grader or design review.

Use this calculator as a quick check while building a control loop, sizing a motor, or grading a physics problem, and cross-reference the result with the omega and torque in your drivetrain before committing a part.

Once you know the spin-up alpha and the final omega, cross-check the rotor against its natural frequency with Vibration Natural Frequency Calculator so the operating point does not land near torsional resonance.

Three physical factors dominate the magnitude of alpha, and a few assumptions limit how closely the simple closed-form branches match a real rotating system.

• • All three branches assume constant alpha. Real spin-up profiles include ramp-up and ramp-down segments, friction, and controller dynamics, so the result is a useful average rather than an instantaneous measurement.
• • The dynamic branch uses a single rigid-body I, so a compound shaft, internal moving parts, or a flexible rotor is only approximated. Use a segment-by-segment sum of I for composite bodies.

Treat the simple alpha as a design sanity check rather than a full transient simulation, and use the same branch consistently across a project.

For very high spin rates watch out: a_t = alpha * r assumes the point stays at a constant radius, not true for a translating or deforming body.

According to OpenStax University Physics Volume 1, Section 10.4 (Moment of Inertia and Rotational Kinetic Energy), the moment of inertia of a solid uniform disk about its central axis is 1/2 m r^2, of a thin hoop is m r^2, of a solid sphere is 2/5 m r^2, and of a thin rod about one end is 1/3 m L^2.

A long shaft that spins up with a high alpha will also wind up along its length, so the resulting twist in radians can be predicted with Angle of Twist Calculator for the same torque, length, and shear modulus.