Angular Velocity and Angular Acceleration Calculator

What is angular velocity?

Angular velocity is the angle a rotating object sweeps per unit time: ω = θ / t. Its SI unit is the radian per second (rad/s). In everyday use revolutions per minute (rpm) is common. The conversion is 1 rpm = 2π / 60 rad/s, so 3000 rpm = 314.16 rad/s.

What is angular acceleration?

Angular acceleration is the change in angular velocity per unit time: α = (ω − ω₀) / t. Its unit is rad/s². It is positive for a motor running up and negative for a wheel being braked.

Rotation with constant angular acceleration

The equations of straight line motion carry over unchanged. Replace distance with angle, velocity with angular velocity and acceleration with angular acceleration:

• ω = ω₀ + α × t
• θ = ω₀ × t + ½ × α × t²
• θ = (ω₀ + ω) / 2 × t

The number of turns is the angle divided by 2π.

Link to linear quantities

A point at distance r from the axis has linear speed v = ω × r, tangential acceleration a = α × r and centripetal acceleration a = ω² × r. All three grow with distance from the axis.

Example

A motor starts from rest and reaches 3000 rpm in 10 seconds. The final angular velocity is 3000 × 2π / 60 = 314.16 rad/s and the angular acceleration is 314.16 / 10 = 31.42 rad/s². The average speed is 1500 rpm, so the motor makes 250 turns in that time.

Limits

The relations hold while angular acceleration is constant. The link between torque, moment of inertia and acceleration (τ = I × α) is outside the scope of this tool.