The Angular Velocity Calculator converts rotation angle and elapsed time into angular velocity in rad/s, RPM, and degrees per second. Connects angular speed with linear velocity at the rim — used in mechanical engineering, motor design, robotics, and rotational dynamics problems.
3.141593
rad/s
30
RPM
0.5
Hz
2
s
6.283185
rad
3.141593
rad/s
30
RPM
0.5
Hz
2
s
6.283185
rad
RPM on a tachometer, rad/s in a physics textbook, degrees per second in a game engine — rotational speed is described in many units depending on the field. The calculator for angular velocity finds ω from angle and time, converts between all common angular speed units, and calculates the linear (tangential) velocity at any radius on the rotating object.
Average angular velocity is simply angle divided by time:
ω = θ / t
where θ is the angle in radians and t is the elapsed time in seconds. The result is in rad/s — the SI unit. For circular motion with period T: ω = 2π/T = 2πf, where f is frequency in Hz. The relationship between angular and linear (tangential) velocity at radius r: v = ω × r. A car wheel of radius 0.33 m rotating at 500 RPM: ω = 500 × 2π/60 = 52.36 rad/s; rim speed = 52.36 × 0.33 = 17.3 m/s ≈ 62 km/h. The angular velocity unit converter handles all unit conversions between rad/s, RPM, deg/s, and Hz.
Converting between the most common angular velocity units:
These conversions appear constantly in motor specifications (rated speed in RPM), control systems (setpoints in rad/s), and robotics (joint velocities in deg/s). Use this online calculator to find angular velocity in all units simultaneously.
Angular velocity quantifies rotation rate across scales from molecular motors to wind turbines:
The angular acceleration calculator and circular motion calculators provide the complete rotational dynamics toolkit.
At any angular velocity ω, a point at radius r experiences centripetal acceleration aₙ = ω²r. This is the acceleration that must be provided by the material strength of the rotating object. For a flywheel or turbine blade spinning at high ω, aₙ grows as ω² — if the hoop stress exceeds the material's tensile strength, the object disintegrates. The burst speed of a rotating disc scales as √(σ_UTS / ρ) where σ_UTS is ultimate tensile strength and ρ is density, which is why high-strength, low-density materials like carbon fiber and titanium alloys dominate high-speed rotating machinery.
The calculator first converts the input angle from degrees to radians:
$$\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$$
Then it computes angular velocity:
$$\omega = \frac{\theta_{rad}}{t}$$
From ω, the derived quantities are:
$$\text{Frequency} = \frac{\omega}{2\pi} \quad \text{(Hz)}$$
$$\text{RPM} = \frac{\omega \times 60}{2\pi}$$
$$\text{Period} = \frac{2\pi}{\omega} \quad \text{(seconds per revolution)}$$
These conversions assume uniform (constant-speed) rotation over the given time interval.
A higher angular velocity means faster rotation. Common reference points: a clock second hand has ω ≈ 0.105 rad/s (1 RPM), a car engine idles at about 70–80 rad/s (700–800 RPM), and a typical hard drive spins at about 754 rad/s (7200 RPM). The period tells you how long one complete revolution takes, while frequency counts how many revolutions occur per second.
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A complete 360° turn in 2 s gives ω = π rad/s ≈ 3.14 rad/s, which is 30 RPM or 0.5 Hz.
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100 full rotations per second (6000 RPM) yields ω ≈ 628.3 rad/s, typical of a high-speed turbine.
Angular speed is the magnitude (scalar) of how fast an object rotates, while angular velocity is a vector that also includes the direction of rotation (determined by the right-hand rule). For most calculations, the terms are used interchangeably.
Multiply RPM by 2π/60. For example, 3000 RPM = 3000 × 2π/60 = 314.16 rad/s.
Radians simplify many formulas. The arc length formula s = rθ and derivatives of trigonometric functions only work directly with radians. Radians are also dimensionless, making them compatible with the SI unit system.
Earth rotates 360° in approximately 86,164 seconds (one sidereal day), giving ω ≈ 7.292 × 10⁻⁵ rad/s.
Linear (tangential) velocity equals angular velocity times the radius: v = ωr. Points farther from the axis move faster linearly even though they share the same angular velocity.
Yes. A negative angular velocity indicates rotation in the opposite direction to the defined positive sense. By convention, counterclockwise is usually positive.
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