Angular Velocity Calculator
Convert between RPM, radians per second, degrees per second, and Hz. Also calculates linear velocity (v = ωr) and centripetal acceleration for a given radius. See also our Centrifugal Force Calculator and Speed Calculator.
How to Calculate Angular Velocity
Angular velocity (ω) measures how fast an object rotates — the rate of change of angular position. While linear velocity tells you how fast something moves in a straight line (m/s), angular velocity tells you how fast it spins (rad/s). Every rotating object — from a spinning top to a planet — has an angular velocity. It is the rotational analog of linear velocity.
The SI unit of angular velocity is radians per second (rad/s), but engineers commonly use RPM (revolutions per minute) and physicists sometimes use degrees per second. The conversions are: ω (rad/s) = 2π × RPM / 60 = 2π × f (Hz) = degrees/s × π/180. One complete revolution equals 2π radians or 360 degrees, so 1 RPM = 2π/60 ≈ 0.1047 rad/s.
Angular velocity connects to linear velocity through the radius: v = ωr. A point on the rim of a wheel moves faster than a point near the center, even though both have the same angular velocity. This relationship is crucial for calculating belt speeds, gear ratios, tire speeds, and the tangential velocity of any rotating component. The centripetal acceleration is a = ω²r = v²/r.
Angular Velocity Formulas
Angular Velocity:
ω = Δθ/Δt (rad/s)
ω = 2πf = 2π/T = 2π×RPM/60
Conversions:
1 RPM = 2π/60 = 0.10472 rad/s
1 rad/s = 60/(2π) = 9.5493 RPM
1 rad/s = 180/π = 57.296 °/s
1 Hz = 2π rad/s = 60 RPM
Linear Velocity:
v = ωr (tangential velocity)
Centripetal Acceleration:
a = ω²r = v²/r
Angular Acceleration:
α = Δω/Δt (rad/s²)
Example Calculation
A motor spins at 1000 RPM. A pulley with radius 0.5 m is attached. Find the angular velocity and belt speed:
Given: RPM = 1000, r = 0.5 m
ω = 2π × 1000/60 = 104.72 rad/s
= 6000 °/s = 16.67 Hz
Linear (belt) speed: v = ωr = 104.72 × 0.5 = 52.36 m/s
= 188.5 km/h = 117.1 mph
Period: T = 2π/ω = 0.06 s (60 ms per revolution)
Centripetal acceleration: a = ω²r = 10966 m/s² = 1118 g
Angular Velocity Reference Table
| Object | RPM | rad/s | °/s |
|---|---|---|---|
| Clock second hand | 1 | 0.1047 | 6 |
| Clock minute hand | 0.0167 | 0.00175 | 0.1 |
| Vinyl record (33⅓) | 33.33 | 3.49 | 200 |
| CD/DVD | 200-500 | 21-52 | 1200-3000 |
| Washing machine | 1200 | 125.7 | 7200 |
| Car engine (idle) | 800 | 83.8 | 4800 |
| Car engine (redline) | 6000 | 628.3 | 36000 |
| Helicopter rotor | 300-500 | 31-52 | 1800-3000 |
| Jet turbine | 10000-20000 | 1047-2094 | 60000-120000 |
| Dental drill | 400000 | 41888 | 2.4×10⁶ |
| Earth rotation | 0.000694 | 7.27×10⁻⁵ | 0.00417 |
Frequently Asked Questions
What is angular velocity?
Angular velocity (ω) is the rate at which an object rotates around an axis, measured in radians per second (rad/s). It describes how quickly the angular position changes: ω = Δθ/Δt. Unlike linear velocity which has units of distance/time, angular velocity has units of angle/time. It is a vector quantity — its direction is along the axis of rotation (determined by the right-hand rule). All points on a rigid rotating body share the same angular velocity.
What is the difference between RPM and rad/s?
RPM (revolutions per minute) counts complete rotations per minute — it is intuitive and widely used in engineering. Rad/s (radians per second) is the SI unit used in physics equations. One revolution = 2π radians, and there are 60 seconds per minute, so: ω (rad/s) = RPM × 2π/60. RPM is preferred for motors, engines, and machinery specifications. Rad/s is required for physics formulas like F = mω²r, τ = Iα, and v = ωr.
How does angular velocity relate to linear velocity?
Linear (tangential) velocity equals angular velocity times radius: v = ωr. This means points farther from the rotation axis move faster in a straight line, even though all points rotate at the same angular rate. A point on the rim of a 1-meter wheel spinning at 10 rad/s moves at 10 m/s, while a point 0.1 m from center moves at only 1 m/s. This relationship is fundamental to gear design, belt drives, and understanding tire speed.
What is angular acceleration?
Angular acceleration (α) is the rate of change of angular velocity: α = Δω/Δt, measured in rad/s². It is the rotational analog of linear acceleration. A motor speeding up from 0 to 3000 RPM in 2 seconds has α = (314.16 - 0)/2 = 157.08 rad/s². Newton's second law for rotation is τ = Iα, where τ is torque and I is moment of inertia. Angular acceleration determines how quickly a rotating system can speed up or slow down.
Why do we use radians instead of degrees?
Radians are the natural unit for angles in physics because they simplify formulas. With radians: arc length s = rθ, v = ωr, a = αr — no conversion factors needed. With degrees, you would need π/180 factors everywhere. One radian is the angle where arc length equals radius (~57.3°). The full circle is 2π radians. Calculus with trigonometric functions also requires radians: d/dx[sin(x)] = cos(x) only when x is in radians.
How do gear ratios affect angular velocity?
When two gears mesh, their linear velocities at the contact point are equal: ω₁r₁ = ω₂r₂. The gear ratio is ω₁/ω₂ = r₂/r₁ = N₂/N₁ (where N is number of teeth). A small gear driving a large gear reduces speed but increases torque (and vice versa). A 20-tooth gear driving a 60-tooth gear gives a 3:1 reduction — output speed is 1/3 of input, but torque is 3× greater. This is how transmissions trade speed for force.
Applications of Angular Velocity
Angular velocity is essential in mechanical engineering (motor and turbine design, gear systems, flywheels), automotive (engine RPM, wheel speed, transmission ratios), aerospace (propeller and turbine speeds, satellite rotation), robotics (joint velocities, servo control), and physics (planetary motion, gyroscopes, particle accelerators). Understanding angular velocity and its relationship to linear velocity, torque, and energy is fundamental to designing any rotating system.