Angular Velocity Calculator

Calculate angular velocity, period, frequency, or linear velocity for rotating objects using ω = 2πf = 2π/T.

🔄 Rotation 📐 ω = 2πf ⚙️ Angular
Input type
Value
⚠️ Enter valid positive numbers.

Understanding Angular Velocity

Angular velocity (ω) measures how fast an object rotates, expressed in radians per second (rad/s). One complete revolution = 2π radians, so ω = 2πf = 2π/T, where f is frequency (Hz) and T is period (seconds). In RPM (revolutions per minute): ω (rad/s) = RPM × 2π/60.

Angular velocity is the rotational analogue of linear velocity. Just as linear velocity v tells you how fast a point moves in a straight line, angular velocity ω tells you how fast an angle changes. The relationship between linear and angular velocity at radius r is v = ω·r — outer edges of a rotating disk move faster than inner edges.

Angular velocity is a vector quantity pointing along the axis of rotation (right-hand rule: curl fingers in direction of rotation, thumb points along ω). This vector nature explains gyroscope stability and precession — changing ω requires a torque, and the direction of ω change determines precession direction.

Angular velocity appears in nearly all rotating machinery: motors (measured in RPM), centrifuges (thousands of RPM), turbines, wheels, gyroscopes, planets, and galaxies. The tangential acceleration of a point on a rotating body is a_t = α·r (where α = dω/dt is angular acceleration), while centripetal acceleration is a_c = ω²r.

Formula Reference Table

Solve ForFormulaNotes
Angular velocity (ω)ω = 2π/T = 2πfrad/s
From RPMω = RPM × 2π/601 RPM = 0.1047 rad/s
Linear velocityv = ω·rm/s at radius r
PeriodT = 2π/ωseconds per revolution
Frequencyf = ω/(2π)Hz = rev/s
Centripetal accela_c = ω²r = v²/rInward acceleration

3 Worked Examples

Example 1
Car Wheel at 60 mph

A car wheel has diameter 0.65 m (r = 0.325 m). At 60 mph (26.8 m/s), find angular velocity.

  • v = ω·r → ω = v/r = 26.8/0.325 = 82.5 rad/s
  • f = ω/(2π) = 82.5/6.283 = 13.1 Hz
  • RPM = 13.1 × 60 = 788 RPM
✓ ω = 82.5 rad/s (788 RPM)
Example 2
Hard Drive Platters at 7200 RPM

Find angular velocity and rim speed of a 3.5-inch HDD (r = 44.5 mm = 0.0445 m) spinning at 7200 RPM.

  • ω = 7200 × 2π/60 = 753.9 rad/s
  • v_rim = ω × r = 753.9 × 0.0445 = 33.5 m/s = 121 km/h
✓ ω = 754 rad/s; rim speed = 33.5 m/s
Example 3
Earth's Rotation

Earth rotates once per 24 hours (86,400 s). Find ω and equatorial surface speed.

  • ω = 2π/T = 2π/86,400 = 7.27×10⁻⁵ rad/s
  • v = ω × r_Earth = 7.27×10⁻⁵ × 6.371×10⁶ = 463 m/s (1,668 km/h)
✓ ω = 7.27×10⁻⁵ rad/s; equatorial speed = 463 m/s

Real-World Applications

⚙️
Motors & Engines
Motor speed is specified in RPM. Converting to ω (rad/s) is required for power calculations (P = τ·ω) and centripetal force calculations for rotating components.
🏎️
Wheels & Tires
Wheel RPM determines car speed: v = ω × r = (RPM × 2π/60) × r_tire. Speedometer calibration depends on tire diameter — changing tire size without recalibrating causes speedometer error.
💿
Hard Drives & Optical Media
HDD platters spin at 5,400–15,000 RPM. Data access time depends on how long before the right sector rotates under the read head — directly determined by ω.
🌪️
Turbines & Generators
Synchronous generators must maintain precise ω to produce 50 Hz (ω = 314 rad/s) or 60 Hz (ω = 377 rad/s) electricity. Any deviation causes frequency problems on the power grid.
🌍
Planetary Science
Earth's angular velocity (7.27×10⁻⁵ rad/s) determines day length, equatorial bulge, and Coriolis effect. The Moon's decreasing orbital angular momentum (due to tidal friction) causes Earth's rotation to slow ≈ 1.4 ms/century.

Common Mistakes to Avoid

⚠️
Mixing degrees and radians

ω = 2π/T gives rad/s. Do not use degrees/s in equations — all physics formulas (centripetal force, rotational KE) require rad/s. Convert: 1 rev = 360° = 2π rad; 1 deg/s = π/180 rad/s.

⚠️
Confusing RPM with rad/s

RPM (revolutions per minute) ≠ rad/s. 1 RPM = 2π/60 ≈ 0.1047 rad/s. Car engines at 3,000 RPM: ω = 3000 × 0.1047 = 314 rad/s.

⚠️
Using v instead of ω in rotational formulas

Rotational KE = ½Iω², not ½Iv². Torque-power: P = τω, not τv. Always use ω (rad/s) in rotational mechanics formulas.

⚠️
Forgetting v = ωr direction

v = ω × r is a cross product — the linear velocity is tangential (perpendicular to the radius), not radial. The speed magnitude |v| = ω|r|, but the direction is tangential.

⚠️
Assuming uniform ω when it varies

If angular velocity changes with time, the object has angular acceleration α = dω/dt. Don't use constant ω equations if the rotation is speeding up or slowing down — use rotational kinematics instead.

Frequently Asked Questions

What is the difference between angular velocity and angular speed? ▾
Angular speed is the magnitude |ω| (rad/s). Angular velocity is the full vector, including direction (along the rotation axis, determined by the right-hand rule). For most scalar calculations, they are used interchangeably, but for gyroscope and rigid body dynamics, the vector nature matters.
How do I convert between RPM, Hz, and rad/s? ▾
RPM → rad/s: ω = RPM × 2π/60. RPM → Hz: f = RPM/60. Hz → rad/s: ω = 2πf. rad/s → RPM: RPM = ω × 60/(2π). Example: 1500 RPM = 25 Hz = 157.1 rad/s.
What is angular acceleration? ▾
α = dω/dt = Δω/Δt (rad/s²) — the rate of change of angular velocity. The rotational analogue of linear acceleration. Torque τ = Iα where I is moment of inertia. SI unit: rad/s².
Why do outer edges of a disk move faster? ▾
v = ωr. For a rigid rotating object, all points share the same ω. But r increases from center to edge, so v increases proportionally. A CD with ω = 50 rad/s: inner edge (r = 25 mm) moves at 1.25 m/s; outer edge (r = 58 mm) at 2.9 m/s.
What is precession? ▾
When a gyroscope (spinning top, gyroscope) has an external torque applied, instead of falling over (as a stationary object would), it precesses — the spin axis slowly rotates around the torque direction. Precession rate Ω = τ/(Iω). Gyroscopes in spacecraft and aircraft navigation exploit this effect.
How does Earth's angular velocity affect daily life? ▾
Earth's rotation (ω = 7.27×10⁻⁵ rad/s) creates the Coriolis effect — deflecting winds and ocean currents (right in NH, left in SH). Hurricane rotation, the long-range path of artillery shells, and the direction water drains are all influenced by Earth's rotation, though the toilet-drain myth is false at small scales.
What is moment of inertia and why does it interact with ω? ▾
Moment of inertia I (kg·m²) is the rotational analogue of mass — it measures resistance to rotational acceleration. Angular momentum L = Iω. Conservation of L explains why a figure skater spins faster when pulling in arms (I decreases, ω increases proportionally).
What is the relationship between angular velocity and centripetal acceleration? ▾
a_c = ω²r (equivalent to v²/r since v = ωr). The centripetal acceleration at radius r depends on ω² and r. At the equator (r = 6.371×10⁶ m), Earth's ω creates a_c = (7.27×10⁻⁵)² × 6.371×10⁶ = 0.0337 m/s² — this slightly reduces effective g from 9.8 to ~9.766 m/s² at the equator.

Related Calculators

🔄 Centripetal Force ⚙️ Torque 💫 Angular Momentum 🎡 Moment of Inertia 🚀 Orbital Velocity

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

Atwood Machine Calculator →Binding Energy Per Nucleon Calculator →Buoyancy Force Calculator →Physics Formula Explorer →