ω = Δθ / Δt — angular speed in rad/s, RPM and deg/s.
The Average Angular Speed Calculator applies the kinematic definition ω = Δθ / Δt to compute the average angular speed of any rotating object. Enter the angle swept in revolutions, degrees or radians and the elapsed time, and the calculator returns ω in rad/s, RPM and deg/s simultaneously.
Optionally enter a radius to also compute the equivalent linear (tangential) speed v = ω × r.
Average Angular Speed is calculated from angle swept Δθ and elapsed time Δt: ω̄ = Δθ / Δt. The Average Angular Speed Calculator reports this in rad/s, RPM, deg/s.
Industrial motors average 100–3600 RPM. Bicycle cranks average 80–100 RPM. Turbojet shafts average 8000–20000 RPM. Hard-disk platters average 5400–15000 RPM.
Average Angular Speed is a statistical / derived quantity rather than a measured distance ÷ time. It describes the average behaviour of a population — gas molecules, orbiting bodies, fluid parcels or rotating points — at equilibrium.
The Average Angular Speed formula is ω̄ = Δθ / Δt. Symbolically: ω̄ = Δθ / Δt.
The formula has these rearrangements that solve for any unknown:
The output unit depends on the input units. SI inputs (kelvin, kg/mol or kg, m³/s, m²) produce rad/s.
To calculate average angular speed:
Worked example: A flywheel turns 50 revolutions in 4 s. Δθ = 50 × 2π = 314.16 rad. ω̄ = 314.16 / 4 = 78.54 rad/s = 750 RPM.
Three steps:
Enter angle swept (rad, rev or deg) and elapsed time. The calculator converts between rad/s, RPM and deg/s and links to linear speed v = ω r.
Substitute the physical inputs directly into the formula. Unlike distance / time tools, Average Angular Speed doesn't need a measured trip — it derives from a state variable (temperature, mass, radius).
Worked example: A flywheel turns 50 revolutions in 4 s. Δθ = 50 × 2π = 314.16 rad. ω̄ = 314.16 / 4 = 78.54 rad/s = 750 RPM.
Switch input units freely; the calculator does the conversion before substituting.
Rearrange the formula to solve for any input given the output. The calculator inverts ω̄ = Δθ / Δt for you.
Worked example: Solving for time given ω̄ and Δθ: t = Δθ / ω̄. To complete 1000 revolutions at 200 RPM, t = 1000 × 60 / 200 = 300 s.
This is useful for planning — e.g. finding the temperature required to reach a target average angular speed, or the orbital radius for a target rad/s speed.
For a gearbox, output ω̄ = input ω̄ × gear ratio. Average across multiple stages = product of ratios.
For example, comparing two different operating conditions side-by-side highlights the inverse-square / square-root scaling that governs rotational motion.
Convert minutes / hours to seconds (×60, ×3600) before plugging into ω̄. RPM already factors per-minute.
Where a derived time (e.g. orbital period, mean-free-path) is requested, convert units in the calculator's results panel rather than in the input form.
For a shaft that accelerates and then runs steady, split into legs: total angle = Σ ωᵢ tᵢ. The session average is total Δθ / total Δt.
Segment-by-segment analysis is most useful when the input variable changes — e.g. temperature ramp, orbital perihelion → aphelion, pipe diameter step-down.
Average Angular Speed is normally reported in rad/s. Common alternative units:
The calculator handles all conversions automatically.
Average Angular Speed is a scalar — magnitude only. Velocity is a vector that adds direction.
For a rotating body in a thermal / isotropic situation, the average velocity is zero (directions cancel out) even though the average speed is large. Average Angular Speed reflects the typical magnitude that matters for kinetic energy, mean free path or transport.
Average Angular Speed is the population / time-averaged value. Instantaneous average angular speed is the value at one moment for one rotating body.
For gas molecules, instantaneous speeds follow the Maxwell–Boltzmann distribution; the average is the central tendency, not the value any individual molecule has.
Constant average angular speed means the value doesn't change with time. Average Angular Speed can equal that constant if conditions are steady (constant T, constant r, constant Q), but the moment one input drifts, the average drifts with it.
For practical purposes treat Average Angular Speed as a snapshot of the system's current state — re-evaluate whenever an input changes.
An angular-speed-time graph behaves like a linear v-t graph: the area under the ω(t) curve equals total angle Δθ.
The "graph" for a physics-style average average angular speed is usually a distribution plot rather than a v(t) trace — the average corresponds to the area-weighted centroid of the distribution.
Replace velocity with signed angular velocity. Positive areas are one rotation direction; negative areas reverse rotation.
For isotropic systems, plotting one Cartesian component of velocity yields a Gaussian centred at zero whose width sets the magnitude of Average Angular Speed.
There are several common mistakes when computing average angular speed. Click each card below to expand the explanation.
Practice the following worked average angular speed problems. Click "Show Solution" to reveal the step-by-step answer.
Δθ = 1200 × 2π = 7540 rad. ω̄ = 7540 / 30 = 251.3 rad/s = 2400 RPM.
1750 × 2π / 60 = 183.3 rad/s.
ω = 18 × 2π/60 = 1.885 rad/s. v = ω r = 1.885 × 60 = 113.1 m/s (407 km/h).
Gears share linear belt velocity v at the pitch line. ω = v / r → smaller r → larger ω. The ratio of speeds equals the inverse ratio of radii.
Average angular speed ω = Δθ ÷ Δt, where Δθ is the angle swept (radians, degrees or revolutions) and Δt is the elapsed time. SI unit is radians per second (rad/s).
Multiply RPM by 2π/60. So 60 RPM = 60 × 0.10472 ≈ 6.28 rad/s. Conversely, 1 rad/s ≈ 9.5493 RPM.
Linear (tangential) speed v = ω × r, where r is the radius. A wheel rotating at 100 rad/s with a 0.3 m radius has a linear speed of 30 m/s at its edge.
Angular speed (ω) is the scalar magnitude. Angular velocity is the vector version, with direction given by the right-hand rule perpendicular to the plane of rotation.
Earth: 7.27 × 10⁻⁵ rad/s (one rotation in 24 h). Car engine at 3000 RPM: 314 rad/s. Hard-drive at 7200 RPM: 754 rad/s. Dentist's drill: ~40 000 rad/s. Centrifuge: up to 100 000 rad/s.
Same magnitude, different concept. Speed is a scalar (magnitude only); velocity is a vector (magnitude + direction). For pure rotation about a fixed axis, the magnitudes are identical.