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Angular Velocity Formula

Rotation and circular motion formulasVectorClass 11Updated 1 Nov 2026

ω=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t}
SI unit
rad/s
Dimensions
[M⁰ L⁰ T⁻¹]
where
  • ω angular velocity rad/s
  • Δθ angular displacement rad
  • Δt time taken s

The angular velocity formula is ω=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t}: the angle an object turns through, in radians, divided by the time it takes. For steady rotation you can also write ω=2πT=2πf\omega = \frac{2\pi}{T} = 2\pi f, where T is the time for one turn and f is the number of turns per second.

Angular velocity tells you how quickly something spins, whatever its size. Every point on a rigid wheel has the same ω, while points farther from the axle move faster in metres per second, as v=ωrv = \omega r shows. It is a core idea of the Class 11 rotation chapter and of the circular motion formulas section on this site.

Quick facts

  • Formula

    ω=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t}

  • Steady spin

    ω=2πT=2πf\omega = \frac{2\pi}{T} = 2\pi f

  • Link to speed

    v=ωrv = \omega r

  • SI unit

    radian per second (rad/s)

  • Dimensional formula

    [M⁰ L⁰ T⁻¹]

  • Type

    Vector along the axis (right hand rule)

  • From rpm

    ω=2πN60\omega = \frac{2\pi N}{60}

Each part of the angular velocity formula

Δθ\Delta\theta is the angular displacement: the angle swept by a line drawn from the axis to the moving point. It must be in radians. One full turn is 2π2\pi rad, about 6.28 rad, and 1 rad is about 57.3°. Δt\Delta t is the time in seconds. The ratio is the average angular velocity; shrink the interval to zero and you get the instantaneous value ω=dθdt\omega = \frac{d\theta}{dt}.

Ways to write ω
If you knowUseNote
Angle and timeω=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t}Angle in rad
Period TTω=2πT\omega = \frac{2\pi}{T}T in s
Frequency ffω=2πf\omega = 2\pi ff in Hz
Speed vv and radius rrω=vr\omega = \frac{v}{r}v in m/s, r in m
rpm NNω=2πN60\omega = \frac{2\pi N}{60}N in rev/min

For the time and frequency forms, see the period formula and the frequency formula.

Turning the formula into an answer

  1. Write down which quantities the question gives: angle, period, rpm or linear speed.
  2. Convert degrees to radians (multiply by π/180) and revolutions to radians (multiply by 2π).
  3. Pick the matching row from the table above and substitute with units.
  4. If the question then asks for a speed, use v=ωrv = \omega r with r in metres.
A radian is arc length over radius, so rad/s has the same dimensions as 1/s. Write rad/s anyway; examiners expect it.

Worked examples

Example 1: a wheel turning steadily

A bicycle wheel turns through 12 rad in 4 s. Find its angular velocity.

ω=124=3 rad/s\omega = \frac{12}{4} = 3\ \text{rad/s}

Example 2: a ceiling fan in rpm

A ceiling fan turns at 300 rpm. Its blade tip is 0.6 m from the axis. Find ω and the tip speed.

ω=2π×30060=10π≈31.4 rad/s\omega = \frac{2\pi \times 300}{60} = 10\pi \approx 31.4\ \text{rad/s}

v=ωr=31.4×0.6≈18.8 m/sv = \omega r = 31.4 \times 0.6 \approx 18.8\ \text{m/s}

Example 3: the spinning Earth

Earth turns once in about 24 h. Find its angular velocity.

T = 24 × 3600 = 86400 s. ω=2π86400≈7.27×10−5 rad/s\omega = \frac{2\pi}{86400} \approx 7.27 \times 10^{-5}\ \text{rad/s}.

Using the true spin period of about 23.93 h gives 7.29×10−57.29 \times 10^{-5} rad/s, a value you may also meet.

Angular velocity versus linear velocity

Two children on a merry-go-round complete each lap together, so they share one ω. The child at the rim covers a bigger circle in the same time, so her linear speed is higher. That is the meaning of v=ωrv = \omega r. The linear velocity formula page treats the speed side in more depth.

Squaring the relation gives the inward acceleration of circular motion, ac=ω2ra_c = \omega^2 r, which leads to the centripetal acceleration formula. When ω itself changes, the rate of change is the angular acceleration α, and multiplying ω by the moment of inertia gives angular momentum.

Direction, unit and dimensions

Angular velocity is a vector. Curl the fingers of your right hand the way the body turns and your thumb points along ω⃗\vec{\omega}. For a clock face seen from the front, the hands turn clockwise, so ω points into the wall.

Because the radian is dimensionless, ω carries only an inverse time, written [M⁰ L⁰ T⁻¹]. Frequency and angular velocity share these dimensions but differ by the factor 2π2\pi, so do not swap them. The dimensional formula of angular velocity page shows the full derivation.

Common errors in Class 11 tests

  • Putting degrees into v=ωrv = \omega r. The relation is true only when the angle is in radians.
  • Using rpm as if it were rad/s. A wheel at 60 rpm has ω = 2π rad/s, not 60.
  • Thinking ω differs across a rigid body. It is the same for every point; only v changes with r.
  • Confusing ω with f. ω = 2πf, so a 50 Hz motor runs at about 314 rad/s.

Frequently asked questions

What is the formula of angular velocity?

Angular velocity is ω = Δθ / Δt, the angle in radians divided by time. For uniform rotation, ω = 2π / T = 2πf, and it links to linear speed through v = ωr.

Which unit should I write for angular velocity?

Write radian per second, rad/s. The radian is a pure ratio, so in base dimensions ω reduces to [M⁰ L⁰ T⁻¹], an inverse second.

What is the formula for angular velocity in terms of rpm?

Multiply the rpm by 2π and divide by 60: ω = 2πN / 60. For example, 120 rpm gives 4π rad/s, about 12.6 rad/s.

Is angular velocity a vector?

Yes. It points along the axis of rotation, and the right hand rule gives which way.

What is the difference between angular velocity and angular frequency?

For uniform circular motion and for simple harmonic motion they have the same value and unit, 2πf. Angular velocity describes an actual rotation, while angular frequency is the rate at which the phase of an oscillation advances.