The angular velocity formula

Angular velocity, ω, is how fast something turns: the angle it sweeps out each second, measured in radians per second (rad/s). The basic angular velocity formula is ω = θ ÷ t. For an object moving in a circle it can also be written ω = v ÷ r, where v is the speed and r the radius, or ω = 2π ÷ T = 2πf, where T is the time for one turn and f is the frequency. A wheel turning once a second has ω = 2π ≈ 6.28 rad/s.

The angular velocity formula in every form

They all say the same thing. One full turn is 2π radians, so turning once every T seconds means sweeping 2π radians in T seconds: ω = 2π ÷ T. Since f = 1 ÷ T, that is also 2πf. AQA's A-level data booklet gives ω = v ÷ r and ω = 2πf under circular motion.

The angular velocity symbol and units

The symbol for angular velocity is ω, the lower case Greek letter omega. Its SI unit is the radian per second, written rad/s or rad s⁻¹. The radian is the natural unit for angles in physics: 2π rad = 360°, so 1 rad ≈ 57.3°. Always use radians in these formulas. If an angle is given in degrees, convert it first by multiplying by π ÷ 180.

Angular velocity and angular speed

At A-level the two words are used for the same number. AQA's specification lists the magnitude of angular speed, ω = v ÷ r = 2πf, and says the direction of angular velocity will not be considered. At university, angular velocity is a vector: OpenStax's University Physics explains that it points along the axis of rotation, with its direction given by the right-hand rule. Curl the fingers of your right hand the way the object turns, and your thumb points along ω.

Linear velocity and angular velocity

Every point on a turning wheel has the same angular velocity: they all go round once in the same time. But points further from the centre travel further in that time, so they move faster. The link is:

To turn linear velocity into angular velocity, divide by the radius: ω = v ÷ r. A point on the rim of a 0.3 m radius wheel moving at 6 m/s has ω = 6 ÷ 0.3 = 20 rad/s, and a point halfway out, at 0.15 m, moves at only 20 × 0.15 = 3 m/s.

Converting rpm to rad/s

Rotation speeds are often given in revolutions per minute, rpm. Each revolution is 2π radians and each minute is 60 seconds, so:

A motor spinning at 3000 rpm has ω = 3000 × 2π ÷ 60 = 314 rad/s. To go back from rad/s to rpm, multiply by 60 ÷ 2π, about 9.55.

The Circular Motion Calculator finds angular velocity from speed and radius or from the period, and works out speed, period, centripetal acceleration and force from it, with every step shown.

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Circular Motion Calculator

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Angular velocity worked examples

Example 1: from an angle and a time

A turntable rotates through 12 radians in 4 seconds. Find its angular velocity.

Example 2: a fan, from rpm

A ceiling fan spins at 900 rpm. Its blades reach 0.30 m from the centre. Find the angular velocity and the speed of a blade tip.

Keep the unrounded ω, 94.248 rad/s, for the second step.

Example 3: a clock's minute hand

A minute hand is 12 cm long. Find its angular velocity and the speed of its tip.

It turns once every hour, so T = 3600 s:

The tip moves at v = ωr = 1.745 × 10⁻³ × 0.12 = 2.09 × 10⁻⁴ m/s, about 0.2 mm per second. The hour hand turns 12 times more slowly, at 1.45 × 10⁻⁴ rad/s.

Example 4: a car wheel

A car travels at 18 m/s on wheels of radius 0.30 m. Find the angular velocity of a wheel, its frequency and its rpm.

If the tyre does not slip, the rim moves at the car's speed, so:

Then f = ω ÷ 2π = 60 ÷ 2π = 9.55 Hz, which is 9.55 × 60 = 573 rpm.

Example 5: centripetal force from ω

A 0.2 kg ball on a string moves in a horizontal circle of radius 0.5 m at 6 rad/s. Find the tension in the string, ignoring gravity.

The tension provides the centripetal force, F = mω²r:

Here is a quick check before moving on.

Angular frequency

In oscillations and waves, ω = 2πf is called the angular frequency. Nothing has to go round in a circle: a mass on a spring or a pendulum bob just moves back and forth. AQA's booklet uses ω in the simple harmonic motion equations, such as a = −ω²x and x = A cos(ωt). The link is that SHM looks like the shadow of a point moving round a circle at a steady angular velocity ω.

For example, a mass on a spring that oscillates with a period of 0.40 s has angular frequency ω = 2π ÷ 0.40 = 15.7 rad/s.

Angular frequency vs angular velocity

They share a symbol, a unit and the formula 2πf. Angular velocity describes something actually rotating. Angular frequency describes how quickly a repeating motion goes through its cycle, measured as if each cycle were a turn of 2π radians.

Centripetal acceleration and force with ω

Writing v = ωr into the circular motion equations gives versions in terms of angular velocity. AQA's data booklet gives both forms:

Use the ω form when you know the angular velocity or the period, which saves converting to v first.

Common mistakes

• Using degrees. ω must be in rad/s. A full turn is 2π rad, not 360.
• Not converting rpm. Multiply by 2π and divide by 60 before using any formula.
• Mixing up f and ω. f is turns per second; ω is radians per second. ω = 2πf, so ω is always about 6.28 times bigger.
• Using the diameter instead of the radius in v = ωr.
• Thinking outer points turn faster. They move faster, but every point on a rigid object has the same ω.
• Rounding ω too early in a two-step problem.

Practise circular motion in a short timed paper

A-level Physics: Year 2 and University Physics: Year 1 each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so circular motion and oscillations turn up regularly but not in every paper.

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