acalculator

What is the angular velocity?

Choose what you know about the turning, then type the angle and time, the speed and radius, the rpm, or the time for one turn. The angular velocity calculator gives ω in rad/s, °/s, rpm and hertz, and the speed and acceleration at a radius.

Your numbers

Units
From
Answer
ω = 45 rad/s (429.718 rpm)

The angular velocity is 45 rad/s, or 429.718 rpm.

Angular velocity (rad/s)
45
Angular velocity (°/s)
2,578.31
Revolutions per minute (rpm)
429.718
Turns per second (Hz)
7.16197
Time for one turn (s)
0.139626
Tangential speed (m/s)
4.5
Centripetal acceleration (m/s²)
202.5

Answer: ω = 45 rad/s (429.718 rpm). The angular velocity is 45 rad/s, or 429.718 rpm.

How to calculate

Angular velocity calculator: ω = Δθ ÷ Δt from an angle and time, ω = v ÷ r from a speed and radius, or from rpm or the time of one turn, in rad/s, °/s, rpm and Hz.

Example with the default inputs (From Angle and time, Angle turned (Δθ) 1,350, Angle unit rad, Time (Δt) 30 s, Radius (r) 0.328083989501312 ft): The angular velocity is 45 rad/s, or 429.718 rpm.

Method: ω = Δθ ÷ Δt; ω = v ÷ r; ω = 2π × rpm ÷ 60; ω = 2π ÷ T; v = rω; a = rω².

  • The angular velocity is steady over the time given (an average otherwise).
  • The radius is measured from the axis of rotation to the point that moves.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. From Angle and time, Angle turned (Δθ) 1,350, Angle unit rad, Time (Δt) 30 s, Radius (r) 0.3281 ft gives Angular velocity (rad/s) 45, Tangential speed (v) 4.5, Centripetal acceleration (m/s²) 202.5, Revolutions per minute (rpm) 429.718346, Answer ω = 45 rad/s (429.718 rpm).Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §6.3 Centripetal Force (a_c = v² ÷ r = rω²), https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force (retrieved 2026-10-19)
  2. From Turns per minute, Rotation rate (n) 250, Rate unit rpm (rev/min) gives Angular velocity (rad/s) 26.179939, Turns per second (Hz) 4.166667, Time for one turn (T) 0.24.Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19)
  3. From Speed and radius, Tangential speed (v) 30, Radius (r) 0.9843 ft gives Angular velocity (rad/s) 100, Revolutions per minute (rpm) 954.929659, Centripetal acceleration (m/s²) 3,000.Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Equation 10.4 (v_t = rω)
  4. From Time for one turn, Time for one turn (T) 86,164.09054 gives Angular velocity (rad/s) 0.000073, Angular velocity (°/s) 0.004178.Source: NASA JPL Solar System Dynamics, Astrodynamic Parameters (mean sidereal day 86,164.09054 s), https://ssd.jpl.nasa.gov/astro_par.html (retrieved 2026-10-19); OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19)
  5. From Angle and time, Angle turned (Δθ) 90, Angle unit °, Time (Δt) 2 s gives Angular velocity (rad/s) 0.785398, Angular velocity (°/s) 45, Time for one turn (T) 8.Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.3 and 10.4; Example 10.1: a flywheel at 45.0 rad/s turns 1,350 rad in 30 s, and a point 0.1 m out moves at 4.5 m/s; Example 10.2: 250 rpm is 26.2 rad/s), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19)

How it works

Pick what you know:

  • Angle and time: ω = Δθ ÷ Δt.
  • Speed and radius: ω = v ÷ r, from the tangential speed v of a point at radius r.
  • Turns per minute: a rotation rate n in rpm, rev/s, °/s or rad/s: ω = n × 2π ÷ 60 for rpm, n × 2π for rev/s, n × π ÷ 180 for °/s.
  • Time for one turn: ω = 2π ÷ T.

The page gives ω in rad/s and °/s, the turns per minute (ω × 60 ÷ 2π) and per second (ω ÷ 2π, the frequency in Hz), and the time for one turn, T = 2π ÷ ω.

With a radius r (optional, except with a speed), it also gives the tangential speed v = rω and the centripetal acceleration a = rω², in SI units.

Units (exact): 1 rev = 2π rad; 1° = π ÷ 180 rad; 1 ms = 0.001 s; 1 min = 60 s; 1 h = 3,600 s; 1 d = 86,400 s; 1 mm = 0.001 m; 1 cm = 0.01 m; 1 km = 1,000 m; 1 in = 0.0254 m; 1 ft = 0.3048 m; 1 mi = 1,609.344 m; 1 km/h = 5 ÷ 18 m/s; 1 mph = 0.44704 m/s; 1 ft/s = 0.3048 m/s.

Rules

  • Angle from 10⁻¹² to 10¹⁵ and rotation rate from 10⁻¹² to 10¹², each in the unit chosen beside it; time and period from 10⁻¹² s to 10¹² s; speed from 10⁻⁹ m/s to 10⁹ m/s; radius from 10⁻⁹ m to 10⁹ m. Limits include their ends; a value typed in another unit is converted first.
  • With a speed, the radius is needed; without it there is no answer.
  • A result too large or too small to show as a number gives no answer. Within these limits it does not happen.

Output format

The answer line reads ω = a rad/s (b rpm). It and every number show to 6 significant figures, rounded half up, with trailing zeros dropped and thousands commas, and below 10⁻⁶ or from 10¹⁵ up as a × 10ⁿ. Typed values are read back as the exact decimals you typed, and every unit size is an exact fraction times a power of π (1 rpm = π ÷ 30 rad/s), so each result is an exact fraction until it is multiplied or divided by π once (or not at all, as in ω = v ÷ r).

Worked examples by hand

A flywheel. OpenStax Example 10.1: 1,350 rad in 30 s gives ω = 1,350 ÷ 30 = 45 rad/s = 45 × 60 ÷ 2π = 429.718 rpm. A point 0.1 m from the axis moves at v = 0.1 × 45 = 4.5 m/s with a = 0.1 × 45² = 202.5 m/s².

From rpm. OpenStax Example 10.2: 250 rpm = 250 × 2π ÷ 60 = 26.1799 rad/s, which is 250 ÷ 60 = 4.16667 turns per second, and one turn takes 60 ÷ 250 = 0.24 s.

From a speed. A wheel of radius 0.3 m at 30 m/s: ω = 30 ÷ 0.3 = 100 rad/s = 954.930 rpm, and the rim's acceleration is 0.3 × 100² = 3,000 m/s².

The Earth. One turn in a sidereal day, 86,164.09054 s: ω = 2π ÷ 86,164.09054 = 7.29212 × 10⁻⁵ rad/s, or 360 ÷ 86,164.09054 = 0.00417807°/s.

A quarter turn. 90° in 2 s: ω = (π ÷ 2) ÷ 2 = 0.785398 rad/s = 45°/s, and a full turn takes 360 ÷ 45 = 8 s.

Other questions people ask

How do you calculate angular velocity?

Divide the angle turned by the time: ω = Δθ ÷ Δt, in radians per second. A flywheel that turns 1,350 rad in 30 s has ω = 1,350 ÷ 30 = 45 rad/s.

How do you convert rpm to rad/s?

Multiply by 2π (radians in one turn) and divide by 60 (seconds in a minute): ω = rpm × 2π ÷ 60. So 250 rpm = 26.18 rad/s, and 1 rad/s = 9.5493 rpm.

What is the relation between angular velocity and linear speed?

A point at radius r moves along its circle at v = rω, so ω = v ÷ r. A wheel of radius 0.3 m rolling at 30 m/s turns at ω = 30 ÷ 0.3 = 100 rad/s, or 954.93 rpm.

How do you find angular velocity from the period?

One turn is 2π radians, so ω = 2π ÷ T. The Earth turns once in a sidereal day of 86,164.09 s, so ω = 2π ÷ 86,164.09 = 7.2921 × 10⁻⁵ rad/s.

What is the difference between angular velocity and angular frequency?

For steady rotation they have the same value, ω = 2πf, in rad/s. The frequency f is the number of turns per second, in hertz. Angular velocity also has a direction along the axis (by the right-hand rule).

What is the centripetal acceleration at a radius?

a = rω², toward the axis. A point 0.1 m from the axis of a flywheel at 45 rad/s has a = 0.1 × 45² = 202.5 m/s², about 21 g.