Angular velocity calculator

What is angular velocity?

How fast something turns: the angle swept per unit of time, ω = Δθ ÷ Δt, measured in radians per second (OpenStax University Physics). One revolution is 2π radians, or 360°.

How do I convert rpm to rad/s?

Multiply by 2π and divide by 60: ω = rpm × 2π ÷ 60 ≈ rpm × 0.10472. OpenStax’s bicycle wheel at 250 rpm turns at 250 × 2π ÷ 60 = 26.2 rad/s. To go back, rpm = rad/s × 60 ÷ 2π ≈ rad/s × 9.549.

How is angular velocity related to linear speed?

A point at distance r from the axis moves at v = r ω (ω in rad/s). In an OpenStax example a flywheel turning at 45.0 rad/s has a tangential speed of 45.0 × 0.10 m = 4.5 m/s at a radius of 10 cm, and in 30 s it turns 1,350 rad — about 215 revolutions.

What is the centripetal acceleration?

Anything moving in a circle accelerates toward the center at a = v² ÷ r = r ω². For the same flywheel, a point 10 cm out has 0.10 × 45.0² = 202.5 m/s², about 20.6 g. Enter a radius and the calculator shows both.

Is angular velocity the same as angular frequency?

For steady rotation they are the same number in rad/s. The frequency in revolutions per second (hertz) is ω ÷ 2π, and the time for one turn is 2π ÷ ω.

Why does a point farther from the axis move faster?

Every point on a rigid spinning object turns through the same angle in the same time, so all share one angular velocity. A point farther out has to cover a bigger circle in that time, so its linear speed v = r ω is larger. Doubling the radius doubles the speed but leaves ω unchanged.

What is the angular velocity of the Earth?

One turn relative to the stars takes a sidereal day, about 86,164 seconds (23 h 56 min 4 s), so ω = 2π ÷ 86,164 ≈ 7.29 × 10⁻⁵ rad/s. That is a little more than one turn per 24 hours, because the Sun-to-Sun day includes the extra angle the Earth moves along its orbit.

Which direction is positive?

By the usual convention counterclockwise rotation, as seen from above the plane of rotation, is positive and clockwise is negative. The magnitude (the speed of rotation) is always positive; the sign only tells the direction.

How the calculation works

Angular velocity ω is the angle turned divided by the time taken. The calculator works in radians per second internally and converts from and to the other units: revolutions per minute (rpm), degrees per second and revolutions per second. One revolution is 2π radians, which is 360°, so rpm × 2π ÷ 60 gives rad/s. With a radius it also gives the linear speed of a point on the rim, v = r ω, and its acceleration toward the axis, a = r ω². The formulas assume rigid rotation about a fixed axis at a steady rate; if the speed is changing, the result is the average over the interval.

Worked example

A disk spins at 7,200 rpm. In radians per second that is 7200 × 2π ÷ 60 = 754 rad/s, or 43,200 degrees per second, and one turn takes 8.3 milliseconds. A point 5 cm (0.05 m) from the axis moves at v = 0.05 × 754 = 37.7 m/s, and its acceleration toward the axis is 0.05 × 754² = 28,424 m/s², which is about 2,898 times standard gravity (g = 9.80665 m/s²). This is why small, fast rotors need strong mounts.

Common mistakes

• Using rpm in v = r ω. The relations v = r ω and a = r ω² hold only when ω is in radians per second. Convert from rpm first; skipping the factor 2π ÷ 60 makes the speed wrong by a factor of about ten.
• Using the diameter as the radius. The radius is measured from the axis to the point, which is half the diameter for the rim of a wheel.
• Mixing up angular and linear speed. Two points on the same wheel have the same ω but different v. If a problem gives the speed of the car, the wheel's ω is v ÷ r, assuming no slipping.

Sources and related calculators

Definitions and formulas: OpenStax University Physics Volume 1, chapter 10 (Fixed-Axis Rotation), sections on angular velocity and on rotation with constant angular acceleration; the radian as a derived SI unit is described in NIST SP 811. Related: centripetal force calculator, angular momentum calculator and acceleration calculator.