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Walk-through

How to Use This Calculator

3 steps

1

Choose how you know your rotation

Pick the mode that matches what you already have: RPM (revolutions per minute, e.g. from a motor or turntable spec), rotation period (seconds for one full turn), or linear speed and radius (for a point moving along the edge of a spinning object).

2

Enter your value and read ω

Type in your number and the calculator instantly returns the angular velocity (ω) in radians per second, along with the equivalent RPM and degrees per second. If you also entered a radius, the tangential (linear) speed at that radius is shown too.

3

Cross-check with the other tabs

Use From Linear to solve ω = v / r directly from a speed and a radius, or Conversions to instantly translate any single rad/s, RPM, or deg/s value into the other two — useful for checking a spec sheet or homework answer.

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Reference

Formula & Methodology

1 formula

Angular velocity

ω = 2πf = 2π·RPM/60 = v/r = 2π/T

Angular velocity (ω, in radians per second) measures how fast something rotates. It equals 2π times the rotation frequency f (in revolutions per second), or 2π·RPM/60 when RPM is revolutions per minute, or the linear (tangential) speed v divided by the radius r at which that speed is measured, or 2π divided by the rotation period T (seconds per revolution). All four forms describe the same physical quantity and are mathematically interchangeable.

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Glossary

Key Terms Explained

6 terms

Angular velocity (ω) The rate at which an object rotates around an axis, measured in radians per second (rad/s). It is a vector quantity in full physics treatments, but for a single spinning object its magnitude alone is usually what's reported.

Radian The SI unit of angle. One full revolution equals 2π radians (≈ 6.2832 rad), so 1 radian ≈ 57.3°. Radians are the natural unit for angular velocity because they make the relationship v = ωr work without extra conversion factors.

RPM Revolutions per minute — a common non-SI unit for rotational speed, especially for motors, engines, and drives. RPM = ω × 60 / (2π), or equivalently ω = RPM × 2π / 60.

Period (T) The time, in seconds, for one complete revolution. Period and angular velocity are inversely related: ω = 2π / T, and T = 2π / ω.

Frequency (f) The number of revolutions completed per second, measured in hertz (Hz). Frequency and angular velocity are related by ω = 2πf, and f = ω / (2π). RPM is simply frequency expressed per minute instead of per second.

Tangential (linear) speed The straight-line speed of a specific point on a rotating object, measured in the same units as ordinary speed (e.g. m/s). It depends on both the angular velocity and the distance from the axis: v = ωr. Every point on a rigid rotating body shares the same ω, but points farther from the axis move faster in a straight line.

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Scenarios

Real-World Examples

3 worked examples

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Motor spec sheet

60 RPM turntable

RPM 60 Radius 0

60 RPM means one revolution per second. Converting: ω = 60 × 2π / 60 = 2π ≈ 6.283 rad/s. This is the classic textbook conversion — 60 RPM always equals 2π rad/s, regardless of the object's size.

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Bicycle wheel

Point on the rim

Linear speed (v) 3.14 m/s Radius (r) 0.5 m

A point on the rim of a 0.5 m radius wheel moving at 3.14 m/s implies ω = v/r = 3.14/0.5 = 6.28 rad/s — almost identical to the 60 RPM example above, showing that RPM and a matching linear-speed/radius pair describe the same rotation.

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Unit conversion

rad/s to deg/s

Angular velocity (ω) 6.283 rad/s Convert to deg/s

Multiply by 180/π: 6.283 × 180 / π ≈ 360 deg/s. That makes sense — 2π rad/s is exactly one full 360° rotation every second, so the two units agree perfectly at this value.

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Deep Dive

RPM, rad/s, and linear speed — one conversion

3 chapters · 4 min read

Angular velocity shows up under three different names depending on the field: engineers quote RPM, physicists write ω in rad/s, and anyone standing next to a spinning wheel cares about how fast a point on its edge is actually moving. They're all the same underlying quantity, just expressed in different units or from a different vantage point.

How the Angular Velocity Calculator works

The calculator accepts whichever value you have — RPM, a rotation period, or a linear speed with a radius — and converts it to angular velocity in radians per second using the identity ω = 2πf = 2π·RPM/60 = v/r = 2π/T. Radians are used internally because they're the only angle unit that makes v = ωr true without a conversion constant: a point at radius r on an object spinning at ω rad/s moves along the circle at exactly v = ωr meters per second.

Inputs and what they mean

RPM (revolutions per minute) is the most common spec-sheet unit for motors, fans, and drives — default 60. Rotation period (T, seconds per revolution) is the inverse of frequency and is handy when you've timed a rotation with a stopwatch. Linear speed (v, m/s) and radius (r, m) together describe a specific point on a rotating body; radius is optional in the RPM and period modes but unlocks the linear-speed readout when supplied. Whichever mode you use, the result panel always reports the same ω in rad/s, RPM, and degrees per second so you can cross-check against whatever unit your source material uses.

Limits and edge cases

Angular velocity as computed here describes rigid-body rotation at a constant rate — it doesn't account for angular acceleration, wobble, or non-circular motion. A radius of zero or a negative value has no physical meaning for the linear-speed link and is rejected. For very high speeds (e.g. turbine or centrifuge RPM), double-check that your RPM figure is genuinely revolutions per minute and not already converted to Hz or rad/s, since mixing units at that stage is the most common source of a 60× or 2π-scale error.

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Questions

Frequently Asked Questions

6 questions

How do I convert RPM to rad/s?+

Multiply by 2π and divide by 60: ω (rad/s) = RPM × 2π / 60. For example, 60 RPM = 60 × 2π / 60 = 2π ≈ 6.283 rad/s. The Angular Velocity Calculator does this automatically — just select RPM mode and enter your value.

How is angular velocity linked to linear (tangential) speed?+

Through v = ωr, where v is the linear speed of a point at distance r from the axis, and ω is the angular velocity in rad/s. Every point on a rotating rigid body shares the same ω, but points farther from the axis move faster in a straight line because their v is larger for the same ω.

What units does angular velocity use?+

The SI unit is radians per second (rad/s). It's also commonly expressed in RPM (revolutions per minute) for machinery, or degrees per second for graphics and robotics work. All three are exact conversions of the same quantity — this calculator's Conversions tab switches between them instantly.

How do I find angular velocity from the rotation period?+

Divide 2π by the period: ω = 2π / T, where T is the time in seconds for one full revolution. A shorter period means a faster spin and a larger ω — for example, T = 1 second gives ω = 2π ≈ 6.283 rad/s, the same result as 60 RPM.

How do I convert rad/s to degrees per second?+

Multiply by 180/π, since one radian equals 180/π ≈ 57.3 degrees. So deg/s = rad/s × 180/π. As a sanity check, 2π rad/s (one full revolution per second) always converts to exactly 360 deg/s.

How is angular velocity related to frequency?+

ω = 2πf, where f is the rotation frequency in revolutions per second (hertz). Conversely, f = ω / (2π). RPM is just frequency measured per minute instead of per second, so RPM = f × 60.

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Angular Velocity Calculator

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