RPM to Angular Frequency Calculator

Calculator Inputs

Rotational Speed

rpm

Example: 1200 rpm (can be negative).

Rounding 0 decimals1 decimals2 decimals3 decimals4 decimals5 decimals6 decimals7 decimals8 decimals9 decimals10 decimals11 decimals12 decimals

Choose output precision for exports.

Gear Ratio Option

Apply gear ratio (output RPM = input RPM × ratio)

Use ratio > 1 to speed up, < 1 to reduce.

Gear Ratio

Example: 0.5 halves speed; 2 doubles speed.

Calculate Reset

Example Data Table

Formula Used

Angular frequency converts rotations per minute into radians per second:

• f (Hz) = RPM / 60
• ω (rad/s) = 2π × f = 2π × RPM / 60
• ω (deg/s) = ω × (180/π)
• T (s) = 1 / |f| (period magnitude)

If you enable gear ratio, the calculator first computes RPM_out = RPM_in × ratio, then applies the same conversion.

RPM to Angular Frequency Guide

1) Why convert RPM to angular frequency

Many machines display RPM, but most formulas use angular frequency ω in rad/s. Converting helps when calculating torque, angular acceleration, vibration response, and rotational energy. It also lets you compare motors, fans, and spindles using one consistent unit.

2) Key conversion constant

One revolution equals 2π radians, and one minute equals 60 seconds. The constant 2π/60 ≈ 0.104719755 converts directly from RPM to rad/s. Example: 60 RPM ≈ 6.283 rad/s, and 1200 RPM ≈ 125.664 rad/s. The factor uses π, so results are stable with standard double precision in browsers today.

3) Understanding rad/s versus Hz

Frequency f in hertz counts cycles per second, while ω expresses the same rate in radians per second. They are linked by ω = 2πf. Because f = RPM/60, you can convert RPM → Hz by dividing by 60, then multiply by 2π for ω. Handy checks: 1 Hz = 60 RPM, 10 Hz = 600 RPM, and 1 Hz = 6.283 rad/s.

4) Using gear ratio for driven shafts

Belt drives and gearboxes change speed between shafts. If your ratio means output speed divided by input speed, use RPM_out = RPM_in × ratio. A ratio of 0.5 halves speed; a ratio of 2 doubles it. The calculator applies the ratio first, then converts to ω.

5) Period and direction notes

The period T is the time for one cycle and uses T = 1/|f|. RPM may be negative to indicate direction, and ω keeps the sign. Period uses magnitude because time between cycles is always positive. With RPM = 0, period is undefined and shown as a dash.

6) Precision and rounding for reports

Choose decimals to match your instrument resolution. Use 2–4 decimals for quick checks, and 6–8 for calibration notes. Extra decimals do not improve accuracy when the RPM reading itself has uncertainty.

7) Common engineering and lab uses

Angular frequency appears in rotor dynamics, resonance checks, and control tuning. A spindle at 3600 RPM corresponds to about 376.991 rad/s. If a rig needs 100 rad/s, the required speed is roughly 955 RPM. Exporting CSV or PDF keeps calculations attached to maintenance logs and experiment records.

FAQs

1) What is the fastest way to convert RPM to rad/s?

Multiply RPM by 0.104719755. This constant equals 2π/60, converting revolutions per minute into radians per second directly.

2) Why does the formula include 2π?

One full rotation is 2π radians. RPM counts rotations, so multiplying by 2π converts rotations into radians before converting minutes into seconds.

3) How do I include a gearbox or pulley ratio?

Enable the gear ratio option and enter output/input speed. The calculator first computes effective RPM = RPM × ratio, then converts the effective RPM to ω.

4) Can RPM be negative?

Yes. Negative RPM can represent reverse direction. The ω value will keep the sign, while the period output uses |RPM| because cycle time is direction-independent.

5) What happens if RPM is zero?

With RPM = 0, frequency is 0 and angular frequency is 0. The period is undefined because you cannot divide by zero, so the calculator displays a dash for period.

6) Which output should I use for vibration problems?

Use ω (rad/s) for equations written in radians, and Hz for charts or sensor readouts. They describe the same speed, linked by ω = 2πf.