Angular velocity from RPM:
formula and worked example

Why convert RPM to angular velocity

RPM is the everyday way to describe how fast something spins: motors, fans, drills, car engines. Physics and engineering formulas, though, are built on angular velocity in radians per second, because radians make the link between rotation and straight-line motion clean.

The conversion is the gateway to everything else: rim speed, centripetal force, kinetic energy of a flywheel and mechanical power all use rad/s. Getting it right once saves repeated errors downstream.

The conversion and where the factors come from

• RPM: how many complete turns happen in one minute.
• 2π: the number of radians in one full revolution, about 6.2832.
• 60: seconds in a minute, which turns per-minute into per-second.

Combined, 2π ÷ 60 is approximately 0.10472, so one RPM equals about 0.1047 rad/s. That single constant is convenient for mental estimates: 1,000 rpm is about 105 rad/s.

If you want a frequency in hertz instead, divide RPM by 60, since hertz are revolutions per second. Angular velocity is then 2π times that frequency.

A shaft turning at 1,800 rpm

1. 1Convert to revolutions per second: 1,800 ÷ 60 = 30 rev/s.
2. 2Multiply by 2π radians per revolution: 30 × 6.2832 = 188.496.
3. 3Write the unit: 188.496 rad/s.
4. 4Cross-check: 1,800 × 0.10472 ≈ 188.5, which agrees.

This figure of 1,800 rpm is common for four-pole AC motors on a 60 Hz supply, which is why the example is a sensible one.

What to do with rad/s

Once you have angular velocity, linear speed at any radius is straightforward: speed equals ω × r. For a point on a 0.3 m radius wheel on this shaft, the speed is 188.496 × 0.3 ≈ 56.5 m/s.

• Rim speed: v = ω × r, with radius in metres.
• Mechanical power: torque × ω, so 40 N·m at 188.5 rad/s is about 7.5 kW.
• Kinetic energy of a rotating body: half of its moment of inertia times ω².

The radian is dimensionless, which is why rad/s sometimes appears as 1/s. Keep the rad label in working to avoid confusion with frequency in hertz.

Scaling with speed

A 10% increase in RPM, to 1,980, yields 207.345 rad/s, exactly 10% more. The relationship is a pure proportion, so any percentage change in RPM carries straight through.

What matters is what you then do with it. Rim speed rises by the same 10%, but centripetal acceleration and stress in a spinning part depend on ω², so they rise by 21%. A small RPM overspeed is a larger mechanical load.

Conversion mistakes to avoid

• Forgetting to divide by 60 and ending up 60 times too large.
• Plugging RPM straight into power or rim-speed formulas as if it were rad/s.
• Using degrees per second where radians are required.
• Mixing up angular velocity in rad/s with frequency in hertz, which differ by a factor of 2π.

Quick estimates in the workshop

A drill at 3,000 rpm turns at about 314 rad/s. A 10 mm bit has a radius of 0.005 m, so its edge speed is about 1.6 m/s. Doing this in your head tells you whether a cutting speed is plausible before you commit to the job.

For belts and pulleys, convert both shafts, then use the fact that rim speed matches at the belt. The ratio of angular speeds is the inverse of the ratio of radii.

The reverse conversion is just as handy. To turn rad/s back into rpm, multiply by 60 and divide by 2π, or simply multiply by about 9.549. A flywheel spec of 100 rad/s therefore corresponds to roughly 955 rpm. Motor data sheets quote rpm for convenience, so converting is usually the first step in checking power and rim-speed figures.