RPM to Radians per Second Explained

A motor label says 3,000 rpm, but your physics formula wants radians per second. What number goes in? The answer is 314.16 rad/s, and one small factor gets you there every time. This guide shows that factor, a ready chart, and the two formulas that need rad/s most.

RPM to Rad/s Conversion Chart

The chart below lists common rotation rates in four units. Every row uses the same rule: multiply rpm by 0.10472 to get rad/s. Degrees per second are simply rpm times 6.

Look at the 60 rpm row first. One turn per second equals 2 pi, or 6.2832, radians per second. That single row is a handy anchor for checking any answer by eye.

Five terms appear again and again in rotation problems. Here is what each one means in plain words.

Angular velocity (omega)

How fast an object turns, measured as angle covered per unit of time.

Radian (rad)

The angle where the arc length equals the radius. One full turn is 2 pi radians.

Revolutions per minute (rpm)

The number of full turns completed in one minute.

Tangential speed (v)

The straight-line speed of a point on the spinning object, in meters per second.

Torque

The turning force on a shaft, measured in newton meters (N m).

Why Multiply RPM by 2 Pi Over 60?

You multiply by 2 pi because one turn holds 2 pi radians. You divide by 60 because one minute holds 60 seconds. Together they turn rpm into rad/s.

Write the units out, and the logic becomes clear. Start with rev/min, multiply by 2 pi rad per rev, then multiply by 1 min per 60 s. The rev and min labels cancel, leaving rad/s.

Working Through 3,000 RPM

Take a motor spinning at 3,000 rpm. First, 3,000 x 2 pi gives 18,849.6 radians per minute. Next, divide by 60, and you get 314.16 rad/s.

The angular velocity converter shows six significant figures, so it displays 314.159 rad/s. Both numbers agree; the chart simply rounds to two decimals.

OpenStax uses the same chain in a bicycle example. A rear wheel reaching 250 rpm spins at 26.2 rad/s, which matches 250 x 0.10472.

Going From Rad/s Back to RPM

Multiply rad/s by 60, then divide by 2 pi. The shortcut is to multiply by 9.5493, since one rad/s equals about 9.55 rpm.

Say a physics problem gives 100 rad/s. Multiply by 9.5493, and the shaft turns at about 954.93 rpm. That number is easier to compare with a motor label or a tool rating.

Revolutions per second sit between the two units. Divide rpm by 60 to get rev/s, and 3,000 rpm becomes 50 rev/s. Each revolution is 2 pi radians, so 50 x 6.2832 gives the same 314.16 rad/s.

That link also explains frequency. A rotation of 50 turns per second is a frequency of 50 hertz. Physics books write the rule as omega = 2 pi f, where f is in hertz.

Where Do Degrees per Second Fit In?

Degrees per second equal rpm times 6. One turn is 360 degrees, and 360 divided by 60 seconds is exactly 6.

A clock gives a clean check. The second hand makes one turn per minute, so it moves at 1 rpm. That is 6 degrees per second, or 0.10472 rad/s.

The minute hand is 60 times slower. It covers 0.1 degrees per second, which is about 0.001745 rad/s. Small numbers like these are why the converter also lists per-hour and per-day units.

To swap between degree and radian rates, use 1 rad/s = 57.2958 deg/s. That factor comes straight from the angle units themselves. Our guide to degrees vs radians in trigonometry explains where 57.2958 comes from.

A motion sensor that reports degrees per second needs one extra step. Physics formulas for speed and power expect radians per second. Convert before you plug in, or the result will be off by a factor of 57.3.

How Does Angular Velocity Become Linear Speed?

Multiply the angular velocity in rad/s by the radius in meters. The formula v = omega r gives the straight-line speed of a point on the spinning edge.

OpenStax calls this the tangential speed. HyperPhysics adds a useful point: every spot on a rigid spinning object shares the same angular velocity. The linear speed, however, grows with distance from the axis.

A Saw Blade at 3,000 RPM

Picture a blade with a radius of 0.125 meters, spinning at 3,000 rpm. Multiply 314.16 rad/s by 0.125 m, and the tip moves at 39.27 m/s. A point halfway out moves at only half that speed.

The same idea works for wheels. A wheel with a 0.35 meter radius turning at 250 rpm has an edge speed of about 9.16 m/s. The rad/s step must come first, because rpm times meters gives no useful unit.

How Do Torque and RPM Give Power?

Power equals torque times angular velocity, with omega in rad/s. OpenStax writes it as P = tau x omega, and newton meters times rad/s gives watts.

Try a motor making 20 N m of torque at 3,000 rpm. Convert first: 3,000 rpm is 314.16 rad/s. Then 20 x 314.16 gives about 6,283 watts, or 6.28 kilowatts.

The formula also runs backward. OpenStax describes a boat engine producing 90,000 watts at 300 rpm. Divide 90,000 by 31.42 rad/s, and the propeller shaft carries about 2,865 N m of torque.

Some engine ratings use horsepower instead of watts. NIST lists one mechanical horsepower as 745.7 watts, so 6,283 watts is about 8.43 hp. Our article on horsepower vs watts covers that conversion in full.

Which Unit Slips Cause Wrong Answers?

The biggest slip is putting rpm straight into a formula built for rad/s. The answer comes out about 9.55 times too large, with no warning sign.

A second slip is forgetting the 60. Multiplying 3,000 rpm by 2 pi gives 18,849.6, which is radians per minute, not per second. The result looks plausible, so the error hides easily.

A third slip mixes degrees and radians. A sensor reading of 90 deg/s is only 1.5708 rad/s. Treating it as 90 rad/s inflates every later step by a factor of 57.3.

The fix is simple. Write the units beside each number and cancel them on paper. When the final label reads rad/s, the value is ready for v = omega r or P = tau x omega.

Questions People Ask About Angular Velocity

Where These Numbers Come From

This article is general physics and unit-conversion education. Worked values are rounded from exact calculations with pi. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.