Angular Velocity to Radians per Second

This calculator converts an angular velocity value into radians per second (rad/s). You enter a number and choose its unit (degrees per second, revolutions per minute, revolutions per second, or radians per minute), and the calculator returns the equivalent value in rad/s, which is the standard SI unit for rotational speed.

Formula

The conversion from any angular velocity unit to radians per second uses the definition of a radian and the relationships between time units. The general formula is:

ω_rad/s = ω_given × (π / 180) for degrees per second, or ω_rad/s = ω_rpm × (2π / 60) for revolutions per minute, or ω_rad/s = ω_rps × 2π for revolutions per second, or ω_rad/s = ω_rad/min / 60 for radians per minute.

Where:

• ω_rad/s is the angular velocity in radians per second (rad/s).
• ω_given is the input value in the chosen unit.
• π ≈ 3.14159265358979.
• 2π is the number of radians in one full revolution.

All conversions are exact multiplications by constants; no measurement uncertainty is introduced by the conversion itself.

Worked Example

Using the default values: value = 180, unit = degrees per second.

1. Convert degrees to radians: 180° × (π / 180) = π radians.
2. Since the time unit is already seconds, the result is π rad/s.
3. Numerically, π ≈ 3.1416 rad/s (rounded to 4 decimal places).

The calculator shows 3.1416 rad/s, matching this calculation.

Why Radians per Second?

Radians per second is the SI unit for angular velocity. It appears in physics and engineering formulas because the radian is a dimensionless unit defined as the ratio of arc length to radius. This makes many equations simpler—for example, linear speed v = rω, centripetal acceleration a = rω², and rotational kinetic energy K = ½Iω² all work directly in rad/s without extra conversion factors.

In contrast, degrees per second or revolutions per minute (RPM) are common in everyday contexts—car engines, hard drives, fans—but they introduce arbitrary constants (360, 60) into formulas. Converting to rad/s before calculations avoids errors and keeps units consistent.

Common Conversion Factors

Here are the exact factors you will use most often:

• 1 revolution per second = 2π rad/s ≈ 6.2832 rad/s
• 1 revolution per minute (RPM) = 2π/60 rad/s ≈ 0.10472 rad/s
• 1 degree per second = π/180 rad/s ≈ 0.017453 rad/s
• 1 radian per minute = 1/60 rad/s ≈ 0.016667 rad/s

These factors are exact because they come from the definitions of the units, not from measurements.

How to Read the Result

The result tells you how many radians the rotating object sweeps through in one second. One full revolution equals 2π radians, so a value of 6.2832 rad/s means exactly one revolution per second. A value of 0.10472 rad/s means one revolution per minute.

For practical interpretation:

• If you know the radius r in meters, the linear speed of a point on the rim is v = r × ω (in m/s).
• If you need frequency in hertz (cycles per second), divide ω by 2π.
• If you need period in seconds, take 2π divided by ω.

Common Mistakes and Misconceptions

One frequent error is confusing angular velocity with angular frequency. In simple harmonic motion, angular frequency (often denoted ω) is also measured in rad/s, but it represents phase change per second, not physical rotation. The units are the same, but the physical meaning differs.

Another mistake is mixing up RPM and revolutions per second. RPM is per minute, so dividing by 60 is required. For example, 120 RPM is 2 revolutions per second, which is 4π rad/s ≈ 12.566 rad/s—not 120 × 2π.

Also, remember that degrees are not SI. If you plug degrees per second into a physics formula without converting, your answer will be off by a factor of 180/π. Always convert to rad/s first.

Limits of the Conversion

This conversion is purely mathematical and works for any positive or negative value, including zero. It does not account for acceleration, direction changes, or the physical limits of any real object. The radian is dimensionless, so the result is valid for any scale—from subatomic particles to galaxies.

One limitation: the calculator assumes a constant angular velocity. If the rotation is speeding up or slowing down, the instantaneous value at a specific moment is what you would enter, not an average. The conversion itself remains valid for any instantaneous value.

Frequently Asked Questions

How do I convert RPM to rad/s manually?

Multiply the RPM value by 2π and divide by 60. For example, 3000 RPM × (2π / 60) = 314.16 rad/s. This works because one revolution is 2π radians and one minute has 60 seconds.

Is rad/s the same as Hz?

No. Hertz (Hz) measures cycles per second, where one cycle is one full revolution. Rad/s measures radians per second. Since one revolution is 2π radians, 1 Hz = 2π rad/s. They are proportional but not equal.

What is the angular velocity of the Earth in rad/s?

The Earth rotates once every 24 hours, so its angular velocity is 2π radians divided by 86,400 seconds, which is approximately 7.27 × 10⁻⁵ rad/s. This is a common example in physics textbooks.

Can angular velocity be negative?

Yes. Negative angular velocity simply indicates rotation in the opposite direction (clockwise vs. counterclockwise). The conversion to rad/s works the same way for negative values; the sign is preserved.

Why use radians instead of degrees in physics?

Radians simplify equations because they are dimensionless and directly relate arc length to radius. Many derivative and integral relationships, such as d(sin θ)/dθ = cos θ, only hold when θ is in radians. Using degrees would introduce constant factors everywhere.

Related

• CST to EST Converter: Time Difference Explained
• Triangular Prism Volume from Base Area
• Compression Ratio Calculator SBC: How to Use It

in val | Value | - | 180 | 0
sel unit | Unit | deg/s=1, rpm=2, rps=3, rad/min=4 | 1
out rad_s | Radians per second | rad/s | (unit == 1 ? val * pi / 180 : unit == 2 ? val * 2 * pi / 60 : unit == 3 ? val * 2 * pi : val / 60) | 4
out deg_s | Degrees per second | deg/s | rad_s * 180 / pi | 4
out rpm | Revolutions per minute | rpm | rad_s * 60 / (2 * pi) | 4