# Radian Per Second Converters

> Convert radians per second to and from RPM, degrees per second, hertz, and other angular velocity units

**Category:** Conversion
**Keywords:** rad/s, radian per second, angular velocity, rpm, degrees per second, hertz, conversion
**URL:** https://complete.tools/radian-per-second-converters

## How it calculates

The converter uses precise mathematical relationships between angular velocity units, with radians per second as the base unit. The fundamental conversion formulas are derived from the definition of a radian and the relationship between angular and linear measurements.

Key conversion formulas from radians per second:

1. To Revolutions per Minute (RPM):
   RPM = rad/s x 60 / (2 x pi) = rad/s x 30/pi
   Since one complete revolution equals 2 pi radians, and there are 60 seconds in a minute.

2. To Degrees per Second:
   deg/s = rad/s x 180/pi
   Based on the relationship that pi radians equals 180 degrees.

3. To Hertz (Hz):
   Hz = rad/s / (2 x pi)
   One hertz represents one complete cycle (2 pi radians) per second.

4. To Revolutions per Second:
   rev/s = rad/s / (2 x pi)
   Identical to hertz since one revolution equals one cycle.

5. To Milliradians per Second:
   mrad/s = rad/s x 1000
   Simple metric prefix conversion.

6. To Gradians per Second:
   grad/s = rad/s x 200/pi
   Based on 400 gradians in a full circle versus 2 pi radians.

The tool first converts any input unit to radians per second, then applies the appropriate factor to convert to the target unit, ensuring consistent accuracy across all conversions.

## Who should use this

Mechanical engineers designing rotating machinery such as motors, turbines, and gear systems where specifications must be converted between RPM and rad/s for calculations; electrical engineers working with AC circuits and oscillating systems who need to convert between angular frequency in rad/s and frequency in hertz; aerospace engineers calculating the angular velocity of aircraft components, satellite attitude control systems, and gyroscopic instruments; robotics engineers programming servo motors and joint movements that require precise angular velocity specifications; physicists conducting experiments involving circular motion, pendulum dynamics, or wave mechanics where rad/s is the standard theoretical unit; automotive engineers analyzing crankshaft speeds, wheel rotation, and drivetrain components; students studying classical mechanics, electromagnetism, or signal processing who need to verify their unit conversions; and manufacturing technicians calibrating CNC machines and industrial equipment with rotational components.

## Worked examples

Example 1: Converting motor speed from rad/s to RPM
A motor controller outputs angular velocity as 150 rad/s. To find the equivalent RPM:
RPM = 150 x 60 / (2 x pi) = 150 x 30 / pi = 4500 / pi = 1432.39 RPM
This is useful when specifying motor performance in datasheets.

Example 2: Converting oscillation frequency to angular velocity
A signal oscillates at 60 Hz. To find the angular velocity in rad/s:
rad/s = 60 x 2 x pi = 376.99 rad/s
This conversion is essential for calculating phase relationships in AC circuits.

Example 3: Converting robotic arm joint speed
A robotic arm joint moves at 45 degrees per second. To convert to rad/s:
rad/s = 45 x pi / 180 = 0.7854 rad/s
Engineers use this for trajectory planning and dynamics calculations.

Example 4: High-speed turbine analysis
A gas turbine operates at 15,000 RPM. Converting to rad/s:
rad/s = 15000 x 2 x pi / 60 = 15000 x pi / 30 = 1570.80 rad/s
This value is used in stress analysis and vibration calculations.

Example 5: Gyroscope calibration
A gyroscope measures 2.5 rad/s. Converting to degrees per second:
deg/s = 2.5 x 180 / pi = 143.24 deg/s
Navigation systems often display angular rates in degrees per second for readability.

## Limitations

This converter tool has several technical limitations to consider. First, it assumes constant angular velocity and does not account for angular acceleration or time-varying rotational speeds. Second, the precision is limited to eight decimal places, which may be insufficient for extremely high-precision scientific applications. Third, the tool performs pure mathematical conversions and does not account for physical constraints such as maximum rotational speeds of materials, centrifugal forces, or mechanical resonance frequencies. Fourth, while the tool handles both positive and negative values (representing opposite rotation directions), it treats angular velocity as a scalar quantity and does not consider the full vector nature of angular velocity in three-dimensional rotation. Fifth, the conversions assume ideal conditions and do not factor in real-world effects like slip, friction, or measurement uncertainty. Finally, for very large or very small values, numerical precision may be affected by floating-point representation limits.

## FAQs

**Q:** Why is radians per second the SI unit for angular velocity?


**A:** Radians per second is the SI unit because the radian is a dimensionless ratio (arc length divided by radius), making rad/s naturally compatible with other SI units in physics equations. This allows direct use in formulas without additional conversion factors, particularly in calculus and dynamics equations.


**Q:** What is the difference between angular velocity and angular frequency?


**A:** Angular velocity and angular frequency are mathematically identical (both measured in rad/s), but they are used in different contexts. Angular velocity describes physical rotation of objects, while angular frequency describes the rate of change of the phase angle in oscillating systems like waves and AC circuits.


**Q:** How do I convert between RPM and Hz?


**A:** RPM and Hz differ by a factor of 60, since RPM is revolutions per minute and Hz is cycles (revolutions) per second. To convert: Hz = RPM / 60, or RPM = Hz x 60.


**Q:** Can negative angular velocity values be converted?


**A:** Yes, negative values can be input and converted. Negative angular velocity indicates rotation in the opposite direction (clockwise versus counterclockwise when viewed from a specific reference point).


**Q:** Why do motor specifications use RPM instead of rad/s?


**A:** RPM is more intuitive for practical applications because it represents complete rotations in a familiar time unit (minutes). Engineers and technicians can more easily visualize and compare motor speeds in RPM. However, engineering calculations typically convert to rad/s for mathematical analysis.


**Q:** What is the relationship between linear velocity and angular velocity?


**A:** Linear velocity (v) at a point on a rotating object equals angular velocity (omega) multiplied by the radius (r) from the rotation axis: v = omega x r. This requires omega in rad/s to produce v in meters per second when r is in meters.

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