Angular Velocity Converter
Our free signal & frequency converter handles angular velocity conversions. See tables, ratios, and examples for quick reference.
Angular Velocity Converter
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Formula: omega(rad/s) = RPM x 2pi / 60 | deg/s = rad/s x 180/pi | f(Hz) = omega / 2pi
Worked example โ 3000 RPM = 314.159 rad/s = 50 Hz
Formula
omega(rad/s) = RPM x 2pi / 60 | deg/s = rad/s x 180/pi | f(Hz) = omega / 2pi
All angular velocity conversions use radians per second as the base unit. One revolution equals 2 pi radians. RPM converts by multiplying by 2 pi / 60. Degrees convert using the factor 180 / pi. Frequency in hertz equals revolutions per second, which is angular velocity divided by 2 pi.
Worked Examples
Example 1: Engine RPM to Rad/s
Problem:Convert a car engine running at 3000 RPM to rad/s and Hz.
Solution:rad/s = 3000 x 2 x pi / 60 = 3000 x 6.2832 / 60 = 314.159 rad/s Hz (rps) = 3000 / 60 = 50 Hz Period = 1 / 50 = 0.02 seconds per revolution
Result:3000 RPM = 314.159 rad/s = 50 Hz
Example 2: Hard Drive Spin Speed
Problem:A hard drive spins at 7200 RPM. What is its angular velocity in degrees per second?
Solution:rad/s = 7200 x 2 x pi / 60 = 753.982 rad/s deg/s = 753.982 x 180 / pi = 43,200 deg/s This means the platters rotate 120 times per second.
Result:7200 RPM = 753.98 rad/s = 43,200 deg/s
Frequently Asked Questions
What is angular velocity and how is it measured?
Angular velocity measures how fast an object rotates or revolves around an axis, expressed as the angle covered per unit time. The SI unit is radians per second (rad/s), where one full revolution equals 2 pi radians (approximately 6.2832 rad). In engineering and everyday use, RPM (revolutions per minute) is more common. Angular velocity differs from linear velocity in that it describes rotation rather than straight-line motion. The relationship between them is v = omega times r, where v is linear velocity and r is the radius.
How do I convert between RPM and radians per second?
To convert RPM to radians per second, multiply by 2 pi and divide by 60. The formula is omega = RPM x 2 x pi / 60. For example, 3000 RPM = 3000 x 6.2832 / 60 = 314.16 rad/s. To convert back, multiply rad/s by 60 and divide by 2 pi. RPM is commonly used for motors, engines, and hard drives, while rad/s is preferred in physics and engineering calculations because it simplifies many equations.
What is the relationship between angular velocity and frequency?
Angular velocity (omega) and frequency (f) are related by omega = 2 pi times f. Frequency in hertz equals revolutions per second. The period (T) is the reciprocal of frequency: T = 1/f = 2 pi / omega. For example, an object rotating at 10 Hz has an angular velocity of 62.83 rad/s and a period of 0.1 seconds. In signal processing, angular frequency (omega) is used extensively in Fourier analysis and is measured in radians per second.
How does angular velocity apply to real-world applications?
Angular velocity is critical in many fields. Car engines typically run at 1000-7000 RPM. Hard disk drives spin at 5400 or 7200 RPM. A washing machine spin cycle operates at 800-1600 RPM. The Earth rotates at approximately 0.00007292 rad/s (one revolution per sidereal day of 23 hours 56 minutes). In mechanical engineering, angular velocity determines gear ratios, centrifugal forces, and power transmission. In electronics, angular frequency describes AC current oscillation (50 or 60 Hz mains frequency equals 314.16 or 376.99 rad/s).
How do you get the rim speed of a rotating part from its angular velocity?
Tangential speed is v = omega x r, with omega in radians per second and r the radius in metres. This is the one relationship that forces you to use radians rather than degrees or revolutions, because the radian is defined so that arc length equals angle times radius. A wheel at 3,000 RPM turns at 3,000 x 2 pi / 60 = 314.16 rad/s, so a point 0.15 m from the axis moves at 314.16 x 0.15 = 47.1 m/s while the axle itself is stationary.
Why are radians per second and hertz not the same thing even though both are per second?
They count different things. Hertz counts complete revolutions or cycles per second, while radians per second counts angle swept, and one revolution is 2 pi radians. So 1 Hz equals 6.2832 rad/s, and omega = 2 pi f. Dropping that factor of 2 pi is the most common error in rotational work, and it hides easily because the radian is dimensionless, so both quantities carry units of inverse seconds and a dimensional check will not catch the mistake.
How do you find the centrifugal loading produced by a given rotation speed?
Radial acceleration is a = omega squared x r, which can also be written v squared / r. Because omega is squared, doubling the speed quadruples the load. A washing machine drum of 0.25 m radius at 1,600 RPM has omega = 1,600 x 2 pi / 60 = 167.55 rad/s, giving 167.55 squared x 0.25 = about 7,020 m/s2, roughly 716 times standard gravity. That is why spin speed, not drum size, dominates bearing loads and water extraction in a spin cycle.
What is angular acceleration and what units does it use?
Angular acceleration is the rate of change of angular velocity, measured in radians per second squared. It is alpha = change in omega divided by the time taken. An engine spinning up from rest to 3,000 RPM in 2 seconds goes from 0 to 314.16 rad/s, so alpha is 157.1 rad/s2. Multiplying angular acceleration by the moment of inertia gives the torque required, which is the rotational analogue of force equals mass times acceleration.
How does a gear ratio change angular velocity and torque?
For meshing gears, the product of angular velocity and radius matches at the contact point, so output speed equals input speed divided by the ratio of output to input teeth. A 3:1 reduction driven at 3,000 RPM delivers 1,000 RPM at the output. Since mechanical power is torque multiplied by angular velocity in rad/s, and an ideal gearbox conserves power, the torque rises by the same factor of three. Real gearboxes lose a few percent to friction, so the torque gain is slightly less.
Why is Earth's rotation quoted as one turn per 23 hours 56 minutes rather than 24 hours?
That figure is the sidereal day, the time for Earth to complete one rotation relative to the distant stars, about 86,164 seconds. Its angular velocity is 2 pi / 86,164 = 7.292 x 10^-5 rad/s. The 24-hour solar day is longer because Earth also moves along its orbit, so it must turn nearly one extra degree to bring the Sun back to the same position. Satellite tracking and telescope drives use the sidereal value; clocks use the solar one.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator ยท Editorial policy
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