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Frequency To Radians Per Second Calculator

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A Frequency to Radians Per Second Calculator is a scientific tool that converts a standard frequency measurement, given in Hertz (Hz), into its equivalent value in radians per second (rad/s). Frequency in Hertz tells us the number of full cycles or revolutions an object completes in one second. Radians per second, also known as angular frequency or angular velocity, measures the rate of rotation in terms of angular displacement. This conversion is fundamental in physics and engineering, especially in the study of wave mechanics, circular motion, and alternating current (AC) electrical systems, where radians are the natural unit for describing oscillations.

formula of Frequency To Radians Per Second Calculator

The formula to convert a frequency from Hertz to radians per second is direct and is based on the geometric definition of a circle.

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Radians per Second (ω) = 2 * π * Frequency (f)

Where:

  • ω (Omega): The angular speed, in radians per second (rad/s). This is often also called angular frequency.
  • π (Pi): The mathematical constant, with an approximate value of 3.14159. The factor of 2π is used because one full cycle or revolution covers an angle of 2π radians.
  • f: The frequency, in Hertz (Hz), which represents cycles per second.

Common Frequency to Radian Conversions

This table provides a quick reference for the radians per second equivalents of common frequencies, particularly those found in mechanical and electrical systems.

Frequency (f) in HzCalculationRadians per Second (ω) (approx.)
1 Hz2 * π * 16.28 rad/s
10 Hz2 * π * 1062.83 rad/s
50 Hz2 * π * 50314.16 rad/s
60 Hz2 * π * 60376.99 rad/s
100 Hz2 * π * 100628.32 rad/s
440 Hz (Note A4)2 * π * 4402,764.6 rad/s

Example of Frequency To Radians Per Second Calculator

The standard frequency of the AC power supply in North America is 60 Hz. An electrical engineering student needs to find the angular frequency in radians per second for use in circuit analysis equations.

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Step 1: Identify the known frequency.

  • Frequency (f): 60 Hz

Step 2: Apply the conversion formula.
Radians per Second (ω) = 2 * π * f
Radians per Second (ω) = 2 * π * 60 ≈ 376.99 rad/s

Therefore, the angular frequency of a 60 Hz power supply is approximately 376.99 radians per second.

Most Common FAQs

Why is 2π used in the conversion formula?

The number 2π is used because it represents the total number of radians in one full circle or one complete cycle. Since frequency in Hertz is a measure of "cycles per second," multiplying it by "2π radians per cycle" correctly cancels out the "cycles" unit, leaving a final result in "radians per second."

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What is the difference between Hertz and radians per second?

Hertz (Hz) and radians per second (rad/s) are two different ways to describe the same phenomenon of oscillation or rotation. Hertz counts the number of discrete, full cycles that happen in a second. Radians per second describes this as a continuous rate of rotation, measuring the angular distance traveled per second. While they are related, radians per second is often the more mathematically convenient unit in physics and engineering equations.

Can I use this calculator to convert RPM to radians per second?

Yes, but you must first convert the Revolutions Per Minute (RPM) into Hertz (Hz). To do this, you simply divide the RPM value by 60, since there are 60 seconds in a minute. Once you have the frequency in Hz, you can then use the standard formula to find the value in radians per second.

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