What's the difference between angular frequency (ω) and frequency (f)?

Frequency (f) measures the number of complete cycles or revolutions per second (measured in Hertz). Angular frequency (ω) measures the rate of rotation in radians per second. They are related by the formula ω = 2πf, because one full cycle is equal to 2π radians.

Is angular frequency a vector?

In more advanced physics, angular frequency can be treated as a vector quantity (ω⃗), where its direction points along the axis of rotation. However, for many applications, including this calculator, it is treated as a scalar representing the magnitude of the rotation rate.

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Defining Rotational Rate: Angular Frequency

Angular frequency (also known as angular speed, radial frequency, or radian frequency, and symbolized by the Greek letter omega, ω) is a scalar measure of rotation rate. It refers to the angular displacement per unit of time. It's a crucial concept in physics and engineering, especially when describing oscillatory motion, such as waves, pendulums, or alternating current (AC) circuits.

🧮 How to Use the Calculator

You can calculate angular frequency from two different starting points:

From Frequency:
- Select the "From Frequency" mode.
- Enter the frequency (f) of the oscillation in Hertz (Hz). 1 Hz equals one cycle per second.
From Time Period:
- Select the "From Time Period" mode.
- Enter the time period (T) in seconds. This is the time it takes to complete one full oscillation or cycle.
Calculate: Click the "Calculate" button or press Enter to see the result.

The calculator will display the angular frequency in radians per second (rad/s).

The Relationship and Formulas

Angular frequency (ω), frequency (f), and time period (T) are all intrinsically linked.

Formula from Frequency (f):
ω = 2πf
This formula is used because one full cycle (or revolution) corresponds to 2π radians. So, the total angle per second (angular frequency) is 2π times the number of cycles per second (frequency).
Formula from Time Period (T):
ω = 2π / T
Since frequency is the inverse of the time period (f = 1/T), we can substitute this into the first formula to get the second one.

The standard unit for angular frequency is radians per second (rad/s).

💡 Real-World Applications

Electrical Engineering: It's used to describe alternating current (AC) circuits. For example, North American power grids operate at a frequency of 60 Hz, which corresponds to an angular frequency of about 377 rad/s.
Physics: Angular frequency is essential in describing simple harmonic motion, such as a swinging pendulum or a mass on a spring, and in the analysis of all types of waves (e.g., sound, light, water).
Mechanical Systems: It is used to describe the rotational speed of engines, turbines, and other rotating machinery, often being converted to or from Revolutions Per Minute (RPM).

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Angular Frequency Calculator