In wave mechanics, any given wave contains parameters such as frequency, time period, wavelength, amplitude, etc. The angular frequency is an important computation for an object having periodic motion. For example such as a ball at the end of a rope being swung around in a circle, then the rate at which the ball sweeps through a complete 360Â°. In this article, we will discuss the angular frequency formula with examples. Let us learn the interesting concept!

## Angular Frequency Formula

### What is the angular frequency?

To understand this quantity, we need to understand a natural quantity as a time period. The time period of an oscillating object is the amount of time it takes to complete one oscillation. For example, the period for the motion of the Earth around the Sun is 365 days.

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The time period is the time taken by a complete cycle of the wave to pass a point, frequency is the number of the complete cycle of waves passing a point in unit time. Angular frequency is the angular displacement of any element of the wave per unit time. Also, in wave terminology for a sinusoidal wave, the angular frequency refers to the angular displacement of any element of the wave per unit time. It is represented by \( \omega \)

A frequency is a rate, so, the dimensions of this quantity are radians per unit time. The units will depend on the specific problem taken. If we are talking about the rotation of a merry-go-round, then we may want to talk about angular frequency in radians per minute. On the other hand, the angular frequency of the Moon around the Earth will be in radians per day.

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### Angular Frequency Formula

To know about the rate at which the rotations are occurring, we need to find the angular frequency. The frequency of rotation i.e. how many rotations take place in a certain amount of time can be computed as:

f = \( \frac{1}{T}\)

In the case of the Earth, one rotation takes 365 days, thus

f =\( \frac{1}{365}\)

The formula for angular frequency is the oscillation frequency ‘f’ measured in oscillations per second, multiplied by the angle through which the body moves. The angular frequency formula for an object which completes a full oscillation or rotation is computed as:

\( \omega = 2\pi f \)

Also in terms of the time period, we compute angular frequency as:

\( \omega =\frac{2\pi}{T} \)

For example, the angular frequency for the above case will be

\( \omega =\frac{2\pi}{365} \)

Thus the Earth moves through angle \( 2\pi \) radians in 365 days.

Angular frequency is a scalar quantity, it means it is just a magnitude. But, sometimes we talk about angular velocity, which is a vector. Therefore, the angular velocity formula is the same as the equation for angular frequency.

Its SI unit is rad/sec.

Where,

\( \omega \) | angular frequency of the wave |

T | the time period of the wave |

f | ordinary frequency of the wave |

## Solved Examples onÂ Angular Frequency Formula

Q.1: Calculate the angular frequency of an oscillating object with a time period of 30 seconds.

Solution: As given,

T= 30 sec

Applying the formula,

\( \omega =\frac{2\pi}{T} \)

Substituting the values,

\( \omega =\frac{2\pi}{30}\)

= \( \frac{360Â°}{30}\)

\( \omega = 12Â Â° sec^{-1} \)

Therefore angular frequency is 12Â° /sec

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