Angular frequency - this is how I understood it

The word ‘frequency’ is used in different contexts.

Sometimes it is used as a number of regularly occurring events in unit time, like this pedestrian signal which flashes every second or this tuning fork which vibrates 512 times in one second and so on.

Sometimes time is not considered to specify frequency like these poles for railway electric wires placed after every 50 meters.

In physics, frequency is defined as regularly occurring events in a unit of time. This needle oscillates three times in 1 second . this pendulum repeats its path every 2 seconds, this AC current which changes direction twice every second.

All these examples can be classified as regular frequency.

Regular or linear frequency (f)counts the number of complete cycles in a given period of time. Its units are therefore cycles per second (cps), also called hertz (Hz).

There is another term to measure frequency - angular frequency.

"Regular" and angular frequencies are alternative units of measurement for describing how fast an object rotates or oscillates.

Any sinusoidal process, for example, an alternating current, the movement of a spring, up and down motion of the floating ball . involves the regular oscillation of a physical parameter, such as voltage or current or physical position. Each cycle of oscillation is identical in form with the previous one. The oscillations follow the sine or cosine law.

In an earlier video, we explored the motion of an oscillating needle.

We also discussed , motion of a pen rotating at constant speed and the nature of its projection, a shadow which executes simple harmonic motion.

In both scenarios, we can use regular frequency to describe these repetitions.

But what about angular frequency ?

Movement of this oscillating needle or shadow of the pen, does not seem to have any angle associated with it.

Well, there is no satisfactory mathematical way of analyzing and computing things that have an up-and-down or back-and-forth movement. However, we do have simple but powerful ways of analyzing rotating things, and the two are closely related.

This sinusoidal oscillation is an incidental by-product of the circular motion. The magnitude of oscillation of this shadow at any instant of time is derived from the cos of the angle at which the pen is located.

We borrow these mathematical methods and apply them wherever they seem to work. The up-and-down, wave shaped graph related to sin( θ ) and cos theta is one such mathematical method.

In both scenarios, it is the mathematical calculation expressed in terms of angles which helps us find out the value of certain physical quantities in this case horizontal distance.

As per our derived equation, The scientists could have kept the term 2 pi divided by period T as it is. Instead they decided to introduce a new term - angular frequency and use it everywhere instead .

In order to find the value of physical quantity, we have to use the term angular frequency. Regular frequency does not help much in this regard.

Let us observe this shadow in slow motion.

Shadow starts from the center and comes back to the same position in half second.
It completes one cycle in half a second. It completes the second cycle in another half second.

Frequency of this oscillatory motion of the shadow is 2 Hertzs as it completes two cycles in 1 second.

Let us consider the rotation of the pen along with the motion of the shadow.

In the first cycle of the shadow, the pen also rotates a complete circle that is 2 pi radians. In the second cycle of the shadow, the pen covers another 2 pi radians. Pen has covered the total radial distance of 4 pi radians in 1 second.

Angular frequency of the pen is 4 Pi radians per second. It is also the angular frequency of the shadow which is oscillating even though there is no angle associated with this motion as such.

Angular frequency (ω), measures angular displacement per unit time. Its units are radians per second.

As its units are different, regular frequency and angular frequency are not the same.

Angular frequency helps us to derive values of physical quantities related to phenomena which are oscillatory in nature.

Do you think angular frequency is associated with this flashing signal ? Not really.

There is another term which uses the same symbol - angular velocity.

"angular frequency" is mostly associated with oscillating motion while angular velocity is used about rotating objects like this spinning top or moving in an orbital path around some point.

Hope this helps.

Thank you.