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shinobi20

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I have seen so many questions and confusion about the difference between angular velocity/speed and angular frequency. Usually, answers were always given in the context of uniform circular motion (angular speed) and simple harmonic oscillation (angular frequency), but this is what causes the confusion (I think).Let ##\omega_v## be the angular velocity and ##\omega_f## be the angular frequency.

Let's consider a nonuniform circular motion (not constant tangential speed) and recall that the relationship between tangential speed ##v## and angular velocity ##\omega_v## is given by ##v=R\omega_v## where ##R## is the radius of the circular shape made by the rotation. Now, since ##v## changes throughout the rotation ##\omega_v## also changes, for example a rock tied to a string being swirled vertically therefore its angular speed is lowest at the top and highest at the bottom. The angular frequency then measures the "speed" with which the rotation takes to cover a full cycle (let's assume in this case it is ##2\pi##) for a given period ##T##, say ##T=2## s, so that ##\omega_f = \pi## rad/s which is a constant throughout the rotation. Throughout the rotation, ##\omega_v## could be ##2## rad/s, ##1.5## rad/s, ##4## rad/s, ##5## rad/s, or even ##\pi## rad/s which is just a coincidence, but this shows how different ##\omega_v## from ##\omega_f##. It is when there is uniform circular motion that ##\omega_v = \omega_f##.

I think a clearer definition so as to make the two distinct would be,

Definition. The angular velocity is the instantaneous rate of change of the angular displacement where the angular speed is just the magnitude of angular velocity.

Definition. Angular frequency is the speed with which the rotation covers the whole cycle for a given period.

Note: Angular frequency is a constant of a given oscillation or rotation whereas angular speed can vary (nonuniform circular motion).

Can anyone comment on my thoughts on this?

Let's consider a nonuniform circular motion (not constant tangential speed) and recall that the relationship between tangential speed ##v## and angular velocity ##\omega_v## is given by ##v=R\omega_v## where ##R## is the radius of the circular shape made by the rotation. Now, since ##v## changes throughout the rotation ##\omega_v## also changes, for example a rock tied to a string being swirled vertically therefore its angular speed is lowest at the top and highest at the bottom. The angular frequency then measures the "speed" with which the rotation takes to cover a full cycle (let's assume in this case it is ##2\pi##) for a given period ##T##, say ##T=2## s, so that ##\omega_f = \pi## rad/s which is a constant throughout the rotation. Throughout the rotation, ##\omega_v## could be ##2## rad/s, ##1.5## rad/s, ##4## rad/s, ##5## rad/s, or even ##\pi## rad/s which is just a coincidence, but this shows how different ##\omega_v## from ##\omega_f##. It is when there is uniform circular motion that ##\omega_v = \omega_f##.

*"So even though the angular speed might change from this value to that value in a cycle, it will always take the same "speed" (angular frequency) for it to cover a cycle."*I think a clearer definition so as to make the two distinct would be,

Definition. The angular velocity is the instantaneous rate of change of the angular displacement where the angular speed is just the magnitude of angular velocity.

Definition. Angular frequency is the speed with which the rotation covers the whole cycle for a given period.

Note: Angular frequency is a constant of a given oscillation or rotation whereas angular speed can vary (nonuniform circular motion).

Can anyone comment on my thoughts on this?

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