# Angular frequency=2*pi*frequency

This is not a homework problem, but something that came to my attention.

I attached a pic.
The thin horizontal line in the pic represents the axis of rotation of pin. So the 1st one is rotation from +30 to -30 degrees and the second from 10 to -10 degrees.
Let's assume both are rotating at 3Hz.

So my question is that the equation ω=2*∏*f doesn't hold for this case right?
If both are rotating at 3Hz, then the drawing on the left must be rotating at a larger angular speed because the amplitude is bigger.

Is ω=2*∏*f only for something that experiences a cycle of 360 degrees?

For the figure on the left, one cycle is 60 degrees, so 3 cycles is 180 degrees. So that means the angular speed is 180 degrees/s.

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No, they have the same angular frequency. Perhaps you are thinking that both the rotations take the same amount of time - which is not true. The pin could complete a whole circle of 360 degrees at the same angular frequency - the angle through which it turns has nothing to do with how often it retraces it's path . The left pin has covered 60 degrees, which is 1/6th of a full cycle - hence it takes half a second. The right pin has covered 20 degrees, which is 1/18th of a full cycle - hence it takes 1/6th of a second.

I'm not sure I understood your question properly though, so let someone smarter answer.

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No, they have the same angular frequency. Perhaps you are thinking that both the rotations take the same amount of time - which is not true. The pin could complete a whole circle of 360 degrees at the same angular frequency - the angle through which it turns has nothing to do with how often it retraces it's path . The left pin has covered 60 degrees, which is 1/6th of a full cycle - hence it takes half a second. The right pin has covered 20 degrees, which is 1/18th of a full cycle - hence it takes 1/6th of a second.

I'm not sure I understood your question properly though, so let someone smarter answer.
The problem is really confusing me.
I thought angular frequency was how many radians/degrees it travels per second. If both have the same frequency (cycles/second) and the angular amplitude for the one on the left is bigger, wouldn't that one have to rotate at a faster speed to achieve the same cycles per second?

Yes, angular frequency is how many degrees the particle travels per second. So with a frequency of 3 Hz, you get an angular frequency of 2 * pi * 3 = 18.84 radians per second.

Now let's say a particle traverses one radian (180 degrees) - that means it takes it 18.84 seconds. If another particle traverses 2 radians (360 degrees), then it would take it 18.84 * 2 = 37.68 seconds. Different times taken though they had the same angular frequency.

I am pretty sure that the ω=2*∏*f only applies to continuous rotations. In your setup, the angular displacement is discontinuous at the max and mins. The setups are not rotating at the same speed=>they have diff. angular frequencies.

I don't get it - how are they discontinuous? Hasn't the OP shown us parts of a rotation - one which covered 60 degrees, and the other 20 degrees?

I don't get it - how are they discontinuous? Hasn't the OP shown us parts of a rotation - one which covered 60 degrees, and the other 20 degrees?
He showed us the "whole" rotation, not parts. Refer to the OP where it mentioned "axis of rotation." It's oscillating between the + and - angles.

I tend to sense some confusion here. Is the rotation in a plane, or is the rotation about the center line and is in fact rotating on the surface of a cone. How you picture the problem makes a difference.

I tend to sense some confusion here. Is the rotation in a plane, or is the rotation about the center line and is in fact rotating on the surface of a cone. How you picture the problem makes a difference.

Basically, the picture I attached is a top view of a thin panel. So if you look from the top view it's rotating from a positive angle to a negative angle. Does that make sense?