Angular Frequency and Angular Velocity

[mnandlall]

Both angular frequency in Simple Harmonic Motion and angular velocity in Rotational Motion are given the same symbol, "omega" (w). I was wondering if these two values are equivalent. For instance, if there was a cylinder attached to spring and rolling down a ramp, would the angular frequency of the harmonic motion equal the angular speed of the rotating cylinder. I am thinking that this is not true, because the angular frequency is constant while the angular velocity will be changing throughout the motion. Does any relationship exist between the two values?

 

[Dale]

There is a relationship between the two, but it is mathematical rather than physical. If you have something in uniform circular motion about the origin then you can write the location as:
r cos(wt+c) x + r sin(wt+c) y
where r is the radius of the circle, w is the angular velocity, c is the phase (determined by the position at t=0), and x and y are unit vectors along the x and y-axis respectively.

You can see then that simple harmonic motion is mathematically just the x component of uniform circular motion.

More generally, any time there is a an expression of the form cos(at+b) we call the "a" term angular frequency and the "b" term phase.

 

[Claude Bile]

There is one important difference, angular frequency is a scalar, while angular velocity is a vector. Otherwise, they are equivalent.

Claude.