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