I'm trying to find a formula representing the angular displacement of an object fixed on a circle (as in, it cannot move any closer or further away) with a constant linear velocity. (Essentially any motion perpendicular to the circle is completely ignored).
Here's a diagram of the situation as I understand it.
I know that the perpendicular motion component of a linear force is equal to the sine of the difference between it's relative angle and the angle of the point on the circle on which it's applied, which lets me work out the angular velocity at any given point.
What I want to do is take that knowledge and derive a formula that will allow me to work out the angular position of the object after a given time interval. My knowledge of calculus is fairly rusty and my attempts to use this function as the ds/dt and then integrate from their back up to a more usable form have proven to be less than accurate.
If I take this information and produce an iterative graph of the angular velocity and displacement over time, I get graphs that look like (given that the angle of the linear motion is $\frac{\pi}{4}$).
