I've attached a diagram of the described situation.
R is the radius pointing to A resp. A'. R = 16"
r is the radius pointing to B. r = 15"
When the wheel performs the angle , the the point A moves back.
You've got right triangles so you can calculate :
. Thus = 40.728°
The perimeter of the (active) wheel =
Now calculate the number of revolutions per second. I've got 8.8837885 rps. (Please check my result carefully. I'm the king of mistakes here!).
That means the wheel performs 8.8837885 * 360° = 3198.16° per second.
The time which is needed to perform the angle :
(By the way: Why do you want a time exact to billionst second?)