When I did the calculation in the simulation, I already assume the particle will move alone the circular path. The tangential component of "g" is the only source of acceleration along the circular path. Actually, it was transform to angular acceleration.
The equation of motion is d[sup]2[/sup]?/dt[sup]2[/sup]= -g*sin?/L , where g*sin? is the tangential component of "g".
The centripetal force is provided from normal force to keep it move in circular path (to restrict it's path or change moving direction only, the centripetal force did not change the magnitude of the velocity).
If you need to know the detail of the motion before it reach the end of the first circular, it is exactly the same as a pendulum. Please check out [url=http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1116]Force analysis of a pendulum[/url] and [url=http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1068.0]Pendulum[/url].
It is assumed the same time step dt=0.05s in the above simulation, and the program calculate new position from velocity of the particle(also update particle velocity from the external force at each time step).
We did not calculate the time it will reach the end of first circle in advance, what we did was checking at each time step if the particle pass the end point.
May I know why you think you need to know how to calculate the time for the particle to reach at the end of first circle? May be there are better way to solve your problem.