physics - Vertical circular motion : time(x/y) versus velocity equation -

I wanted to simulate the following through the animation:
a certain velocity at the lowest point of a ball underneath Starts from
rolling in a vertical circular loop and its velocity permit.

For this, I wanted to find the velocity / X / Y vs. time equation. If the ball has mass: 5 kg, radius of circular loop = 10 meters, and the starting speed of the ball is 200 meters, its velocity and position (x, y) will be after 5 seconds?

Thank you.

Sliding case ball with a point-particle

In this case we are not worried about the rotational energy and it is assuming that the ball is actually a point particle. Then, to stay on top of the ball, the position of the central force must be satisfied:

  m * v_top ^ 2 / r = m * g  

Such a

  v_top = sqrt (r * g)  

is determined by the minimum initial velocity:

  1/2 * m * v0 ^ 2 & gt; = 1/2 * m * v_top ^ 2 + m * g * 2 * r v0> = sqrt (5 * r * g)   
 It is similar to peat, except that it is To remain on top forgot the position of the central force. 
 
 Next, the acceleration for the track is given by touch: 
 
  a = - g * sin (theta)  
 
 but  a = r * alpha = r * d ^ 2 (theta) / dt ^ 2  Where there is alpha rotating acceleration, we will 
 
  r * d ^ 2 ( Theta) / dt ^ 2 = g * sin (theta)  
 
 However, I do not know about the analytical solution for this differential equation and also with finding mathematica Was stumbling. You can not move  dt  s to the other side and integrated because it is a function of theta t. I recommend solving it with a numerical means such as or maybe. I have solved this by using the Mathematica for the parameters given by you, but the ball is moving so fast, it does not really move very slowly when I reduce the initial speed, then I have time As a work, theta was able to sharpen and slow down by plotting. 
 
 It is definitely possible to add a finite ball radius, rotational energy and other things like friction, but I will worry about being able to solve this first case before moving forward because it only It gets more complex than here. By the way, with friction you will have to select some kinetic coefficient of friction for the given materials, which will definitely be proportional to the normal force applied on the ball on the track, which can be solved by summing it with force components. Do not forget to include the radius of the radius and the central force position. 
 
 If you have not done this kind of physics before, then I definitely need to start with an initial good book on physics (with calculus) through it, you only need to be disturbed with those sections Which apply to mechanics, although this is probably a very big part of the book, although moving ahead like some of the resources can be a better route.