What are the conditions for a complete circular motions? Im a high school student and last week I learnt about vertical circular motion was thinking about this today, that what conditions are to be met in order to complete circular motion. First I thought the tension at the top most position must be greater than zero, then I thought that the tangential velocity at the top must be greater than zero, because then even if the tension becomes zero as soon as the object pulls it due to its speed a new tension arise and so it will keep going on a circular motion. Are there a big list of other things that need to be met to complete circular motion? And which is correct out of my thoughts, the tension ? Or velocity ? Or both? 
 A: Under circular motion, at constant velocity, you always have the constraint that the centripetal force, $F_c$, has the value $F_c=mv^2/r$, where $m$ is the mass of the object and $v$ its tangential velocity. The tension at the top and at the bottom will be different. At the top, $F_c=T+mg$, at the bottom will be $F_c=T-mg$. Thus, at the top: $T=mv^2/r-mg$ and at the bottom $T=mv^2/r+mg$ You can see a nice image with the different forces at http://www.ic.sunysb.edu/Class/phy141md/lib/exe/fetch.php?media=phy141:lectures:ballonstring.png
A: Julian wrote a good answer - I am writing a different one just because when you are learning you sometimes need to see the same thing from two different angles.
When an object is performing circular motion, "something" must be keeping it in the circular orbit. You already learnt that this requires a constant acceleration towards the center of the circle.
Now when you let a mass swing around on a string in a vertical plane, something interesting happens: as the mass reaches the bottom of the circle, it will have to go faster (because of all the potential energy that was converted to kinetic energy - namely $2\cdot m \cdot g \cdot r$. But even at the top it must have some tangential velocity to stay in orbit. How much?
Well, there is the force of gravity already helping to "pull" the mass toward the center - so the question is what is the minimum velocity we need at the top?
The centripetal force is of course $F_c = \frac{mv^2}{r}$. If the only force at the top of the circular path is gravity, then the string is briefly slack, and the velocity is given by
$$\frac{mv^2}{r} = m \cdot g\\
v = \sqrt{g \cdot r}$$
It is OK for the masss to go faster - but if it goes slower (at the top) it will not remain in the circular orbit.
So your intuition was pretty good. You need the tension in the string to be "greater or equal" to zero, and you need a minimum velocity at the top that depends on the radius of the circular path and the acceleration of gravity.
