Answer
See the detailed answer below.
Work Step by Step
To understand this situation, we need to draw the force diagram exerted on the ball while it is moving in a vertical circle.
At the top of the circle, the two forces exerted on the ball are both downward, its own weight, and the tension in the string.
$$\sum F_y=F_G+T=ma_r=\dfrac{mv^2}{R}$$
$$F_G+T=\dfrac{mv^2}{R}$$
where $R$ is the radius of the circle which is the length of the string, $m$ is the ball's mass, and $v$ is its speed.
Thus, at a minimum speed, $T=0\;\rm N$
if we are rotating at a minimum speed that just allows it to complete the vertical loop, the tension force in the string attached to it is then zero.
$$\overbrace{F_G}^{=mg} =\dfrac{mv^2}{R}\tag 1$$
Thus, the minimum velocity required for the ball to complete the vertical loop is given by
$$ { \color{red}{\bf\not}mg} =\dfrac{\color{red}{\bf\not}mv^2}{R}$$
$$v =\sqrt{Rg}$$
Any speed less than that, the ball will be under the free-fall acceleration with some horizontal velocity component which allowed it to move as a projectile that is fired from the top of the vertical circle.
