Water contained in a bucket is whirled in a vertical circle using...

At the highest point of the vertical circle, the forces acting on the water are its weight $mg$ acting downwards and the normal reaction $N$ exerted by the bottom of the bucket, also acting downwards.

The net downward force provides the necessary centripetal force for circular motion:

$N + mg = \dfrac{mv^2}{r}$

For the water not to fall out, it must maintain contact with the bucket, which means the normal reaction $N$ must be greater than or equal to zero, i.e., $N \ge 0$.

$\dfrac{mv^2}{r} - mg \ge 0$

$mg \le \dfrac{mv^2}{r}$

This implies that $mg$ is not greater than $\dfrac{mv^2}{r}$.