A bucket full of water is rotated in a vertical circle of radius 0.873m. What must be the minimum speed of the pail at the top of the circle is no water is to spill out?

Suppose mass of water is m, radius of bucket r, and velocity v

At the top, downward force on water is m while upward force is the centrifugal force = mv^2/r

Hence in this case minimum value of centrifugal force must be mg so that water is not spilled

That is : Minimum value of mv^2/r = mg

Minimum value of v^2 = gr

You can find minimum value of v now.

Suppose mass of water is m, radius of bucket r, and velocity v

At the top, downward force on water is m while upward force is the centrifugal force = mv^2/r

Hence in this case minimum value of centrifugal force must be mg so that water is not spilled

That is : Minimum value of mv^2/r = mg

Minimum value of v^2 = gr

You can find minimum value of v now.