Question

Water in a bucket is whirled in a vertical circle with a string attached to to it. The water does not fall down even when the bucket is inverted at the top of its path. We conclude that in this position
(a) mg=mv²/r (b) mg is greater than mv²/r
(c) mg is not greater than mv²/r
(d) mg is not less than mv²/r

Answer 1

c) mg is not greater than mv2/r
As........
The centrifugal force is the force you would feel if you were in the rest frame of the water. There would be this force pushing you into the bucket as the rest of the world was rotating. The centripetal force of the bucket is keeping the bucket in the circle of rotation. The water wants to keep moving tangent to the circle of rotation but the bucket pushes the water back toward the center, thereby changing its direction. 
Regards

Answer 2

When the bucket is whirled in a vertical circle with a string attached to it, the water may or may not fall down during its motion around the circle. It totally depends upon the Velocity of the body. If the velocity of the body bucket will be greater than the √rg then the water from the bucket will not fall otherwise it will fall.

This means,

  v ≥ √rg

⇒ v² ≥ rg

Multiplying both the sides by mass m, which is the mass of the water with the bucket.

∴ mv² ≥ mgr

⇒ mv²/r ≥ mg

This tells us that mg must be less than centripetal force that water from the bucket will not fall.

Hence, Option (c). is correct.

Hope it helps.