Answer:

Introduction:
When a bucket containing water rotates about its vertical axis at a constant speed, it produces a specific type of flow pattern known as a vortex. Vortex flow is characterized by the movement of fluid particles in a circular or spiral path around a central axis. Depending on the speed of rotation, the bucket can produce either a free vortex or a forced vortex.

Explanation:

1. Free Vortex:
A free vortex is a type of vortex flow in which the fluid particles rotate purely due to the action of their own inertia. In a free vortex, the fluid particles move in concentric circles around a central point without any external force or pressure. The velocity of the fluid particles decreases as the radius increases, following the principle of conservation of angular momentum. In the case of a rotating bucket, if the speed of rotation is relatively low, the water in the bucket will form a free vortex.

2. Forced Vortex:
A forced vortex is a type of vortex flow in which the fluid particles are forced to rotate by an external force or pressure. In a forced vortex, the velocity of the fluid particles is not solely determined by their inertia but is also influenced by external factors. In the case of a rotating bucket, if the speed of rotation is relatively high, the water in the bucket will form a forced vortex. The centrifugal force generated by the rotation of the bucket causes the water to move in a spiral path towards the outer edges of the bucket.

3. Both of the Above:
The answer to the given question is option C) both of the above, depending on the speed of rotation. The type of vortex flow produced by the rotating bucket will vary depending on the speed at which it rotates. A lower speed will result in a free vortex, while a higher speed will result in a forced vortex. Thus, the same system can exhibit both types of vortex flows depending on the conditions.

Conclusion:
In conclusion, when a bucket containing water rotates about its vertical axis at a constant speed, it can produce both a free vortex and a forced vortex depending on the speed of rotation. A lower speed will result in a free vortex, while a higher speed will result in a forced vortex. The type of vortex flow is determined by the balance between inertia and external forces acting on the fluid particles.