What are the two physics ideas related to spinning a bucket of water vertically?
The important concepts are centripetal force, and weight force. How these two forces interact during the circular motion. You could also consider energy conservation/conversion during the motion
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You could investigate the ideas of centripetal force, and weight force, these are the most important concepts in this situation. You could have a look at the Hyperphysics pages on circular motion especially this one
In part, yes. For example at the top of the motion, the weight of the bucket/water also acts downwards (i.e. centripetally) so the tension force only needs to be large enough to make the total downward force equal to the centripetal force requirement: F_{tension}\;+\;F_{weight}\;=\;F_{centripetal}
At the bottom, the situation will be different, because the weight is no longer acting centripetally…
and so at the bottom, the tension force and weight force will still continue to provide centripetal force? In my report should I state what force are causing the centripetal force?
The weight can’t provide any centripetal force at the bottom, as it acts away from the centre of the motion. The tension has to provide all the centripetal force, and also balance out the weight.
Also, I am confused on how Newton’s second law would apply to spinning a bucket in a vertical circle. I know that there is centripetal acceleration involved and this acceleration is caused by a net force acting the same direction as the acceleration (inwards) so it is the centripetal force. By substituting the centripetal acceleration equation into F= ma, centripetal force can be found ? Fc = mv2/r
Yes that is correct. Newton’s second law states that an unbalanced force causes an acceleration. In this case the acceleration is centripetal.
Then how does the water stay in the bucket? does the centripetal force need to be greater or equal to the weight force?
Yes, you are again correct. The “critical speed” at the top of the circle is when F_c\;=\;mg
Yes, there is a normal force acting on the water, it is the reaction force from the bucket.
I am not 100% sure what you mean by “water forces”. The force from the bucket will be transmitted through all particles in the water, and the force of gravity will also be experienced by all the water particles.
Words like centripetal and centrifugal only describe the direction of force, not the nature of the force. Centrifugal force is in fact the reaction force to the centripetal force (the bucket exerts a centripetal force on the water, the water exerts an equal and opposite reaction/normal force on the bucket). Physics teachers and examiners try to avoid the word centrifugal as it implies that the force on an object in circular motion is outwards, which it obviously isn’t. My advice is to avoid this word and use “reaction/normal” instead 
Yes, that’s right. The force of the bucket on the water, combined with the water’s weight force, provides the required centripetal force. At the critical speed, when F_c\;=mg, the reaction force will drop to zero as the weight provides all the required centripetal force. Any slower, and the weight of the water is greater than the required centripetal force, and the water will fall out of the bucket.