Centripetal and Centrifugal force

Question

If water stays in a pail of water that is whirled around a circular path, the water stays in the pail. But is it because of centripetal force or inertia? I'm getting confused by all the different answers i get from different sources.

Answer 1

If a body of mass m hanged on a string is moving, let uniformly, on a circle fixed relatively to the ground, then an observer G on the ground uses the 2nd Newton Law : $$ \mathbf{F}=m\cdot \mathbf{a} \tag{01} $$ and finds the relation between the force $\mathbf{F}$ and the acceleration $\mathbf{a}$. For observer G there exists a "real" force, the tension of the string. This force is the centripetal force which is pulling the body continuously to the centre. Of course $\mathbf{a}$ is the centripetal acceleration. Observer G is justified to use the Law since he(or she) is on an inertial system of reference (called also Newtonian system).

For a massless observer B on the body there exists also the "real" force $\mathbf{F}^{\prime}=\mathbf{F}$, the tension of the string. The body is motionless relatively to him so $\mathbf{a}^{\prime}=\mathbf{0}$. Observer B meets a contradiction using 2nd Newton Law : $$ \mathbf{F}^{\prime}=m\cdot \mathbf{a}^{\prime}=\mathbf{0}, \quad \textbf{FALSE} \tag{02} $$ This is due to the fact that the system of observer B is accelerated relatively to the inertial system G, so it's not an inertial system and B is not justified to use the 2nd Newton Law. In order to use the Law it's necessary to introduce a "virtual" force $\mathbf{A}^{\prime}=-m\cdot \mathbf{a}$ and apply the Law: $$ \mathbf{F}^{\prime}+\mathbf{A}^{\prime}=\mathbf{F}-m\cdot \mathbf{a}=\mathbf{0}, \quad \textbf{TRUE} \tag{03} $$ This force $\mathbf{A}^{\prime}$ is a so-called inertial force and in our case is the centrifugal force which observers on the body feel to push them away from the center. Generally inertial forces appear in non-inertial systems and they are introduced in order to apply correctly the Newton Laws in these systems.

Answer 2

Note that you have to swing the pail with a certain minimum speed for the water to stay in. That minimum speed is such that when the pail is at the top of the arc, the rope accelerates the pail downward faster than gravity accelerates the water downward. Otherwise, the water falls out.

Answer 3

If water particles move in a circular path its because of some net force towards the centre. This net force is usually called "centripetal force".

Don't ever put centrifugal force into the description. It is not a force, but just a name of the "feeling" that your body (or in this case water particles) want to move out of the circular motion but can't.