Centrifugal force effect in a rotating frame of reference If the centrepital force doesn't exist in a rotating frame of reference, then in this frame perspective, how can we explain why a ball tied to a string following a circular motion not to be pushed outward if only the centrifugal force that is acting?
 A: 
If the centrepital force doesn't exist in a rotating frame of reference

The centripetal force does exist in the rotating frame. The centripetal force is a real force, so it exists in all frames.
The centrifugal force is an inertial force so it only exists in the rotating frame and not in the inertial frame.
So in an inertial frame the centripetal force exists and causes an inward acceleration of the object.
In the rotating frame both the centripetal and centrifugal forces exist, and they cancel each other out so that the object remains at rest.
A: The best is to forget about the centrifugal force. It is a fictitious (non-existent) force which is introduced to make first Newton's law of motion work in a non-inertial reference frame such as the rotating disc.
Imagine a person is sitting on a rotating disc and holding an accelerometer. The disc rotates at uniform angular velocity $\omega$, which is by definition the same at each point of the disc. The accelerometer would show acceleration magnitude $a = \omega^2 r$, where $r$ is the distance from the center of rotation to the person. The net force magnitude is then $F_\text{rad} = m \omega^2 r$, where $m$ is mass of the person, and its direction points towards the center of rotation, hence the term centripetal force.
However, the person with an accelerometer is at rest looking from the rotating disc perspective. The first Newton's law of motion tells us that the net force on objects in equilibrium is zero. In order to make this law work in the rotating disc frame, a centrifugal force is introduced to cancel (measured) centripetal force. But by doing this, the third Newton's law of motion is violated since the centrifugal force does not have its reaction pair.
A: Centrifugal and centripetal forces are different.
Centripetal force is present when we observe a rotating body from an inertial frame of reference (frame of reference with zero acceleration). For example, we must include centripetal force when we observe a body on Earth from space. Centripetal force always acts inwards.
Centrifugal force is present when we observe a rotating body from a non-inertial frame of reference (frame of reference that is being accelerated). For example, when we view a body on Earth we must include centrifugal force. The object is not pushed outwards due to gravitational force. You can think of centrifugal force as pseudo force. Centrifugal force always acts outwards.
You are observing the ball tied to the string from an inertial frame of reference, therefore only centripetal force acts on it.
