Understanding the centrifugal force

[cytochrome]

I'm learning about mechanics in noninertial frames and I'd like to clarify the apparent centrifugal force.

To set things straight,

Centrifugal force = Fcf = m(Ω×r)×Ω

where Ω×r is the instaneous velocity at a point on the outside of a circle (or other rotating path) and Ω is the angular velocity?

By looking at a picture, this would have the centrifugal force pointing out radially from the rotating path (which makes sense when imagining a centrifuge for example).

Is this force called "fictitious" because there is actually no force (as observed from a rotating frame of reference, or an inertial frame?) that is causing an object to accelerate in this radial direction? What is the force exactly?

For example - Imagine you are spinning a ball on a string around in circles and the string breaks, causing the ball to fly outward perpandicular to the direction of motion. Is the centrifugal force responsible for this?

How does this force relate to observations seen from a noninterial or inertial frame?

 

[K^2]

In inertial frame.

There is no centrifugal force. The reason the string is under tension is because it pulls on the object in order to keep it centripetally accelerating to travel in a circular path. The reason the object flies away when the string breaks is because it continues to travel in the straight path.

In a rotating frame. (Frame rotation matches rotation of the object.)

Object is stationary. It's not moving at all. So the only explanation for the tension in the string is gravity-like force pulling on the object, forcing it away from center. That's the centrifugal force. When the string snaps, that force accelerates the object away from center.The centrifugal force is a fictitious force because it does not exist in an inertial frame of reference. It only manifests when we try to use Newton's Laws to describe motion in an accelerated frame of reference.