Why isn't there a centrifugal force in an inertial frame?

There isn't a centrifugal force in an inertial frame because it's a fictitious force only observed in rotating, non-inertial frames.

In physics, an inertial frame of reference is one in which Newton's first law of motion – an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force – is valid. In such a frame, all forces are real and result from physical interactions. The centrifugal force, however, is a fictitious or pseudo force. It doesn't result from any physical interaction between objects, but rather from the acceleration of the frame of reference.

To understand this, consider a simple example of a ball tied to a string and being whirled around in a circle. From the perspective of an observer standing outside the circle (an inertial frame), the only force acting on the ball is the tension in the string, which pulls the ball towards the centre, providing the centripetal force necessary for circular motion. There's no outward force pushing the ball away from the centre.

However, from the perspective of an observer moving with the ball (a rotating, non-inertial frame), it appears as if there's an outward force pushing the ball away from the centre. This is the centrifugal force. It's not a real force, but rather an apparent effect of the frame's rotation. It's a correction factor that the observer in the rotating frame must introduce to account for the fact that their frame is accelerating.

In summary, the centrifugal force is a fictitious force that only appears in rotating, non-inertial frames of reference. In an inertial frame, all forces are real and result from physical interactions, and there's no need for the concept of a centrifugal force. This is why there isn't a centrifugal force in an inertial frame.