There is no such thing as “centrifugal” force.
The mud stuck on a rotating wheel is undergoing circular motion. At any instant in time as the wheel rotates, the mud stuck on the side has a velocity vector pointed straight outward tangential to the wheel…and yet it still goes around in a circle. Since the velocity is constantly changine, since the direction is constantly changing (“speed” may or may not be constant), there must be some type of force acting on the mud to cause it to accelerate. This force is called centripetal force, the force causing the mud to contantly accelerate toward the inside (axis of rotation) of the wheel.
This foce is not magic, something must be applying the force on the mud. You could call it the “stickyness” factor of the mud or a type of friction adhearing the mud on the wheel. In reality, it is due to thesruface tension of water [but that is a topic for another day].
This centripetal force is applied on the mud so that is experiences centripetal acceleration causing the musts velocity to constantly change as it goes around the wheel, without this force, the mud would just fly off in a straight line.
At some point, when the wheel is turning fast enoug, not enough centripetal force is being applied to the mud in order to keep it adhearing to the wheel and continue going around in a circle.
Centripetal force = m(v^2 / r),
where m is the mass [of the mud], v is the tangential speed [of the mud], and r is the radius of the rotation.
As you can see, a larger centripetal force is required if the wheel is spinning very fast, so if the wheel is spining so fast that the mud does not get enough centripetal force, it will fly off. Another fact would be a smaller radius of rotation, the smaller the radius, the higher the force needed.
It is inertia, but it depends on the “stickiness” of the mud and the rotational speed of the wheel at its circumference. Mud will not fly off if it is sticky and near the hub. Once the attraction of the mud to the surface is overcome by the centrifugal force, it will fly off.
The answer is: centrifugal force. Basically, as a wheel spins in creates centripital force, which creates energy inwards (towards the center of the wheel). The opposite reaction is centrifugal force, which is an outwards exertion that flings the mud off the wheel.
If you want a more concise explanation, and formulas:
Mud flies off a rapidly spinning tire because the resultant force is not sufficient to keep it moving in a circular path. In this case, the force that plays a major role is the adhesion between the mud and the tire.
Inertia causes it to try to remain in the same speed and the mud wants to move off as the wheel spins.
Well the same way you can’t stay on a merry go round run amuck. The centrifugal force causes you to move away from the center.