Centrifugal Force: The Pseudo-Force in a Rotating Frame

This section treats centrifugal force as its own dedicated topic, because it is one of the most commonly misunderstood ideas in the whole of circular-motion dynamics -- and the key to understanding it correctly is to be absolutely clear about which reference frame is being used to describe the motion.

Section 4.11 established that a body moving in a circle, viewed from an ordinary, inertial (non-accelerating) frame such as the ground, requires a real, physical, inward-directed net force -- the centripetal force -- to keep it turning; without such a force, by Newton's first law, the body would simply continue moving in a straight line, flying off along the tangent to the circle rather than curving around it. In this ground frame, there is no outward force acting on the body at all; the only real force doing anything is the inward centripetal one.

Now consider describing the exact same physical situation instead from a rotating frame -- a frame of reference that is itself spinning along with the body, so that the body appears to be sitting still (at rest) within this frame, rather than moving in a circle. Newton's laws, strictly speaking, are only guaranteed to hold true in an inertial frame (Section 4.2); a rotating frame is a non-inertial frame, since it is constantly accelerating (changing direction) even if it spins at constant angular speed. To go on using Newton's second law formally within this rotating frame regardless -- so that the body, which is genuinely at rest within this frame, can correctly be treated as being "in equilibrium," with the net force on it appearing to be zero -- an extra, outward-directed pseudo-force must be added to the real forces already present. This pseudo-force is called the centrifugal force ("centre-fleeing"), and it is assigned exactly the magnitude needed to balance the real centripetal force:

It is essential to be clear that centrifugal force is a fictitious (pseudo) force: it does not arise from any genuine physical interaction between the body and some other object (unlike gravity, tension, friction, or the normal reaction, each of which can be traced to a definite source), and, unlike a true Newton's-third-law force, it has no reaction-force partner acting back on anything else -- which is exactly what marks it out as fictitious rather than real. It exists, mathematically, only because the observer has chosen to describe the motion from within a non-inertial, rotating frame; switch back to the ordinary ground frame, and the centrifugal force simply disappears from the description entirely, with no physical effect left unaccounted for.

The everyday feeling of "being pushed outward" -- felt by a passenger in a car taking a sharp turn, pressed toward the outer door, or by a rider standing at the rim of a spinning merry-go-round -- is exactly this situation. Described correctly from the ground (inertial) frame, there is no real outward force acting on the passenger at all: by their own inertia, the passenger's body simply tends to keep moving in a straight line, while the car door (or the friction of the merry-go-round's floor, or a hand rail) supplies the genuine, real, inward-directed centripetal force needed to pull the passenger along the actual curved path the vehicle is following; what the passenger feels as an outward push against the door is really just the door pushing inward on them, resisted by their own inertia. Described instead from the rotating frame attached to the car or merry-go-round itself -- the frame the passenger's own body effectively "lives in" -- the passenger can be treated as stationary only by additionally including the outward centrifugal pseudo-force, which then appears to be exactly what is pressing them against the door. Both descriptions are self-consistent and give the same final physical outcome; they simply attribute the "push outward" to different causes because they are using two genuinely different reference frames -- one inertial, one not. …