Rotating Axes: Coriolis and Centrifugal

Two observers watch the same particle. One uses fixed axes bolted to the floor. The other stands on a turntable with its own painted x and y axes. Their acceleration measurements do not agree. The difference is physical, and its two famous names are centrifugal and Coriolis. Both begin with the fact that the second observer's axes are spinning. A reference frame means axes together with a clock. Capital X and capital Y belong to the fixed frame, with origin O, and nothing in that frame turns. The rotating frame has origin B. Its little x and little y directions are painted on the surface, so they turn with the disc at angular velocity omega. Now let P wander outward while the turntable keeps turning. Its position from O is r sub P. The moving origin sits at r sub B. The displacement from B to P, measured in the painted coordinates, is rho. Geometry gives r P equals r B plus rho. These are the same three arrows beside the equation: first r P, then r B, then rho, joined head to tail. Differentiate that identity in the fixed frame. Rho is written in the painted basis, and the basis turns while its components change. The derivative therefore acts on both the numbers and the axes. The task is now precise: differentiate this moving triangle honestly, twice. The extra acceleration terms will come from the turning basis.

Everything hard about rotating axes begins with one question: how fast does a painted axis change as its disc turns? The unit vector i hat has length one, so its tip stays on the circle. Its turned-through angle is theta. Only after that geometry is fixed do its components read cosine theta, sine theta, zero. Differentiate those components. The chain rule gives theta dot times the bracket minus sine theta, cosine theta, zero. That bracket is j hat, the second painted axis. It is exactly ninety degrees ahead of i hat on the same disc. Theta dot is the spin rate omega, so the derivative of i hat is omega times j hat. In vector form, omega points along the rotation axis. Omega crossed with i hat therefore points along j hat, which gives the same derivative. As the disc turns, i hat never changes length. Its derivative remains sideways, at right angles to i hat, at every instant. That perpendicular sweep is the geometry of the cross product. The same reasoning holds for any vector rigidly attached to the body: a painted axis, a spoke, or a robot arm. Its derivative is omega crossed with that vector. Now take an arbitrary vector A and write it in the painted basis as A x i hat, A y j hat, plus A z k hat. The product rule separates its fixed-frame derivative into two families. One differentiates the components and leaves the basis vectors alone. A rider on the disc calls that first family the relative derivative, because those are the component changes measured against the painted axes. The second family leaves the components alone and differentiates i hat, j hat, and k hat. Each basis derivative contributes omega crossed with the corresponding basis vector. Factor omega cross from that whole family. The remaining bracket is the original vector A, so the family becomes omega cross A. This is the transport theorem. A vector's fixed-frame rate equals its relative rate plus omega cross the vector. Applying this statement twice produces every acceleration term that follows.

One line of geometry and one use of the transport theorem give the velocity relation. Begin with r P equals r B plus rho. Differentiate both sides in the fixed frame. The left side becomes the true velocity v P, and the first term on the right becomes the velocity v B of the moving origin. Rho needs the theorem. The red rho vector lies in the disc. The blue omega vector points along the rotation axis. Their yellow cross product is the extra velocity carried by the turning frame. Omega cross rho is at right angles to rho and omega. It lies flat in the plane of the disc, ninety degrees ahead of rho, with magnitude omega times rho. The complete velocity therefore has three pieces: motion of the origin, motion measured on the disc, and motion carried by the disc's spin. A beetle walks outward along a painted spoke while the turntable spins. The three velocity pieces form one head-to-tail chain. First is v B, the velocity of the frame's origin. A bolted turntable has v B equal to zero, while an aircraft or planet need not. Second is omega cross rho. It is tangential and square to the spoke, the velocity the beetle receives simply because the disc carries it. Third is v rel, directed outward along the spoke. Translate that vector to the tip of omega cross rho, and the two arrows now meet head to tail. The arrow from the chain's first tail to its final tip is v P, the beetle's true velocity over the ground. One more derivative produces the acceleration terms.

Differentiate the velocity relation once more. Each vector on the right is measured against the turning disc, so the transport theorem remains beside the calculation. The origin contributes a B directly. The other two derivatives need the theorem because omega cross rho and v rel are both vectors written in the painted basis. Apply the product rule to omega cross rho. Omega dot becomes alpha, so the first contribution is alpha cross rho. It is tangential while the spin rate is changing. The second contribution contains the fixed derivative of rho. Substitute v rel plus omega cross rho, then distribute omega cross across both terms. The fixed derivative of v rel also uses the theorem. It contributes a rel plus a second omega cross v rel. The distributed line contains one omega cross v rel. The final derivative contains another omega cross v rel. These two identical terms add. Their sum is two omega cross v rel. The factor of two is the Coriolis coefficient, produced by two separate derivatives. Collecting the result gives five acceleration terms, all expressed in quantities available to the rotating observer. First, a B is the acceleration of the frame's origin. A floor-mounted turntable makes it zero, but a vehicle or planet need not. Second, alpha cross rho is the Euler term. It appears while the spin rate changes and vanishes at steady omega. Third, omega cross omega cross rho is centripetal. It points from P toward the rotation axis, with magnitude omega squared times the distance out. Fourth, two omega cross v rel is Coriolis. It needs a turning frame and motion relative to that frame at the same instant. Fifth, a rel is the acceleration measured by instruments fixed to the disc. It is often the unknown the rotating observer must determine. The Coriolis condition is now exact. Relative motion without rotation gives zero. Rotation without relative motion also gives zero. Only the final case has both, so only that case carries a Coriolis acceleration.

Newton's second law is written in the fixed frame, but the rider reports a rel. Solving for that measured acceleration begins with the applied force F while the disc continues to turn. Subtract the origin term, the Euler term, the centripetal term, and the Coriolis term from F. The rule leaves m times a rel beneath it. Every subtracted quantity has units of force and occupies a force slot in the rider's equation. Their minus signs record the acceleration terms moved from the fixed-frame side. Bolt the turntable to the floor and hold a steady omega. Then a B is zero and alpha is zero, so those two terms disappear. The centripetal and Coriolis terms survive. The centripetal term changes sign when it crosses the equality. It becomes plus m omega squared rho, directed outward from the axis and increasing with distance from the centre. This is the centrifugal term. Centrifugal force is not an additional interaction. It is the inward centripetal acceleration moved across the equality and read from the rotating frame. The remaining correction is minus two m omega cross v rel, the Coriolis force. Its direction depends on how the body moves across the rotating disc. The painted launch spoke fixes one initial direction. A single ball leaves the centre with no sideways force, and the same flight is measured from the ground and from the disc. From the ground, the ball advances at constant speed along a straight inertial path. Its position stays on the launch direction while the blue current spoke turns underneath it. The gray launch spoke marks the disc at release. The blue current spoke is the same painted radius later in the flight, so their labels distinguish the two instants. From the disc, the painted spoke stays fixed. The identical ball traces a curved path because every later position is resolved against axes that have turned. The disc-frame path bends to the right of the ball's motion all the way outward. The Coriolis correction persists because the relative velocity persists. No sideways interaction acts on the ball. The curvature belongs to the rotating coordinates, which is why a rider assigns it a Coriolis force. The same correction matters in large weather systems, in flow across a pressure gradient, in long-range gunnery, and in the slow turn of a Foucault pendulum. The final relation has five terms, and each has a distinct physical question attached to it. The first asks how the frame origin moves. The second asks whether omega changes. The third asks how far out the point lies. Coriolis asks for relative motion, and the fifth is the rider's own acceleration. Every term follows from one fact about a painted basis vector: its derivative is omega crossed with itself. Apply that fact twice with the product rule, and the rotating-frame acceleration is complete.