Centripetal Acceleration (a = ω²r)

Because $v = \omega r$, the familiar $a_c = v^2/r$ can be rewritten as $a_c = \omega^2 r$. It is the same acceleration, still pointing at the centre, expressed in the variable that machinery is actually specified in — nobody sells a centrifuge by its rim speed, they sell it by its rpm. The substitution is worth doing explicitly once: $(\omega r)^2/r = \omega^2 r$, and the $r$ that survives is a single power rather than an inverse one.

That change of variable reverses the intuition, and the reversal is the interesting part. At a fixed linear speed $v$, a larger radius means a gentler acceleration; at a fixed rotation rate $\omega$, a larger radius means a harsher one. Both statements are true and they describe different situations. A car taking a wider line through a corner at the same speed is doing the first. A sample moved further out in a spinning rotor is doing the second.

A laboratory centrifuge at 10 000 rpm has $\omega = 10\,000 \times 0.10472 \approx 1047$ rad/s. With a 10 cm rotor radius, $a_c = 1047^2 \times 0.10 \approx 110\,000$ m/s², which is about 11 200 times $g$. That is the whole principle of the instrument: an artificial gravity field thousands of times Earth's, driving particles that would take days to settle out under gravity to the bottom of a tube in minutes.

Centrifuge work has a specific and expensive version of the radius mistake. Relative centrifugal force is quoted as a multiple of $g$, so a protocol calling for "12 000 × g" wants $a_c = 12\,000 \times 9.81 = 117\,700$ m/s², not 12 000 in SI units — and converting that to an rpm setting requires the rotor's radius, which is not the same for every rotor that fits the same machine. Swapping a protocol between a fixed-angle rotor and a swinging-bucket rotor without recomputing the rpm is a standard way to ruin a separation. Worse, $r$ varies along the tube: the top of the sample sits at $r_{min}$ and the bottom at $r_{max}$, so the field is not uniform and published figures normally refer to $r_{max}$. Beyond the laboratory, the same $\omega^2 r$ sets the burst limit of every grinding wheel and flywheel, because the hoop stress it induces climbs with the square of speed and with the square of radius — which is why over-speeding a wheel is so much more dangerous than it sounds.