Which variables of motion are kept constant in a centrifuge? Which circular motion variables are kept constant in a centrifuge? My thinking so far is that magnitude of centripetal acceleration is constant and it follows that centripetal force is also constant. Are tangential velocity and radius constant as well?
 A: The centrifuge starts up, and quickly attains maximum angular velocity, $\omega$, and stays there. There is zero angular acceleration, $\alpha$, while running. It runs for awhile at constant speed, sometimes for hours. Then it is turned-off and quickly decelerated to a stop and the test tubes are removed.
This provides constant radial acceleration due to the angular velocity. When something is spinning at a constant speed, the centripetal force needed to hold the object in its circular path stays constant. This force pushes radially inward. The result is that things within the object, inside it, experience a centrifugal force pushing them radially outward.
The test tube is angled to be nearly horizontal with the open end at lower radius so that the centrifugal force pushes everything to the bottom of the tube. It’s basically like putting the tube in a very high gravity. Substances with higher density will be forced toward the bottom even harder, much harder, than under gravity, and they will displace the stuff with low density. When you take it out, the highest density substance will be on the bottom, and the next highest density above that, etc, with the low-density substances on the top of the tube. In addition to high apparent gravity (centrifugal force) to make the dense stuff nestle its way to the bottom, you have the added benefit that centrifugal force is always higher at higher radii, so the bottom of the tube has slightly higher apparent gravity than the top (the bottom is at a higher radius). This helps a little too. Force $F$, velocity $v$, acceleration $a$, radius $r$, and angular velocity $\omega$, we have (remember angular acceleration $\alpha$ is zero during the bulk of the run):
$$ v= \omega r \text{ , } a= \frac{v^2}{r} = \omega^2 r$$
$$ \implies F = ma = \frac{mv^2}{r}= m \omega^2 r$$
So for example if the radius is 1/4 meter, and $\omega$ is just five revolutions per second, or $10 \pi$ radians per second, we have:
$$\omega = 10 \pi  \text{ , } a=\frac{100 \pi^2}{4} = 250\frac{m}{s^2} =25g$$
$F$ and $a$ point radially. $v$ points tangentially. And technically the $\omega$ vector points along the axis by the right hand rule. With constant $\omega$, the magnitude of the velocity is constant, but the velocity vector keeps changing direction. A changing velocity vector is achieved via acceleration. The vector keeps bending inward toward the center which is the radial inward acceleration.
