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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?

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  • $\begingroup$ If the substances to separate move out- or inward, their tangential velocity and the radius of their path are not constant, though they might not change a lot depending on the geometry of the setup. In this case, also the centripetal force (which depends on the radius) is subject to change. I would assume that except for the acceleration and deceleration at the beginning and end of the centrifugation, the angular velocity should be constant, but it is of course totally possible to build a centrifuge which varies this, too. $\endgroup$
    – nu.
    Aug 20, 2021 at 7:49
  • $\begingroup$ Why does this matter, i.e. what is it relevant for which variables may or may not be constant in a centrifuge? $\endgroup$
    – ACuriousMind
    Aug 20, 2021 at 11:54

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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.

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    $\begingroup$ so its safe to say that all the particles in the test tube, regardless of mass, are experiencing a constant centripetal acceleration? I thought if the centrifuge provides a constant centripetal force, then decreasing mass means increasing Ac whilst increasing mass means decreasing Ac. $\endgroup$
    – yug
    Aug 22, 2021 at 6:25
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    $\begingroup$ .. @AlBrown, consider two masses in circular motion along paths with radius $r$ and $R$. ($R\gt r$). Both have the same angular velocity. That means both rotates keeping aligned. Shouldn't centripetal acceleration of mass on track with radius $R$ greater than that of the other? $\endgroup$
    – ACB
    Aug 22, 2021 at 6:41
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    $\begingroup$ @ACB Just seeing your comment for some reason. Hmm. Yes that’s correct. That’s what was meant by, “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.” There is a gradient to the apparent gravity. $\endgroup$
    – Al Brown
    Aug 22, 2021 at 15:20
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    $\begingroup$ @yug All particles at a given radius experience the same centripetal acceleration, $a$. $~$ So by $F=ma$, the heavier particles experience higher force. This makes denser particles get pushed to the bottom harder. That’s why the dense stuff ends up on the bottom. Did that answer the question? $\endgroup$
    – Al Brown
    Aug 22, 2021 at 15:24

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