How Big Is the Centripetal Acceleration in an Ultracentrifuge?

The term rev/min stands for revolutions per minute. By converting this to radians per second, we obtain the angular velocity $\omega$. Because $r$ is given, we can use the second expression in the equation $a_{\text{c}}=\frac{v^2}{r};a_{\text{c}}={\mathit{r\omega }}^2$ to calculate the centripetal acceleration.

To convert $7\text{.}\text{50}\times {\text{10}}^4\text{rev}/\text{min}$ to radians per second, we use the facts that one revolution is $\mathrm{2\pi }\mathrm{<mtext>rad</mtext>}$ and one minute is 60.0 s. Thus,

Now the centripetal acceleration is given by the second expression in as

Converting 7.50 cm to meters and substituting known values gives

Note that the unitless radians are discarded in order to get the correct units for centripetal acceleration. Taking the ratio of $a_{\text{c}}$ to $g$ yields

This last result means that the centripetal acceleration is 472,000 times as strong as $g$. It is no wonder that such high $\omega$ centrifuges are called ultracentrifuges. The extremely large accelerations involved greatly decrease the time needed to cause the sedimentation of blood cells or other materials.