Calculate the centripetal acceleration of a point 7.50 cm from the axis of an ultracentrifuge spinning at 7.5 × 10^4 rev/min. Determine the ratio of this acceleration to that due to gravity. See Figure 6.9(b).
Strategy
The term rev/min stands for revolutions per minute. By converting this to radians per second, we obtain the angular velocity ω. Because r is given, we can use the second expression in the equation a_c = \frac{v^2}{r}; a_c = rω^2 to calculate the centripetal acceleration.
To convert 7.50×10^4 rev / min to radians per second, we use the facts that one revolution is 2πrad and one minute is 60.0 s. Thus,
ω = 7.50×10^4 \frac{rev}{min} × \frac{2π rad}{ 1 rev} × \frac{1 min}{60.0 s} = 7854 rad/s. (6.19)
Now the centripetal acceleration is given by the second expression in a_c = \frac{v^2}{r} ; a_c = rω^2 as
a_c = rw^2 (6.20)
Converting 7.50 cm to meters and substituting known values gives
a_c = (0.0750 m)(7854 rad/s)^2 = 4.63×10^6 m/s^2. (6.21)
Note that the unitless radians are discarded in order to get the correct units for centripetal acceleration. Taking the ratio of a_c to g yields
\frac{a_c}{g} = \frac{4.63×10^6}{9.80} = 4.72×10^5. (6.22)
Discussion
This last result means that the centripetal acceleration is 472,000 times as strong as g . It is no wonder that such high ω centrifuges are called ultracentrifuges. The extremely large accelerations involved greatly decrease the time needed to cause the sedimentation of blood cells or other materials.