The ultracentrifuge is an important tool for separating and analyzing proteins i

Question

The ultracentrifuge is an important tool for separating and analyzing proteins in biological research. Because of the enormous centripetal accelerations that can be achieved, the apparatus must be carefully balanced so that each sample is matched by another on the opposite side of the rotor shaft. Failure to do so is a costly mistake. Any difference in mass of the opposing samples will cause a net force in the horizontal plane on the shaft of the rotor. Suppose that a scientist makes a slight error in sample preparation, and one sample has a mass 11 { m mg} greater than the opposing sample.

If the samples are 11 cm from the axis if the rotor and the ultracentrifuge spins
at 7.4×104 rpm, what is the magnitude of the net force on the rotor due to the unbalanced samples?

Explanation / Answer

We recall that the centripetal acceleration ac=v2/r, where v is the tangential velocity and r is the radius.

v = 7.4*10^4*2**0.11m/min. That is, every minute, the sample goes around the circumference of a circle 7.4*10^4 times, and the circumference is 2r.

v= 51145.1m/min*1min/60sec = 852.419m/s. (We converted from m/min to m/sec)

So, ac=852.4192/0.11 = 6.61*106m/s/s

And, F=ma. The net difference in masses is 11mg = 1.1*10^-5kg.

So, F = 6.61*10^6*1.1*10^-5 = 72.6618N

The magnituted of the net force is about 72.7N.

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