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 mg greater than the opposing sample.
If the samples are 10 cm from the axis if the rotor and the ultracentrifuge spins at 7.5×104 rpm , what is the magnitude of the net force on the rotor due to the unbalanced samples? I got 82 N but it was wrong

Explanation / Answer

As we know from Newton's second law that force F=mass*acceleration

and the centrifugal acceleration is a_c = w^2*r

Here in the question, r= 10 cm or 0.10 m

w = 7.5*10^4 rpm =7.5*10^4*2pi/60 = 7853.98 rad/s

hence acceleration a_c =7853.98^2*0.10 = 6168502.74 m/s^2

Mass m = 11*10^-6 kg

So force F = ma = 11*10^-6*6168502.74 =  67.85 N

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