Final answer:
To match the terminal velocity under a centrifuge to that under gravity, equate the gravitational force at terminal velocity to the centrifugal force in the centrifuge and solve for angular velocity, and convert it to frequency.
Step-by-step explanation:
To determine the frequency at which a centrifuge must rotate so that the terminal velocity of the particles is equal to their terminal velocity under the force of gravity, we need to equate the centrifugal force experienced by the particles in the centrifuge to the gravitational force they experience when falling at terminal velocity.
For a particle falling with terminal velocity, the downward gravitational force (weight) is balanced by the upward drag force and the buoyant force. The drag force in the case of spherical particles in the liquid is described by Stokes Law, which is Fs = 6πrηv, where r is the radius of the particle, η is the coefficient of viscosity, and v is the velocity of the particle.
In a centrifuge, the centrifugal force will be equal to Fc = mω2r, where m is the mass of the particle, ω is the angular velocity of the centrifuge, and r is the radius at which the particle is revolving (centrifuge radius).
To find the frequency at which the particles should be centrifuged, we set the centrifugal force equal to the force under gravitational terminal velocity and solve for the angular velocity, ω. This ω is then converted to frequency, f = ω/(2π).