The analytical ultracentrifuge olympics

Classical Mechanics Level 5

One way that scientists determine the mass of a biological particle is analytical centrifugation. A preparation (several picomoles) of the particle is layered on top of a viscous fluid such as sucrose solution which is then spun at high speeds. The particle experiences several forces: centripetal force (which pushes it down the gradient), buoyancy (which pushes it up the gradient), and drag (that slows its motion). After a short time, these forces balance and the particle travels down the gradient at a constant rate.

Suppose a particle if found to travel at the rate $$v$$ through sucrose. Now, we measure the sedimentation rate of a dimer (two of the particles stuck together). This new particle travels down the gradient at the rate $$v^\prime$$, what is $$v^\prime /v$$?

Notes and assumptions

• Assume the particles are spherical.
• Assume that the dimer is formed by melting down two particles and forming a sphere with twice the volume of the monomer.
• For simplicity, assume that the force pulling particles down the gradient is provided by a strong, constant gravitational field.
• The friction the particle feels is Stokes drag $$F \sim rv$$ where $$r$$ is the radius of the particle.
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