Consider a macromolecule (i.e. a very large molecule with a molecular weight of several millions) immersed in an incompressible fluid of density p at the absolute temperature T. The volume U occupied by one such molecule can be considered known since the volume occupied by a mole of macromolecules can be determined by volume measurements on a solution of macromolecules. A dilute solution of this type is now placed in an ultracentrifuge rotating with a high angular velocity w. In the frame of reference rotating with the centrifuge, any particle of mass m at rest with respect to this frame is then acted upon by an outward centrifugal force mw2r, where r dentoes the distance of the particle from the axis of rotation. (a) What is the net force acting in this frame of reference on a macromolecule of mass m, if the bouyancy effect of the surrounding fluid is taken into account? (b) Suppose that equilibrium has been attained in this frame of reference so that the mean number n(r)dr (per unit volume) of macromolecules located at a distance from the axis of rotation between r and r + dr is independent of time. Apply the canonical distribution to find (to within a constant of proportionality) the number n (r) dr as a function of r. (c) Measurements of the relative number n (r) of molecules as a function of r can be made by measuring the absorption of light by the solution. Show how such measurements can be used to deduce the mass m of a macromolecule.