An incompressible fluid with density p is in a horizontal test rube of inner cross-sectional area. The test tube spins in a horizontal circle in an ultracentrifuge at an angular speed w. Gravitational forces are negligible. Consider a volume element of the fluid of area A and thickness dr' a distance r' from the rotation axis. The pressure on its inner surface is p and on its outer surface is p + dp.
(a) Apply Newton's second law to the volume element to show that dp = pw2r'dr'.
(b) If the surface of the fluid is at radius r0 where the pressure is Po> show that the pressure p at a distance r > r0 is p = Po + pw2 (r2 – r0)/2.
(c) An object of volume V and density P ob has its center of mass at a distance R cmob, from the axis. Show that the net horizontal force on the object is pVw2Rem, where Rem is the distance from the axis to the center of mass of the displaced fluid.
(d) Explain why the object will move inward if pRcm> Pob, Rcmoo and outward if pRom < pooRcmoo.
(e) For small objects of uniform density, Rem = R.m.t,. What happens to a mixture of small objects of this kind with different densities in an ultracentrifuge?