- 42

- 0

**1. Homework Statement**

An incompressible fluid with density ρ is in a horizontal test tube of inner cross-sectional area A. The test tube spins in a horizontal circle in an ultracentrifuge at an angular speed ω. Gravitational forces are negligible. Consider a volume element of the fluid of area A and thickness dr' at a distance r' from the rotation axis. The pressure on its inner surface is p and outer surface is p+dp.

(a) Show that dp=ρω2r'dr'.

(b) If the surface of the fluid is at a radius ro where the pressure is po, show that the pressure p at a distance r≥ro is p=po+ρω2(r2-ro2)/2.

(c) An object of volume V and density ρob has its centre of mass at a distance Rcmob from the axis. Show that the net horizontal force on the object is ρVω2Rcm, where Rcm is the distance from the axis to the center of mass of the displaced fluid.

(d) Explain why the object will move inward if ρRcm>ρobRcmob and outward if ρRcm<ρobRcmob.

(e) For small objects of uniform density, Rcm = Rcmob. What happens to a mixture of small objects of this kind with different densities in an unltracentrifuge?

**2. Homework Equations**

ω=v/r

A= π*r^2

F=ma

p=F/A

**3. The Attempt at a Solution**

I have really tried everything I could but i simply dont understand the way how to tackle these problems. I would really like to know how these answers come about. Can anybody please help me with discussing this??

I would really appreciate it!