What do Rcm and Rcmob represent in this fluid mechanics problem?

[iitjee10]

Q2. 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 r_o where the pressure is p_o, show that the pressure p at a distance r≥r_o is p=p_o+ρω2(r²-r_o²)/2.
(c) An object of volume V and density ρ_ob has its centre of mass at a distance R_cmob from the axis. Show that the net horizontal force on the object is ρVω²R_cm, where R_cm is the distance from the axis to the center of mass of the displaced fluid.
(d) Explain why the object will move inward if ρR_cm>ρ_obR_cmob and outward if ρR_cm<ρ_obR_cmob.
(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?

This was a problem given in Sears and Zemansky (University Physics)

I have solved parts (a) and (b) of this problem completely. But I could not understand (rather visualise) what R_cm, and R_cmob stand for so was unable to proceed further.

Any help in visualising these terms would be of a great help

 

[tiny-tim]

(c) An object of volume V and density ρ_ob has its centre of mass at a distance R_cmob from the axis. Show that the net horizontal force on the object is ρVω²R_cm, where R_cm is the distance from the axis to the center of mass of the displaced fluid.
(d) Explain why the object will move inward if ρR_cm>ρ_obR_cmob and outward if ρR_cm<ρ_obR_cmob.
(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?
…
But I could not understand (rather visualise) what R_cm, and R_cmob stand for so was unable to proceed further.

Hi iitjee10!

It's not clear, but I think ρ_ob is not constant (but ρ, the density of the fluid, is) …

so ρ_ob = mass_ob/V.

So the c.o.m. of the object is not at the centroid (where the c.o.m. of the displaced fluid would have been).

 

[iitjee10]

The density of the object has to be constant otherwise the problem here will become hopelessly complicated. This is what I think.

 

[iitjee10]

Anyone?

 

[Doc Al]

But I could not understand (rather visualise) what R_cm, and R_cmob stand for so was unable to proceed further.

As explained in the problem statement, one is the distance to the center of mass of the displaced fluid while the other is the distance to the center of mass of the object. They are being very precise and picky!

The density of the object has to be constant otherwise the problem here will become hopelessly complicated. This is what I think.

Not really. For example, part C really has nothing to do with the density of the object, just the force on it from the surrounding fluid.