A doubt on fluid mechanics

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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 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 ρRcmobRcmob and outward if ρRcmobRcmob.
(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 Rcm, and Rcmob stand for so was unable to proceed further.

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

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  • #2
tiny-tim
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(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 ρRcmobRcmob and outward if ρRcmobRcmob.
(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 Rcm, and Rcmob stand for so was unable to proceed further.
Hi iitjee10! :smile:

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

so ρob = massob/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). :smile:
 
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The density of the object has to be constant otherwise the problem here will become hopelessly complicated. This is what I think.
 
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Anyone?
 
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Doc Al
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But I could not understand (rather visualise) what Rcm, and Rcmob 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.
 

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