An incompressible fluid with density is in a horizontal

Chapter 12, Problem 12.87

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An incompressible fluid with density \(\rho\) 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 . Gravitational forces are negligible. Consider a volume element of the fluid of area A and  thickness \(d r^{\prime}\) a distance \(r^{\prime}\) 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 \(d p=\rho \omega^{2} r^{\prime} d r^{\prime}\). (b) If the surface of the fluid is at a radius \(r_{0}\) where the pressure is \(p_{0}\), show that the pressure p at a distance \(r \geq r_{0} \text { is } p=p_{0}+\rho \omega^{2}\left(r^{2}-r_{0}^{2}\right) / 2\). (c) An object of volume V and density \(\rho_{\mathrm{o b}}\) has its center of mass at a distance \(R_{\mathrm{cmob}}\) from the axis. Show that the net horizontal force on the object is \(\rho V \omega^{2} R_{\mathrm{cm}}\), where \(R_{\mathrm{cm}}\) is the distance from the axis to the center of mass of the displaced fluid. (d) Explain why the object will move inward if \(\rho R_{\mathrm{cm}}>\rho_{\mathrm{ob}} R_{\mathrm{cmob}}\) and outward if \(\rho R_{\mathrm{cm}}<\rho_{\mathrm{ob}} R_{\mathrm{cmob}}\). (e) For small objects of uniform density, \(R_{\mathrm{cm}}=R_{\mathrm{cmob}}\). What happens to a mixture of small objects of this kind with different densi- ties in an ultracentrifuge?