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Pressure of liquid given radius, help please!

  1. Aug 11, 2012 #1
    A test tube filled with water is being spun around in an ultracentrifuge with angular velocity. The test tube is lying along a radius and the free surface of the water is at radius r(o).

    Show that the pressure at radius r within the test tube is:

    p = .5(p)(angular velocity)^2(r^(2) -r(o)^2)

    where p is the density of the water. Ignore gravity and atmospheric pressure.


    p = p - g(density)(height)


    gravity or centripetal acceleration, a= r(angular velocity)^2

    height or depth of water, h = r- r(o)

    this only gets me to p= p + density*r*angular velocity^2(r-r(0))

    I'm not sure where the rest comes from!
     
  2. jcsd
  3. Aug 11, 2012 #2

    Doc Al

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    Staff: Mentor

    At any depth 'below' the surface, the pressure has to provide enough force to accelerate all the fluid 'above' it. Hint: Set up an integral.
     
  4. Aug 11, 2012 #3
    Doc Al, I'm not sure I understand what I should be taking the integral of. Could you explain further?
     
  5. Aug 11, 2012 #4

    Doc Al

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    Staff: Mentor

    Write an expression for the net force on an infinitesimal slice (thickness dr) of the fluid in the tube; then integrate from r(0) to r to find the total force, and then the pressure, at any point along the tube.
     
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