Pressure in spinning water column in ultracentrifuge

[harlow_barton]

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!

 

[Doc Al]

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.

 

[harlow_barton]

Doc Al, I'm not sure I understand what I should be taking the integral of. Could you explain further?

 

[Doc Al]

Doc Al, I'm not sure I understand what I should be taking the integral of. Could you explain further?

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.