Pressure of liquid given radius, help please!

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

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.