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Answer on Differential Equations Question for Julia

Question #14968
Find the form of the surface of fluid inside separating centrifuge while working and angular velocity of the fluid to reach the given height H.
Expert's answer
Let's look at the problem in Cartesian coordinates.
Let (0, 0) be the
coordinate of the center of centrifuge, w be the angular velocity
of
centrifuge.
Any section (that crosses the origin and is perpendicular to XoY
plane) of the surface
has a form of parabola:

Z - Z0 = w^2 * x^2 /
(2g).

When we substitute x -> sqrt(x^2 + y^2) we get the equation of
the surface:

Z - Z0 = w^2 * (x^2 + y^2) / (2g).

Then

H = Z0
+ r^2 * w^2 / (2g),
where r is the radius of centrifuge.

As h = 1/2 *
(Z0 + H), so
w = 2/r * sqrt(g*(H - h)) - angular velocity.

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