A test tube filled with water is being spun around in an ultracentrifuge with an

Question

A test tube filled with water is being spun around in an ultracentrifuge with angular velocity w(omega). The test tube is lying along a radius and the free surface of the water is at radius r0.

a) Show that the pressure at radius r within the test tube is:     P(pressure) = 0.5 * p(rho) * w^2(omega) *(r^2 - r0^2)

where p(rho) is the density of the water. Ignore gravity and ignore atmoshperic pressure.

I have attempted this problem many times, I am thinking it has something to do with Bernoullis equation but I can't figure it out quite any help would be great.

Explanation / Answer

The total pressure (P) at radius r is:

P = p*w^2*integral(r)dr evaluated between the limits r0 and r

P = 0.5*p*w^2*(r^2 - r0^2)

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