Fluid in rotating tube with different initial levels

[Soren4]

Homework Statement

Homework Equations

Fluid in rotation

The Attempt at a Solution

This exercise is quite different from the classic one of fluidi in rotation. Before rotation starts the height in one branch is bigger than in the other, so I do not really know how to approach the problem.

My main difficulty is: how can I determine the constant $C$ in the following expression in this case?

$$p(r,z)=-\rho g z+\frac{1}{2} \rho \omega^2 r^2+C$$

(The frame of reference considered has the $z$ axis towards up and placed on axis of rotation, $r$ is the radial coordinate)

The fact is that I do not really know how to impose the condition for determinimg $C$ as a function of $\omega$ (which is what I want to determine). I think that, once $C$ is determined the rest of exercise is straightforward.

So how can I determine $C$?

 

[haruspex]

Homework Statement

View attachment 103167

Homework Equations

Fluid in rotation

The Attempt at a Solution

This exercise is quite different from the classic one of fluidi in rotation. Before rotation starts the height in one branch is bigger than in the other, so I do not really know how to approach the problem.

My main difficulty is: how can I determine the constant $C$ in the following expression in this case?

$$p(r,z)=-\rho g z+\frac{1}{2} \rho \omega^2 r^2+C$$

(The frame of reference considered has the $z$ axis towards up and placed on axis of rotation, $r$ is the radial coordinate)

The fact is that I do not really know how to impose the condition for determinimg $C$ as a function of $\omega$ (which is what I want to determine). I think that, once $C$ is determined the rest of exercise is straightforward.

So how can I determine $C$?

In condition a), what is the height of the fluid in the open section? What is the pressure at its top?

 

[rude man]

The fact is that I do not really know how to impose the condition for determinimg $C$ as a function of $\omega$ (which is what I want to determine). I think that, once $C$ is determined the rest of exercise is straightforward.

You can't use Bernoulli when the tube is rotated.
Instead:
What total centripetal force is needed to maintain the system in equilibrium when p_A = 0.8e5 Pa?
How is this force going to be provided?