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VortexLattice
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Homework Statement
We have a U tube, like this one:
With the a nonviscous, incompressible fluid at height h at equilibrium. We're interested in finding the frequency of small oscillations about the equilibrium. The tubes have area A (though I'm guessing this falls out in the end) and are L apart from each other (this definitely falls out).
Homework Equations
At equilibrium:
[itex]f_{app} = \frac{1}{\rho}\nabla p [/itex]
Where f_app is an applied force per unit mass, [itex]\rho[/itex] is the density (constant here), and p is the pressure.
The Attempt at a Solution
There are two forces at play: The applied gravitational force per unit mass, [itex]f_{app} = -g\hat{z}[/itex], and the pressure force (I think). At equilibrium, they should be equal. I'm going to say z is in the vertical direction, and z = 0 at equilibrium. Using the equation above and solving for p, we find that [itex]p = -\rho g z[/itex].
This is good because it's a linear term in z. Now, I know I need to apply Newton's 2nd law and get something of the form [itex]\ddot{z} = -k^2 z[/itex]. We have p, and force on a surface is pA, where A is the area. So I almost have it, but I'm having trouble putting it all together.
I'm also confused because it seems like when I try to sum all the forces for Newton's 2nd law, I have the pressure force and the gravitational force. But the gravitational force doesn't have z in it... I guess I could do that substitution trick where I let x = z - g, but it just seems like I'm doing something simple wrong.
Can anyone help?? Thanks!