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private_donkey
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I am wondering if my reasoning is correct for determining the energy due to the buoyancy of a deformed water surface.
Essentially, one has a floating cylinder that depresses the surface of a liquid in an infinite tank as seen in the figure. I want to compare the energy of a flat surface with the energy of the deformed surface due to the displacement of the liquid by air (I know there are other energies involved but for the time being I am only looking at the energy due to the buoyancy of the displace liquid due to the deformed surface), not including the liquid displaced by the cylinder.
My reasoning is
[tex] \begin{align}<br /> E_{deformed} - E_0 &= \int_0^{-h} -F_{buoyancy} dz \\<br /> &= \int_0^{-h} -P A dz \\<br /> &= \rho g \int_0^{h} z \pi f(z)^2 dz<br /> \end{align}[/tex]
where [itex]\rho[/itex] is the density of the liquid. P is pressue, given by [itex]P = \rho g z[/itex], A is area, and thus A dz is volume of the displace liquid. [itex]A[/itex] can be given by [itex]A = \pi f(z)^2[/itex], where [itex]f(z)[/itex] describes the radius from the centre of the cylinder to the edge of the meniscus to make infinitesimal disks as a function of z. g is gravity accel.
Is this reasoning correct?
Essentially, one has a floating cylinder that depresses the surface of a liquid in an infinite tank as seen in the figure. I want to compare the energy of a flat surface with the energy of the deformed surface due to the displacement of the liquid by air (I know there are other energies involved but for the time being I am only looking at the energy due to the buoyancy of the displace liquid due to the deformed surface), not including the liquid displaced by the cylinder.
My reasoning is
[tex] \begin{align}<br /> E_{deformed} - E_0 &= \int_0^{-h} -F_{buoyancy} dz \\<br /> &= \int_0^{-h} -P A dz \\<br /> &= \rho g \int_0^{h} z \pi f(z)^2 dz<br /> \end{align}[/tex]
where [itex]\rho[/itex] is the density of the liquid. P is pressue, given by [itex]P = \rho g z[/itex], A is area, and thus A dz is volume of the displace liquid. [itex]A[/itex] can be given by [itex]A = \pi f(z)^2[/itex], where [itex]f(z)[/itex] describes the radius from the centre of the cylinder to the edge of the meniscus to make infinitesimal disks as a function of z. g is gravity accel.
Is this reasoning correct?
