Shape of surface of fluid in a rotating tank

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SUMMARY

The shape of the surface of a fluid in a rotating tank, specifically mercury, can be modeled using principles of fluid dynamics. The surface profile is determined by the rotation rate and the density of the fluid, leading to a concave shape that can be approximated as either spherical or parabolic. As the rotation rate increases, the depth of the fluid surface also increases, while a decrease in fluid density results in a more pronounced curvature. Relevant equations include the relationship between tangential velocity (v), radius (r), and angular velocity (ω), expressed as v = rω.

PREREQUISITES
  • Understanding of fluid dynamics principles
  • Familiarity with rotational motion concepts
  • Knowledge of basic calculus for deriving equations
  • Experience with mathematical modeling of physical systems
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  • Research the derivation of the shape of rotating fluid surfaces
  • Study the properties of liquid mirrors and their applications
  • Explore the mathematical modeling of parabolic and spherical surfaces
  • Learn about the effects of angular velocity on fluid behavior
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Students in physics or engineering, particularly those studying fluid dynamics, mechanical engineers, and anyone interested in the applications of rotating fluids in optical systems.

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Homework Statement


A concave astronomical telescope mirror may be made by rotating a circular tank of mercury. Find an expression for the shape of the surface in terms of the density of mercury, the radius from the centre, and the rotation rate.


Homework Equations


v = r \omega


The Attempt at a Solution


Blah, no idea where to start. Like the problem says, the surface would be concave. Spinning faster would make it deeper, as would lowering the density of the fluid in the tank. The shape would be either spherical or paraboloid elliptic. My guess would be parabolic. There would be some pressure variation outwards from the centre, maybe? I don't know. If someone could just throw a couple relevant equations my way, that would be helpful.
 
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