Difficulty: Easy
Correct Answer: a paraboloid
Explanation:
Introduction:The shape of a rotating liquid surface is a classic application of hydrostatics in a rotating reference frame. Recognizing the resulting geometry is important for metering, mixing, and rotating-tank experiments.
Given Data / Assumptions:
Concept / Approach:
In a rotating frame, the liquid experiences an effective body force with radial component proportional to r * omega^2 and vertical component due to gravity. The condition for a free surface is that pressure remains atmospheric along it; integrating the equilibrium equations gives a quadratic dependence of elevation on radius.
Step-by-Step Solution:
1) Set differential condition: dp = rho * (g dz - omega^2 r dr).2) On the free surface, p is constant (atmospheric).3) Integrate: dz/dr = (omega^2 r) / g.4) Integrate again: z = (omega^2 / (2 g)) * r^2 + constant, which is a parabola in r.5) Rotating this curve about the axis yields a paraboloid of revolution.Verification / Alternative check:
Laboratory demonstrations show a deepened rim and depressed center forming a smooth parabolic surface; measuring z versus r^2 gives a straight line, confirming the parabolic relation.
Why Other Options Are Wrong:
A triangle: Free surfaces are smooth and cannot form polygonal facets in this context. An ellipse: Would require a quadratic constraint in both axes not produced by this balance. None of these: Incorrect because the surface is demonstrably parabolic.
Common Pitfalls:
Confusing cross-sectional parabolic shape with full 3D form; neglecting that the result is a paraboloid, not just a plane curve.
Final Answer:
a paraboloid
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