Rotating Liquid in a Cylindrical Vessel When a cylindrical container filled with liquid is revolved at a constant angular speed, the free surface assumes a paraboloid of revolution due to the balance of pressure, gravity, and centrifugal effects.

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:

• Rigid-body rotation of a liquid in a vertical cylindrical vessel.
• Steady angular speed; negligible wave motion.
• Newtonian fluid; free surface exposed to atmosphere.

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:

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