In a closed cylindrical vessel completely filled with liquid and rotating as a forced vortex about its vertical axis, the total hydrostatic-dynamic thrust acting on the top (lid) varies with which power of the cylinder radius?

Introduction / Context:
When a completely filled cylindrical vessel of liquid is rotated about its vertical axis, the liquid executes a forced vortex motion (rigid-body rotation). Engineers often need to estimate the total thrust on the top cover or bottom due to the combined hydrostatic and centrifugal effects. Understanding how this thrust scales with the vessel radius is essential for lid design and fastening strength.

Given Data / Assumptions:

• Cylindrical vessel of radius R, fully filled with an incompressible liquid of density rho.
• Uniform angular velocity omega about the vertical axis; free surface does not exist (vessel is closed and full).
• Gravity acts downward; analysis on a horizontal top lid.

Concept / Approach:
In a forced vortex, the pressure varies radially according to dp/dr = rho * omega^2 * r. Integrating from the axis to radius r gives a pressure rise proportional to r^2. The total thrust on the lid is obtained by integrating this pressure over circular rings of area 2 * pi * r * dr, which introduces another factor of r. Consequently, the final thrust scales with R^4.

Step-by-Step Solution:
Radial pressure distribution: p(r) = p(0) + (rho * omega^2 / 2) * r^2.Elemental force on ring: dF = p(r) * dA = p(r) * 2 * pi * r * dr.Integrate from r = 0 to R: F = ∫[0→R] [p(0) + (rho * omega^2 / 2) * r^2] * 2 * pi * r dr.The variable part integrates as ∝ ∫ r^3 dr, yielding a term ∝ R^4.

Verification / Alternative check:
Dimensional reasoning confirms that pressure rise includes r^2; integrating over area adds another r^2 cumulatively, leading to R^4 dependence for the rotational contribution to thrust.

Why Other Options Are Wrong:
radius, (radius)^2, (radius)^3: each underestimates the radial build-up compounded by area integration.None of these: incorrect because (radius)^4 is correct.

Common Pitfalls:

• Confusing static hydrostatic thrust scaling (which involves R^2) with rotational forced-vortex thrust scaling (R^4).
• Forgetting that the vessel is completely full, so no parabolic free surface forms; still, the forced vortex pressure field applies.

Final Answer:
(radius)^4