Closed cylindrical vessel completely filled with liquid – scaling of total pressure on the top plate For a closed cylindrical tank completely filled with liquid at a uniform internal gauge pressure p, how does the total force on the circular top (roof) vary with the vessel radius r?

Introduction / Context:
Design of pressure vessels and storage tanks requires estimating the net force on heads (end plates). For a cylindrical vessel completely filled with liquid and pressurized uniformly, the total force on the top plate depends on the plate area and the internal pressure. This question checks your scaling intuition with radius.

Given Data / Assumptions:

• Uniform internal gauge pressure p acts on the inside surfaces.
• The top plate is a flat circular area of radius r.
• Hydrostatic gradients are either negligible (small height) or already included in p; the key point is the uniform pressure assumption.

Concept / Approach:
Total force on a surface under uniform pressure equals pressure multiplied by area. For a circle, area A = π r^2. Therefore, the resultant force scales with r^2. This scaling result holds irrespective of the absolute value of p as long as p does not itself depend on r.

Step-by-Step Solution:
Step 1: Identify area A = π r^2.Step 2: Write force F = p * A.Step 3: Substitute A → F = p * π r^2.Step 4: Conclude F ∝ r^2 for constant p.

Verification / Alternative check:
Dimensional analysis shows p has units N/m^2 and area has m^2; their product gives newtons (N), as expected for force. Doubling r quadruples the area and therefore quadruples the total force at the same p.

Why Other Options Are Wrong:

• Inverse powers of r (options b and d): contradict F = p * π r^2.
• r^4 dependence: would arise in bending/deflection formulas, not in pressure resultant.
• Independent of r: incorrect because area depends on r.

Common Pitfalls:
Mixing up pressure (intensive) and force (extensive). Pressure stays the same if r changes, but the total force on a larger area increases with r^2.

Final Answer:
directly proportional to r^2