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Hydrostatic force

Problem 5.223 · easy

A vertical plate shaped like a semicircle of radius 3 m, flat side up is submerged in water with its top at the surface. Find the force of the water on one side. (Water weighs $ \displaystyle 9800\ \text{N}/\text{m}^3 $.)

Let y be depth below the surface. A horizontal strip at depth y has width w(y) = 2*sqrt(9 - y**2) and pressure 9800·y.
Reviewed
\[ \int\limits_{0}^{3} 19600 y \sqrt{9 - y^{2}}\, dy = 176400 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer $ 176400 \approx 176400\ \text{N} $

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The setup correctly models the geometry (width as a function of depth for a semicircle with flat side up) and the physics (hydrostatic pressure). The integration bounds and result are correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly models the geometry (width as a function of depth for a semicircle with flat side up) and the physics (hydrostatic pressure). The integration bounds and result are correct.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly models the geometry and physics: the width function $w(y) = 2\sqrt{9-y^2}$ is correct for a semicircle of radius 3 with the flat side at the surface, and the pressure-depth relationship is standard. The integration limits and result are correct.
gpt-oss:20b: pass 2026-10-04

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/hydrostatic_force, checked 2026-10-04 with SymPy 1.14.0.