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Hydrostatic force

Problem 5.434 · easy

A vertical plate shaped like a rectangle 2 m wide and 4 m tall is submerged in water with its top 2 m below 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 and pressure 9800·y.
Reviewed
\[ \int\limits_{2}^{6} 19600 y\, dy = 313600 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( 313600 \approx 313600\ \text{N} \)

✓ 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 identifies the depth variable, the constant width, and the integration limits corresponding to the plate's position. The calculation is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The setup correctly identifies the depth variable, the constant width, and the integration limits corresponding to the plate's position. The calculation is correct.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The setup correctly identifies the pressure as a function of depth and the width of the plate. The limits of integration (2 to 6) correctly correspond to the top and bottom of the submerged plate. The calculation is correct.

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 2026-10-09 with SymPy 1.14.0.