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Centers of mass and centroids

Problem 5.351 · easy

Find the center of mass of the system: \( \displaystyle m_1 = 6 \) at \( \displaystyle x_1 = 3 \), \( \displaystyle m_2 = 3 \) at \( \displaystyle x_2 = 4 \), \( \displaystyle m_3 = 4 \) at \( \displaystyle x_3 = 1 \), \( \displaystyle m_4 = 5 \) at \( \displaystyle x_4 = 2 \).

\[ 4 \cdot 1 + 5 \cdot 2 + 3 \cdot 4 + 6 \cdot 3 = 44 \]
The moment about the origin, Σ mᵢxᵢ.✓ Proved
\[ \frac{22}{9} \]
Divide by the total mass 18.✓ Proved

Answer \( \bar{x} = \frac{22}{9} \)

Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
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	recomputed in floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution calculates the total moment (44) but divides by 9 instead of the total mass (18). The total mass is 6+3+4+5=18, so the center of mass i

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution calculates the total moment (44) but divides by 9 instead of the total mass (18). The total mass is 6+3+4+5=18, so the center of mass i
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the total moment and divides by the total mass to find the center of mass. The arithmetic is correct.
gpt-oss:20b: pass 2026-10-07

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/center_of_mass, checked 2026-10-07 with SymPy 1.14.0.