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Centers of mass and centroids

Problem 5.345 · easy

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

\[ 3 \left(-2\right) + 1 \cdot 1 + 3 \cdot 2 + 6 \cdot 3 = 19 \]
The moment about the origin, Σ mᵢxᵢ.✓ Proved
\[ \frac{19}{13} \]
Divide by the total mass 13.✓ Proved

Answer \( \bar{x} = \frac{19}{13} \)

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: pass — The solution correctly calculates the total moment and total mass, and applies the center of mass formula. The arithmetic is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the total moment and total mass, and applies the center of mass formula. The arithmetic is correct.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly calculate or state the total mass (M = 13), which is a necessary step to justify the denominator in the final division. While the arithmetic is correct, omitting the calculation of the total mass makes the derivation incomplete and skips a critical part of the center of mass formula.
gpt-oss:20b: fail (error) 2026-10-07 — The moment calculation uses the wrong masses at the wrong positions; the terms should be 3*2 + 3*(-2) + 6*3 + 1*1, not 3*(-2)+1*1+3*2+6*3.

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.