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Centers of mass and centroids

Problem 5.281 · easy

Find the center of mass of the system: $ \displaystyle m_1 = 5 $ at $ \displaystyle x_1 = -1 $, $ \displaystyle m_2 = 2 $ at $ \displaystyle x_2 = 1 $, $ \displaystyle m_3 = 1 $ at $ \displaystyle x_3 = -5 $, $ \displaystyle m_4 = 2 $ at $ \displaystyle x_4 = 2 $.

\[ 5 \left(-1\right) + 1 \left(-5\right) + 2 \cdot 1 + 2 \cdot 2 = -4 \]
The moment about the origin, Σ mᵢxᵢ.✓ Proved
\[ - \frac{2}{5} \]
Divide by the total mass 10.✓ Proved

Answer $ \bar{x} = - \frac{2}{5} $

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 moment and divides by the total mass to find the center of mass. The steps are logically sound and the final answer is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the moment and divides by the total mass to find the center of mass. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution calculates the moment and divides by the total mass, but it fails to explicitly state the formula for the center of mass ($\bar{x} = \frac{\sum m_i x_i}{\sum m_i}$) or define the total mass. While the arithmetic is correct, the logical step connecting the moment to the final answer is missing, making the derivation incomplete and potentially confusing for a student.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.