Centers of mass and centroids
Problem 5.278 · easy
Find the center of mass of the system: \( \displaystyle m_1 = 6 \) at \( \displaystyle x_1 = -2 \), \( \displaystyle m_2 = 1 \) at \( \displaystyle x_2 = 3 \), \( \displaystyle m_3 = 2 \) at \( \displaystyle x_3 = 4 \), \( \displaystyle m_4 = 3 \) at \( \displaystyle x_4 = 1 \).
- \[ 6 \left(-2\right) + 1 \cdot 3 + 3 \cdot 1 + 2 \cdot 4 = 2 \]The moment about the origin, Σ mᵢxᵢ.✓ Proved
- \[ 1 \cdot \frac{1}{6} = \frac{1}{6} \]Divide by the total mass 12.✓ Proved
Answer \( \bar{x} = \frac{1}{6} \)
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: passqwen3.6:27b-mlx: fail (error) — The total mass is 12, not 6. The solution incorrectly divides the moment by 6 instead of the actual total mass, leading to an incorrect center of mass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The total mass is 12, not 6. The solution incorrectly divides the moment by 6 instead of the actual total mass, leading to an incorrect center of mass.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The total mass is calculated as 12 in the text, but the denominator used in the final step is 6. The correct center of mass is 2/12 = 1/6, but the solution claims to divide by 6, implying a total mass of 6, which is incorrect.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.