Centers of mass and centroids
Problem 5.352 · easy
Find the center of mass of the system: \( \displaystyle m_1 = 6 \) at \( \displaystyle x_1 = 3 \), \( \displaystyle m_2 = 1 \) at \( \displaystyle x_2 = 0 \), \( \displaystyle m_3 = 6 \) at \( \displaystyle x_3 = 1 \), \( \displaystyle m_4 = 1 \) at \( \displaystyle x_4 = 6 \).
- \[ 1 \cdot 0 + 6 \cdot 1 + 1 \cdot 6 + 6 \cdot 3 = 30 \]The moment about the origin, Σ mᵢxᵢ.✓ Proved
- \[ \frac{15}{7} \]Divide by the total mass 14.✓ Proved
Answer \( \bar{x} = \frac{15}{7} \)
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 solution calculates the moment as 30, but the correct moment is 36 (6*3 + 1*0 + 6*1 + 1*6). Consequently, the final answer 15/7 is incorrect; it should be 18/7.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution calculates the moment as 30, but the correct moment is 36 (6*3 + 1*0 + 6*1 + 1*6). Consequently, the final answer 15/7 is incorrect; it should be 18/7.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution calculates the moment as 30, but the correct moment is 36 (6*3 + 1*0 + 6*1 + 1*6 = 18 + 0 + 6 + 6). Consequently, the final answer 15/7 is incorrect; it should be 36/14 = 18/7.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.