Centers of mass and centroids
Problem 5.349 · easy
Find the center of mass of the system: \( \displaystyle m_1 = 2 \) at \( \displaystyle x_1 = 4 \), \( \displaystyle m_2 = 3 \) at \( \displaystyle x_2 = -3 \), \( \displaystyle m_3 = 2 \) at \( \displaystyle x_3 = -5 \), \( \displaystyle m_4 = 4 \) at \( \displaystyle x_4 = 1 \).
- \[ 2 \left(-5\right) + 3 \left(-3\right) + 4 \cdot 1 + 2 \cdot 4 = -7 \]The moment about the origin, Σ mᵢxᵢ.✓ Proved
- \[ - \frac{7}{11} \]Divide by the total mass 11.✓ Proved
Answer \( \bar{x} = - \frac{7}{11} \)
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: fail (error) — The first equation omits the term for m₁=2 at x₁=4, using 2*(-5) instead of 2*4. The moment calculation is therefore incorrect, leading to an incorrect center of mass.qwen3.6:27b-mlx: pass — The solution correctly calculates the total moment and divides by the total mass to find the center of mass. The steps are logically sound and the arithmetic is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — The first equation omits the term for m₁=2 at x₁=4, using 2*(-5) instead of 2*4. The moment calculation is therefore incorrect, leading to an incorrect center of mass.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 steps are logically sound and the arithmetic is correct.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the moment and total mass, leading to the correct center of mass. The steps are logically sound and algebraically 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.