Centers of mass and centroids
Problem 5.421 · easy
Find the center of mass of the system: \( \displaystyle m_1 = 3 \) at \( \displaystyle x_1 = 1 \), \( \displaystyle m_2 = 6 \) at \( \displaystyle x_2 = -5 \).
- \[ 6 \left(-5\right) + 3 \cdot 1 = -27 \]The moment about the origin, Σ mᵢxᵢ.✓ Proved
- \[ -3 \]Divide by the total mass 9.✓ Proved
Answer \( \bar{x} = -3 \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: pass — The solution correctly applies the center of mass formula and performs the arithmetic accurately.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the center of mass formula and performs the arithmetic accurately.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — 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.
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 2026-10-09 with SymPy 1.14.0.