Centers of mass and centroids
Problem 5.486 · easy
Find the center of mass of the system: \( \displaystyle m_1 = 3 \) at \( \displaystyle x_1 = 3 \), \( \displaystyle m_2 = 5 \) at \( \displaystyle x_2 = -2 \), \( \displaystyle m_3 = 4 \) at \( \displaystyle x_3 = -4 \).
- \[ 4 \left(-4\right) + 5 \left(-2\right) + 3 \cdot 3 = -17 \]The moment about the origin, Σ mᵢxᵢ.✓ Proved
- \[ - \frac{17}{12} \]Divide by the total mass 12.✓ Proved
Answer \( \bar{x} = - \frac{17}{12} \)
✓ 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 identifies the total mass and the moment, and the final division yields the correct center of mass.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the total mass and the moment, and the final division yields the correct center of mass.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the moment and divides by the total mass to find the center of mass.
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 2026-10-11 with SymPy 1.14.0.