Centers of mass and centroids
Problem 5.212 · easy
Find the center of mass of the system: \( \displaystyle m_1 = 5 \) at \( \displaystyle x_1 = 3 \), \( \displaystyle m_2 = 5 \) at \( \displaystyle x_2 = -3 \).
- \[ 5 \left(-3\right) + 5 \cdot 3 = 0 \]The moment about the origin, Σ mᵢxᵢ.✓ Proved
- \[ 0 \]Divide by the total mass 10.✓ Proved
Answer \( \bar{x} = 0 \)
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 (misleading) — The second sentence incorrectly states that dividing 0 by the total mass yields 0=0; it should conclude that the center of mass ±x = 0/10 = 0. The step misleads the reader about the actual computation.qwen3.6:27b-mlx: fail (misleading) — The solution omits the definition of total mass (M = 10) and the formula for the center of mass, making the division in step 2 appear unmotivated. A student might not understand why the moment is divided by 10 or how the final result is derived from the previous lines.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The solution omits the definition of total mass (M = 10) and the formula for the center of mass, making the division in step 2 appear unmotivated. A student might not understand why the moment is divided by 10 or how the final result is derived from the previous lines.gpt-oss:20b: fail (misleading) 2026-10-04 — The second sentence incorrectly states that dividing 0 by the total mass yields 0=0; it should conclude that the center of mass ±x = 0/10 = 0. The step misleads the reader about the actual computation.qwen3.6:27b-mlx: fail (style) 2026-10-04 — The solution lacks the explicit definition of the center of mass formula, making the jump from the moment calculation to the final answer opaque. It should state that x_bar = (sum of moments) / (total mass).gpt-oss:20b: fail (misleading) 2026-10-04 — The second sentence incorrectly states "0 = 0" after dividing by the total mass; it should show the division 0/10 = 0 to correctly compute 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 2026-10-04 with SymPy 1.14.0.