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Centers of mass and centroids

Problem 5.344 · easy

Find the center of mass of the system: \( \displaystyle m_1 = 3 \) at \( \displaystyle x_1 = -4 \), \( \displaystyle m_2 = 1 \) at \( \displaystyle x_2 = 6 \).

\[ 3 \left(-4\right) + 1 \cdot 6 = -6 \]
The moment about the origin, Σ mᵢxᵢ.✓ Proved
\[ - \frac{3}{2} \]
Divide by the total mass 4.✓ Proved

Answer \( \bar{x} = - \frac{3}{2} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution calculates the moment as -6, but the total mass is 4. Dividing -6 by 4 yields -3/2, which matches the stated answer, but the intermediate step '3*(-4) + 1*6 = -6' is correct, and '-6/4 = -3/2' is correct. Wait, let me re-read. 3*(-4) = -12. 1*6 = 6. Sum = -6. Total mass = 3+1=4. -6/4 = -3/2. The math is correct. Why would it fail? Ah, the prompt says 'Find the center of mass... Stated answer: x_bar = -3/2'. The solution lines are: 1. 3*(-4) + 1*6 = -6. 2. -3/2 = -3/2. Line 2 is just an identity, it doesn't show the division by total mass explicitly in the equation, though the comment says 'Divide by the total mass 4'. The equation '-3/2 = -3/2' is not a derivation step, it's a tautology. It fails to show the calculation -6/4 = -3/2. This is misleading because it hides the actual arithmetic step required to get the answer from the moment.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution calculates the moment as -6, but the total mass is 4. Dividing -6 by 4 yields -3/2, which matches the stated answer, but the intermediate step '3*(-4) + 1*6 = -6' is correct, and '-6/4 = -3/2' is correct. Wait, let me re-read. 3*(-4) = -12. 1*6 = 6. Sum = -6. Total mass = 3+1=4. -6/4 = -3/2. The math is correct. Why would it fail? Ah, the prompt says 'Find the center of mass... Stated answer: x_bar = -3/2'. The solution lines are: 1. 3*(-4) + 1*6 = -6. 2. -3/2 = -3/2. Line 2 is just an identity, it doesn't show the division by total mass explicitly in the equation, though the comment says 'Divide by the total mass 4'. The equation '-3/2 = -3/2' is not a derivation step, it's a tautology. It fails to show the calculation -6/4 = -3/2. This is misleading because it hides the actual arithmetic step required to get the answer from the moment.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The stated answer is incorrect; the correct center of mass is -1.5, but the solution claims -3/2 (which is -1.5) while the calculation in line 1 yields -6, and -6/4 is indeed -1.5. Wait, -3/2 is -1.5. Let me re-read. m1=3 at -4, m2=1 at 6. Moment = 3(-4) + 1(6) = -12 + 6 = -6. Total mass = 4. x_bar = -6/4 = -3/2. The stated answer is -3/2. The solution calculates -6 then divides by 4 to get -3/2. The math is correct. Why did I think it was wrong? Ah, I misread the stated answer as something else or just double checked. -3/2 is correct. The solution is correct. Wait, let me look closer at the prompt's 'Stated answer'. It says \bar{x} = - \frac{3}{2}. The solution derives -3/2. This is correct. Is there a subtle error? No. The setup is correct. The arithmetic is correct. The conclusion is correct. I will mark it as pass.
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.