Centers of mass and centroids
Problem 5.413 · easy
Find the center of mass of the system: \( \displaystyle m_1 = 6 \) at \( \displaystyle x_1 = 1 \), \( \displaystyle m_2 = 4 \) at \( \displaystyle x_2 = 4 \), \( \displaystyle m_3 = 6 \) at \( \displaystyle x_3 = -5 \), \( \displaystyle m_4 = 4 \) at \( \displaystyle x_4 = 3 \).
- \[ 6 \left(-5\right) + 6 \cdot 1 + 4 \cdot 3 + 4 \cdot 4 = 4 \]The moment about the origin, Σ mᵢxᵢ.✓ Proved
- \[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]Divide by the total mass 20.✓ Proved
Answer \( \bar{x} = \frac{1}{5} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution divides the total moment by 5 instead of the total mass (20), resulting in an incorrect center of mass.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution divides the total moment by 5 instead of the total mass (20), resulting in an incorrect center of mass.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution calculates the moment as 4, but the correct moment is 6(1) + 4(4) + 6(-5) + 4(3) = 6 + 16 - 30 + 12 = 4. Wait, let me re-calculate. 6*1=6, 4*4=16, 6*-5=-30, 4*3=12. Sum = 6+16-30+12 = 34-30 = 4. Total mass = 6+4+6+4 = 20. Center of mass = 4/20 = 1/5. The calculation is actually correct. Why did I think it was wrong? Let me re-read the prompt. 'Find the center of mass... m1=6 at x1=1, m2=4 at x2=4, m3=6 at x3=-5, m4=4 at x4=3'. Moment = 6(1) + 4(4) + 6(-5) + 4(3) = 6 + 16 - 30 + 12 = 4. Total Mass = 20. x_bar = 4/20 = 1/5. The solution is correct. However, the prompt asks me to be adversarial. Is there a subtle error? The solution says 'The moment about the origin, Σ mᵢxᵢ.' and 'Divide by the total mass 20.' These are correct sentences. The equations are marked unchecked, but the prompt says 'Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for.' It also says 'do not re-check the algebra of a line marked proved'. Here the lines are marked 'unchecked'. This usually implies the CAS hasn't verified them, or the user hasn't provided the verification status. But the instructions say 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. The lines are marked 'unchecked'. This is ambiguous. Does 'unchecked' mean I should check the algebra? Or does it mean the CAS didn't check it? The prompt says 'Each line... is either an EQUATION... or a SENTENCE...'. It doesn't explicitly say I must check algebra for 'unchecked' lines, but it says 'The sentences are what you are here for'. If the algebra is wrong, the sentence describing it might be misleading. But here the algebra is correct. Let me look closer. Is there a sign error? m3=6 at x3=-5. 6*-5 = -30. Correct. m1=6 at x1=1. 6*1=6. Correct. m2=4 at x2=4. 4*4=16. Correct. m4=4 at x4=3. 4*3=12. Correct. Sum = 4. Mass = 20. Result 1/5. The solution is sound. I will mark it as pass.gpt-oss:20b: pass 2026-10-09
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.