Centers of mass and centroids
Problem 5.275 · medium
A rod on \( \displaystyle 0 \le x \le 4 \) has density \( \displaystyle \rho(x) = x \). Find its center of mass.
- \[ \int\limits_{0}^{4} x\, dx = 8 \]The mass.✓ Proved
- \[ \int\limits_{0}^{4} x^{2}\, dx = \frac{64}{3} \]The moment about x = 0.✓ Proved
- \[ \frac{8}{3} \]x̄ = moment / mass.✓ Proved
Answer \( \bar{x} = \frac{8}{3} \)
✓ 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 |
| 3 | ✓ 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 | numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the formulas for mass and moment, computes the integrals correctly, and derives the center of mass. The comments accurately describe the physical quantities.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the formulas for mass and moment, computes the integrals correctly, and derives the center of mass. The comments accurately describe the physical quantities.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the formulas for mass and moment, computes the integrals accurately, and derives the correct center of mass.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.