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Work: springs, pumping and cables

Problem 5.412 · easy

A force of 12 N holds a spring stretched \frac{1}{5} m beyond its natural length. How much work does it take to stretch it from \frac{1}{5} m to \frac{3}{10} m beyond its natural length?

Hooke's law: F(x) = kx, with x the stretch beyond natural length.
Reviewed
\[ 60 \]
k from the given force and stretch (N/m).✓ Proved
\[ \int\limits_{\frac{1}{5}}^{\frac{3}{10}} 60 x\, dx = \frac{3}{2} \]
W = ∫ F(x) dx.✓ Proved

Answer \( \frac{3}{2} \approx 1.5\ \text{J} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies Hooke's Law, calculates the spring constant k=60 N/m, and sets up the definite integral for work with the correct bounds. The final calculation is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies Hooke's Law, calculates the spring constant k=60 N/m, and sets up the definite integral for work with the correct bounds. The final calculation is correct.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies Hooke's Law to determine the spring constant k=60 N/m and sets up the definite integral for work with the correct bounds and integrand. The calculation is accurate.

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/work, checked 2026-10-09 with SymPy 1.14.0.