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Work: springs, pumping and cables

Problem 5.264 · easy

A force of 30 N holds a spring stretched \frac{1}{5} m beyond its natural length. How much work does it take to stretch it from \frac{2}{5} m to \frac{1}{2} m beyond its natural length?

Hooke's law: F(x) = kx, with x the stretch beyond natural length.
Reviewed
\[ 150 \]
k from the given force and stretch (N/m).✓ Proved
\[ \int\limits_{\frac{2}{5}}^{\frac{1}{2}} 150 x\, dx = \frac{27}{4} \]
W = ∫ F(x) dx.✓ Proved

Answer \( \frac{27}{4} \approx 6.75\ \text{J} \)

✓ 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 applies Hooke's Law to determine the spring constant and sets up the definite integral for work with the correct bounds and integrand.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies Hooke's Law to determine the spring constant and sets up the definite integral for work with the correct bounds and integrand.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies Hooke's Law to determine the spring constant and sets up the definite integral for work with the correct bounds and integrand.
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/work, checked 2026-10-05 with SymPy 1.14.0.