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Spring-mass systems

Problem 6.665 · medium

A weight of \( \displaystyle 32 \) lb stretches a spring \( \displaystyle \frac{2}{3} \) ft. There is no damping. It starts \( \displaystyle 1 \) ft below equilibrium with velocity \( \displaystyle 1 \) ft/s (down is positive). Find the position x(t).

\[ \frac{32}{2 \cdot \frac{1}{3}} = 48 \]
Hooke's law: k = weight / stretch (lb/ft).✓ Proved
\[ \frac{32}{32} = 1 \]
m = weight / g, with g = 32 ft/s².✓ Proved
The motion satisfies m x″ + c x′ + k x = 0: 1x″ + 0x′ + 48x = 0 (x measured downward from equilibrium).
Reviewed
\[ 4 \sqrt{3} \sin{\left(4 \sqrt{3} t \right)} + 48 \cos{\left(4 \sqrt{3} t \right)} + \frac{d^{2}}{d t^{2}} \left(\frac{\sqrt{3} \sin{\left(4 \sqrt{3} t \right)}}{12} + \cos{\left(4 \sqrt{3} t \right)}\right) = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \frac{\sqrt{3} \sin{\left(4 \sqrt{3} t \right)}}{12} + \cos{\left(4 \sqrt{3} t \right)} \right|_{\substack{ t=0 }} = 1 \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \left(\frac{\sqrt{3} \sin{\left(4 \sqrt{3} t \right)}}{12} + \cos{\left(4 \sqrt{3} t \right)}\right) \right|_{\substack{ t=0 }} = 1 \]
... with the right velocity.✓ Proved

Answer \( x(t) = \frac{\sqrt{3} \sin{\left(4 \sqrt{3} t \right)}}{12} + \cos{\left(4 \sqrt{3} t \right)} \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	✓ 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	mpmath's ODE solver integrates the spring from t = 0

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly derives the parameters k and m, sets up the differential equation with appropriate initial conditions, and verifies that the stated answer satisfies both the ODE and the initial conditions.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly derives the parameters k and m, sets up the differential equation with appropriate initial conditions, and verifies that the stated answer satisfies both the ODE and the initial conditions.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the parameters k and m, sets up the differential equation with correct initial conditions, and verifies that the stated answer satisfies both the ODE and the initial conditions.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15

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/spring_mass, checked 2026-10-19 with SymPy 1.14.0.