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Spring-mass systems

Problem 6.818 · medium

A weight of \( \displaystyle 4 \) lb stretches a spring \( \displaystyle \frac{1}{2} \) ft. There is no damping. It is released from rest \( \displaystyle \frac{1}{4} \) ft below equilibrium. Find the position x(t).

\[ \frac{4}{1 \cdot \frac{1}{2}} = 8 \]
Hooke's law: k = weight / stretch (lb/ft).✓ Proved
\[ \frac{4}{32} = \frac{1}{8} \]
m = weight / g, with g = 32 ft/s².✓ Proved
The motion satisfies m x″ + c x′ + k x = 0: 1/8x″ + 0x′ + 8x = 0 (x measured downward from equilibrium).
\[ 2 \cos{\left(8 t \right)} + \frac{\frac{d^{2}}{d t^{2}} \frac{\cos{\left(8 t \right)}}{4}}{8} = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \frac{\cos{\left(8 t \right)}}{4} \right|_{\substack{ t=0 }} = \frac{1}{4} \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \frac{\cos{\left(8 t \right)}}{4} \right|_{\substack{ t=0 }} = 0 \]
... with the right velocity.✓ Proved

Answer \( x(t) = \frac{\cos{\left(8 t \right)}}{4} \)

Lines: 5 proved, 1 not checked. 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. 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	Not checked	—	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: fail (error) — The initial condition is mis‑interpreted: the mass starts 1/4 ft below equilibrium, so x(0) should be –1/4, not +1/4. Consequently the proposed solution x(t)=cos(8t)/4 does not satisfy the correct initial displacement.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-08
gpt-oss:20b: fail (error) 2026-11-08 — The initial condition is mis‑interpreted: the mass starts 1/4 ft below equilibrium, so x(0) should be –1/4, not +1/4. Consequently the proposed solution x(t)=cos(8t)/4 does not satisfy the correct initial displacement.
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly derives the parameters k and m, sets up the differential equation with appropriate initial conditions, and verifies that the proposed solution satisfies both the ODE and the initial conditions.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly derives the parameters k and m, sets up the differential equation with appropriate initial conditions, and verifies that the proposed solution satisfies both the ODE and the initial conditions.
gpt-oss:20b: pass 2026-11-07

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 2026-11-08 with SymPy 1.14.0.