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Spring-mass systems

Problem 6.639 · medium

A \( \displaystyle 4 \) kg mass hangs from a spring with constant \( \displaystyle 16 \) N/m. There is no damping. It is released from rest \( \displaystyle \frac{1}{3} \) m below equilibrium. Find the position x(t).

The motion satisfies m x″ + c x′ + k x = 0: 4x″ + 0x′ + 16x = 0 (x measured downward from equilibrium).
Reviewed
\[ \frac{16 \cos{\left(2 t \right)}}{3} + 4 \frac{d^{2}}{d t^{2}} \frac{\cos{\left(2 t \right)}}{3} = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \frac{\cos{\left(2 t \right)}}{3} \right|_{\substack{ t=0 }} = \frac{1}{3} \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \frac{\cos{\left(2 t \right)}}{3} \right|_{\substack{ t=0 }} = 0 \]
... with the right velocity.✓ Proved

Answer \( x(t) = \frac{\cos{\left(2 t \right)}}{3} \)

Lines: 3 proved, 1 reviewed. 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	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
4	✓ 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 sets up the differential equation, verifies the proposed solution satisfies the ODE, and checks the initial conditions. The logic is sound and complete.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly sets up the differential equation, verifies the proposed solution satisfies the ODE, and checks the initial conditions. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly sets up the differential equation for undamped harmonic motion, derives the correct natural frequency, and verifies that the proposed solution satisfies both the differential equation and the initial conditions.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution verifies that the proposed answer satisfies the differential equation and initial conditions, but it fails to derive the solution or justify why this specific function is the unique solution. It skips the essential steps of finding the general solution (x(t) = c1 cos(2t) + c2 sin(2t)) and applying initial conditions to determine the constants.
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 2026-10-18 with SymPy 1.14.0.