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Spring-mass systems

Problem 6.728 · medium

A \( \displaystyle 4 \) kg mass hangs from a spring with constant \( \displaystyle 4 \) N/m. There is no damping. It starts \( \displaystyle \frac{1}{4} \) m below equilibrium with velocity \( \displaystyle 1 \) m/s (down is positive). Find the position x(t).

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

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

✓ Nihil obstat Lines: 3 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	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 that the proposed function satisfies the ODE, and confirms it meets the initial conditions.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly sets up the differential equation, verifies that the proposed function satisfies the ODE, and confirms it meets the initial conditions.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly sets up the differential equation, verifies the proposed solution satisfies the ODE, and confirms the initial conditions match the problem statement.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly sets up the differential equation, solves it, and verifies the initial conditions. The logic is sound.
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.