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Spring-mass systems

Problem 6.719 · medium

A \( \displaystyle 5 \) kg mass hangs from a spring with constant \( \displaystyle 5 \) N/m. There is no damping. It is released from rest \( \displaystyle \frac{1}{2} \) m below equilibrium. Find the position x(t).

The motion satisfies m x″ + c x′ + k x = 0: 5x″ + 0x′ + 5x = 0 (x measured downward from equilibrium).
\[ \frac{5 \cos{\left(t \right)}}{2} + 5 \frac{d^{2}}{d t^{2}} \frac{\cos{\left(t \right)}}{2} = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \frac{\cos{\left(t \right)}}{2} \right|_{\substack{ t=0 }} = \frac{1}{2} \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \frac{\cos{\left(t \right)}}{2} \right|_{\substack{ t=0 }} = 0 \]
... with the right velocity.✓ Proved

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

Lines: 3 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	Not checked	—	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: fail (error) — The solution verifies that the stated 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. Merely checking a candidate answer does not constitute finding the position x(t) from first principles, and the setup implies a derivation was expected.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution verifies that the stated 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. Merely checking a candidate answer does not constitute finding the position x(t) from first principles, and the setup implies a derivation was expected.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution verifies that a candidate function satisfies the differential equation and initial conditions, but it fails to establish that this solution is unique. In a calculus context, one must derive the general solution (using the characteristic equation) and apply initial conditions to find the specific constants, rather than just checking a guess.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution verifies that the stated 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. Merely checking a candidate answer is not a valid derivation method in this context, and the solution omits the calculation of the natural frequency and the general solution form.
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.