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Spring-mass systems

Problem 6.492 · hard

A $ \displaystyle 4 $ kg mass hangs from a spring with constant $ \displaystyle 64 $ N/m. A damping force of $ \displaystyle 40 $ times the velocity acts on it. It starts $ \displaystyle \frac{1}{4} $ m below equilibrium with velocity $ \displaystyle -2 $ m/s (down is positive). Find the position x(t).

The motion satisfies m x″ + c x′ + k x = 0: 4x″ + 40x′ + 64x = 0 (x measured downward from equilibrium).
\[ 40 \frac{d}{d t} \frac{e^{- 8 t}}{4} + 4 \frac{d^{2}}{d t^{2}} \frac{e^{- 8 t}}{4} + 16 e^{- 8 t} = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \frac{e^{- 8 t}}{4} \right|_{\substack{ t=0 }} = \frac{1}{4} \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \frac{e^{- 8 t}}{4} \right|_{\substack{ t=0 }} = -2 \]
... with the right velocity.✓ Proved

Answer $ x(t) = \frac{e^{- 8 t}}{4} $

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 establish that this is the unique solution. For a second-order linear ODE, uniqueness requires showing the general solution form (e.g., via the characteristic equation) and demonstrating that the given function is the only one fitting the initial conditions, rather than just checking a candidate.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution verifies that the stated answer satisfies the differential equation and initial conditions, but it fails to establish that this is the unique solution. For a second-order linear ODE, uniqueness requires showing the general solution form (e.g., via the characteristic equation) and demonstrating that the given function is the only one fitting the initial conditions, rather than just checking a candidate.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution verifies that the stated answer satisfies the differential equation and initial conditions, but it fails to derive the general solution or justify why this specific form is the unique solution. It presents a 'plug-and-chug' verification as a complete solution, omitting the characteristic equation analysis (which reveals critical damping) and the general form $x(t) = (c_1 + c_2 t)e^{-8t}$, making it impossible for a student to learn how to solve such problems generally.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly sets up the differential equation with the given parameters and verifies that the proposed solution satisfies both the equation and the initial conditions.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.