Spring-mass systems
Problem 6.644 · hard
A \( \displaystyle 2 \) kg mass hangs from a spring with constant \( \displaystyle 2 \) N/m. A damping force of \( \displaystyle 4 \) times the velocity acts on it. It starts \( \displaystyle 1 \) 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: 2x″ + 4x′ + 2x = 0 (x measured downward from equilibrium).
- \[ 2 \left(1 - t\right) e^{- t} + 4 \frac{d}{d t} \left(1 - t\right) e^{- t} + 2 \frac{d^{2}}{d t^{2}} \left(1 - t\right) e^{- t} = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \left(1 - t\right) e^{- t} \right|_{\substack{ t=0 }} = 1 \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \left(1 - t\right) e^{- t} \right|_{\substack{ t=0 }} = -2 \]... with the right velocity.✓ Proved
Answer \( x(t) = \left(1 - t\right) e^{- t} \)
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: fail (error) — The proposed function x(t)=(1‑t)e^{‑t} does not satisfy the differential equation 2x″+4x′+2x=0; substituting it yields (4t‑8)e^{‑t}≠0. Hence the solution is incorrect.qwen3.6:27b-mlx: pass — The setup correctly models the damped harmonic oscillator with the given parameters and initial conditions. The proposed solution satisfies the differential equation and initial conditions.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The setup correctly models the damped harmonic oscillator with the given parameters and initial conditions. The proposed solution satisfies the differential equation and initial conditions.gpt-oss:20b: fail (error) 2026-10-18 — The proposed function x(t)=(1‑t)e^{‑t} does not satisfy the differential equation 2x″+4x′+2x=0; substituting it yields (4t‑8)e^{‑t}≠0. Hence the solution is incorrect.qwen3.6:27b-mlx: pass 2026-10-16 — 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 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The differential equation is correctly derived from the physical parameters and initial conditions. The proposed solution satisfies both the ODE and the initial conditions.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.