Spring-mass systems
Problem 6.500 · hard
A \( \displaystyle 5 \) kg mass hangs from a spring with constant \( \displaystyle 20 \) N/m. A damping force of \( \displaystyle 20 \) times the velocity acts on it. 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: 5x″ + 20x′ + 20x = 0 (x measured downward from equilibrium).Reviewed
- \[ \frac{20 \left(2 t + 1\right) e^{- 2 t}}{3} + 20 \frac{d}{d t} \frac{\left(2 t + 1\right) e^{- 2 t}}{3} + 5 \frac{d^{2}}{d t^{2}} \frac{\left(2 t + 1\right) e^{- 2 t}}{3} = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\left(2 t + 1\right) e^{- 2 t}}{3} \right|_{\substack{ t=0 }} = \frac{1}{3} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \frac{\left(2 t + 1\right) e^{- 2 t}}{3} \right|_{\substack{ t=0 }} = 0 \]... with the right velocity.✓ Proved
Answer \( x(t) = \frac{\left(2 t + 1\right) e^{- 2 t}}{3} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly sets up the differential equation with the given parameters and verifies that the stated answer satisfies both the equation and the initial conditions.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly sets up the differential equation with the given parameters and verifies that the stated answer satisfies both the equation and the initial conditions.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the differential equation with the given parameters and verifies that the stated answer satisfies both the equation and the initial conditions.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution merely verifies that the stated answer satisfies the differential equation and initial conditions, but it does not derive the solution. A complete solution must show the derivation (e.g., solving the characteristic equation to find the general solution and applying 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.