Spring-mass systems
Problem 6.498 · medium
A \( \displaystyle 1 \) kg mass hangs from a spring with constant \( \displaystyle 9 \) 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: 1x″ + 0x′ + 9x = 0 (x measured downward from equilibrium).Reviewed
- \[ 3 \sin{\left(3 t \right)} + \frac{9 \cos{\left(3 t \right)}}{4} + \frac{d^{2}}{d t^{2}} \left(\frac{\sin{\left(3 t \right)}}{3} + \frac{\cos{\left(3 t \right)}}{4}\right) = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\sin{\left(3 t \right)}}{3} + \frac{\cos{\left(3 t \right)}}{4} \right|_{\substack{ t=0 }} = \frac{1}{4} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \left(\frac{\sin{\left(3 t \right)}}{3} + \frac{\cos{\left(3 t \right)}}{4}\right) \right|_{\substack{ t=0 }} = 1 \]... with the right velocity.✓ Proved
Answer \( x(t) = \frac{\sin{\left(3 t \right)}}{3} + \frac{\cos{\left(3 t \right)}}{4} \)
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, verifies the proposed solution satisfies the ODE, and confirms the initial conditions match the problem statement.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — 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 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the differential equation, verifies the general solution satisfies the ODE, and confirms the initial conditions are met. The logic is sound and complete.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The proposed solution x(t) = (1/3)sin(3t) + (1/4)cos(3t) yields an initial position of 1/4 m, but the problem states the mass starts 1/4 m below equigpt-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.