Spring-mass systems
Problem 6.809 · medium
A \( \displaystyle 5 \) kg mass hangs from a spring with constant \( \displaystyle 45 \) N/m. There is no damping. It is released from rest \( \displaystyle 1 \) m below equilibrium. Find the position x(t).
- The motion satisfies m x″ + c x′ + k x = 0: 5x″ + 0x′ + 45x = 0 (x measured downward from equilibrium).Reviewed
- \[ 45 \cos{\left(3 t \right)} + 5 \frac{d^{2}}{d t^{2}} \cos{\left(3 t \right)} = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \cos{\left(3 t \right)} \right|_{\substack{ t=0 }} = 1 \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \cos{\left(3 t \right)} \right|_{\substack{ t=0 }} = 0 \]... with the right velocity.✓ Proved
Answer \( x(t) = \cos{\left(3 t \right)} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 for simple harmonic motion, verifies the proposed solution satisfies the ODE and initial conditions, and implicitly handles the sign convention (downward positive) consistent with the initial displacement.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly sets up the differential equation for simple harmonic motion, verifies the proposed solution satisfies the ODE and initial conditions, and implicitly handles the sign convention (downward positive) consistent with the initial displacement.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly sets up the differential equation for undamped harmonic motion, identifies the natural frequency, and verifies that the proposed solution satisfies both the equation and the initial conditions.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly sets up the differential equation for simple harmonic motion, identifies the natural frequency, and verifies that the proposed solution satisfies both the equation and the initial conditions.gpt-oss:20b: pass 2026-11-07
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 2026-11-08 with SymPy 1.14.0.