Spring-mass systems
Problem 6.810 · medium
A \( \displaystyle 1 \) kg mass hangs from a spring with constant \( \displaystyle 16 \) N/m. There is no damping. 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: 1x″ + 0x′ + 16x = 0 (x measured downward from equilibrium).Reviewed
- \[ - 8 \sin{\left(4 t \right)} + 16 \cos{\left(4 t \right)} + \frac{d^{2}}{d t^{2}} \left(- \frac{\sin{\left(4 t \right)}}{2} + \cos{\left(4 t \right)}\right) = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\left(-1\right) \sin{\left(4 t \right)}}{2} + \cos{\left(4 t \right)} \right|_{\substack{ t=0 }} = 1 \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \left(\frac{\left(-1\right) \sin{\left(4 t \right)}}{2} + \cos{\left(4 t \right)}\right) \right|_{\substack{ t=0 }} = -2 \]... with the right velocity.✓ Proved
Answer \( x(t) = - \frac{\sin{\left(4 t \right)}}{2} + \cos{\left(4 t \right)} \)
✓ 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 undamped harmonic motion, verifies that the proposed solution satisfies the ODE, and confirms that the initial conditions for position and velocity are met.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly sets up the differential equation for undamped harmonic motion, verifies that the proposed solution satisfies the ODE, and confirms that the initial conditions for position and velocity are met.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly sets up the differential equation, verifies the proposed solution satisfies the ODE, and confirms the initial conditions. The logic is sound and complete.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly sets up the differential equation for undamped harmonic motion, derives the correct general solution form, and applies the initial conditions accurately to determine the constants. The final answer matches the stated answer.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.