Spring-mass systems
Problem 6.816 · medium
A weight of \( \displaystyle 4 \) lb stretches a spring \( \displaystyle 2 \) ft. There is no damping. It starts \( \displaystyle \frac{1}{2} \) ft below equilibrium with velocity \( \displaystyle 1 \) ft/s (down is positive). Find the position x(t).
- \[ \frac{4}{2} = 2 \]Hooke's law: k = weight / stretch (lb/ft).✓ Proved
- \[ \frac{4}{32} = \frac{1}{8} \]m = weight / g, with g = 32 ft/s².✓ Proved
- The motion satisfies m x″ + c x′ + k x = 0: 1/8x″ + 0x′ + 2x = 0 (x measured downward from equilibrium).Reviewed
- \[ \frac{\sin{\left(4 t \right)}}{2} + \cos{\left(4 t \right)} + \frac{\frac{d^{2}}{d t^{2}} \left(\frac{\sin{\left(4 t \right)}}{4} + \frac{\cos{\left(4 t \right)}}{2}\right)}{8} = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\sin{\left(4 t \right)}}{4} + \frac{\cos{\left(4 t \right)}}{2} \right|_{\substack{ t=0 }} = \frac{1}{2} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \left(\frac{\sin{\left(4 t \right)}}{4} + \frac{\cos{\left(4 t \right)}}{2}\right) \right|_{\substack{ t=0 }} = 1 \]... with the right velocity.✓ Proved
Answer \( x(t) = \frac{\sin{\left(4 t \right)}}{4} + \frac{\cos{\left(4 t \right)}}{2} \)
✓ Nihil obstat Lines: 5 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ 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 derives the parameters k and m, sets up the differential equation with correct initial conditions, and verifies the final answer satisfies both the ODE and the initial conditions.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly derives the parameters k and m, sets up the differential equation with correct initial conditions, and verifies the final answer satisfies both the ODE and the initial conditions.gpt-oss:20b: pass 2026-11-08qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly derives the parameters k and m, sets up the differential equation with appropriate initial conditions, and verifies that the stated answer satisfies both the equation and the initial conditions.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-mlx: pass 2026-11-07gpt-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.