Spring-mass systems
Problem 6.811 · medium
A weight of \( \displaystyle 16 \) lb stretches a spring \( \displaystyle \frac{1}{2} \) 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).
- \[ \frac{16}{1 \cdot \frac{1}{2}} = 32 \]Hooke's law: k = weight / stretch (lb/ft).✓ Proved
- \[ \frac{16}{32} = \frac{1}{2} \]m = weight / g, with g = 32 ft/s².✓ Proved
- The motion satisfies m x″ + c x′ + k x = 0: 1/2x″ + 0x′ + 32x = 0 (x measured downward from equilibrium).Reviewed
- \[ 4 \sin{\left(8 t \right)} + 16 \cos{\left(8 t \right)} + \frac{\frac{d^{2}}{d t^{2}} \left(\frac{\sin{\left(8 t \right)}}{8} + \frac{\cos{\left(8 t \right)}}{2}\right)}{2} = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\sin{\left(8 t \right)}}{8} + \frac{\cos{\left(8 t \right)}}{2} \right|_{\substack{ t=0 }} = \frac{1}{2} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \left(\frac{\sin{\left(8 t \right)}}{8} + \frac{\cos{\left(8 t \right)}}{2}\right) \right|_{\substack{ t=0 }} = 1 \]... with the right velocity.✓ Proved
Answer \( x(t) = \frac{\sin{\left(8 t \right)}}{8} + \frac{\cos{\left(8 t \right)}}{2} \)
✓ 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 that the proposed solution 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 that the proposed solution 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 proposed solution satisfies both the ODE and the initial conditions.gpt-oss:20b: pass 2026-11-07qwen3.6:27b-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 proposed solution satisfies both the ODE 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.