Spring-mass systems
Problem 6.495 · medium
A \( \displaystyle 2 \) kg mass hangs from a spring with constant \( \displaystyle 8 \) N/m. There is no damping. It starts \( \displaystyle \frac{1}{3} \) 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: 2x″ + 0x′ + 8x = 0 (x measured downward from equilibrium).Reviewed
- \[ - 8 \sin{\left(2 t \right)} + \frac{8 \cos{\left(2 t \right)}}{3} + 2 \frac{d^{2}}{d t^{2}} \left(- \sin{\left(2 t \right)} + \frac{\cos{\left(2 t \right)}}{3}\right) = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. - \sin{\left(2 t \right)} + \frac{\cos{\left(2 t \right)}}{3} \right|_{\substack{ t=0 }} = \frac{1}{3} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \left(- \sin{\left(2 t \right)} + \frac{\cos{\left(2 t \right)}}{3}\right) \right|_{\substack{ t=0 }} = -2 \]... with the right velocity.✓ Proved
Answer \( x(t) = - \sin{\left(2 t \right)} + \frac{\cos{\left(2 t \right)}}{3} \)
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 with the proper sign convention (down is positive) and verifies that the stated answer satisfies both the ODE and the initial conditions.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly sets up the differential equation with the proper sign convention (down is positive) and verifies that the stated answer satisfies both the ODE and the initial conditions.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the differential equation, verifies the proposed solution satisfies the ODE and initial conditions, and the final answer matches the derived solution.gpt-oss:20b: fail (error) 2026-10-15 — The proposed solution does not satisfy the differential equation 2x″+8x=0; the computed second derivative gives a non‑zero left‑hand side, so the model is incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer x(t) = -sin(2t) + (1/3)cos(2t) does not satisfy the initial condition x(0) = 1/3; it yields x(0) = 1/3, which is correct, but the derivative x'(t) = -2cos(2t) - (2/3)sin(2t) yields x'(0) = -2, which is also correct. Wait, let me re-evaluate. x(0) = 0 + 1/3 = 1/3. Correct. x'(0) = -2(1) - 0 = -2. Correct. The solution satisfies the ODE and ICs. Why did I think it failed? Let me check the ODE again. m=2, k=8. omega = sqrt(8/2) = 2. General solution x(t) = A cos(2t) + B sin(2t). x(0) = A = 1/3. x'(t) = -2A sin(2t) + 2B cos(2t). x'(0) = 2B = -2 => B = -1. So x(t) = (1/3)cos(2t) - sin(2t). This matches the stated answer. The solution is actually correct. I will mark it as pass.gpt-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.