Spring-mass systems
Problem 6.640 · medium
A \( \displaystyle 1 \) kg mass hangs from a spring with constant \( \displaystyle 4 \) N/m. There is no damping. It is released from rest \( \displaystyle \frac{1}{2} \) m above equilibrium. Find the position x(t).
- The motion satisfies m x″ + c x′ + k x = 0: 1x″ + 0x′ + 4x = 0 (x measured downward from equilibrium).
- \[ - 2 \cos{\left(2 t \right)} + \frac{d^{2}}{d t^{2}} \left(- \frac{\cos{\left(2 t \right)}}{2}\right) = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\left(-1\right) \cos{\left(2 t \right)}}{2} \right|_{\substack{ t=0 }} = - \frac{1}{2} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \frac{\left(-1\right) \cos{\left(2 t \right)}}{2} \right|_{\substack{ t=0 }} = 0 \]... with the right velocity.✓ Proved
Answer \( x(t) = - \frac{\cos{\left(2 t \right)}}{2} \)
Lines: 3 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The initial condition is stated as 1/2 m above equilibrium, but with x measured downward, this corresponds to x(0) = -1/2. The proposed solution x(t) = -cos(2t)/2 yields x(0) = -1/2, which is correct. However, the standard convention for 'position' usually implies the coordinate value. Let's re-read carefully. 'Released from rest 1/2 m above equilibrium'. If x is downward, above is negative. So x(0) = -0.5. The solution gives x(0) = -0.5. The velocity is 0. The equation is x'' + 4x = 0. The general solution is A cos(2t) + B sin(2t). x(0) = A = -0.5. x'(0) = 2B = 0 => B=0. So x(t) = -0.5 cos(2t). The stated answer is -cos(2t)/2. This matches. Why did I think it failed? Let me check the sign again. 'Above equilibrium' -> negative x if x is down. x(0) = -1/2. Solution x(t) = -1/2 cos(2t). At t=0, x=-1/2. Correct. The solution seems correct. Wait, is there a subtle error? The problem asks for x(t). The solution provides it. The steps verify it satisfies the ODE and ICs. Is the setup sentence misleading? 'x measured downward from equilibrium'. This is a valid choice. The ICs are applied correctly based on this choice. The answer is correct. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The initial condition is stated as 1/2 m above equilibrium, but with x measured downward, this corresponds to x(0) = -1/2. The proposed solution x(t) = -cos(2t)/2 yields x(0) = -1/2, which is correct. However, the standard convention for 'position' usually implies the coordinate value. Let's re-read carefully. 'Released from rest 1/2 m above equilibrium'. If x is downward, above is negative. So x(0) = -0.5. The solution gives x(0) = -0.5. The velocity is 0. The equation is x'' + 4x = 0. The general solution is A cos(2t) + B sin(2t). x(0) = A = -0.5. x'(0) = 2B = 0 => B=0. So x(t) = -0.5 cos(2t). The stated answer is -cos(2t)/2. This matches. Why did I think it failed? Let me check the sign again. 'Above equilibrium' -> negative x if x is down. x(0) = -1/2. Solution x(t) = -1/2 cos(2t). At t=0, x=-1/2. Correct. The solution seems correct. Wait, is there a subtle error? The problem asks for x(t). The solution provides it. The steps verify it satisfies the ODE and ICs. Is the setup sentence misleading? 'x measured downward from equilibrium'. This is a valid choice. The ICs are applied correctly based on this choice. The answer is correct. I will mark it as pass.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The initial condition is stated as 1/2 m above equilibrium, which corresponds to x(0) = -1/2 if x is measured downward. However, the proposed solutiogpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The initial condition is 1/2 m above equilibrium, which corresponds to x(0) = -1/2 given the downward-positive convention. The proposed solution x(t) = -cos(2t)/2 yields x(0) = -1/2, which is correct. However, the problem asks to 'Find the position', implying a derivation. The solution merely verifies the answer without deriving it from the differential equation and initial conditions. More critically, the prompt asks to review the *solution* provided. The solution consists of a setup sentence and three verification steps. While the verification is correct, it is not a solution method. A student would learn that verifying an answer is equivalent to solving the problem, which is misleading pedagogy. However, looking closer at the prompt's constraints: 'Check that... each conclusion follows from the lines before it'. The solution does not derive the answer; it assumes it and checks it. This is not a valid derivation. But is it an 'error' in the math? No. Is it 'misleading'? Yes, it teaches that checking is solving. Let's look for a harder error. Wait, the problem statement says 'released from rest 1/2 m above equilibrium'. If x is measured downward, above equilibrium is negative. So x(0) = -1/2. The solution has x(0) = -1/2. This is consistent. The differential equation is correct. The verification is correct. The only issue is that it's a verification, not a derivation. In many contexts, 'Find x(t)' requires derivation. If I must find what is wrong, the lack of derivation is the main flaw. However, often these 'adversarial' reviews look for subtle sign errors or physics mistakes. Let's re-read carefully. '1 kg mass', 'k=4', 'no damping'. omega = sqrt(k/m) = 2. x(t) = A cos(2t) + B sin(2t). x(0) = -1/2 => A = -1/2. x'(0) = 0 => B = 0. So x(t) = -1/2 cos(2t). The stated answer is correct. The solution verifies it. Is there a hidden error? No. The 'error' is likely that the solution is incomplete/not a derivation. But usually, 'fail' is reserved for incorrect math. Let's look at the severity options. 'Misleading' fits best if the math is right but the method is bad. 'Error' if the math is wrong. The math is right. The method is 'guess and check'. This is often acceptable in multiple choice but not in 'Find x(t)'. I will mark it as misleading because it fails to demonstrate the solution process.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.