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Spring-mass systems

Problem 6.631 · medium

A \( \displaystyle 2 \) kg mass hangs from a spring with constant \( \displaystyle 72 \) 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: 2x″ + 0x′ + 72x = 0 (x measured downward from equilibrium).
\[ - 36 \cos{\left(6 t \right)} + 2 \frac{d^{2}}{d t^{2}} \left(- \frac{\cos{\left(6 t \right)}}{2}\right) = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \frac{\left(-1\right) \cos{\left(6 t \right)}}{2} \right|_{\substack{ t=0 }} = - \frac{1}{2} \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \frac{\left(-1\right) \cos{\left(6 t \right)}}{2} \right|_{\substack{ t=0 }} = 0 \]
... with the right velocity.✓ Proved

Answer \( x(t) = - \frac{\cos{\left(6 t \right)}}{2} \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution defines x as positive downward and states the initial position is 1/2 m above equilibrium, which implies x(0) = -1/2. However, the prop

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution defines x as positive downward and states the initial position is 1/2 m above equilibrium, which implies x(0) = -1/2. However, the prop
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — 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(6t)/2 yields x(0) = -1/2, which is correct. However, the problem asks to 'Find the position', implying a derivation is needed, not just verification. More critically, the solution presented is just a verification of a given answer, not a derivation. But looking closer at the prompt's constraints: 'Each line... is either an EQUATION... or a SENTENCE'. The solution provided is a verification script. Is it 'wrong'? The prompt asks to review the solution. The solution claims to find x(t) but only verifies it. This is a methodological failure (not solving, just checking). However, usually 'find' implies derivation. Let's look for a mathematical error. x(0) = -1/2. x'(0) = 0. The general solution is x(t) = A cos(6t) + B sin(6t). x(0)=A=-1/2. x'(0)=6B=0 => B=0. So x(t) = -1/2 cos(6t). The stated answer is correct. The solution lines are: 1. Setup. 2. Check ODE. 3. Check IC1. 4. Check IC2. This is a valid 'verification' solution if the goal was to check. But the problem says 'Find'. A solution that only verifies is incomplete/misleading as a 'find' solution. However, often in these automated checks, if the math is correct, it passes. Let's look for a subtle error. 'x measured downward'. '1/2 m above equilibrium'. Above is negative if down is positive. So x(0) = -0.5. The solution uses -cos(6t)/2. At t=0, this is -0.5. This is correct. Is there a sign error in the ODE? mx'' + kx = 0. 2x'' + 72x = 0. x'' + 36x = 0. omega = 6. Correct. The solution is mathematically sound as a verification. But is it a 'solution' to 'Find'? It's a 'check'. If I must find what is wrong: The solution does not derive the answer, it only verifies it. This is a 'style' or 'misleading' issue regarding what constitutes a solution. But wait, look at line 1: '2x'' + 0x' + 72x = 0'. This is correct. Line 2 checks the ODE. Line 3 checks x(0). Line 4 checks x'(0). Everything is correct. Why would it fail? Perhaps the 'Stated answer' is wrong? No, it's correct. Perhaps the setup direction? If x is upward, x(0)=0.5, x(t) = 0.5 cos(6t). The solution defines x downward. This is a valid choice. The result is consistent with the choice. I will mark it pass because the math is correct and consistent, even if the method is verification rather than derivation. Wait, the instructions say 'find what is wrong'. If there is nothing wrong, say pass. Is there anything wrong? No. The solution correctly identifies the model, parameters, and verifies the unique solution satisfying the ICs. It is a valid (if minimal) solution path.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution defines x as positive downward but applies the initial condition x(0) = -1/2 for a release 1/2 m above equilibrium. Since 'above' corresponds to negative x in this coordinate system, the initial position should be x(0) = -1/2, which matches the answer, BUT the text in line 3 claims 'starts in the right place' without explicitly connecting the sign convention to the physical position, which is confusing. However, the primary error is that the problem asks to 'Find the position', implying a derivation is needed, but the solution merely verifies a given answer. More critically, if x is measured downward, 'above equilibrium' means x(0) = -0.5. The solution sets x(0) = -1/2. This is consistent. Wait, let me 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 has x(0) = -1/2. This is correct. Why did I think it was wrong? Let me check the sign of the answer. x(t) = -0.5 cos(6t). At t=0, x=-0.5. Correct. Velocity is 0. Correct. The differential equation is correct. The verification steps are correct. Is there a subtle error? The prompt says 'Find the position x(t)'. The solution just verifies. This is a 'style' or 'misleading' issue regarding the method (verification vs derivation), but technically the math is sound. However, usually, these adversarial reviews look for sign errors. Let's look at the coordinate system again. 'x measured downward'. '1/2 m above equilibrium'. Above is opposite to downward. So x(0) should be negative. x(0) = -1/2. The solution uses -1/2. This is correct. Is there any other interpretation? If x was upward, x(0) would be +1/2. The solution explicitly states x is downward. So the sign is correct. The only potential issue is that it doesn't derive the solution, just checks it. But the instructions say 'review worked calculus solutions'. If the solution is just a check, is it 'wrong'? The verdict 'pass' is for sound solutions. This solution is mathematically sound. But wait, look at line 1: '2x'' + 0x' + 72x = 0'. This is correct. Line 2 checks the DE. Line 3 checks IC position. Line 4 checks IC velocity. The logic is: 'If it satisfies the DE and ICs, it is the solution.' This is a valid method (uniqueness theorem). So it is sound. Why would it fail? Maybe the prompt implies a derivation is required? No, 'Find the position' can be satisfied by identifying the correct function. Let's look closer at the numbers. m=2, k=72. omega = sqrt(72/2) = sqrt(36) = 6. Correct. Amplitude A = 1/2. x(t) = A cos(omega t + phi). x(0) = -1/2 => A cos(phi) = -1/2. v(0) = 0 => -A omega sin(phi) = 0 => sin(phi) = 0 => phi = 0 or pi. If phi=0, cos(0)=1, x(0)=A=1/2. But we need -1/2. So phi must be pi. cos(6t + pi) = -cos(6t). So x(t) = 1/2 * (-cos(6t)) = -1/2 cos(6t). The solution is correct. Is there a 'misleading' aspect? The solution presents the answer as given and checks it. It doesn't show how to get it. This might be 'style' or 'misleading' if a student thinks this is how one 'finds' a solution. But strictly, the math is correct. Let's look for a hidden error. 'x measured downward'. '1/2 m above'. x(0) = -0.5. Solution uses -0.5. Correct. Is it possible the problem implies x is upward? Usually, standard convention is upward positive, but the solution explicitly defines downward. If the solution defines its own convention, it must be consistent. It is. I will mark it as pass unless I find a contradiction. Wait, look at the stated answer: x(t) = - cos(6t)/2. This is -0.5 cos(6t). Correct. I see no error. I will mark pass.
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.