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Spring-mass systems

Problem 6.726 · medium

A \( \displaystyle 2 \) kg mass hangs from a spring with constant \( \displaystyle 2 \) N/m. There is no damping. It is released from rest \( \displaystyle 1 \) m below equilibrium. Find the position x(t).

The motion satisfies m x″ + c x′ + k x = 0: 2x″ + 0x′ + 2x = 0 (x measured downward from equilibrium).
\[ 2 \cos{\left(t \right)} + 2 \frac{d^{2}}{d t^{2}} \cos{\left(t \right)} = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \cos{\left(t \right)} \right|_{\substack{ t=0 }} = 1 \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \cos{\left(t \right)} \right|_{\substack{ t=0 }} = 0 \]
... with the right velocity.✓ Proved

Answer \( x(t) = \cos{\left(t \right)} \)

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: fail (error) — The proposed solution x(t) = cos(t) does not satisfy the differential equation 2x'' + 2x = 0, which requires angular frequency omega = 1, but the equation simplifies to x'' + x = 0, so omega is indeed 1. Wait, let me re-evaluate. m=2, k=2. Equation: 2x'' + 2x = 0 => x'' + x = 0. General solution x(t) = A cos(t) + B sin(t). IC: x(0) = 1 => A=1. x'(0) = 0 => B=0. So x(t) = cos(t) is actually correct. Why did I think it was wrong? Let me check the algebra again. 2*(-cos(t)) + 2*cos(t) = 0. Yes, it works. The solution is correct. The verdict should be pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-20 — The proposed solution x(t) = cos(t) does not satisfy the differential equation 2x'' + 2x = 0, which requires angular frequency omega = 1, but the equation simplifies to x'' + x = 0, so omega is indeed 1. Wait, let me re-evaluate. m=2, k=2. Equation: 2x'' + 2x = 0 => x'' + x = 0. General solution x(t) = A cos(t) + B sin(t). IC: x(0) = 1 => A=1. x'(0) = 0 => B=0. So x(t) = cos(t) is actually correct. Why did I think it was wrong? Let me check the algebra again. 2*(-cos(t)) + 2*cos(t) = 0. Yes, it works. The solution is correct. The verdict should be pass.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to solve the differential equation generally; it merely verifies that x(t) = cos(t) is a solution. It does not demonstrate that this is the unique solution satisfying the initial conditions, nor does it derive the solution from the ODE.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to solve the differential equation to find the general form and apply initial conditions. It merely verifies that x(t) = cos(t) is a solution, ignoring that the actual solution is x(t) = cos(t/sqrt(2)) due to the mass and spring constant values.
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.