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Spring-mass systems

Problem 6.720 · medium

A \( \displaystyle 4 \) kg mass hangs from a spring with constant \( \displaystyle 100 \) N/m. There is no damping. It starts \( \displaystyle \frac{1}{3} \) m below equilibrium with velocity \( \displaystyle 1 \) m/s (down is positive). Find the position x(t).

The motion satisfies m x″ + c x′ + k x = 0: 4x″ + 0x′ + 100x = 0 (x measured downward from equilibrium).
\[ 20 \sin{\left(5 t \right)} + \frac{100 \cos{\left(5 t \right)}}{3} + 4 \frac{d^{2}}{d t^{2}} \left(\frac{\sin{\left(5 t \right)}}{5} + \frac{\cos{\left(5 t \right)}}{3}\right) = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \frac{\sin{\left(5 t \right)}}{5} + \frac{\cos{\left(5 t \right)}}{3} \right|_{\substack{ t=0 }} = \frac{1}{3} \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \left(\frac{\sin{\left(5 t \right)}}{5} + \frac{\cos{\left(5 t \right)}}{3}\right) \right|_{\substack{ t=0 }} = 1 \]
... with the right velocity.✓ Proved

Answer \( x(t) = \frac{\sin{\left(5 t \right)}}{5} + \frac{\cos{\left(5 t \right)}}{3} \)

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 (misleading) — The solution verifies that the stated answer satisfies the differential equation and initial conditions, but it fails to derive the solution from the problem statement. This teaches the student that verification is equivalent to solving, omitting the crucial step of finding the general solution and applying initial conditions.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-20 — The solution verifies that the stated answer satisfies the differential equation and initial conditions, but it fails to derive the solution from the problem statement. This teaches the student that verification is equivalent to solving, omitting the crucial step of finding the general solution and applying initial conditions.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly sets up the differential equation, verifies that the proposed function satisfies the ODE, and confirms the initial conditions. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly sets up the differential equation for undamped harmonic motion, solves for the natural frequency, and applies the initial conditions accurately. The final answer matches the derived solution.
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.