Home›Calculus 3›Spring-mass systems›Problem 6.499

Spring-mass systems

Problem 6.499 · hard

A \( \displaystyle 2 \) kg mass hangs from a spring with constant \( \displaystyle 18 \) N/m. A damping force of \( \displaystyle 6 \) times the velocity acts on it. It starts \( \displaystyle 1 \) 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″ + 6x′ + 18x = 0 (x measured downward from equilibrium).
Reviewed
\[ 18 \left(- \frac{\sqrt{3} \sin{\left(\frac{3 \sqrt{3} t}{2} \right)}}{9} + \cos{\left(\frac{3 \sqrt{3} t}{2} \right)}\right) e^{- \frac{3 t}{2}} + 6 \frac{d}{d t} \left(- \frac{\sqrt{3} \sin{\left(\frac{3 \sqrt{3} t}{2} \right)}}{9} + \cos{\left(\frac{3 \sqrt{3} t}{2} \right)}\right) e^{- \frac{3 t}{2}} + 2 \frac{d^{2}}{d t^{2}} \left(- \frac{\sqrt{3} \sin{\left(\frac{3 \sqrt{3} t}{2} \right)}}{9} + \cos{\left(\frac{3 \sqrt{3} t}{2} \right)}\right) e^{- \frac{3 t}{2}} = 0 \]
This x(t) satisfies it ...✓ Proved
\[ \left. \left(\frac{- \sqrt{3} \sin{\left(\frac{3 \sqrt{3} t}{2} \right)}}{9} + \cos{\left(\frac{3 \sqrt{3} t}{2} \right)}\right) e^{- \frac{3 t}{2}} \right|_{\substack{ t=0 }} = 1 \]
... starts in the right place ...✓ Proved
\[ \left. \frac{d}{d t} \left(\frac{- \sqrt{3} \sin{\left(\frac{3 \sqrt{3} t}{2} \right)}}{9} + \cos{\left(\frac{3 \sqrt{3} t}{2} \right)}\right) e^{- \frac{3 t}{2}} \right|_{\substack{ t=0 }} = -2 \]
... with the right velocity.✓ Proved

Answer \( x(t) = \left(- \frac{\sqrt{3} \sin{\left(\frac{3 \sqrt{3} t}{2} \right)}}{9} + \cos{\left(\frac{3 \sqrt{3} t}{2} \right)}\right) e^{- \frac{3 t}{2}} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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: pass
qwen3.6:27b-mlx: pass — The differential equation is correctly set up with the given parameters and sign conventions. The proposed solution satisfies the ODE and the initial conditions, as verified by the algebraic checks.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The differential equation is correctly set up with the given parameters and sign conventions. The proposed solution satisfies the ODE and the initial conditions, as verified by the algebraic checks.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the differential equation with the given parameters and initial conditions, and the provided function is verified to satisfy both the ODE and the initial conditions.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14
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.