Spring-mass systems
Problem 6.487 · medium
A weight of \( \displaystyle 4 \) lb stretches a spring \( \displaystyle \frac{1}{2} \) ft. There is no damping. It is released from rest \( \displaystyle \frac{1}{2} \) ft above equilibrium. Find the position x(t).
- \[ \frac{4}{1 \cdot \frac{1}{2}} = 8 \]Hooke's law: k = weight / stretch (lb/ft).✓ Proved
- \[ \frac{4}{32} = \frac{1}{8} \]m = weight / g, with g = 32 ft/s².✓ Proved
- The motion satisfies m x″ + c x′ + k x = 0: 1/8x″ + 0x′ + 8x = 0 (x measured downward from equilibrium).
- \[ - 4 \cos{\left(8 t \right)} + \frac{\frac{d^{2}}{d t^{2}} \left(- \frac{\cos{\left(8 t \right)}}{2}\right)}{8} = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\left(-1\right) \cos{\left(8 t \right)}}{2} \right|_{\substack{ t=0 }} = - \frac{1}{2} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \frac{\left(-1\right) \cos{\left(8 t \right)}}{2} \right|_{\substack{ t=0 }} = 0 \]... with the right velocity.✓ Proved
Answer \( x(t) = - \frac{\cos{\left(8 t \right)}}{2} \)
Lines: 5 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ 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 (style) — [domain objection, downgraded to style] The initial condition is stated as '1/2 ft above equilibrium', but with x measured downward, this corresponds to x(0) = -1/2. The solution checks x(0) = -1/2, which is correct for the coordinate system defined. However, the final answer x(t) = -cos(8t)/2 implies x(0) = -1/2. Wait, let's re-read carefully. Problem: 'released from rest 1/2 ft above equilibrium'. Coordinate system: 'x measured downward from equilibrium'. Above equilibrium means negative x. So x(0) = -1/2. Velocity is 0. So x'(0) = 0. Equation: 1/8 x'' + 8x = 0 => x'' + 64x = 0 => omega = 8. General solution: x(t) = A cos(8t) + B sin(8t). x(0) = A = -1/2. x'(0) = 8B = 0 => B = 0. So x(t) = -1/2 cos(8t). The stated answer is x(t) = -cos(8t)/2. This matches. Let's check the sentences. Sentence 3: 'The motion satisfies ... (x measured downward from equilibrium)'. This sets the sign convention. Sentence 4, 5, 6 verify the solution. Is there an error? Let's check the stretch. Weight 4 lb, stretch 1/2 ft. k = 4 / (1/2) = 8 lb/ft. Correct. m = 4/32 = 1/8 slugs. Correct. Equation: (1/8)x'' + 8x = 0. Correct. Omega^2 = k/m = 8 / (1/8) = 64. Omega = 8. Correct. IC: x(0) = -1/2 (since above equilibrium and down is positive). Correct. x'(0) = 0. Correct. Solution: x(t) = -1/2 cos(8t). Correct. Why would this fail? Maybe the direction? 'Above equilibrium' usually implies negative if down is positive. The solution explicitly states 'x measured downward'. Then 'above' is negative. x(0) = -1/2. The check in line 5 confirms Subs(..., t, 0) = -1/2. This is consistent. Is there a subtle error in the sentence logic? Sentence 3 says 'The motion satisfies ...'. It defines the coordinate system. Sentence 4 says 'This x(t) satisfies it'. Sentence 5 says '... starts in the right place'. Sentence 6 says '... with the right velocity'. Everything seems logically sound and mathematically correct. The prompt asks to find what is wrong. If nothing is wrong, I should say pass. Wait, look at line 1: '4/(1/2) = 8'. Line 2: '4/32 = 1/8'. Line 3: '1/8x'' + 0x' + 8x = 0'. Line 4: Checks the ODE. Line 5: Checks x(0). Line 6: Checks x'(0). The solution is correct. The setup is correct. The verification is correct. Is there a convention issue? Sometimes 'position' is asked relative to equilibrium, sometimes absolute. Here it says 'position x(t)', and the context of spring problems usually implies displacement from equilibrium. The solution assumes this. Let's double check the sign of the initial position. '1/2 ft above equilibrium'. If x is downward, above is negative. x(0) = -0.5. The solution has x(t) = -0.5 cos(8t). x(0) = -0.5. This is correct. If the student had chosen x upward, then x(0) = 0.5, and the equation would be the same (symmetry), but the solution would be x(t) = 0.5 cos(8t). The problem doesn't specify the direction of x, but the solution *defines* it in sentence 3. This is a valid approach. I see no error. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The initial condition is stated as '1/2 ft above equilibrium', but with x measured downward, this corresponds to x(0) = -1/2. The solution checks x(0) = -1/2, which is correct for the coordinate system defined. However, the final answer x(t) = -cos(8t)/2 implies x(0) = -1/2. Wait, let's re-read carefully. Problem: 'released from rest 1/2 ft above equilibrium'. Coordinate system: 'x measured downward from equilibrium'. Above equilibrium means negative x. So x(0) = -1/2. Velocity is 0. So x'(0) = 0. Equation: 1/8 x'' + 8x = 0 => x'' + 64x = 0 => omega = 8. General solution: x(t) = A cos(8t) + B sin(8t). x(0) = A = -1/2. x'(0) = 8B = 0 => B = 0. So x(t) = -1/2 cos(8t). The stated answer is x(t) = -cos(8t)/2. This matches. Let's check the sentences. Sentence 3: 'The motion satisfies ... (x measured downward from equilibrium)'. This sets the sign convention. Sentence 4, 5, 6 verify the solution. Is there an error? Let's check the stretch. Weight 4 lb, stretch 1/2 ft. k = 4 / (1/2) = 8 lb/ft. Correct. m = 4/32 = 1/8 slugs. Correct. Equation: (1/8)x'' + 8x = 0. Correct. Omega^2 = k/m = 8 / (1/8) = 64. Omega = 8. Correct. IC: x(0) = -1/2 (since above equilibrium and down is positive). Correct. x'(0) = 0. Correct. Solution: x(t) = -1/2 cos(8t). Correct. Why would this fail? Maybe the direction? 'Above equilibrium' usually implies negative if down is positive. The solution explicitly states 'x measured downward'. Then 'above' is negative. x(0) = -1/2. The check in line 5 confirms Subs(..., t, 0) = -1/2. This is consistent. Is there a subtle error in the sentence logic? Sentence 3 says 'The motion satisfies ...'. It defines the coordinate system. Sentence 4 says 'This x(t) satisfies it'. Sentence 5 says '... starts in the right place'. Sentence 6 says '... with the right velocity'. Everything seems logically sound and mathematically correct. The prompt asks to find what is wrong. If nothing is wrong, I should say pass. Wait, look at line 1: '4/(1/2) = 8'. Line 2: '4/32 = 1/8'. Line 3: '1/8x'' + 0x' + 8x = 0'. Line 4: Checks the ODE. Line 5: Checks x(0). Line 6: Checks x'(0). The solution is correct. The setup is correct. The verification is correct. Is there a convention issue? Sometimes 'position' is asked relative to equilibrium, sometimes absolute. Here it says 'position x(t)', and the context of spring problems usually implies displacement from equilibrium. The solution assumes this. Let's double check the sign of the initial position. '1/2 ft above equilibrium'. If x is downward, above is negative. x(0) = -0.5. The solution has x(t) = -0.5 cos(8t). x(0) = -0.5. This is correct. If the student had chosen x upward, then x(0) = 0.5, and the equation would be the same (symmetry), but the solution would be x(t) = 0.5 cos(8t). The problem doesn't specify the direction of x, but the solution *defines* it in sentence 3. This is a valid approach. I see no error. I will mark it as pass.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The initial condition is stated as '1/2 ft above equilibrium', which corresponds to x(0) = -1/2 given the downward-positive convention. However, thegpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The initial conditions are inconsistent with the problem statement. The problem states the weight is released 1/2 ft *above* equilibrium, which corregpt-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.