Spring-mass systems
Problem 6.666 · hard
A weight of \( \displaystyle 8 \) lb stretches a spring \( \displaystyle \frac{1}{2} \) ft. A damping force of \( \displaystyle 4 \) times the velocity acts on it. It starts \( \displaystyle \frac{1}{2} \) ft above equilibrium with velocity \( \displaystyle 1 \) ft/s (down is positive). Find the position x(t).
- \[ \frac{8}{1 \cdot \frac{1}{2}} = 16 \]Hooke's law: k = weight / stretch (lb/ft).✓ Proved
- \[ \frac{8}{32} = \frac{1}{4} \]m = weight / g, with g = 32 ft/s².✓ Proved
- The motion satisfies m x″ + c x′ + k x = 0: 1/4x″ + 4x′ + 16x = 0 (x measured downward from equilibrium).
- \[ 8 \left(- 6 t - 1\right) e^{- 8 t} + 4 \frac{d}{d t} \frac{\left(- 6 t - 1\right) e^{- 8 t}}{2} + \frac{\frac{d^{2}}{d t^{2}} \frac{\left(- 6 t - 1\right) e^{- 8 t}}{2}}{4} = 0 \]This x(t) satisfies it ...✓ Proved
- \[ \left. \frac{\left(- 6 t - 1\right) e^{- 8 t}}{2} \right|_{\substack{ t=0 }} = - \frac{1}{2} \]... starts in the right place ...✓ Proved
- \[ \left. \frac{d}{d t} \frac{\left(- 6 t - 1\right) e^{- 8 t}}{2} \right|_{\substack{ t=0 }} = 1 \]... with the right velocity.✓ Proved
Answer \( x(t) = \frac{\left(- 6 t - 1\right) e^{- 8 t}}{2} \)
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 (error) — The initial condition for position is incorrect. The problem states the weight starts 1/2 ft above equilibrium, and since down is positive, x(0) should be -1/2. However, the solution claims x(0) = -1/2 is 'starting in the right place' but the derived function x(t) = (-6t - 1)e^{-8t}/2 yields x(0) = -1/2. Wait, let me re-read. x(0) = (-1)/2 = -1/2. This matches '1/2 ft above equilibrium' if down is positive. So the IC is correct. Let me check the ODE setup. m=1/4, c=4, k=16. Characteristic eq: 1/4 r^2 + 4r + 16 = 0 => r^2 + 16r + 64 = 0 => (r+8)^2 = 0. Critical damping. Solution form: x(t) = (c1 + c2 t)e^{-8t}. x(0) = c1 = -1/2. x'(t) = c2 e^{-8t} - 8(c1 + c2 t)e^{-8t}. x'(0) = c2 - 8c1 = 1. c2 - 8(-1/2) = 1 => c2 + 4 = 1 => c2 = -3. So x(t) = (-1/2 - 3t)e^{-8t} = -(1 + 6t)e^{-8t}/2 = (-6t - 1)e^{-8t}/2. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's stated answer. Stated answer: x(t) = ((-6t - 1)e^{-8t})/2. My derivation matches. The solution steps are correct. The sentence in step 3 sets up the ODE correctly. Step 5 checks x(0)=-1/2. Step 6 checks x'(0)=1. Everything seems correct. Is there a subtle error? 'A damping force of 4 times the velocity'. c=4. Correct. 'Weight 8 lb stretches 1/2 ft'. k = 8/(1/2) = 16. Correct. m = 8/32 = 1/4. Correct. ODE: 1/4 x'' + 4 x' + 16 x = 0. Correct. The solution provided in the text is mathematically sound and matches the problem statement. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The initial condition for position is incorrect. The problem states the weight starts 1/2 ft above equilibrium, and since down is positive, x(0) should be -1/2. However, the solution claims x(0) = -1/2 is 'starting in the right place' but the derived function x(t) = (-6t - 1)e^{-8t}/2 yields x(0) = -1/2. Wait, let me re-read. x(0) = (-1)/2 = -1/2. This matches '1/2 ft above equilibrium' if down is positive. So the IC is correct. Let me check the ODE setup. m=1/4, c=4, k=16. Characteristic eq: 1/4 r^2 + 4r + 16 = 0 => r^2 + 16r + 64 = 0 => (r+8)^2 = 0. Critical damping. Solution form: x(t) = (c1 + c2 t)e^{-8t}. x(0) = c1 = -1/2. x'(t) = c2 e^{-8t} - 8(c1 + c2 t)e^{-8t}. x'(0) = c2 - 8c1 = 1. c2 - 8(-1/2) = 1 => c2 + 4 = 1 => c2 = -3. So x(t) = (-1/2 - 3t)e^{-8t} = -(1 + 6t)e^{-8t}/2 = (-6t - 1)e^{-8t}/2. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's stated answer. Stated answer: x(t) = ((-6t - 1)e^{-8t})/2. My derivation matches. The solution steps are correct. The sentence in step 3 sets up the ODE correctly. Step 5 checks x(0)=-1/2. Step 6 checks x'(0)=1. Everything seems correct. Is there a subtle error? 'A damping force of 4 times the velocity'. c=4. Correct. 'Weight 8 lb stretches 1/2 ft'. k = 8/(1/2) = 16. Correct. m = 8/32 = 1/4. Correct. ODE: 1/4 x'' + 4 x' + 16 x = 0. Correct. The solution provided in the text is mathematically sound and matches the problem statement. I will mark it as pass.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The initial condition for position is incorrect. The problem states the weight starts 1/2 ft above equilibrium, and since down is positive, x(0) should be -1/2. However, the solution claims the verified value -1/2 means it 'starts in the right place' without explicitly acknowledging the sign convention, but more critically, the stated answer x(t) = (-6t - 1)e^{-8t}/2 yields x(0) = -1/2, which is correct for 'above equilibrium' if down is positive. Wait, let me re-read carefully. Problem: 'starts 1/2 ft above equilibrium'. Down is positive. So x(0) = -1/2. The solution checks x(0) = -1/2 and says 'starts in the right place'. This is correct. Let me check the velocity. v(0) = 1 ft/s. The derivative of (-6t-1)e^{-8t}/2 at t=0 is [ -6*e^0 + (-1)*(-8)*e^0 ] / 2 = (-6 + 8)/2 = 1. This is also correct. The differential equation setup: m=1/4, c=4, k=16. 1/4 x'' + 4 x' + 16 x = 0. Characteristic eq: 1/4 r^2 + 4 r + 16 = 0 => r^2 + 16 r + 64 = 0 => (r+8)^2 = 0. Critical damping. Solution form: x(t) = (c1 + c2 t) e^{-8t}. x(0) = c1 = -1/2. x'(t) = c2 e^{-8t} - 8(c1 + c2 t) e^{-8t}. x'(0) = c2 - 8c1 = 1 => c2 - 8(-1/2) = 1 => c2 + 4 = 1 => c2 = -3. So x(t) = (-1/2 - 3t) e^{-8t} = -(1/2 + 3t) e^{-8t} = (-1 - 6t) e^{-8t} / 2. The stated answer is (-6t - 1) e^{-8t} / 2. This matches. The solution seems correct. Why did I think it was wrong? Let me re-read the prompt's 'Stated answer'. x(t) = ((-6t - 1) e^{-8t}) / 2. My derivation: x(t) = (-1/2 - 3t) e^{-8t} = (-1 - 6t)/2 * e^{-8t}. Yes, it matches. The sentences are: 3. Setup is correct. 4,5,6 are verification steps. Sentence 3 says 'x measured downward from equilibrium'. This is consistent with 'down is positive'. Sentence 5 says 'starts in the right place'. x(0)=-1/2. Problem says '1/2 ft above'. Above is negative in this coordinate system. So -1/2 is correct. Sentence 6 says 'right velocity'. v(0)=1. Problem says 'velocity 1 ft/s'. It doesn't specify direction for velocity, but usually 'velocity 1 ft/s' implies the vector value. If it meant speed, it would say speed. Given 'down is positive', a velocity of 1 ft/s is downward. The calculation holds. Is there a subtle error? 'A damping force of 4 times the velocity acts on it.' This usually means F_d = -4v or F_d = -4x'. The equation m x'' + c x' + k x = 0 assumes the damping force opposes motion, so it is -c x'. The term in the ODE is +c x' on the LHS, which corresponds to a force -c x' on the RHS (or included in the sum of forces = 0). This is standard. The setup is correct. The verification is correct. The answer is correct. I will mark it as pass.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The initial condition check in line 5 is incorrect. The problem states the weight starts 1/2 ft above equilibrium, and since down is positive, x(0) should be -1/2. However, the proposed solution yields x(0) = -1/2, which matches. Wait, let me re-read carefully. The problem says 'starts 1/2 ft above equilibrium'. Down is positive. So x(0) = -1/2. The solution checks Subs(..., t, 0) = -1/2. This is correct. Let me check the velocity. x'(t) = d/dt [(-6t-1)e^{-8t}/2] = 1/2 [ -6e^{-8t} + (-6t-1)(-8)e^{-8t} ] = 1/2 e^{-8t} [ -6 + 48t + 8 ] = 1/2 e^{-8t} [ 48t + 2 ]. At t=0, x'(0) = 1/2 [2] = 1. This matches the given velocity of 1 ft/s. The differential equation is 1/4 x'' + 4x' + 16x = 0 => x'' + 16x' + 64x = 0. Characteristic eq: r^2 + 16r + 64 = 0 => (r+8)^2 = 0 => r = -8. General solution x(t) = (C1 + C2 t) e^{-8t}. x(0) = C1 = -1/2. x'(t) = C2 e^{-8t} - 8(C1 + C2 t) e^{-8t}. x'(0) = C2 - 8C1 = 1 => C2 - 8(-1/2) = 1 => C2 + 4 = 1 => C2 = -3. So x(t) = (-1/2 - 3t) e^{-8t} = -(1/2 + 3t) e^{-8t} = (-1 - 6t) e^{-8t} / 2. The stated answer is correct. The solution steps are correct. Why did I think it was wrong? Let me re-read the prompt's instruction. 'Find what is wrong'. Is there a subtle error? Line 1: k = 16. Line 2: m = 1/4. Line 3: 1/4 x'' + 4x' + 16x = 0. This is correct. Line 4: Checks ODE. Line 5: Checks x(0). Line 6: Checks x'(0). Everything seems correct. Is there a sign error in the problem statement interpretation? 'starts 1/2 ft above equilibrium'. Down is positive. So position is -1/2. Correct. 'velocity 1 ft/s'. Down is positive. So velocity is +1. Correct. The solution is sound. Wait, look at line 5 again. `Subs((-6*t - 1)*exp(-8*t)/2, t, 0) = -1/2`. The text says '... starts in the right place ...'. The problem says 'starts 1/2 ft above equilibrium'. If down is positive, above is negative. So x(0) = -1/2. The check confirms x(0) = -1/2. This is consistent. Look at line 6. `Subs(Derivative(...), t, 0) = 1`. The problem says velocity 1 ft/s. The check confirms x'(0) = 1. This is consistent. Is there any other issue? Maybe the units? lb, ft, s. g=32. Correct. Damping force 4 times velocity. c=4. Correct. Weight 8 lb. k=16. Correct. Mass 8/32 = 1/4. Correct. The solution appears completely correct. I will mark it as 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 2026-10-19 with SymPy 1.14.0.