Exploring the Lewis Ratchet in Python - Axiomatic Proof

A simple explanation:

The Lewis Ratchet: Moving a Raft Without Paddles

Imagine you’re sitting on a floating raft in the middle of a calm lake. There’s no wind, no current, and no paddles. But, beside you, there’s a pile of heavy rocks.

At first, the raft is completely still. But what if I told you that by throwing these rocks across the raft, we could slowly move the entire thing in one direction? That’s exactly what we’re going to do.

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Step 1: Throwing the Rocks (Pushing the Raft Backward)
We start by picking up one rock and throwing it to the other side of the raft.

- The moment we push the rock forward, the raft moves slightly backward.
- Once the rock lands, the raft settles in its new position.
- We repeat this for every rock in the pile, throwing them one by one.

By the time we’ve thrown all of them, the raft has moved backward across the water.

💡 Why does this happen?
- It’s like when you push off a wall in a swimming pool—you go backward.
- Since we pushed the rock forward, the raft had to move in the opposite direction.

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Step 2: Walking to Get the Rocks (Still Moving Backward!)
Now all the rocks are at the far end of the raft, and we need to go get them.

- As we start walking forward, something surprising happens…
- The raft moves even further backward!
- That’s because every step we take pushes against the raft, shifting it back just a little more.

💡 Why is this still helping us?
- Normally, if you walk on solid ground, you don’t push it anywhere.
- But here, the raft floats freely, so pushing with our feet makes it drift slightly backward..
- So even while just walking forward, we’re still moving in the direction we want!

When we reach the far end, we pick up the rocks, one by one, and prepare to return.

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Step 3: Walking Back with the Rocks (Pushing the Raft Forward)
Now we walk back to our original spot, but there’s a difference—we’re carrying the rocks now.

- This time, the raft moves forward with each step we take.
- Since we’re now heavier, the push against the raft is stronger than before.
- If we rush back too fast, we might undo all the progress we made earlier!

💡 How do we prevent losing too much ground?
- Instead of walking normally, we move slowly and carefully to reduce how much the raft shifts forward.
- This way, we keep most of the progress from the earlier steps.

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Step 4: Repeat the Process
Now that we’re back at the starting position, holding the rocks again, we can repeat the cycle:

1️⃣ Throw the rocks forward again -> The raft moves backward.
2️⃣ Walk forward to retrieve them -> The raft moves backward again.
3️⃣ Walk back with the rocks (slowly!) -> The raft moves slightly forward, but not enough to undo the progress.

Each time we do this, the raft moves farther and farther backward across the water.

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The Trick: The Ratchet Effect
Think of a ratchet wrench—it lets you turn a bolt in one direction but prevents it from slipping backward.

This process works the same way:
✅ Throwing the rocks and walking forward both move the raft in the right direction.
⚠️ Walking back with the rocks tries to push the raft forward, but we control it.

By repeating this cycle, we keep making progress in one direction, little by little, without a paddle, without an engine—just by using physics! 🚀

And that’s the Lewis Ratchet in action.

The Lewis Ratchet: A Momentum-Based Displacement System

The Lewis Ratchet describes a controlled displacement phenomenon where an object—such as a raft—accumulates small, incremental shifts due to carefully applied momentum transfers. In this case, a person on a raft throws masses sequentially, leveraging the conservation of momentum to shift the raft in a desired direction. However, when retrieving the masses, time-adjusted steps ensure that no additional unintended displacement occurs.

Key Phases of the Lewis Ratchet in Action:

1. Momentum Transfer Through Throws:

   • Each mass thrown exerts a recoil force, causing the raft to move incrementally backward.
   • Over multiple throws, this results in this scenario in a net displacement of -62.42 m.
   • Energy dissipates due to drag forces, gradually bringing the raft to rest after each impulse.

2. Controlled Retrieval Without Displacement:

   • Walking across the raft to pick up the masses is carefully timed.
   • Steps are adjusted dynamically to cancel out momentum effects, ensuring zero unintended movement.
   • This allows the system to maintain its accumulated displacement without reversal.

3. Walking Back with the Masses:

   • The person returns with the retrieved masses, applying the same time-adjusted walking strategy.
   • Again, no net displacement occurs—proving that momentum transfer is entirely dictated by the throwing phase.

Why It’s Called a Ratchet?

A traditional ratchet allows movement in one direction only, preventing backward motion. The Lewis Ratchet follows this principle by permitting unidirectional displacement through carefully managed impulse transfers while blocking unintended drift during retrieval.

This simulation serves as a powerful teaching tool, demonstrating principles of Newtonian mechanics, momentum conservation, drag forces, and controlled motion in a dynamic environment. 🚀

https://github.com/mikelewis1971/The-Lewis-Ratchet

Expected Output:

"Lewis Ratchet.py" 

=== Raft & Person Dynamics Simulation ===

Scenario: A person on a raft throws masses sequentially and then carefully walks

to retrieve and carry the masses back. We follow each dynamic event in time,

tracking energy dissipation and displacement.

============================================

Starting Parameters:

  Raft mass          = 160.0 kg

  Ball mass          = 5.0 kg (each), 4 balls total

  Throwing speed     = 5.0 m/s

  Drag coefficient   = 0.05

============================================

=== Throw 1 ===

[Throw Event] m_ball = 5.00 kg, v_throw = 5.00 m/s

  Impulse delivered: 25.0000 kg·m/s

  Initial raft velocity (recoil): -0.1562 m/s

  --> Throw resolved in 100.0000 s, displacement = -15.6060 m

      Energy dissipated: 0.0095 J (Initial KE was 1.9531 J)

=== Throw 2 ===

[Throw Event] m_ball = 5.00 kg, v_throw = 5.00 m/s

  Impulse delivered: 25.0000 kg·m/s

  Initial raft velocity (recoil): -0.1562 m/s

  --> Throw resolved in 100.0000 s, displacement = -15.6060 m

      Energy dissipated: 0.0095 J (Initial KE was 1.9531 J)

=== Throw 3 ===

[Throw Event] m_ball = 5.00 kg, v_throw = 5.00 m/s

  Impulse delivered: 25.0000 kg·m/s

  Initial raft velocity (recoil): -0.1562 m/s

  --> Throw resolved in 100.0000 s, displacement = -15.6060 m

      Energy dissipated: 0.0095 J (Initial KE was 1.9531 J)

=== Throw 4 ===

[Throw Event] m_ball = 5.00 kg, v_throw = 5.00 m/s

  Impulse delivered: 25.0000 kg·m/s

  Initial raft velocity (recoil): -0.1562 m/s

  --> Throw resolved in 100.0000 s, displacement = -15.6060 m

      Energy dissipated: 0.0095 J (Initial KE was 1.9531 J)

--- Summary of Throw Events ---

Total raft displacement due to throws: -62.4238 m

Total time for throw events: 400.0000 s

Total energy dissipated during throws: 0.0380 J

[Walking Event] Person mass = 100.00 kg, walking 10.00 m in 10 steps

  Each step (relative distance) = 1.0000 m, adjusting time to prevent raft motion.

    Step 1: Adjusted time = 3.1623 s to minimize raft motion.

    Step 2: Adjusted time = 3.1623 s to minimize raft motion.

    Step 3: Adjusted time = 3.1623 s to minimize raft motion.

    Step 4: Adjusted time = 3.1623 s to minimize raft motion.

    Step 5: Adjusted time = 3.1623 s to minimize raft motion.

    Step 6: Adjusted time = 3.1623 s to minimize raft motion.

    Step 7: Adjusted time = 3.1623 s to minimize raft motion.

    Step 8: Adjusted time = 3.1623 s to minimize raft motion.

    Step 9: Adjusted time = 3.1623 s to minimize raft motion.

    Step 10: Adjusted time = 3.1623 s to minimize raft motion.

  [Walking Complete] Total raft displacement from walking: 0.0000 m in 31.6228 s

--- Summary of Walking Event ---

Raft displacement while walking carefully: 0.0000 m

Total time required for careful walking: 31.6228 s

Simulation Complete: Lewis Ratchet Demonstration

✅ Throwing Dynamics:

• Each throw caused a backward displacement of -15.606 m.
• After four throws, the raft moved -62.42 m.
• Total energy dissipated due to drag: 0.038 J.

✅ Walking Forward to Retrieve Masses:

• Required 10 careful steps with adjusted timing to prevent raft motion.
• Total retrieval time: 31.62 s.
• Raft displacement: 0.000 m (Success! No unintended movement).

✅ Walking Back While Carrying the Masses:

• Again, required 10 careful steps.
• Total return time: 31.62 s.
• Raft displacement: 0.000 m (No unintended movement).

✅ Overall Simulation Summary:

• Total raft displacement: -62.42 m (due to throwing, as expected).
• Total time elapsed: 463.25 s.

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Visualization Recap

📊 Raft Displacement Over Throws:

• Shows how the raft moves backward with each throw.

🔥 Energy Dissipation Plot:

• Illustrates the accumulated energy lost to drag.

🚶‍♂️ Walking Time Adjustments Plot:

• Forward & return walk times aligned for zero raft displacement.

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1. Axioms and Setup

1. Axiom (Newton’s Laws)

   • Newton’s First Law: A body remains in uniform motion or at rest unless acted upon by a net external force.
   • Newton’s Second Law: The acceleration $\vec{a}$ of a body of mass $m$ is given by $m \, \vec{a} = \sum \vec{F}_{\text{ext}}$.
   • Newton’s Third Law: Every force has an equal and opposite reaction force (internal to the system).

2. Axiom (Conservation of Momentum in an Isolated System)
   In the absence of external forces, the total linear momentum of a closed system remains constant. Equivalently, the center of mass of the system moves at a constant velocity (possibly zero if initially at rest).

3. Axiom (Drag as an External Force)
   We assume a “drag force” or “frictional force” from the surrounding medium (e.g., water). This force depends on velocity, e.g.

   $$\vec{F}_{\text{drag}} = -k \, \vec{v} \; \Big(\text{or } -\tfrac12 \rho C_d A \, \|\vec{v}\| \,\vec{v}\Big),$$

   for some constant(s) $k$ (or $\rho, C_d, A$). Crucially, drag is external to the raft-plus-person system.

4. Initial Conditions

   • The raft-plus-person system is initially at rest (total momentum = 0).
   • The environment exerts no net force unless/until motion occurs (i.e., drag is zero or negligible if velocity is zero).
   • Internal masses (balls, or parts of the person) can be moved or thrown inside the system.

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2. Statement of the Problem

We want to see how a person on a raft, by internal actions only (walking around or throwing masses), might produce a net horizontal displacement.

• Question: In the absence of any external force (no drag, frictionless water), can there be a net displacement of the raft’s center of mass after all internal motions settle?
• Lewis Ratchet: If there is a drag force, does the repeated sequence of throws/mass shifts plus slow walks yield an accumulated net displacement (a “ratchet” effect)?

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3. Proof Without Drag

1. Consider the system: raft + person + internal masses. Let this system be closed (no external forces except for negligible buoyancy in the vertical direction).
2. By Axiom 2 (Conservation of Momentum), any internal transfer of momentum (e.g., throwing a ball forward) imparts an equal and opposite momentum to the raft.
3. Symmetry: If we wait for all motion to settle (velocities returning to zero) and no external forces act, the net change in the system’s center of mass is zero.
   • Reason: The momentum that shot the ball forward and caused the raft to recoil is later “undone” when the ball is re-collected or any internal shifting ends.
4. Conclusion: No net shift of the center of mass can occur if the system is truly isolated (frictionless, drag‐free). This follows directly from Newton’s laws and the conservation of momentum principle.

Hence, in a frictionless world:

$$\Delta \vec{R}_{\text{COM}} = 0 \quad (\text{if no external force}).$$

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4. Proof With Drag (The “Lewis Ratchet”)

1. External Drag Force Exists: Introduce a velocity-dependent drag force $\vec{F}_{\text{drag}}$, which acts opposite to the raft’s motion through the water.

   • This drag is external to the raft‐person system.
   • The moment the raft (or the thrown ball, or the person walking on the raft) has any net velocity, drag dissipates kinetic energy from the system.

2. Throwing Phase

   • The person throws a mass (ball) forward at velocity $\vec{v}_{\text{throw}}$. By Newton’s Third Law, the raft recoils with momentum $m_{\text{ball}} \, \vec{v}_{\text{throw}}$ in the opposite direction.
   • However, drag opposes the raft’s recoil and reduces the raft’s velocity over time, dissipating some of the system’s mechanical energy to heat/turbulence.
   • Eventually, the raft comes to rest again (relative to the water) but at a small shifted position from the starting point. Because the system lost energy to drag, it cannot “undo” that displacement simply by reversing the internal process.

3. Walking Phase

   • Next, the person tries to walk across the raft to retrieve the thrown masses. If they do this very slowly, the drag force can counter the small reaction forces that would otherwise move the raft.
   • Careful control of each step’s acceleration can keep the raft from noticeably drifting. In effect, the person plus raft experiences minute velocity changes, each quickly damped by drag.
   • As a result, the raft’s final position at the end of the walk might remain almost the same as it was after the throwing phase.

4. Carrying Masses Back

   • The person, now holding the thrown masses, returns. Again, if the walk is done slowly enough, drag can prevent further net shifts.
   • The key here is that any attempt at a “fast” walk would push the raft backward or forward, but with careful steps, drag kills each incremental motion before it accumulates.

5. Net Effect

   • Because drag has dissipated energy and “locked in” each incremental shift (from throws, small unbalanced pushes, etc.), the raft may end up at a net displacement from its original position.
   • Internal steps alone cannot recoup the lost energy or re-shift the system back to the starting point, because that would require accelerating the raft in the reverse direction—again opposed by drag.

6. Conclusion:

   • The presence of drag (an external force) breaks the perfect momentum‐conservation symmetry for the raft‐person subsystem.
   • Energy is removed (dissipated), so the system can “ratchet” from one offset state of rest to another.
   • Once the system stops, it cannot spontaneously revert to the original position because the energy needed to move the center of mass back has already been lost.

Formally, once you include any non‐conservative external force (like friction or drag), the total momentum of the raft‐person subsystem alone is no longer conserved. Momentum is “exchanged” with the surrounding medium. This mechanism underlies the Lewis Ratchet: it allows net displacement via carefully orchestrated internal actions plus the dissipative effect of drag.

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5. Summary of the Axiomatic Argument

1. Ideal, Frictionless Case:
   • No net external force → total momentum of the raft‐person system is conserved → the center of mass does not move from its initial position once all internal movements settle.
2. With Drag:
   • Drag is external and non-conservative; it counteracts motion and dissipates energy.
   • Repetitive actions that impart momentum to the raft can “lock in” partial displacements each time because the system cannot fully recover the lost energy to move back.
   • Hence, a net displacement accumulates—a ratchet effect—provided the actions are carefully timed and slow walking ensures minimal “undoing” of that net offset.

This explains why the simulation reproduces a net shift in the raft whenever drag is active and shows no net shift when drag is absent. It also formally aligns with Noether’s theorem: in a frictionless environment (perfect symmetry, no external forces), you cannot spontaneously break momentum conservation or shift the center of mass. Once that symmetry is broken (by an external drag force), the door is open for dissipative “ratcheting” displacements.

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Hence, the ‘Lewis Ratchet’ is proven feasible by:

1. Identifying drag as the external force that breaks strict momentum conservation in the raft‐person subsystem,
2. Showing that internal mass shifting plus drag dissipation leads to net shifts,
3. Demonstrating via simulation that each incremental displacement is “locked in” by energy dissipation, leaving the system at a new offset equilibrium after each cycle.