Sovereignty Module: Multiply the Force
Complete Mechanical Advantage: Levers, Pulleys, Gears, and Simple Machines Guide
Simple machines multiply human strength. One person with proper mechanical advantage can move loads that would otherwise require ten. This campaign covers every force-multiplying device buildable from basic materials.
Chapter 1: The Six Simple Machines
| Machine | Mechanical Advantage | How It Works | Example |
|---|---|---|---|
| Lever | Distance ratio (effort arm ÷ load arm) | Rotates around fulcrum | Pry bar, wheelbarrow, oar |
| Wheel and axle | Wheel radius ÷ axle radius | Large wheel turns small axle | Windlass, steering wheel, doorknob |
| Pulley | Number of supporting ropes | Redirects and multiplies force | Block and tackle, well windlass |
| Inclined plane | Length ÷ height | Spreads lift over longer distance | Ramp, road switchback |
| Wedge | Length ÷ width at base | Concentrates force to thin edge | Axe, knife, chisel, nail |
| Screw | Circumference ÷ pitch (thread spacing) | Inclined plane wrapped in circle | Vise, jack, press, drill |
Chapter 2: Lever Classes
| Class | Fulcrum Position | Advantage | Examples | Force Multiplication |
|---|---|---|---|---|
| First class | Between effort and load | Can be >1 or <1 | Crowbar, seesaw, scissors | Effort arm ÷ load arm |
| Second class | Load between fulcrum and effort | Always >1 (force multiplier) | Wheelbarrow, nutcracker, bottle opener | Effort arm ÷ load arm |
| Third class | Effort between fulcrum and load | Always <1 (speed multiplier) | Fishing rod, tweezers, human forearm | Trades force for speed/reach |
Lever calculation: If effort arm is 6 feet and load arm is 1 foot, mechanical advantage = 6:1. A 50-lb push moves a 300-lb rock. Longer lever = more advantage (limited by lever strength).
Chapter 3: Pulley Systems
| System | Pulleys | Mechanical Advantage | Rope Needed | Friction Loss |
|---|---|---|---|---|
| Single fixed | 1 (fixed) | 1:1 (direction change only) | 1x height | ~5% |
| Single movable | 1 (movable) | 2:1 | 2x height | ~5% |
| Double (gun tackle) | 2 (1 fixed + 1 movable) | 2:1 or 3:1 | 2-3x height | ~10% |
| Triple (luff tackle) | 3 (2 fixed + 1 movable) | 3:1 or 4:1 | 3-4x height | ~15% |
| Block and tackle (4) | 4 (2 fixed + 2 movable) | 4:1 | 4x height | ~20% |
| Block and tackle (6) | 6 (3 fixed + 3 movable) | 6:1 | 6x height | ~25% |
| Compound (theoretical) | Multiple stages | Multiplied (e.g., 3×3=9:1) | Varies | High (cumulative) |
Rule: Count the number of rope segments supporting the movable block = mechanical advantage. A 4:1 system means 50 lbs of pull lifts 200 lbs (minus friction).
Chapter 4: Windlass and Capstan
| Device | Function | Mechanical Advantage | Construction |
|---|---|---|---|
| Windlass (horizontal drum) | Lifts loads vertically (wells, construction) | Handle radius ÷ drum radius | Drum on axle + crank handle |
| Capstan (vertical drum) | Pulls loads horizontally (ships, logs) | Lever arm length ÷ drum radius | Vertical drum + push bars |
| Spanish windlass | Tightens rope/tourniquet | Stick length ÷ rope diameter | Stick twisted in loop of rope |
| Differential windlass | Very high MA from small difference | Large drum radius ÷ (large - small radius) | Two different diameter drums on same axle |
Windlass example: Drum diameter 4 inches (radius 2 inches). Handle length 12 inches. MA = 12 ÷ 2 = 6:1. A 30-lb crank effort lifts a 180-lb bucket of water from a well.
Chapter 5: Inclined Plane and Wedge
| Application | Rise | Run | Mechanical Advantage | Effort Required |
|---|---|---|---|---|
| Loading ramp (gentle) | 3 ft | 12 ft | 4:1 | 1/4 of lifting force |
| Loading ramp (steep) | 3 ft | 6 ft | 2:1 | 1/2 of lifting force |
| Road switchback | 100 ft | 1,000 ft | 10:1 | 1/10 of direct climb |
| Splitting wedge | 6 inches long, 1 inch wide | - | 6:1 | Concentrates blow 6x |
| Axe head | 4 inches long, 1/2 inch edge | - | 8:1 | Concentrates blow 8x |
| Screw jack (1-inch pitch, 6-inch handle) | - | - | 37.7:1 | Lifts tons with hand force |
Screw MA calculation: MA = (2 × π × handle length) ÷ pitch. A screw with 1-inch pitch and 6-inch handle: MA = (2 × 3.14 × 6) ÷ 1 = 37.7:1. A 20-lb push on the handle generates 754 lbs of force.
Chapter 6: Gear Ratios
| Configuration | Ratio | Effect | Application |
|---|---|---|---|
| Large gear drives small gear | >1 (speed increase) | Output spins faster, less torque | Bicycle high gear, drill press |
| Small gear drives large gear | <1 (torque increase) | Output spins slower, more force | Winch, clock mechanism, mill |
| Same size gears | 1:1 | Direction change only | Transfer between shafts |
| Compound gears (multiple stages) | Multiplied | Extreme ratios possible | Clock (60:1), crane (100:1+) |
| Worm gear | Very high (20:1 to 100:1) | Self-locking, high torque | Vises, tuning pegs, jacks |
Gear ratio: Count teeth. 40-tooth gear driving 10-tooth gear = 4:1 speed increase (or 1:4 torque increase if reversed). Compound: 4:1 × 4:1 = 16:1 total ratio.
Reference Card
- Lever: longer effort arm = more force multiplication. MA = effort arm ÷ load arm.
- Pulleys: count ropes supporting the load = mechanical advantage. 4 ropes = 4:1.
- Trade-off: more MA = more rope to pull (or more cranking). Force × distance is constant.
- Windlass: handle length ÷ drum radius = MA. Longer handle = easier lifting.
- Inclined plane: length ÷ height = MA. Longer ramp = less effort (but more distance).
- Screw: MA = (2π × handle) ÷ pitch. Highest MA of any simple machine. Lifts tons.
- Gears: tooth count ratio determines speed vs torque trade-off. Compound for extreme ratios.
- Friction: real-world MA is always less than theoretical. Allow 5-10% loss per pulley/bearing.