Math & Physics Core

Arithmetic You'll Actually Use

• Percentages: Discounts, interest rates, mixture ratios, battery charge levels
• Ratios: Fuel mixing (50:1), concrete ratios (1:2:3), recipe scaling
• Unit conversion: Fahrenheit to Celsius, gallons to liters, feet to meters
• Averages: Rainfall averages, fuel consumption, daily usage
• Area and volume: Garden bed square footage, water tank capacity, paint needed

Key Formulas

Area of rectangle = Length × Width

Area of circle = π × r² (π ≈ 3.14, r = radius)

Volume of rectangle = Length × Width × Height

Volume of cylinder = π × r² × Height

Circumference = 2 × π × r

Example: Circular garden bed, 4 foot radius. Area = 3.14 × 4² = 50 sq ft. At 1 plant per sq ft, you need 50 plants.

Unit Conversions

Ratio Calculations

Common in fuel mixing, concrete, dilutions:

Parts needed = (Total amount ÷ Sum of ratio parts) × Ratio number

Example: 2-stroke fuel at 50:1 ratio for 1 gallon (128 oz) gas.
Oil needed = 128 ÷ 50 = 2.56 oz of oil per gallon of gas.

Example: Concrete 1:2:3 (cement:sand:gravel) for 6 cubic feet total.
Sum = 1+2+3 = 6. Cement = (6÷6)×1 = 1 cu ft. Sand = 2 cu ft. Gravel = 3 cu ft.

Force and Leverage

Mechanics basics for tools and construction:

• Lever: Long handle = more force. Distance from pivot matters.
• Pulley: More pulleys = less force needed, but more rope pulled.
• Inclined plane: Ramp makes lifting easier but requires more distance.
• Gear ratio: Small gear driving big gear = more torque, less speed.

A 2-foot wrench applies twice the torque of a 1-foot wrench with the same force.

Simple Machines

Structural Basics

Why things stand or fall:

• Triangle is strongest shape: Doesn't deform under load. Use in bracing.
• Compression vs tension: Concrete strong in compression, weak in tension. Steel strong in both.
• Load path: Weight must transfer to ground through solid path. Break the path = collapse.
• Span limits: Longer spans need deeper beams or more support. Double span = need much stronger beam.

Don't modify structural walls or beams without understanding load paths. Houses have collapsed from removing "just one wall."

Thermodynamics Basics

• Heat flows hot to cold: Insulation slows this. That's all insulation does.
• Thermal mass: Dense materials (concrete, stone) store heat. Cool during day, release at night.
• Phase change: Water absorbs huge energy to boil. Sweating cools you. Evaporative cooling works.
• Insulation R-value: Higher = better insulation. R-13 wall, R-38 ceiling typical.

Water Pressure

Pressure (psi) ≈ Height (feet) × 0.43

• Water tower 100 ft tall = ~43 psi at ground
• Every 2.3 feet of elevation = 1 psi
• Household pressure 40-60 psi typical

Gravity feed from elevated tank: 10 ft elevation = 4.3 psi. Enough for basic flow, but not strong pressure.

Electrical Basics

Power (Watts) = Voltage (V) × Current (Amps)

Current (Amps) = Power (Watts) ÷ Voltage

• 1500W space heater on 120V = 12.5 amps (most circuits are 15A)
• Don't exceed 80% of circuit capacity continuously (12A on 15A circuit)
• Long extension cords = voltage drop = less power at device

Electricity kills. If you don't understand it, don't work on it. This section is for understanding, not training.

Fuel and Energy

BTU = British Thermal Unit. 1 BTU = energy to heat 1 lb water by 1°F.

Body Measurements

Use your body when you don't have tools:

• Hand span: Tip of thumb to tip of pinkie, fingers spread. Measure yours.
• Arm span: Fingertip to fingertip ≈ your height
• Pace: Normal walking step. Count steps to estimate distance.
• Knuckle: Middle knuckle to wrist crease ≈ 3 inches (check yours)
• Finger joints: Each joint ≈ 1 inch (varies by person)

Measure your body references now and remember them. They're always available.

Estimation Rules

• 5-second rule for lightning: Count seconds between flash and thunder. Divide by 5 = miles away.
• Fist at arm's length: ≈ 10 degrees of sky
• Thumb at arm's length: ≈ 2 degrees of sky
• Hand width at arm's length: ≈ 15-20 degrees

Drill: Calculate Materials Needed

Garden bed: 8 feet long, 4 feet wide, 12 inches deep. Fill with soil mix.

1. Volume = 8 × 4 × 1 = 32 cubic feet
2. Soil sold in 2 cubic foot bags. Need 32 ÷ 2 = 16 bags
3. Or bulk: 27 cubic feet = 1 cubic yard. Need about 1.2 cubic yards.

Drill: Scale a Recipe

Recipe serves 4, you need to serve 10. Original calls for 2 cups flour.

1. Scale factor = 10 ÷ 4 = 2.5
2. Flour needed = 2 × 2.5 = 5 cups
3. Apply to all ingredients: multiply each by 2.5