Overview
This lecture explains how to convert between derived units, especially when dealing with square (area) and cubic (volume) units and how metric prefixes affect conversions in multiple dimensions.
Derived Units and Prefixes
- Derived units are created by combining base units, often resulting in units for area (square) or volume (cubic).
- Metric prefixes (like centi-, milli-, kilo-) change when applied to squared or cubed units.
- When converting, raise the linear prefix power to match the dimensionality (square/cube the exponent).
Volume and Area Conversion Examples
- 1 cm³ (cubic centimeter) = 10⁻⁶ m³ (cubic meters), since (10⁻² m)³ = 10⁻⁶ m³.
- For any length, cube (or square) the conversion factor to get the relationship for cubic (or square) units.
- Example: 1 mm³ = (10⁻³ m)³ = 10⁻⁹ m³.
- For area: 1 cm² = (10⁻² m)² = 10⁻⁴ m².
Metric Base Units vs. Metric System Units
- The SI system defines volume from base units (e.g., cubic meters), while the metric system defines liters directly.
- Prefixes for liters work as expected: 1 mL = 10⁻³ L; 1 GL = 10⁹ L.
- 1 dm³ (cubic decimeter) = 1 L (liter); 1 cm³ = 1 mL (milliliter).
Step-by-Step Conversion Process
- Write the linear metric relationship (e.g., 1 mm = 10⁻³ m).
- Cube (or square) both sides for cubic (or square) conversions.
- Set up unit cancellation: place unwanted units opposite each other to cancel.
- Multiply and divide as appropriate; adjust exponents based on dimensional changes.
Non-Metric Unit Conversions
- 1 ft = 12 in; to convert cubic feet to cubic inches, cube 12: 1 ft³ = 1,728 in³.
Practice and Application
- Always determine the relationship between desired units and cube/square as needed.
- Use the dimensionality to adjust the exponents for correct conversions.
Key Terms & Definitions
- Derived Unit — A unit made from combining base units (e.g., m², m³).
- Metric Prefix — A modifier indicating powers of ten (e.g., centi- = 10⁻²).
- Cubic Unit — A unit resulting from length cubed, measuring volume (e.g., m³).
- Square Unit — A unit resulting from length squared, measuring area (e.g., m²).
- Liter (L) — Metric unit for volume, defined as 1 dm³.
Action Items / Next Steps
- Practice converting between units with different prefixes and dimensions.
- Memorize that 1 L = 1 dm³ and 1 mL = 1 cm³.
- Complete the example problems provided in the lecture.