Overview
This lecture covers dimensional analysis, a systematic method for converting units using conversion factors, with examples on length, area, volume, and speed conversions.
Dimensional Analysis Basics
- Dimensional analysis uses units and conversion factors to convert one quantity to another.
- Always set up the conversion so that unwanted units cancel, leaving only the desired unit.
Single Conversion Example: cm to inches
- Write the given value and multiply by the conversion factor as a fraction.
- Place the given unit in the denominator of the conversion factor to cancel it.
- Example: (15, \text{cm} \times \frac{1, \text{in}}{2.54, \text{cm}} = 5.9, \text{in}) (rounded to 2 significant figures).
Multiple Conversion Factors: meters to inches
- Use sequential conversion factors for units not directly related.
- Example: (8.00, \text{m} \to \text{cm} \to \text{in}):
- (8.00, \text{m} \times \frac{1, \text{cm}}{10^{-2}, \text{m}} \times \frac{1, \text{in}}{2.54, \text{cm}} = 315, \text{in}) (rounded to 3 significant figures).
- Enter (1 \times 10^{-2}) carefully on your calculator.
Area Conversion: in² to cm²
- Calculate area in initial units first (e.g., area of a square with 2-inch sides = (4.0, \text{in}^2)).
- Use the squared conversion factor: (\frac{2.54, \text{cm}}{1, \text{in}}) must be squared.
- Example: (4.0, \text{in}^2 \times \left(\frac{2.54, \text{cm}}{1, \text{in}}\right)^2 = 26, \text{cm}^2).
Volume Conversion: gallons to cubic inches
- Use a sequence of conversions (gallons → liters → milliliters → cubic inches).
- Example: (1.5, \text{gal} \times \frac{3.7854, \text{L}}{1, \text{gal}} \times \frac{1000, \text{ml}}{1, \text{L}} \times \frac{1, \text{in}^3}{16.4, \text{ml}} = 350, \text{in}^3).
Gas Mileage Conversion: mi/gal to km/L
- Start with given ratio and convert numerator and denominator units.
- Example: (\frac{254, \text{mi}}{11.2, \text{gal}} \to \frac{254, \text{mi} \times 1.6093, \text{km}/\text{mi}}{11.2, \text{gal} \times 3.7854, \text{L}/\text{gal}} = 9.64, \text{km/L}).
Speed Conversion: m/s to mi/hr
- Convert both meters to miles and seconds to hours through intermediate units.
- Example: (515, \text{m/s} \to \frac{515, \text{m} \times 1, \text{km}/10^3,\text{m} \times 1, \text{mi}/1.6093,\text{km} \times 60,\text{s}/1,\text{min} \times 60,\text{min}/1,\text{hr}}{1} = 1150, \text{mi/hr}).
Key Terms & Definitions
- Dimensional Analysis — A method using unit relationships to convert between different measurement units.
- Conversion Factor — A fraction representing the relationship between two units, used to change one unit into another.
- Significant Figures (sig figs) — The digits in a value that are known with certainty plus one estimated digit.
Action Items / Next Steps
- Practice setting up and solving unit conversions for length, area, volume, and rates.
- Carefully enter scientific notation values into calculators as instructed.