Dimensional Analysis Lecture Notes

Introduction to Dimensional Analysis

• Dimensional analysis is a systematic approach to solving problems.
• Key considerations:
  • Identify the problem's goal.
  • Determine starting information.
  • Develop a strategy for reaching the solution.
  • Analyze whether the solution makes sense.

Concepts and Conversion Factors

• Conversion Factors:
  • Core to dimensional analysis.
  • Example: 1 meter = 1 x 10^6 micrometers.
  • Known relationships do not affect significant figures (sig figs).

Length Conversion Examples

• 1 foot = 12 inches
• 1 inch = 2.54 centimeters
• 1 mile = 5280 feet

Mass Conversion Examples

• 1 pound = 453.59 grams
• 1 pound = 16 ounces

Volume Conversion Examples

• 1 liter = 1.0567 quarts
• 1 milliliter = 1 cubic centimeter

Temperature Conversion

• Fahrenheit to Celsius: [ °C = \frac{5}{9}(°F - 32) ]
• Celsius to Kelvin: [ K = °C + 273.15 ]

Energy Conversion

• 1 kilocalorie = 1 Calorie (food energy)
• 1 kilocalorie = 4.184 kilojoules

Pressure Conversion

• 1 mmHg = 1 torr
• 760 torr = 1 atmosphere

Sample Problem: Gold Leaf Thickness

• Problem: Find average thickness of gold sheet in nanometers.
• Given:
  • Mass = 200 milligrams (mg)
  • Dimensions: 2.4 feet x 1.0 feet
  • Density = 19.32 grams per cubic centimeter (g/cm³)

Steps to Solution

1. Convert Mass: 200 mg to grams = 0.200 grams
2. Convert Dimensions to centimeters:
   • 2.4 feet to 73.152 cm
   • 1.0 feet to 30.48 cm
3. Calculate Volume: [ \text{Volume} = \frac{\text{mass}}{\text{density}} = \frac{0.200 g}{19.32 g/cm³} = 0.01035 cm³ ]
4. Solve for Thickness:
   • [ \text{Thickness (cm)} = \frac{\text{Volume}}{\text{Length} \times \text{Width}} ]
   • Result: 4.6 x 10^-6 cm
5. Convert to Nanometers:
   • Convert cm to meters to nanometers.
   • Result: 46 nm

Participation Questions

1. Pressure Conversion: Convert given pressure units to desired units using known conversion factors.
2. Temperature Conversion: Convert 315 K to degrees Fahrenheit using both conversion formulas.