Lecture on Dimensional Analysis

Key Concepts

• Dimensional analysis involves converting units from one type to another using conversion factors.
• Setting up conversion fractions allows for the desired units to remain while undesired units cancel out.

Example Problems

Problem 1: Seconds in a Year

• Objective: Convert years to seconds.
• Steps:
  1. Convert years to days: 1 year = 365 days (365.25 for average including leap years)
  2. Convert days to hours: 1 day = 24 hours
  3. Convert hours to minutes: 1 hour = 60 minutes
  4. Convert minutes to seconds: 1 minute = 60 seconds
  5. Calculate: 1 * 365 * 24 * 60 * 60 = 31,536,000 seconds

Problem 2: Speed Conversion (Miles per Hour to Meters per Second)

• Objective: Convert 45 miles per hour to meters per second.
• Steps:
  1. Convert miles to kilometers: 1 mile = 1.609 kilometers
  2. Convert kilometers to meters: 1 kilometer = 1000 meters
  3. Convert hours to minutes: 1 hour = 60 minutes
  4. Convert minutes to seconds: 1 minute = 60 seconds
  5. Calculation: (45 * 1.609 * 1000) / (60 * 60) = 20.1 meters per second

Problem 3: Density Conversion (Kilograms per Cubic Meter to Grams per Milliliter)

• Objective: Convert density of aluminum (2700 kg/m³) to g/mL.
• Steps:
  1. Convert kilograms to grams: 1 kilogram = 1000 grams
  2. Convert cubic meters to cubic centimeters: 1 m³ = 1,000,000 cm³ (since 1 m = 100 cm, cube both sides)
  3. Convert cubic centimeters to milliliters: 1 cm³ = 1 mL
  4. Calculation: (2700 * 1000) / 1,000,000 = 2.7 g/mL

Problem 4: Area Conversion (Square Inches to Square Feet)

• Objective: Find the area of a doormat in square feet given dimensions in inches.
• Steps:
  1. Calculate area in square inches: Area = Length × Width = 26 in × 18 in = 468 in²
  2. Convert square inches to square feet: 1 ft = 12 in, therefore 1 ft² = 144 in²
  3. Calculation: 468 / 144 = 3.25 ft²

Problem 5: Reading Time Calculation

• Objective: Calculate time taken to read 200 pages based on reading speed.
• Given: 15 pages in 45 minutes
• Steps:
  1. Convert pages to minutes (using given rate): 200 pages × (45 minutes / 15 pages) = 600 minutes
  2. Convert minutes to hours: 1 hour = 60 minutes
  3. Calculation: 600 minutes / 60 = 10 hours

Summary

• Dimensional analysis is a powerful tool for unit conversion by using the method of multiplying by "one" through conversion factors.
• Practice involves setting up problems such that undesired units cancel, leaving only the desired units in the result.