Dimensional Analysis Problems

Problem 1: Converting Years to Seconds

• Objective: Convert 1 year to seconds.
• Steps:
  • Convert years to days: 365 days/year.
  • Convert days to hours: 24 hours/day.
  • Convert hours to minutes: 60 minutes/hour.
  • Convert minutes to seconds: 60 seconds/minute.
• Calculation:
  • Multiply all conversion factors: (1 \times 365 \times 24 \times 60 \times 60 = 31,536,000 \text{ seconds})
  • Alternative for 365.25 days/year: 31,557,600 seconds.

Problem 2: Converting Miles Per Hour to Meters Per Second

• Objective: Convert 45 miles per hour to meters per second.
• Steps:
  • Convert miles to kilometers: 1 mile = 1.609 kilometers.
  • Convert kilometers to meters: 1 kilometer = 1000 meters.
  • Convert hours to minutes and then seconds: 1 hour = 60 minutes, 1 minute = 60 seconds.
• Calculation:
  • Use conversion factors and calculate: (45 \times 1.609 \times 1000 \div 60 \div 60 = 20.1 \text{ meters/second})

Problem 3: Converting Density of Aluminum

• Objective: Convert 2700 kilograms per cubic meter to grams per milliliter.
• Steps:
  • Convert kilograms to grams: 1 kilogram = 1000 grams.
  • Convert cubic meters to cubic centimeters: 1 cubic meter = 1,000,000 cubic centimeters ((1\times10^6)).
  • Convert cubic centimeters to milliliters: 1 cubic centimeter = 1 milliliter.
• Calculation:
  • Multiply and divide to convert: ((2700 \times 1000) \div 1,000,000 = 2.7 \text{ grams/milliliter})
• Note: Common conversion in physics and chemistry.

Problem 4: Area of a Doormat

• Objective: Calculate area in square inches and square feet.
• Given: Length = 26 inches, Width = 18 inches.
• Steps:
  • Calculate area in square inches: (26 \times 18 = 468 \text{ square inches}).
  • Convert square inches to square feet: 12 inches = 1 foot.
  • Use conversion factor for square units: (468 \div 12^2 = 3.25 \text{ square feet}).

Problem 5: Reading Time for a Book

• Objective: Determine how many hours to read 200 pages.
• Given: John reads 15 pages in 45 minutes.
• Steps:
  • Convert pages to minutes: In 45 minutes, John reads 15 pages.
  • Convert minutes to hours: 60 minutes = 1 hour.
• Calculation:
  • Use conversion to find time: ((200 \times 45) \div 15 \div 60 = 10 \text{ hours}).

These problems illustrate how to use dimensional analysis to convert between different units efficiently. Each step involves setting up conversion factors such that units cancel out until only the desired unit remains.