Unit Conversion and Dimensional Analysis

• Unit conversion is the process of converting measurements from one unit to another
• This is done through the use of conversion factors

Conversion Factors

• A conversion factor is a ratio that expresses how many of one unit are equal to another unit
• Conversion factors are used to change the units of a measurement without changing the value

Dimensional Analysis

• Dimensional analysis (also called factor-label method or unit-factor method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value
• It is a technique that involves using the units of measurement to guide calculations
• In unit conversion, dimensional analysis involves multiplying the original measurement by a conversion factor
• The units are treated as algebraic quantities that can be cancelled

Steps for Unit Conversion using Dimensional Analysis

• Identify the original units and the desired units
• Find a conversion factor that relates the original and desired units
• Set up the conversion factor so that the original units cancel out and the desired units remain
• Multiply the original measurement by the conversion factor
• Simplify the expression and express the answer in the desired units

Single-Step Conversions

• Single-step conversions involve converting from one unit to another using a single conversion factor
• Example: Convert 5 meters to feet (1 meter = 3.28084 feet)
  • 5 meters * (3.28084 feet / 1 meter) = 16.4042 feet

Multi-Step Conversions

• Multi-step conversions involve converting from one unit to another using multiple conversion factors
• Example: Convert 10 miles per hour to meters per second (1 mile = 1609.34 meters, 1 hour = 3600 seconds)
  • 10 miles/hour * (1609.34 meters / 1 mile) * (1 hour / 3600 seconds) = 4.4704 meters/second

Common Unit Conversions

• Length:
  • 1 inch = 2.54 centimeters
  • 1 foot = 12 inches
  • 1 yard = 3 feet
  • 1 mile = 5280 feet
  • 1 meter = 100 centimeters
  • 1 kilometer = 1000 meters
• Mass:
  • 1 pound = 16 ounces
  • 1 kilogram = 1000 grams
  • 1 kilogram = 2.20462 pounds
  • 1 metric ton = 1000 kilograms
• Volume:
  • 1 gallon = 4 quarts
  • 1 quart = 2 pints
  • 1 pint = 2 cups
  • 1 liter = 1000 milliliters
  • 1 liter = 1.05669 quarts
• Time:
  • 1 minute = 60 seconds
  • 1 hour = 60 minutes
  • 1 day = 24 hours
  • 1 year = 365.25 days
• Temperature:
  • Celsius to Fahrenheit: °F = (°C * 9/5) + 32
  • Fahrenheit to Celsius: °C = (°F - 32) * 5/9
  • Celsius to Kelvin: K = °C + 273.15

SI Base Units

• The International System of Units (SI) defines seven base units
• Meter (m): unit of length
• Kilogram (kg): unit of mass
• Second (s): unit of time
• Ampere (A): unit of electric current
• Kelvin (K): unit of thermodynamic temperature
• Mole (mol): unit of amount of substance
• Candela (cd): unit of luminous intensity

Derived Units

• Derived units are formed by combining base units according to the algebraic relationships linking the corresponding quantities
• Examples:
  • Area: square meter (m²)
  • Volume: cubic meter (m³)
  • Speed: meter per second (m/s)
  • Acceleration: meter per second squared (m/s²)
  • Force: Newton (N = kg*m/s²)
  • Energy: Joule (J = N*m)
  • Power: Watt (W = J/s)

Prefixes

• SI prefixes are used to denote decimal multiples and submultiples of SI units
• Examples:
  • kilo (k): 10^3
  • mega (M): 10^6
  • giga (G): 10^9
  • milli (m): 10^-3
  • micro (µ): 10^-6
  • nano (n): 10^-9

Area Conversions

• Area is a two-dimensional quantity, so when converting units of area, the conversion factor must be squared
• Example: Convert 1 square meter to square feet (1 meter = 3.28084 feet)
  • 1 m² * (3.28084 ft / 1 m)² = 1 m² * (10.7639 ft² / 1 m²) = 10.7639 ft²

Volume Conversions

• Volume is a three-dimensional quantity, so when converting units of volume, the conversion factor must be cubed
• Example: Convert 1 cubic meter to cubic feet (1 meter = 3.28084 feet)
  • 1 m³ * (3.28084 ft / 1 m)³ = 1 m³ * (35.3147 ft³ / 1 m³) = 35.3147 ft³

Temperature Conversions

• Converting between Celsius and Fahrenheit requires different formulas than other unit conversions because the scales have different zero points
• Celsius to Fahrenheit: °F = (°C * 9/5) + 32
• Example: Convert 25°C to Fahrenheit
  • °F = (25 * 9/5) + 32 = 45 + 32 = 77°F
• Fahrenheit to Celsius: °C = (°F - 32) * 5/9
• Example: Convert 68°F to Celsius
  • °C = (68 - 32) * 5/9 = 36 * 5/9 = 20°C
• Celsius to Kelvin: K = °C + 273.15
• Example: Convert 25°C to Kelvin
  • K = 25 + 273.15 = 298.15 K

Estimation

• Estimation can be used to approximate unit conversions when exact conversion factors are not available or when an approximate answer is sufficient
• Round the original measurement and the conversion factor to make the calculation easier

Significant Figures in Unit Conversions

• When performing unit conversions, the number of significant figures in the final answer should be the same as the number of significant figures in the original measurement
• Conversion factors are usually exact numbers and do not limit the number of significant figures in the final answer