Dimensional Analysis Vocabulary

Principles of Measurement and Units

  • Every measurement consists of two essential elements: a numerical value and a unit.
  • Reporting the unit is just as important as reporting the number itself, as a number without a unit lacks physical context or meaning.
  • Standard guidelines for recording measurements require using full spellings or standard accepted abbreviations for all units.
  • Units serve an operational purpose beyond identification: analyzing units provides a direct clue as to whether multiplication or division should be performed in mathematical calculations.

Fundamentals of Dimensional Analysis

  • Dimensional analysis is a quantitative problem-solving methodology that treats units as algebraic quantities that can be multiplied, divided, and canceled.
  • Definition of a Conversion Factor:
    • A conversion factor is a fraction in which the exact same quantity is expressed one way in the numerator and another way in the denominator.
    • Example of equivalence:     1 in=2.54 cm1\,\text{in} = 2.54\,\text{cm}
    • Conversion factors derived from this equivalence:     1 in2.54 cm\frac{1\,\text{in}}{2.54\,\text{cm}}2.54 cm1 in\frac{2.54\,\text{cm}}{1\,\text{in}}
  • Algebraic behavior of units in calculations:
    • When setting up an equation, units are multiplied or divided along with their corresponding numerical values.
    • Example relationship for total wage calculations:     time worked in hours×hourly rate in dollars/hour=total wages in dollars\text{time worked in hours} \times \text{hourly rate in dollars/hour} = \text{total wages in dollars}
    • Calculation for a student working 30 h30\,\text{h} at an hourly rate of $9/h\$9/\text{h}:     30\,\text{h} \times \frac{\9}{1\,\text{h}} = \270270

Systematic Steps for Dimensional Analysis

  • Step 1: Write down the quantity given or, occasionally, a ratio to be converted.
  • Step 2: Multiply the quantity by one or more conversion factors (rates or ratios), which will change the units given to those required for the final answer.
  • Nature of Conversion Factors:
    • Conversion factors may be given directly in the context of the problem statement.
    • Conversion factors may be standard constants of known value.

Worked Examples and Sample Calculations

  • Converting Minutes to Hours:

    • Problem statement: Convert 5445 min5445\,\text{min} to hours.
    • Conversion factor relation: 60 min=1 h60\,\text{min} = 1\,\text{h}
    • Mathematical setup:     5445 min×1 h60 min=90.75 h5445\,\text{min} \times \frac{1\,\text{h}}{60\,\text{min}} = 90.75\,\text{h}

  • Calculating Required Time from Total Earnings:

    • Problem statement: Calculate the time required for a student aide to earn $378\$378 at $9\$9 per hour.
    • Mathematical setup:     \378 \times \frac{1\,\text{h}}{\9} = 42\,\text{h}

  • Metric Length Conversion (Kilometers to Meters):

    • Problem statement: How many meters are in 5.200 km5.200\,\text{km}?
    • Conversion factor relation: 1 km=1000 m1\,\text{km} = 1000\,\text{m}
    • Mathematical setup:     5.200 km×1000 m1 km=5200 m5.200\,\text{km} \times \frac{1000\,\text{m}}{1\,\text{km}} = 5200\,\text{m}

  • Metric Length Conversion (Micrometers to Meters):

    • Problem statement: How many meters are in 5.200 μm5.200\,\mu\text{m}?
    • Conversion factor relation: 1 μm=10−6 m1\,\mu\text{m} = 10^{-6}\,\text{m}
    • Mathematical setup:     5.200 μm×10−6 m1 μm=5.200×10−6 m5.200\,\mu\text{m} \times \frac{10^{-6}\,\text{m}}{1\,\mu\text{m}} = 5.200 \times 10^{-6}\,\text{m}

  • Multi-Step Time Conversion (Hours to Seconds):

    • Problem statement: Calculate the number of seconds in 5.175 h5.175\,\text{h}.
    • Conversion factor relations: 1 h=60 min1\,\text{h} = 60\,\text{min} and 1 min=60 s1\,\text{min} = 60\,\text{s}
    • Mathematical setup:     5.175 h×60 min1 h×60 s1 min=18630 s5.175\,\text{h} \times \frac{60\,\text{min}}{1\,\text{h}} \times \frac{60\,\text{s}}{1\,\text{min}} = 18630\,\text{s}

  • Multi-Step Metric to Imperial Conversion (Millimeters to Inches):

    • Problem statement: Convert 755 mm755\,\text{mm} to inches, given that there are exactly 2.54 cm2.54\,\text{cm} in 1 in1\,\text{in}.
    • Conversion factor relations: 10 mm=1 cm10\,\text{mm} = 1\,\text{cm} and 2.54 cm=1 in2.54\,\text{cm} = 1\,\text{in}
    • Mathematical setup:     755 mm×1 cm10 mm×1 in2.54 cm=29.7244 in755\,\text{mm} \times \frac{1\,\text{cm}}{10\,\text{mm}} \times \frac{1\,\text{in}}{2.54\,\text{cm}} = 29.7244\,\text{in}

  • Comparative Job Earnings Analysis:

    • Problem statement: A student is offered two summer jobs. Job A pays $500\$500 per week, whereas Job B pays $11.25\$11.25 per hour. Both jobs are 40 h40\,\text{h} per week. Which job pays more money?
    • Job A weekly earnings:     Weekly EarningsJob A=$500\text{Weekly Earnings}_{\text{Job A}} = \$500
    • Job B weekly earnings setup:     40\,\text{h} \times \frac{\11.25}{1\,\text{h}} = \450450
    • Comparison and conclusion: Job A pays $500\$500 per week while Job B pays $450\$450 per week; therefore, Job A pays more money.

  • Area Unit Conversion (Square Yards to Square Feet):

    • Problem statement: Calculate the number of square feet, ft2\text{ft}^2, that are in 12.012.0 square yards, yd2\text{yd}^2.
    • Base relation: 1 yd=3 ft1\,\text{yd} = 3\,\text{ft}
    • Squared conversion factor:     (3 ft1 yd)2=9 ft21 yd2\left(\frac{3\,\text{ft}}{1\,\text{yd}}\right)^2 = \frac{9\,\text{ft}^2}{1\,\text{yd}^2}
    • Mathematical setup:     12.0 yd2×9 ft21 yd2=108 ft212.0\,\text{yd}^2 \times \frac{9\,\text{ft}^2}{1\,\text{yd}^2} = 108\,\text{ft}^2

  • Volume Unit Conversion (Cubic Kilometers to Cubic Centimeters):

    • Problem statement: Earth's volume is 1.08×1012 km31.08 \times 10^{12}\,\text{km}^3. What is this volume in units of cubic centimeters?
    • Base relation: 1 km=103 m=105 cm1\,\text{km} = 10^3\,\text{m} = 10^5\,\text{cm}
    • Cubed conversion factor:     (105 cm1 km)3=1015 cm31 km3\left(\frac{10^5\,\text{cm}}{1\,\text{km}}\right)^3 = \frac{10^{15}\,\text{cm}^3}{1\,\text{km}^3}
    • Mathematical setup:     1.08×1012 km3×1015 cm31 km3=1.08×1027 cm31.08 \times 10^{12}\,\text{km}^3 \times \frac{10^{15}\,\text{cm}^3}{1\,\text{km}^3} = 1.08 \times 10^{27}\,\text{cm}^3

Key Concepts and Section Review

  • Unit Cancellation Rules:
    • In dimensional analysis, units may be canceled like variables in algebra.
    • Placing the units intentionally so that unwanted units cancel out to yield the desired units ensures correct mathematical calculation setup.
  • Classification of Conversion Factors:
    • Constant conversion factors: Fixed standard equivalences, such as the number of cents in a dollar or centimeters in an inch.
    • Variable conversion factors: Situational rates that change depending on context, such as the number of miles traveled by a car per hour or an hourly pay rate; these must be provided in the problem statement.