Dimensional Analysis, Unit Conversions, and Metric Prefixes — Vocabulary

Warm-up: converting 60 mph to ft/s

  • Goal: practice dimensional analysis with two-unit conversions and discuss significant figures
  • Result (per instructor’s method):
    60 mph×5280 ft1 mile×1 hour3600 s=60×52803600 fts≈88 fts.60\ \text{mph} \times \frac{5280\ \text{ft}}{1\ \text{mile}} \times \frac{1\ \text{hour}}{3600\ \text{s}} = 60 \times \frac{5280}{3600}\ \frac{\text{ft}}{\text{s}} \approx 88\ \frac{\text{ft}}{\text{s}}.
  • Significant figures discussion:
    • The instructor asked whether the decimal point after 60 affects significant figures; he stated that the decimal point signals two significant figures in this setup, so the result is given with two sig figs (88 ft/s in this context).
    • Note from the lecturer: initial numbers and how many sig figs are carried depend on how the measurement was recorded (e.g., 60 mph vs 60.0 mph) and how the problem states precision.

Key concepts (summary from lecture)

  • Dimensional analysis / unit cancellation
    • Multiply by conversion factors written as fractions so units cancel across numerator/denominator
    • The unit on the bottom of a fraction is the unit you want to cancel; the unit on the top is the one you want to end up with
    • Example: to cancel miles, put miles on the bottom; to end with feet, put feet on the top
  • Plan before calculation
    • Treat conversion like planning a trip: outline steps, then execute numbers
    • You can plan the order of conversions (distance first, time second, or vice versa); multiplication/division steps are interchangeable as long as units cancel and factors are valid
    • Write down useful conversion factors before calculating, e.g.
      1 mile=5280 ft,1 hour=60 min,1 min=60 s.1\ \text{mile} = 5280\ \text{ft}, \quad 1\ \text{hour} = 60\ \text{min}, \quad 1\ \text{min} = 60\ \text{s}.
  • Metric prefixes and base units (stepwise approach)
    • Use base units as an intermediate step: convert from a prefixed unit to its base unit, then to the target prefixed unit
    • For length prefixes: 1 pm = $10^{-12}$ m; 1 cm = $10^{-2}$ m; 1 m = 100 cm; 1 km = $10^3$ m
    • For area/volume, apply the square/cube appropriately: when converting cm^2 to m^2, square the conversion factor; for cm^3 to m^3, cube it
  • Exact numbers vs measured numbers
    • Metric conversion factors (e.g., 1 cm = $10^{-2}$ m, 1 km = $10^3$ m) are exact and do not limit significant figures
    • Measured values carry uncertainty and determine the sig figs of the final result
  • Squared and cubed units
    • Area conversions require squaring both numerical factors and units (e.g., cm^2 → m^2 → km^2)
    • Volume conversions require cubing (e.g., cm^3 → m^3 → km^3)
  • Practical notes on practice and checks
    • Write out steps to avoid errors and to catch sign/unit mistakes (especially on tests and homework)
    • Check magnitudes for reasonableness (e.g., cardboard density should be less than water, so < 1 g/cm^3)
    • Avoid rounding error by keeping extra digits during intermediate steps and rounding only at the end
    • Use reminders like a quick reference index card for common conversion factors

Worked example 1: miles per hour to feet per second (two-path planning)

  • Plan options
    • Path A: convert distance first (miles → feet) then time (hours → seconds)
    • Path B: convert time first (hours → minutes → seconds) then distance (miles → feet)
    • Result should be the same if all conversions are valid
  • Example setup (Path A example, as described):
    • Known useful factors:
      1 mile=5280 ft,1 hour=3600 s1\ \text{mile} = 5280\ \text{ft}, \quad 1\ \text{hour} = 3600\ \text{s}
    • Dimensional analysis setup written in fractions:
      60 mph×5280 ft1 mile×1 hour3600 s.60\ \text{mph} \times \frac{5280\ \text{ft}}{1\ \text{mile}} \times \frac{1\ \text{hour}}{3600\ \text{s}}.
    • Cancellation: miles cancel with miles in the first fraction; hours cancel with hours in the second fraction; result in ft/s
  • Calculation and result
    • 60×52803600=60×1.4666‾≈88.00‾ ft/s.60 \times \frac{5280}{3600} = 60 \times 1.466\overline{6} \approx 88.0\overline{0} \text{ ft/s}.
    • With appropriate significant figures (as discussed), the final answer is approximately 88 fts.88\ \frac{\text{ft}}{\text{s}}.
  • Notes from the instructor
    • If you wrote decimals after the decimal point (e.g., 60.0 mph), you might carry more sig figs; the point about significant figures depends on the given data and how precision is stated
    • In handwritten work, showing decimal points and units helps avoid sign/scale errors

Worked example 2: 1430 pm (picometers) to centimeters

  • Step 1: convert picometers to meters
    • 1 pm = $10^{-12}$ m
    • 1430 pm×10−12 m1 pm=1.430×10−9 m1430\ \text{pm} \times \frac{10^{-12}\ \text{m}}{1\ \text{pm}} = 1.430 \times 10^{-9} \ \text{m}
  • Step 2: convert meters to centimeters
    • 1 m = 100 cm
    • 1.430×10−9 m×100 cm1 m=1.430×10−7 cm1.430 \times 10^{-9} \ \text{m} \times \frac{100\ \text{cm}}{1\ \text{m}} = 1.430 \times 10^{-7} \ \text{cm}
  • Result and interpretation
    • Radius in centimeters: 1.43×10−7 cm.1.43 \times 10^{-7} \ \text{cm}.
  • Key point
    • For metric prefix conversions, stepping through base units helps reduce mistakes and aligns with the recommended plan

Worked example 3: area conversion with metric prefixes (cm^2 → km^2)

  • Problem context (hypothetical patch area): 19.5 cm^2 to km^2
  • Step 1: convert cm^2 to m^2 (base unit for length)
    • 1 cm = $10^{-2}$ m ⇒ 1 cm^2 = $(10^{-2} \text{ m})^2 = 10^{-4} \text{ m}^2$
    • 19.5 cm2=19.5×10−4 m2=1.95×10−3 m219.5\ \text{cm}^2 = 19.5 \times 10^{-4} \ \text{m}^2 = 1.95 \times 10^{-3} \ \text{m}^2
  • Step 2: convert m^2 to km^2
    • 1 km = $10^3$ m ⇒ 1 km^2 = $(10^3\text{ m})^2 = 10^6 \text{ m}^2$
    • 1.95×10−3 m2=1.95×10−3×10−6 km2=1.95×10−9 km21.95 \times 10^{-3} \ \text{m}^2 = 1.95 \times 10^{-3} \times 10^{-6} \ \text{km}^2 = 1.95 \times 10^{-9} \ \text{km}^2
  • Final answer
    • 19.5  cm2=1.95×10−9 km219.5\ \,\text{cm}^2 = 1.95 \times 10^{-9} \ \text{km}^2
  • Conceptual note
    • For area, square the conversion factors and square the units accordingly; you can apply the length conversion twice (once for each dimension) or square the entire factor

Worked example 4: density conversion (cardboard) from lb/ft^3 to g/cm^3

  • Given: density ≈ 43 lbft343\ \frac{\text{lb}}{\text{ft}^3}
  • Conversion plan (stepwise, base units approach)
    • Step 1: convert pounds to grams
      1 lb=453.592 g1\ \text{lb} = 453.592\ \text{g}
    • Step 2: convert cubic feet to cubic centimeters
    • 1 ft = 30.48 cm ⇒ 1 ft^3 = $(30.48\ \text{cm})^3 = 28\,316.846…\ \text{cm}^3$ (approx 28,317 cm^3)
    • Combine factors to get g/cm^3
  • Calculation setup
    • 43lbft3×453.592 glb×1 ft3(30.48 cm)3=0.689gcm343 \frac{\text{lb}}{\text{ft}^3} \times \frac{453.592\ \text{g}}{\text{lb}} \times \frac{1\ \text{ft}^3}{(30.48\ \text{cm})^3} = 0.689 \frac{\text{g}}{\text{cm}^3}
  • Result and interpretation
    • Density of cardboard ≈ 0.689 gcm30.689\ \frac{\text{g}}{\text{cm}^3} (rounded to ~0.69 g/cm^3 for discussion)
  • Reasonableness check
    • Cardboard density is less than water (~1 g/cm^3), so it should float; the result near 0.69 g/cm^3 aligns with qualitative expectations
  • Practical note from lecture
    • Group activity: practice converting mass density with a density given as pounds per cubic foot
    • Use conversion factors and keep track of units, including squaring for area conversions if needed later

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