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Lecture 10: Dimensional Analysis and Prefix Review

Prefixes - Review of common metric prefixes and their powers of 10

  • tera (T): 10^{12}
  • giga (G): 10^{9}
  • mega (M): 10^{6}
  • kilo (k): 10^{3}
  • hecto (h): 10^{2}
  • deka (da): 10^{1}
  • base unit: 10^{0} = 1
  • deci (d): 10^{-1}
  • centi (c): 10^{-2}
  • milli (m): 10^{-3}
  • micro (µ): 10^{-6}
  • nano (n): 10^{-9}
  • pico (p): 10^{-12}
  • femto (f): 10^{-15}
    • Note: Prefixes for other powers exist, but the lecture emphasizes the common ones listed above.
  • Memorization tips mentioned in the lecture:
    • Memorize at least the following scale factors:
    • k = 10^{3}, d = 10^{-1}, c = 10^{-2}, m = 10^{-3}, n = 10^{-9}
    • If a prefix is unfamiliar, refer to a prefix table (e.g., Table 1.2 referenced in the lecture).
  • Quick relationships to know:
    • 1000 = 10^{3} and 0.001 = 10^{-3}, etc.
    • Prefixes encode the factor by which you scale the base unit when converting between units.
  • Practical implication:
    • When you see a statement like 1\text{ km}, you should remember it is 1000\text{ m}, since 1\text{ km} = 10^{3}\text{ m}.
  • Example checks:
    • 1\text{ cm} = 10^{-2}\text{ m} and 1\text{ m} = 100\text{ cm}; 1\text{ mm} = 10^{-3}\text{ m} and 1\text{ m} = 1000\text{ mm}.

Dimensional Analysis: Base Rule and Notation

  • Key idea: Convert all quantities to a base or common unit, then convert to the desired unit.
  • When converting 5.12\text{ nm} to meters:
    • nm corresponds to 10^{-9}\text{ m}, so 5.12\text{ nm} = 5.12 \times 10^{-9}\text{ m}.
  • Placement of factors when writing conversions:
    • The factor corresponding to the prefix goes in front of the unit that has the prefix.
    • The scaling factor (e.g., 10^{-9}) goes in front of the base unit.
  • Conceptual reminder:
    • Dimensional analysis is about dimensional consistency and unit tracking, not about changing the numerical value arbitrarily.

Two-Step Conversions: A Worked Example

  • Problem: Convert 50\text{ ng} to femtograms (fg).
  • Step 1: Convert to the base unit (grams): 50\text{ ng} \to \text{g}
    • 50\text{ ng} \times \frac{10^{-9}\text{ g}}{1\text{ ng}} = 5.0 \times 10^{-8}\text{ g}
  • Step 2: Convert from base unit to final prefix (fg): 1\text{ g} \to 10^{15}\text{ fg}
    • 5.0 \times 10^{-8}\text{ g} \times \frac{10^{15}\text{ fg}}{1\text{ g}} = 5.0 \times 10^{7}\text{ fg}
  • Answer:
    • 5.0 \times 10^{7}\text{ fg}
  • Concept: To change prefixes, use two steps: base-unit conversion, then prefix conversion.
  • Note on femto (f): Common prefixes to memorize include k, d, c, m, n; refer to a prefix table for unfamiliar prefixes.

Example: Which Quantity is Larger?

  • Given: 7.11 \times 10^{1}\text{ km} and 5.88 \times 10^{6}\text{ cm}
  • Convert both to the same base unit (meters):
    • 7.11 \times 10^{1}\text{ km} = 7.11 \times 10^{4}\text{ m}
    • 5.88 \times 10^{6}\text{ cm} = 5.88 \times 10^{4}\text{ m}
  • Comparison: 7.11 \times 10^{4}\text{ m} > 5.88 \times 10^{4}\text{ m}, so the 7.11 \times 10^{1}\text{ km} quantity is larger.

Practice: Which Quantity is Larger?

  • Example 1: 6.1 \times 10^{2}\ \mu\text{s} vs 92\text{ ms}
    • Convert to seconds:
    • 6.1 \times 10^{2}\ \mu\text{s} = 6.1 \times 10^{2} \times 10^{-6}\text{ s} = 6.1 \times 10^{-4}\text{ s}
    • 92\text{ ms} = 9.2 \times 10^{-2}\text{ s}
    • Comparison: 9.2 \times 10^{-2}\text{ s} > 6.1 \times 10^{-4}\text{ s}; therefore, the ms quantity is larger.
  • Final takeaway: Memorize the scaling factors (k, d, c, m, n) and use base-unit conversions to compare.

Your Turn Problems (Interpretation and Solutions)

  • Problem 1: Compare 874,338\text{ g} with 6.40 \times 10^{8}\text{ mg}
    • Convert the second quantity to grams:
    • 6.40 \times 10^{8}\text{ mg} \times \frac{1\text{ g}}{10^{3}\text{ mg}} = 6.40 \times 10^{5}\text{ g}
    • Compare: 874,338\text{ g} = 8.74338 \times 10^{5}\text{ g} vs 6.40 \times 10^{5}\text{ g}
    • Result: 8.74338 \times 10^{5}\text{ g} > 6.40 \times 10^{5}\text{ g}, so 874,338\text{ g} is larger.

Practice: Basic Prefix and Unit Relationships

  • Practice: 1\text{ cm} = 10^{-2}\text{ m}; 1\text{ m} = ?\text{ cm}
    • Therefore: 1\text{ m} = 100\text{ cm} and 1\text{ cm} = 10^{-2}\text{ m}
  • Practice: 1\text{ mm} = 10^{-3}\text{ m}; 1\text{ m} = ?\text{ mm}
    • Therefore: 1\text{ m} = 1000\text{ mm} and 1\text{ mm} = 10^{-3}\text{ m}
  • Practical note: As a beginner, memorize the scaling factors for each prefix (e.g., 1\text{ m} = 100\text{ cm}, 1\text{ g} = 1000\text{ mg}, etc.) because they are derived from the prefix factors.

Derived Units: Length, Area, and Volume Conversions

  • 1\ \mu\text{m} to meters:
    • 1\ \mu\text{m} = 10^{-6}\text{ m}
  • Squared units (area):
    • (1\ \mu\text{m})^2 = (10^{-6}\text{ m})^2 = 10^{-12}\text{ m}^2
    • Example: 1.5\ \mu\text{m}^2 = 1.5 \times 10^{-12}\text{ m}^2
  • Lengths to volumes (cubic units):
    • 1\text{ km} = 10^{3}\text{ m}
    • (1\text{ km})^3 = (10^{3}\text{ m})^3 = 10^{9}\text{ m}^3
    • Example: 1.4 \times 10^{9}\text{ km}^3 = 1.4 \times 10^{9} \times 10^{9}\text{ m}^3 = 1.4 \times 10^{18}\text{ m}^3
  • Special note on mm^3 to m^3 conversion:
    • 1\text{ mm} = 10^{-3}\text{ m}, so 1\text{ mm}^3 = (10^{-3}\text{ m})^3 = 10^{-9}\text{ m}^3
    • Therefore, to convert any mm^3 quantity to m^3, multiply by 10^{-9}.
    • Example rule: If you have X\text{ mm}^3, then in m^3 you have X \times 10^{-9}\text{ m}^3.
  • Note on consistent units:
    • When combining numbers with units, ensure all terms share a common unit before performing arithmetic.

Speed and Time: From Hours and Miles to Kilometers per Second

  • Example: Convert 70.0\text{ miles/hour} to km/s.
    • Use exact conversions: 1\text{ hour} = 3600\text{ s}, 1\text{ mile} = 1.609\text{ km}.
    • Calculation:
    • 70.0\text{ miles/hour} \times \frac{1.609\text{ km}}{1\text{ mile}} \times \frac{1\text{ hour}}{3600\text{ s}} = \frac{70.0 \times 1.609}{3600}\text{ km/s}
    • Numerically: \approx 0.0313\text{ km/s}
  • Important note on exact numbers and sig figs:
    • 3600 and 1.609 are treated as exact numbers in this calculation, so the only limiting factor for sig figs comes from the given value 70.0 (3 significant figures). Therefore, the final answer should be reported with 3 significant figures: 0.0313\text{ km/s}.
  • Practical takeaway:
    • In unit conversion problems, identify exact numbers from the problem statement (often listed for conversion constants) and apply appropriate fractional multipliers to move from one unit to another.

Quick Reference: Common Conversions to Remember

  • Length:
    • 1\text{ m} = 100\text{ cm}, 1\text{ cm} = 10^{-2}\text{ m}
    • 1\text{ m} = 1000\text{ mm}, 1\text{ mm} = 10^{-3}\text{ m}
  • Volume and area:
    • 1\text{ mm}^3 = 10^{-9}\text{ m}^3
    • 1\text{ km} = 10^{3}\text{ m}, 1\text{ km}^3 = 10^{9}\text{ m}^3
  • Mass and prefixes:
    • Base examples: 1\text{ g} = 1000\text{ mg}, 1\text{ ng} = 10^{-9}\text{ g}
  • Derived-unit practice:
    • When squaring or cubing a unit, apply the exponent to the unit as well, e.g.,
    • (1\ \mu\text{m})^2 = 10^{-12}\text{ m}^2