Chapter 1 Unit Conversions

Review of Order of Operations and Significant Figures

• PEMDAS Sequence: Rounding and tracking significant figures follow the standard order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction (PEMDAS).

• Abbott and Costello Reference: Classical comedy skits, such as Abbott and Costello routines, frequently poke fun at the humorous misunderstandings caused by neglecting the strict order of operations.

• Exact Numbers in Formulas: Constants and integers appearing in mathematical equations (such as the fraction $\frac{1}{2}$ in kinematic equations) are treated as exact numbers. They possess infinite precision (an infinite number of significant figures), meaning they never limit the number of significant figures in a calculated result.

• Addition and Subtraction Rules: In multi-step calculations, even if every input value contains three significant figures, the final result may round to two significant figures due to the decimal-place limiting rule of addition and subtraction.

Fundamentals of the International System of Units (SI)

• Core Unit Conversion Rule: Every unit conversion is executed simply by multiplying a given physical quantity by a factor of $1$ (unity).

• Origin of the SI System: The abbreviation SI stands for Système International (French). It was established when France and Britain were global maritime powers to standardize international trade and commerce across borders.

• MKS Base Units: Classical mechanics primarily utilizes the MKS system, which defined three fundamental base units:

  • Length: Meter ($\text{m}$, lowercase m)

  • Time: Second ($\text{s}$, lowercase s)

  • Mass: Kilogram ($\text{kg}$)

Standard Definitions of Base Physical Quantities

• Definition of Length:

  • Historical Standard: The meter was originally defined as a fixed fraction of the distance from the equator to the North Pole along a meridian passing through Paris (derived from Earth's circumference divided by a specific large integer).

  • Limitations of Earth-Based Standard: Earth's surface consists predominantly of water, and its dynamic, irregular geoid shape limits precision measurement.

  • Modern Standard: Length is now defined relative to the speed of light in a vacuum ($c$).

  • Exact Speed of Light: Light travels at exactly $299,792,458\,\text{m/s}$.

  • Modern Meter Definition: One meter is defined as the exact distance traveled by light in a vacuum during a time interval of $\frac{1}{299,792,458}$ of a second.

  • Visual Scale: A standard $60\,\text{cm}$ ($600\,\text{mm}$) rod is roughly $60\%$ of a meter. If nearly $300,000,000$ meter bars were connected end-to-end, light would traverse the entire length in exactly one second.

  • Abstract Concept of Distance: Distance is defined as the shortest spatial path between two points. Historically, distance was measured using physical reference objects (e.g., counting how many pen lengths span two spatial points).

  • Regional Units: Regional systems created unique spatial units, such as the rai in Thailand for measuring land area, compared to the acre used in Western nations.

• Definition of Time:

  • Conceptual Accounting: Time is a physical accounting system created to order sequential observations on Earth.

  • Solar Cycle (Day): Based on the apparent passage of the Sun rising above the horizon, reaching zenith, dipping below the horizon, and returning.

  • Heliocentric / Seasonal Cycle (Year): In ancient times prior to light pollution, celestial star patterns (constellations forming the zodiac symbols) were tracked to build solar calendars based on Earth's orbital position around the Sun.

  • Lunar Cycle (Month): Based on the Moon's regular phase cycle from new moon to full moon, lasting approximately $28\,\text{days}$.

  • Societal Purpose: Calendars were essential for agrarian and hunter-gatherer societies to optimize crop planting schedules around rain cycles and track seasonal migrations of game animals (e.g., wildebeests).

  • Leap Year Corrections: A true solar year equals approximately $365.24\,\text{days}$. Counting exactly $365\,\text{days}$ per year creates an error of $\approx 0.24\,\text{days/year}$ ($\approx \frac{1}{4}\,\text{day/year}$). Over $100\,\text{years}$, this creates a $25\,\text{day}$ shift (nearly a full month error), disrupting agriculture.

  • Julian Calendar Fix: Julius Caesar introduced the leap year system (giving rise to the month of July, while Augustus gave rise to August), adding one extra day to February every $4\,\text{years}$ to correct for $\approx 0.25\,\text{days/year}$.

  • Modern Atomic Time: Precision navigation and spacecraft tracking (e.g., deep space probes launched to Pluto) require microsecond timing. Modern time is defined by Cesium atomic clocks, measuring the characteristic frequency oscillations of electromagnetic transitions in cesium atoms.

• Definition of Mass:

  • Standard Artifact: Mass was historically defined by an absolute reference object: a physical cylinder made of platinum, stored inside a triple vacuum vault at the International Bureau of Weights and Measures in Paris, France. The mass of this precise cylinder defined exactly $1\,\text{kg}$.

• Comparison of Measurement Systems:

  • MKS System: Meter, Kilogram, Second.

  • CGS System: Centimeter, Gram, Second.

  • British Engineering System / US Customary: Uses feet and inches for length, pounds ($\text{lb}$) for force/weight, and seconds for time. The United States continues using this imperial standard primarily because converting the nation's vast industrial manufacturing machinery to the metric system would be enormously expensive.

Derived Physical Quantities and Examples

• Derived Quantities: Physical quantities formed by multiplying or dividing fundamental base quantities.

• Speed:

  • Formula: $\text{Speed} = \frac{\text{Length}}{\text{Time}} = \frac{\text{Distance}}{\text{Time}}$

  • SI Unit: Meters per second ($\text{m/s}$)

• Volume:

  • Formula: $\text{Volume} = \text{Length}^3 = \text{Length} \times \text{Length} \times \text{Length}$

  • SI Unit: Cubic meters ($\text{m}^3$)

Scales, Orders of Magnitude, and Physical Intuition

Developing physical intuition requires visualizing structural scales across powers of ten ($10^n$):

• Length Orders of Magnitude:

  • Quark: $\sim 10^{-18}\,\text{m}$ (fundamental building block forming protons and neutrons; up and down quarks).

  • Proton / Neutron Diameter: $\sim 10^{-15}\,\text{m}$ (composed of three quarks; a million times larger than a quark).

  • Virus: $\sim 10^{-7}\,\text{m}$ (microscopic particle compared to a living cell; functions like a dust particle on a cell surface).

  • Sheet of Paper Thickness: $\sim 10^{-4}\,\text{m}$. Stacking $1,000$ viruses ($10^3 \times 10^{-7}\,\text{m} = 10^{-4}\,\text{m}$) equals the thickness of one piece of paper.

  • Soccer / Football Field: $\sim 10^2\,\text{m}$ ($100\,\text{m}$).

  • Mount Everest Height: $\sim 10^4\,\text{m}$ ($10,000\,\text{m}$). Equivalent to stacking $100$ football fields end-to-end.

  • Earth Diameter: $\sim 10^7\,\text{m}$.

  • Astronomical Unit ($1\,\text{AU}$): Mean Earth-to-Sun distance $\approx 10^{11}\,\text{m}$. Equivalent to placing $10,000$ Earths side by side ($10^4 \times 10^7\,\text{m} = 10^{11}\,\text{m}$).

  • Distance to Nearest Star: $\sim 10^{16}\,\text{m}$ ($\approx 10,000\,\text{AU}$).

  • Distance to Nearest Galaxy: $\sim 10^{22}\,\text{m}$ ($\approx 1,000,000$ times the distance to the nearest star).

• Mass Orders of Magnitude:

  • Electron: $\sim 10^{-30}\,\text{kg}$ ($9.11 \times 10^{-31}\,\text{kg}$).

  • Proton / Neutron: $\sim 10^{-27}\,\text{kg}$ ($1.67 \times 10^{-27}\,\text{kg}$; $\approx 1,000$ times more massive than an electron).

  • DNA Molecule: $\sim 10^{-17}\,\text{kg}$ (contains $\sim 10^{10}$ protons/neutrons).

  • Bacterium: $\sim 10^{-15}\,\text{kg}$ ($1,000$ times more massive than DNA).

  • Mosquito: $\sim 10^{-5}\,\text{kg}$ ($10^{10}$ times more massive than a bacterium).

  • Cargo Ship: $\sim 10^8\,\text{kg}$.

  • Earth: $\sim 10^{24}\,\text{kg}$ ($10^{16}$ times more massive than a giant ship; dominates regional gravitational attraction).

  • Sun: $\sim 10^{30}\,\text{kg}$ ($\approx 1,000,000$ times more massive than Earth).

  • Milky Way Galaxy: $\sim 10^{41}\,\text{kg}$.

Order of Magnitude Estimations

Order of magnitude estimations rely on basic scaling assumptions to approximate complex astronomical properties:

• Estimating the Number of Stars in a Galaxy:

  • Assumption: The galaxy is primarily composed of stars, and the Sun ($M_{\text{Sun}} \sim 10^{30}\,\text{kg}$) represents an average stellar mass.

  • Calculation:     $\text{Number of Stars} \approx \frac{M_{\text{galaxy}}}{M_{\text{Sun}}} = \frac{10^{41}\,\text{kg}}{10^{30}\,\text{kg}} = 10^{11}\text{ stars}$

  • Result: $\sim 10^{10}$ to $10^{11}$ stars ($10\text{ billion}$ to $100\text{ billion}$ stars).

• Estimating the Mass of the Observable Universe:

  • Hubble Space Telescope deep-field surveys resolve $\approx 2\text{ trillion}$ galaxies ($2 \times 10^{12}\text{ galaxies}$).

  • Calculation assuming an average galaxy mass of $10^{41}\,\text{kg}$:     $M_{\text{universe}} \approx (2 \times 10^{12}) \times 10^{41}\,\text{kg} = 2 \times 10^{53}\,\text{kg} \sim 10^{53}\,\text{kg}$

• Estimating Total Nucleons (Protons/Neutrons) in the Universe:

  • Assumption: Stellar mass is composed primarily of hydrogen matter ($m_p \sim 10^{-27}\,\text{kg}$).

  • Calculation:     $\text{Total Nucleons} \approx \frac{M_{\text{universe}}}{m_p} = \frac{2 \times 10^{53}\,\text{kg}}{10^{-27}\,\text{kg}} = 2 \times 10^{80} \sim 10^{80}\text{ protons and neutrons}$

SI Unit Prefixes and Base Quantities

• Common Metric Prefixes:

  • Yotta ($Y$): $10^{24}$

  • Peta ($P$): $10^{15}$

  • Tera ($T$): $10^{12}$

  • Giga ($G$): $10^9$ (e.g., gigabyte = $10^9\text{ bytes}$)

  • Mega ($M$): $10^6$ (e.g., megahertz = $10^6\text{ Hz}$)

  • Kilo ($k$): $10^3$ ($1,000$)

  • Centi ($c$): $10^{-2}$ ($\frac{1}{100}$)

  • Milli ($m$): $10^{-3}$ ($\frac{1}{1,000}$)

  • Nano ($n$): $10^{-9}$ (nanotechnology operates at the $10^{-9}\,\text{m}$ scale)

  • Yocto ($y$): $10^{-24}$

  • Googol: Defined as $10^{100}$ ($1$ followed by $100$ zeros).

  • Global digital data storage currently exists on the petabyte scale ($10^{15}\text{ bytes}$) and is rapidly approaching the exabyte scale ($10^{18}\text{ bytes}$).

• The Seven Fundamental SI Base Quantities:

  1. Length: Meter ($\text{m}$)

  2. Time: Second ($\text{s}$)

  3. Mass: Kilogram ($\text{kg}$)

  4. Electric Current: Ampere ($\text{A}$) — studied in Physics 46

  5. Temperature: Kelvin ($\text{K}$) — studied in Physics 45

  6. Amount of Substance: Mole ($\text{mol}$) — studied in Chemistry

  7. Luminous Intensity: Candela ($\text{cd}$) — studied in Astronomy

Unit Conversions and the Method of Unity

• Mathematical Basis: Converting units requires multiplying a quantity by a conversion factor equal to $1$ (unity).

• Forming Unity Fractions: Because $1\,\text{kg} = 1,000\,\text{g}$, the ratios $\frac{1\,\text{kg}}{1,000\,\text{g}} = 1$ and $\frac{1,000\,\text{g}}{1\,\text{kg}} = 1$ are both mathematically equal to unity.

• Canceling Units: Select the ratio that places the unwanted starting unit in the opposite position (numerator vs. denominator) so that the unit algebraic labels cancel out.

Step-by-Step Unit Conversion Examples

• Example 1: Convert $100\,\text{g}$ to Kilograms:

  • Equivalence: $1\,\text{kg} = 1,000\,\text{g}$.

  • Setup:     $100\,\text{g} \times \left(\frac{1\,\text{kg}}{1,000\,\text{g}}\right) = 0.1\,\text{kg}$

  • Exam Rule: All worked problems must explicitly show the factor-label fraction setup.

• Example 2: Calculate Seconds in One Millennium (to 3 Significant Figures):

  • Conversion Chain:

  • $1\text{ millennium} = 1,000\text{ years} = 10^3\text{ years}$

  • $1\text{ year} = 365\text{ days}$

  • $1\text{ day} = 24\text{ hours}$

  • $1\text{ hour} = 3,600\text{ seconds}$ ($60\text{ min} \times 60\text{ s}$)

  • Full Setup:     $\text{Seconds} = 10^3\text{ years} \times \left(\frac{365\text{ days}}{1\text{ year}}\right) \times \left(\frac{24\text{ hours}}{1\text{ day}}\right) \times \left(\frac{3,600\text{ seconds}}{1\text{ hour}}\right)$

  • Calculation:     $\text{Seconds} = 10^3 \times 365 \times 24 \times 3,600\,\text{s} = 31,536,000,000\,\text{s} = 3.15 \times 10^{10}\,\text{s}$     (Rounds to $3.14 \times 10^{10}\,\text{s}$ depending on exact sig-fig rounding variations).

• Example 3: Converting Volumetric Units ($1\,\text{km}^3$ to $\text{m}^3$):

  • Common Pitfall: Students frequently multiply by $10^3$ once rather than cubing the entire dimensional conversion factor.

  • Dimensional Logic: A cubic volume consists of three orthogonal linear dimensions ($1\,\text{km}^3 = 1\,\text{km} \times 1\,\text{km} \times 1\,\text{km}$).

  • Setup:     $1\,\text{km}^3 \times \left(\frac{10^3\,\text{m}}{1\,\text{km}}\right)^3 = 1\,\text{km}^3 \times \frac{(10^3)^3\,\text{m}^3}{1^3\,\text{km}^3} = 10^9\,\text{m}^3$

  • Algebraic Property: The exponent $3$ distributes to every numerical term and unit symbol inside the parentheses:     $\left(\frac{10^3\,\text{m}}{1\,\text{km}}\right)^3 = \frac{10^9\,\text{m}^3}{1\,\text{km}^3}$

Questions & Discussion

• Question: Where did the original definition of a meter come from?

  • Answer: The original definition came from measuring Earth's circumference (divided by a specific fixed number) as navigators sailed around the planet. However, because Earth is primarily water and hard to measure with high precision, modern physics redefined the meter using the precise speed of light ($299,792,458\,\text{m/s}$).

• Question: What does MKS stand for?

  • Answer: Meter, Kilogram, Second.

• Question: What does CGS stand for?

  • Answer: Centimeter, Gram, Second. Both MKS and CGS are metric systems, whereas the American system relies on old British imperial engineering units (feet, inches, pounds).

• Question: How can the number of stars in the galaxy be estimated using order-of-magnitude mass values?

  • Answer: By dividing the estimated mass of the galaxy ($10^{41}\,\text{kg}$) by the mass of an average star like the Sun ($10^{30}\,\text{kg}$). Subtracting exponents ($41 - 30 = 11$) yields $\sim 10^{11}$ stars ($10\text{ to }100\text{ billion}$ stars).

• Question: Why must the conversion factor be applied three times when converting cubic kilometers ($\text{km}^3$) to cubic meters ($\text{m}^3$)?

  • Answer: Because a cubic kilometer represents three dimensions ($1\,\text{km} \times 1\,\text{km} \times 1\,\text{km}$). Multiplying by unity three times ($\frac{10^3\,\text{m}}{1\,\text{km}} \times \frac{10^3\,\text{m}}{1\,\text{km}} \times \frac{10^3\,\text{m}}{1\,\text{km}}$) cancels all three kilometer units in the denominator, yielding $10^9\,\text{m}^3$.