Sigfigs, Scientific Notation, Units

Introduction to Measurement in Physics

• Objective of Physics: Physics aims to explain the natural world, encompassing quantities that range from incredibly small to exceedingly large.
• Importance of Measurement: To understand our world, precise measurement of various quantities is essential.
• Three Elements of Proper Measurement:
  1. Precision of Measurement: Each measurement can only be reported to a certain precision, and it is crucial to communicate this by using correct significant figures.
  2. Use of Scientific Notation: To simplify writing large or small numbers that frequently arise in physics, scientists utilize scientific notation.
  3. Standard Units: It is necessary to select widely accepted units for expressing quantities.
  • Speed: meters per second ($m/s$), miles per hour ($mph$)
  • Mass: kilogram ($kg$)

Measurements and Significant Figures

• Precision in Measurement: When measuring quantities (e.g., length, weight), precision varies based on the measuring instrument.
• Measuring Tools:
  • Ruler: Can usually measure to about $\pm 1\text{ mm}$ (precision of $1\text{ mm}$).
  • Digital Calipers: Can measure to within $\pm 0.01\text{ mm}$ (precision of $0.01\text{ mm}$).
• Impact of Skill: The skill of the person measuring can also affect precision.

Conveying Measurement Precision

• Reporting Measurements Properly:
  • Example: Measuring a frog's length using a ruler with $\pm 1\text{ mm}$ precision.
  • Reported as $6.2\text{ cm}$, indicating the actual length falls between $6.15\text{ cm}$ and $6.25\text{ cm}$.
  • Reporting as $6\text{ cm}$ would imply less precision; reporting as $6.213\text{ cm}$ implies greater precision than actually known.
• Significant Figures for Clarity:
  • $6.2\text{ cm}$ has $2$ significant figures (the $6$ and the $2$).
  • Next decimal place (hundredths) is not significant since it is not reliably known.
  • A time measurement of $34.62\text{ s}$ consists of $4$ significant figures.
  • For whole numbers with trailing zeros (e.g., $200\text{ kg}$), scientific notation helps clarify ambiguity.
  • $200\text{ kg}$ is assumed to have at least $2$ significant figures unless stated otherwise.

Calculation and Significant Figures Rules

• Calculating Results: In calculations, the number of significant figures of the result cannot exceed that of the least precise measurement.
• Rules of Significant Figures:
  1. Multiplication/Division: The result should match the number of significant figures of the least precise number used.
  2. Addition/Subtraction: The answer should match the smallest number of decimal places of any measurement in the calculation.
  3. Exact Numbers: These have no uncertainty (e.g., numbers from definitions or counts) and don't affect significant figures in calculations.
  4. Trailing Zeros in Whole Numbers: Interpreted as having at least $2$ significant figures.

Introduction to SI Units

• Importance of Units: A measurement includes both magnitude and unit; for example, asking for weight must specify the unit (pounds, kilograms, etc.).
• English vs. SI Units: While the English system (inches, pounds) is commonly used in daily life, the SI units are essential in scientific contexts.
• Basic SI Units:
  • Time: Seconds ($s$)
  • Length: Meters ($m$)
  • Mass: Kilograms ($kg$)
• Derived SI Units for Motion:
  • Speed is expressed in meters per second ($m/s$), combining length and time.

Prefixes in SI Units

• Using Prefixes: Prefixes allow for easy expression of quantities that are larger or smaller than standard SI units.
• Common Prefixes:
  • nano- (n): $10^{-9}$
  • micro- (\mu): $10^{-6}$
  • milli- (m): $10^{-3}$
  • centi- (c): $10^{-2}$
  • kilo- (k): $10^{3}$
  • mega- (M): $10^{6}$
  • giga- (G): $10^{9}$

Common Conversion Equivalencies

• Importance of Conversions: Familiarity with English units is necessary for various contexts.
• Equivalencies:
  • $1\text{ inch (in)} = 2.54\text{ cm}$
  • $1\text{ foot (ft)} = 0.305\text{ m}$
  • $1\text{ mile (mi)} = 1.609\text{ km}$
  • $1\text{ mile per hour (mph)} = 0.447\text{ m/s}$
  • $1\text{ m} = 39.37\text{ in}$
  • $1\text{ km} = 0.621\text{ mi}$
  • $1\text{ m/s} = 2.24\text{ mph}$
• Conversion Factors: They represent relationships where the numerator and denominator yield a value of $1$, helping to convert between units.

Estimation in Measurement

• Need for Estimation: Sometimes a rough estimate suffices for practical purposes.
• Order-of-Magnitude Estimate: Reported using the symbol $\sim$ to indicate less precision.
• Skill in Estimating: One-significant-figure estimates help gauge rough values.
• Useful Approximate Conversion Factors:
  • $1\text{ in} \approx 2.5\text{ cm}$
  • $1\text{ m} \approx 3\text{ ft}$
  • $1\text{ mi} \approx 1.5\text{ km}$
  • $1\text{ mph} \approx 0.5\text{ m/s}$
  • $1\text{ m/s} \approx 2\text{ mph}$

Example Case: Estimating Walking Speed

• Goal: To estimate walking speed in $m/s$ using rough values based on everyday experience.
• Preparation Steps: Involve familiarizing oneself with context and the necessary calculations.