Unit Summary: Measurements, Scientific Notation, Uncertainty, and Graphs

Units and Measurements

• SI base units: time = s, length = m, mass = kg, temperature = K
• Base unit: defined in terms of a physical event/object; independent of other units; prefixes add scale.
• Prefixes: used to describe a wide range of measurements; e.g., kilo-, milli-, micro- (scale factors).
• Volume and density via derived concepts: volume and density involve combinations of base units.
• Liter and cubic relationships: 1\ \text{L} = 1\ \text{dm}^3; density \rho = \frac{m}{V}; mass can be found from density and volume: m = \rho V.

Base Units and SI Prefixes

• Time base unit: \text{second} = s, based on cesium-133 frequency.
• Length base unit: \text{meter} = m, defined by light travel in vacuum in a specific time.
• Mass base unit: \text{kilogram} = kg, ~2.2 lb.
• Temperature base unit: \text{kelvin} = K; absolute zero: 0\ K\text{ (no kinetic energy)}.
• Other scales: Celsius and Fahrenheit exist for everyday temperature comparisons.
• SI prefixes scale base units (e.g., m, g, L) by powers of 10.

Derived Units

• Not all quantities use base units alone; derived units are combinations of base units.
• Examples: speed = \dfrac{m}{s}, volume = m^3 (or cm^3), density = \dfrac{g}{cm^3}$

Density and Volume: Quick Practice

• Volume from measurements: if final volume is Vf and initial is Vi, then volume of sample: V{sample} = Vf - V_i
• Example: sample in graduated cylinder where initial volume = 10.5\ \text{mL} and final volume = 13.5\ \text{mL}:
  V_{sample} = 13.5\ \text{mL} - 10.5\ \text{mL} = 3.0\ \text{mL}
• Mass from density: m = \rho V; density example: \rho = 2.7\ \text{g/mL}, for V=3.0\ \text{mL}:
  m = 2.7\ \text{g/mL} \times 3.0\ \text{mL} = 8.1\ \text{g}

Example Problem Summary (Density and Mass Check)

• Density equation: \rho = \dfrac{m}{V} \Rightarrow m = \rho V
• Verify by plugging calculated mass back into density equation:
  • If mass and volume yield the given density, the mass is consistent.

Scientific Notation and Dimensional Analysis

• Purpose: express very large or small numbers as a coefficient between 1 and 10 times a power of 10.
• Form: a \times 10^{n} where 1 \le a < 10; exponent sign indicates decimal move direction.
• Examples: 800 = 8.0 \times 10^{2}, 0.0000343 = 3.43 \times 10^{-5}.
• Addition/Subtraction in sci notation:
  • Exponents must be the same; rewrite values to common exponent, then add coefficients.
  • Example: 2.840 \times 10^{18} + 3.60 \times 10^{17} = 2.840 \times 10^{18} + 0.360 \times 10^{18} = 3.200 \times 10^{18}
• Multiplication/Division in sci notation:
  • Multiplication: (a \times 10^{m})(b \times 10^{n}) = (ab) \times 10^{m+n}
  • Division: \dfrac{(a \times 10^{m})}{(b \times 10^{n})} = \left(\dfrac{a}{b}\right) \times 10^{m-n}
• Example problems: convert data to sci notation as needed; (Sun diameter, density of atmosphere) illustrate the rules.

Dimensional Analysis

• Definition: systematic method using conversion factors to move from one unit to another.
• Conversion factor: ratio of equal values with different units (e.g., 1\ \text{dozen} = 12\ \text{eggs}).
• Key rule: a conversion factor must cancel one unit and introduce another.

Dimensional Analysis Example: Egyptian Cubits to Meters

• Problem: convert 6 cubits to meters using chained conversion factors:
  6\ \text{cubits} \times \dfrac{7\ \text{palms}}{1\ \text{cubits}} \times \dfrac{4\ \text{fingers}}{1\ \text{palm}} \times \dfrac{18.75\ \text{mm}}{1\ \text{finger}} \times \dfrac{1\ \text{m}}{1000\ \text{mm}} = 3.150\ \text{m}
• Concept: each factor cancels old units, leaving the desired unit (m).

Uncertainty in Data

• Key concepts: accuracy, precision, error, percent error, significant figures.
• Accuracy: closeness to accepted value.
• Precision: closeness of a set of measurements to each other.
• Error: \text{error} = \text{experimental} - \text{accepted}.
• Percent error: \%
  \text{ error} = \left| \dfrac{\text{error}}{\text{accepted}} \right| \times 100
• Significant Figures: rules to express measured values with appropriate precision:
  • Rule 1: nonzero digits are significant.
  • Rule 2: zeros between nonzero digits are significant.
  • Rule 3: trailing zeros to the right of a decimal are significant.
  • Rule 4: placeholder zeros are not significant (use scientific notation to remove).
  • Rule 5: counting numbers and defined constants have infinite sig figs.
• Rounding rules (for calculations):
  • Rounding for addition/subtraction: use the least number of decimal places from inputs.
  • Rounding for multiplication/division: use the least number of significant figures from inputs.
• Example: sum of lengths (28.0 cm, 23.538 cm, 25.68 cm) to 77.2 cm (one decimal place).

Graphs and Representing Data

• Graphs visually display data to reveal trends; types include:
  • Circle/pie charts: show percentages of a whole.
  • Bar graphs: compare quantities across categories.
  • Line/XY graphs: independent variable on x-axis, dependent on y-axis; slope shows rate.
• Interpolation: estimating values between known data points.
• Extrapolation: estimating values beyond known data by extending the line.

Quick Recap for Last-Minute Review

• Know SI base units: s, m, kg, K; base definitions and absolute zero concept.
• Derived units: \rho = \dfrac{m}{V}, \; V = LWH, \; 1\ \text{L} = 1\ \text{dm}^3$$.
• Scientific notation: 1–10 coefficient; correct handling of exponents in addition, subtraction, multiplication, and division.
• Dimensional analysis: build conversion chains to cancel units and reach the desired unit.
• Uncertainty: distinguish accuracy vs precision; compute percent error; apply sig figs and rounding rules correctly.
• Representing data: choose appropriate graph type; read/interpret graphs; know interpolation vs extrapolation.