SBUR General Physics 1: Measurement and Uncertainty Notes (AY 2025-2026)

Measurement and Uncertainty

• What is Physics?

  • Physics is the study that deals with matter, energy and their transformations, and the interaction between them.
  • As an experimental science, its goal is to understand the natural world.

• Measurements, Units, and Standards

  • Measurement is the art of comparing unknown values to a standard, or the accepted set of values for a particular quantity.
  • Units are the standards in which physical quantities are expressed.
  • SI System: International System of Units. In 1960 the name was changed to SI (Système International d’Unités) and is used today.

• Fundamental and Derived Quantities

  • Fundamental quantities: quantities that cannot be expressed in terms of other quantities; independent quantities. There are 7 fundamental quantities.
  • Derived quantities: quantities derived from fundamental quantities (e.g., area, volume, acceleration, force, pressure).

• Fundamental Quantities (7)

  • Mass (unit: kilogram, kg)
  • Electric current (unit: ampere, A)
  • Time (unit: second, s)
  • Temperature (unit: kelvin, K)
  • Amount of substance (unit: mole, mol)
  • Length (unit: meter, m)
  • Luminous intensity (unit: candela, cd)

• SI Base Units: brief definitions

  • Mass (kg): A kilogram is defined by an international prototype kilogram made of platinum–iridium, kept at the International Bureau of Weights and Measures (BIPM) in Sèvres, France.
  • Electric current (A): An Ampere is the current maintained in two straight wires, placed 1 meter apart in vacuum, which produce a force of 2 × 10^{-7} N per meter of length.
  • Time (s): A second is the specified transition of Cesium-133 atom, during which 9,192,631,770 cycles of microwave radiation are generated by the atom.
  • Temperature (K): The Kelvin scale is based on absolute temperature; 273.15 is the offset relative to Celsius.
  • Amount of substance (mol): One mole contains 6.02 × 10^{23} particles of atoms (Avogadro's number).
  • Length (m): A meter is defined as the distance traveled by light in vacuum in 1/299,792,458 of a second.
  • Luminous intensity (cd): Candela is the luminous intensity of a source emitting with a frequency of 540 × 10^{12} Hz in a specified direction.
  • Note: The first letter of a unit named after a person is uppercase.

• Metric System and Units

  • SI prefixes facilitate dealing with very large or very small numbers. Some prefixes and their powers of ten are listed below.

• Fundamental Quantities vs. Derived Quantities: quick recap

  • Fundamental: cannot be derived from other quantities (7 total).
  • Derived: formed from combinations of fundamental quantities (e.g., area, volume, acceleration, force, pressure).

• Derived Quantities and their Formulas

  • Area: A = l \cdot w
  • Volume: V = l \cdot w \cdot h
  • Acceleration: a = \frac{\Delta v}{\Delta t} = \frac{d^2x}{dt^2} \quad (\text{SI unit: } \mathrm{m/s^2})
  • Force: F = m a
  • Pressure: P = \frac{F}{A}
  • Units: \text{Area} = \mathrm{m^2}, \quad \text{Volume} = \mathrm{m^3}, \quad \text{Force} = \mathrm{N}, \quad \text{Pressure} = \mathrm{Pa}
  • Newton: 1~\mathrm{N} = 1~\mathrm{kg\, m\, s^{-2}}
  • Joule: 1~\mathrm{J} = 1~\mathrm{kg\, m^2 ! s^{-2}}
  • Hertz: 1~\mathrm{Hz} = 1~\mathrm{s^{-1}}
  • Watt: 1~\mathrm{W} = 1~\mathrm{kg\, m^2 ! s^{-3}}
  • Pascal: 1~\mathrm{Pa} = \frac{1~\mathrm{N}}{\mathrm{m^2}}

• Common Conversions and Unit Associations

  • Area conversions:
  • 1~\text{km}^2 = 10^6~\mathrm{m^2}
  • 1~\text{ha} = 10^4~\mathrm{m^2}
  • 1~\text{acre} = 4047~\mathrm{m^2}
  • 1~\text{are} = 100~\mathrm{m^2}
  • Volume: 1~\mathrm{m^3}; 1~\mathrm{cm^3} = 10^{-6}~\mathrm{m^3}
  • Capacity: 1~\text{kL} = 1000~\mathrm{L}, \quad 1~\mathrm{cL} = 0.01~\mathrm{L}
  • Mass and weight: 1~\text{metric ton} = 10^6~\mathrm{g} = 10^3~\mathrm{kg}; 1~\text{cg} = 0.01~\mathrm{g}
  • Fundamental to derived conversions: 1 N, 1 J, 1 Hz, 1 Pa, 1 W as above.
  • Note: 1 N = 1 kg·m/s^2; 1 J = 1 kg·m^2/s^2; 1 Pa = 1 N/m^2; 1 W = 1 m^2·kg/s^3.

• Metric Prefixes (official SI prefixes as of 2022)

  • Large prefixes:
  • quetta (Q) = 10^{30}
  • ronna (R) = 10^{27}
  • yotta (Y) = 10^{24}
  • zetta (Z) = 10^{21}
  • exa (E) = 10^{18}
  • peta (P) = 10^{15}
  • tera (T) = 10^{12}
  • giga (G) = 10^{9}
  • mega (M) = 10^{6}
  • kilo (K) = 10^{3}
  • Mid-range prefixes:
  • hecto (h) = 10^2
  • deca (da) = 10^1
  • Small prefixes:
  • 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}
  • atto (a) = 10^{-18}
  • zepto (z) = 10^{-21}
  • yocto (y) = 10^{-24}
  • Very small and very large prefixes beyond those listed have been proposed for future adoption:
  • ronto (r) = 10^{-27}
  • quecto (q) = 10^{-30}
  • Note: Officially used prefixes start from kilo up to yotta and their lower counterparts; newer prefixes (ronna/quetta and ronto/quecto) have been added to the SI family.

• Scientific Notation

  • Express numbers as: M \times 10^{E} where the mantissa M satisfies 1 \leq M < 10
  • Example: To express measurements compactly and to perform arithmetic with large/small numbers.
  • Mantissa and exponent conventions: mantissa is the coefficient, exponent is the power of ten.

• Scientific Notation Practice (examples)

  • Express in scientific notation:
  • 300 N → 3.00 \times 10^{2}\
  • 250\,000 kg → 2.50 \times 10^{5} \text{ kg}
  • 0.0004 mm → 4.0 \times 10^{-4} \text{ mm} (significant figures depend on measurement precision)
  • Convert to standard form from scientific notation:
  • 0.90 × 10^{−1} cm → 9.0 × 10^{−2} \text{ cm}
  • 2.66 × 10^{4} J → 2.66 × 10^{4} \text{ J} (same form; decimal expansion if required would be 26600 J)
  • 1.5 × 10^{−3} kg → 1.5 × 10^{−3} \text{ kg} (decimal form: 0.0015 kg)

• Significant Figures

  • Significance conveys the accuracy of a measurement.
  • Basic rule: All nonzero digits are significant.
  • Rule 1: Count digits after the first nonzero digit from the left for numbers with decimal point.
  • Examples:
    • 0.01020 → 4 significant figures
    • 0.00500 → 3 significant figures
    • 1000.005 → 7 significant figures
  • Rule 2: In the absence of a decimal point, count from the right up to the first nonzero digit.
  • Examples:
    • 102 004 000 → 6 significant figures
    • 178 000 000 → 3 significant figures
    • 105 010 100 → 7 significant figures
  • Example 2 (from the notes): determine the number of significant figures in:
  • 342 700 m
  • 0.000984 Hz
  • 0.51020 mL
  • 1.00290 A
  • 1000 × 10^−3 Pa

• Dimensional Analysis

  • Steps for unit conversion using dimensional analysis.
  • Example tasks include converting measurements such as kilometers to meters, weeks to hours, grams to kilograms, kph to m/s, liters to cm^3, and temperature conversions between C, K, and F.
  • Example tasks in the slides (Example 3 and Example 4) demonstrate formal dimensional analysis methods and applying conversion factors.

• Rounding Off Numbers

  • Rounding rules:
  • If the next digit is < 5, drop it and leave the previous digit unchanged.
  • If the next digit is ≥ 5, drop it and increase the preceding digit by 1.
  • Example: 1.684 → 1.68; 1.247 → 1.25.

• Uncertainty, Accuracy, and Precision

  • Uncertainty: an interval around a measured value within which repetitions are expected to fall.
  • Accuracy: closeness of a measurement to the true/accepted value.
  • Precision: closeness among several measurements obtained in the same way.
  • Distinctions explained with visual bull’s-eye activity (quick intuition):
  • Accurate and precise: measurements cluster near the true value and with small spread.
  • Accurate but not precise: clustered around true value but spread is wide.
  • Not accurate but precise: measurements are consistently off from true value but tightly clustered.
  • Not accurate and not precise: scatter spread and away from true value.

• Reducing Random vs Systematic Errors

  • Random errors (affect precision):
  • Repeat measurements and take the average.
  • Collect measurements from a larger sample.
  • Use high-precision instruments.
  • Control environmental factors.
  • Systematic errors (affect accuracy):
  • Calibrate measuring instruments regularly.
  • Improve experimental procedures and controls.
  • Establish a control setup or baseline data.
  • Compare obtained values with standard values.

• Examples of Measurement and Uncertainty in Practice

  • Example 5 (Golf ball mass): measurements: 45.89 g, 45.91 g, 46.00 g, 45.94 g, 45.90 g. Accepted value: 45.93 g.
  • Compute mean: approximately 45.93 g; the results demonstrate both accuracy (near the accepted value) and precision (low spread).
  • Example 6 (Different measurement sets for a true value of 10 inches): sets A, B, C, D with various values. The set that balances high accuracy and high precision generally is the one with a mean close to 10 in and with little spread; sets with numbers clustered near 10 inches show higher accuracy and precision. (Discussion notes show the trade-off: some sets are highly accurate but not precise, others precise but not accurate.)

• Additional Practice and Worksheets (from the slides)

  • Worksheet 1: Measurement and Uncertainty (oral)
  • Quiz 1: Measurement and Error Analysis
  • Exit Ticket: Reflect on why accurate measurement matters in experiments and real-world applications.

• Quick Reminders and Study Tips

  • Always identify whether you are dealing with a fundamental or derived quantity.
  • Use the correct SI units and verify the unit consistency when performing calculations.
  • Track significant figures consistently in computation and reporting.
  • Practice dimensional analysis often to avoid unit errors.
  • Use calibration and controls to minimize systematic errors; increase sample size and instrumentation precision to minimize random errors.

• Connections to Real-World Relevance

  • Accurate measurement is critical in medicine, construction, and laboratory experiments.
  • Understanding uncertainties helps in risk assessment, quality control, and decision making in engineering and science.

• Ethical and Practical Implications

  • Calibrating instruments and reporting uncertainties responsibly prevents misinterpretation of data in safety-critical contexts (medicine, aviation, structural engineering).

• Notation and Language Notes

  • Scientific notation: mantissa × 10^E with 1 ≤ mantissa < 10.
  • Common terms: accuracy, precision, uncertainty, error, systematic error, random error.

• Summary Takeaways

  • Measurement is central to physics and science; units and standards define comparability.
  • Fundamental quantities define the base from which derived quantities are built.
  • Uncertainty quantifies confidence in measurements and informs interpretation.
  • Distinguishing accuracy from precision helps diagnose measurement quality.
  • Dimensional analysis, scientific notation, and significant figures are essential tools for clean quantitative work.

Fundamental Quantities and SI Base Units

• The seven fundamental quantities and their SI base units:
  • Mass — unit: kilogram (kg)
  • Electric current — unit: ampere (A)
  • Time — unit: second (s)
  • Temperature — unit: kelvin (K)
  • Amount of substance — unit: mole (mol)
  • Length — unit: meter (m)
  • Luminous intensity — unit: candela (cd)

SI Base Units: Brief Definitions (in practice)

• Mass (kg): defined by the international prototype kilogram kept at BIPM.
• Electric current (A): defined via force between two parallel wires separated by 1 meter in vacuum.
• Time (s): defined using Cesium-133 atomic transition.
• Temperature (K): offset from Celsius (273.15 K = 0 °C).
• Amount of substance (mol): contains 6.02 × 10^{23} particles.
• Length (m): distance light travels in vacuum in 1/299,792,458 s.
• Luminous intensity (cd): luminous intensity corresponding to a specified light frequency.

SI Derived Quantities (with Formulas and Units)

• Area: A = l \cdot w \quad (\mathrm{m^2})
• Volume: V = l \cdot w \cdot h \quad (\mathrm{m^3})
• Acceleration: a = \frac{\Delta v}{\Delta t} = \frac{d^2x}{dt^2} \quad (\mathrm{m\,s^{-2}})
• Force: F = m a \quad (\mathrm{N})
• Pressure: P = \frac{F}{A} \quad (\mathrm{Pa})

Machine-Readable Prefixes and Prefix Tables (Summary)

• Large prefixes: quetta (Q) 10^{30}, ronna (R) 10^{27}, yotta (Y) 10^{24}, zetta (Z) 10^{21}, exa (E) 10^{18}, peta (P) 10^{15}, tera (T) 10^{12}, giga (G) 10^{9}, mega (M) 10^{6}, kilo (K) 10^{3}
• Mid-range: hecto (h) 10^2, deca (da) 10^1
• Small prefixes: deci (d) 10^{-1}, centi (c) 10^{-2}, milli (m) 10^{-3}, micro (\mu) 10^{-6}, nano (n) 10^{-9}, pico (p) 10^{-12}, femto (f) 10^{-15}, atto (a) 10^{-18}, zepto (z) 10^{-21}, yocto (y) 10^{-24}
• Very small/large additions (as of recent SI updates): ronto (r) 10^{-27}, quecto (q) 10^{-30}

Dimensional Analysis and Practice Problems

• Dimensional analysis steps: identify units, use conversion factors, cancel units to obtain target units.
• Examples in the slides cover conversions such as kilometers to meters, weeks to hours, grams to kilograms, and temperature conversions between C, K, and F.

Scientific Notation and Examples

• Notation form: M \times 10^{E} with 1 \le M < 10.
• Examples (from the notes):
  • Express in scientific notation: 300 N → 3.00 \times 10^{2} \mathrm{N}
  • 250 000 kg → 2.50 \times 10^{5} \mathrm{kg}
  • 0.0004 mm → 4.0 \times 10^{-4} \mathrm{mm}
• Convert back from scientific notation:
  • 0.90 × 10^{−1} cm → 9.0 × 10^{−2} \mathrm{cm}
  • 2.66 × 10^4 J → 2.66 × 10^4 \mathrm{J} (or 26600 J)
  • 1.5 × 10^{−3} kg → 1.5 × 10^{−3} \mathrm{kg}$$ (0.0015 kg)

Significant Figures (Overview)

• Definition: Significant figures convey the precision of a measurement.
• Rule 1 (nonzero digits): All nonzero digits are significant.
• Rule 1 examples: 0.01020 (4 sf); 0.00500 (3 sf); 1000.005 (7 sf).
• Rule 2 (leading/trailing zeros): In the absence of a decimal point, count from the right up to the first nonzero digit.
• Examples: 102 004 000 (6 sf); 178 000 000 (3 sf); 105 010 100 (7 sf).
• Practice problems: determine sf in numbers such as 342 700 m, 0.000984 Hz, 0.51020 mL, 1.00290 A, and 1000 × 10^{−3} Pa.

Uncertainty, Accuracy, and Precision (Key Concepts)

• Uncertainty: a measured value is accompanied by an interval around it where repeated measurements are expected to lie.
• Accuracy: closeness of a measurement to the true/accepted value.
• Precision: closeness of repeated measurements to each other when performed in the same way.
• Distinguishing features via the bull’s-eye activity descriptions (accurate vs precise, etc.).

Sources of Error and How to Address Them

• Random errors (affect precision):

  • Repeat measurements and take the average.
  • Increase the number of samples.
  • Use higher-precision instruments.
  • Control environmental factors.

• Systematic errors (affect accuracy):

  • Calibrate instruments regularly.
  • Improve experimental procedures and controls.
  • Establish a baseline/control setup.
  • Compare results with standard values.

• Examples of errors in measurement (from the slides):

  • A thermometer consistently reads 2°C higher than actual.
  • Human delay biases timing in a ruler drop test.
  • Ruler readings of wire diameter show inconsistent values (0.4 cm vs 0.5 cm).
  • Digital balance readings fluctuate near an open window.
  • Timing a fall with a stopwatch yields slightly varying times.

Practice and Review Components

• Worksheet 1: Measurement and Uncertainty (oral)
• Quiz No. 1: Measurement and Error Analysis
• Exit Ticket: Reflect on why accurate measurement matters in this lesson.
• Two Truths and a Lie activity: test understanding of significant figures concepts.

Takeaways for Exam Readiness

• Be able to define and distinguish measurement, units, fundamental vs derived quantities.
• Know the seven SI base quantities and their units, plus the primary definitions for each base unit.
• Be able to write and simplify expressions for area, volume, acceleration, force, and pressure.
• Understand unit conversions, common conversions, and the meaning of prefixes.
• Apply dimensional analysis to convert units and track units through calculations.
• Use scientific notation correctly and convert between standard form and scientific notation with appropriate significant figures.
• Identify and reduce random and systematic errors; describe how accuracy and precision are impacted.
• Interpret data in terms of accuracy, precision, and uncertainty; relate to real-world measurement scenarios.

End of Notes