Measurement and Vectors - GENPHY 1 Topic 1

MEASUREMENT

• Physics is a fundamental science and experimental in nature; it studies matter and energy and their interactions, forming the basis for engineering and technology.
• Measurement is the process of assigning quantities to objects or events.
• A physical quantity is a number used to describe a physical phenomenon (examples: height, weight, age, brightness, pitch).
• When measuring, we compare the quantity with a reference standard to define the unit.
• The International System of Units (SI) was established for consistency and coherence; promoted after the 1968 General Conference on Weights and Measures (CGPM).
  • SI is based on the metric system; used globally with some exceptions (notably the United States).
• The metric system uses base units for length, mass, and volume, and prefixes to express multiples or submultiples of these units.
• Prefixes are powers of ten to facilitate measurement across scales; example prefixes include Tera (T), Giga (G), Mega (M), Kilo (k), Hecto (h), Deca (da), Deci (d), Centi (c), Milli (m), Micro (μ), Nano (n), Pico (p).
• Base units (SI Seven) and what they measure:
  • Meter (m): Length
  • Kilogram (kg): Mass
  • Second (s): Time
  • Kelvin (K): Temperature
  • Ampere (A): Electric current
  • Candela (cd): Luminous intensity
  • Mole (mol): Amount of substance
• Some derived units can be expressed in terms of base units (examples):
  • Newton (N) = kg·m·s⁻² (force)
  • Pascal (Pa) = N/m² (pressure)
  • Hertz (Hz) = s⁻¹ (frequency)
  • Coulomb (C) = A·s (electric charge)
  • Ohm (Ω) = V/A (electrical resistance)
  • Wb (Weber) = V·s (magnetic flux)
  • S (Siemens) = Ω⁻¹ (electrical conductance)
  • T (Tesla) = Wb/m² (magnetic flux density)
  • H (Henry) = Wb/A (inductance)
  • Bq (Becquerel) = s⁻¹ (radioactivity)
  • J (Joule) = N·m = kg·m²·s⁻² (energy/dose units like Sv for dose)
• The British (Imperial) system exists with units like inch, foot, yard, mile, gallon, etc., and is used in a few countries besides the US.
• Converting between units requires known conversion factors; examples given in the material:
  • 57 pounds to kilograms with the note 2.204 lbs = 1 kg → $57 ext{ lb} imes \frac{1 ext{ kg}}{2.204 ext{ lb}} \approx 25.84 ext{ kg}$
  • 23°C to (a) Fahrenheit and (b) Kelvin; standard conversion formulas:
  • Fahrenheit: $F = frac{9}{5}C + 32$
  • Kelvin: $K = C + 273.15$
  • 3.2 meters to inches with the note 39.370 in = 1 m →$3.2 ext{ m} imes 39.370\frac{ ext{in}}{ ext{m}} \approx 125.984 ext{ in}$
  • 2 gallons to liters with the note 3.785 L = 1 gal → $2 ext{ gal} imes 3.785\frac{ ext{L}}{ ext{gal}} \approx 7.57 ext{ L}$
  • 2 hours to seconds with the note 1 h = 3600 s → $2 ext{ h} imes 3600\frac{ ext{s}}{ ext{h}} \approx 7200 ext{ s}$
  • 2 gallons to liters and 39.370 in = 1 m are examples that appear in the text.
• Additional conversions listed as practice: 57 lb to kg; 23°C to Fahrenheit and Kelvin; 3.2 m to inches; 2 gal to L; 2 h to s; 3 nm to μm; 8 mL to L; 53 g to kg.
• Scientific notation (power-of-ten notation) is a convenient way to write very large or very small numbers:
  • 1 000 000 = $10^{6}$
  • 0.000 000 000 001 = $10^{-12}$
• Uncertainty and errors:
  • Uncertainty indicates the maximum likely difference between the measured value and the true value; depends on the measurement technique.
  • Examples of measurement tools and techniques: ruler vs caliper; triple beam balance vs analytical balance; oral thermometer vs infrared thermometer.
  • Accuracy vs precision:
  • Accuracy: how close measurements are to the true value. Example: 56.47 m, 55.30 m, 56.90 m.
  • Precision: how close the measurements are to each other. Example: 56.47 ± 0.02 mm. A high-quality measurement is both precise and accurate.
• This section connects measurement concepts to data quality and reliability in experiments.

THE INTERNATIONAL SYSTEM OF UNITS (SI)

• SI is an agreed-upon system of units to ensure consistency across science and engineering.
• The seven base units (see measurement section) define all other units.
• Base units and their quantities are used to derive other units used in physics and engineering.

THE METRIC SYSTEM AND PREFIXES

• The metric system forms the basis of SI; it uses prefixes to denote multiples of 10 or 1/10.
• Examples of prefixes and their base-10 exponents:
  • Tera (T): $10^{12}$
  • Giga (G): $10^{9}$
  • Mega (M): $10^{6}$
  • Kilo (k): $10^{3}$
  • Hecto (h): $10^{2}$
  • Deca (da): $10^{1}$
  • Base 10 (no prefix): $10^{0} = 1$
  • Deci (d): $10^{-1}$
  • Centi (c): $10^{-2}$
  • Milli (m): $10^{-3}$
  • Micro (μ): $10^{-6}$
  • Nano (n): $10^{-9}$
  • Pico (p): $10^{-12}$

THE SEVEN FUNDAMENTAL UNITS

• Base quantity and unit:
  • Length → Meter (m)
  • Mass → Kilogram (kg)
  • Time → Second (s)
  • Temperature → Kelvin (K)
  • Electric current → Ampere (A)
  • Luminous intensity → Candela (cd)
  • Amount of substance → Mole (mol)

DERIVED UNITS IN BASE UNITS

• Common derived units and their base-unit expressions:
  • Newton (N) = kg·m·s⁻²
  • Pascal (Pa) = N·m⁻²
  • Hertz (Hz) = s⁻¹
  • Coulomb (C) = A·s
  • Ohm (Ω) = V/A
  • Weber (Wb) = V·s
  • Siemens (S) = Ω⁻¹
  • Tesla (T) = Wb/m²
  • Henry (H) = Wb/A
  • Becquerel (Bq) = s⁻¹
  • Joule (J) = N·m = kg·m²·s⁻²
  • Sievert (Sv) = J/kg (dose quantity)
  • Conductance, magnetic etc. are captured by the Ω⁻¹, Wb·m², Wb/A, s⁻¹ relationships above.

THE BRITISH SYSTEM (Limited use)

• Units include Inch, Foot, Yard, Mile, Acre, Pint, Quart, Gallon, Ounce, Pound, Ton, etc.
• Noted as being used only in the United States and a few other countries in the material.

CONVERSION OF UNITS

• Example problems and steps are provided to practice unit conversion:
  • 57 pounds to kilograms with the given conversion 2.204 lbs = 1 kg:
  • $57\text{ lb} \times \frac{1\text{ kg}}{2.204\text{ lb}} \approx 25.84\text{ kg}.$
  • 23°C to Fahrenheit and Kelvin (use standard conversion formulas provided above).
  • 3.2 meters to inches with 1 m = 39.370 in:
  • $3.2\text{ m} \times 39.370\frac{\text{in}}{\text{m}} \approx 125.984\text{ in}.$
  • 2 gallons to liters with 1 gal = 3.785 L:
  • $2\text{ gal} \times 3.785\frac{\text{L}}{\text{gal}} \approx 7.57\text{ L}.$
  • 2 hours to seconds using 1 h = 3600 s:
  • 2\text{ h} \times 3600\frac{\text{s}}{\text{h}} \approx 7200\text{ s}.$n

SCIENTIFIC NOTATION

• A convenient notation for large or small numbers:
  • Large: 1\,000\,000 = 10^{6}
  • Small: 0.000!000!000!001 = 10^{-12}

UNCERTAINTY, ERRORS, ACCURACY, AND PRECISION

• Uncertainty indicates the maximum likely difference between the measured value and the true value; depends on the technique used.
• Examples of measurement tools and comparisons:
  • Ruler vs. caliper
  • Triple beam balance vs. analytical balance
  • Oral thermometer vs. infrared thermometer
• Accuracy vs. precision:
  • Accuracy: closeness of a measurement to the true value; examples show a value near the true value.
  • Precision: closeness of multiple measurements to each other.
  • A high-quality measurement is both precise and accurate (illustrated by the example 56.47 m, 55.30 m, 56.90 m and 56.47±0.02 mm).

VECTORS: BASIC CONCEPTS

• Physical quantities come in two broad classes for dynamics:
  • Scalar quantities: magnitude only (e.g., volume, density, speed, energy, mass, time).
  • Vector quantities: magnitude and direction (e.g., displacement, velocity, acceleration, force, momentum, weight, current).

REPRESENTING A VECTOR

• Arrow form representation:
  • Length of the line indicates magnitude; arrowhead direction indicates direction.
  • Tail to head convention: tail → head marks the direction of the vector.
• Magnitude-Direction form:
  • Example: A = 400\text{ km},\quad 45^{\circ} \text{ North of East}.

CARTESIAN COORDINATE SYSTEM FOR VECTORS

• Vectors can be expressed in Cartesian components: \vec{A} = \langle Ax, Ay \rangle
• Each vector lies in the xy-plane; components can be broken into x- and y-directions.
• First-quadrant convention shown: Q1 (x>0, y>0); Q2 (x

VECTOR ADDITION IN 1D: GRAPHICAL METHOD (HEAD-TO-TAIL)

• When adding parallel vectors, place them head-to-tail in the same order; the resultant is the straight line from the tail of the first to the head of the last.
• Examples from the material (for illustration):
  • 10 km West + 6 km West
  • 25 mm South + 70 mm South
  • 13 m East + 65 m East
  • 20 cm North + 33 cm North
• When vectors are antiparallel, magnitudes subtract: the resultant equals the difference of magnitudes, taking the direction of the larger vector.
  • 10 km East and 6 km West; 25 mm South and 70 mm North; 13 m West and 65 m East; 20 cm North and 33 cm South.

VECTOR ADDITION IN 1D: GRAPHICAL METHOD (DETAILED STEP)

• Head-to-tail construction steps:
  • Choose an appropriate scaled coordinate system.
  • Draw the first vector from the origin.
  • Draw the second from the head of the first, and so on.
  • The resultant is the vector from the tail of the first to the head of the last.
• Practical example in the material shows the process with a sequence of vectors drawn in sequence.

VECTOR ADDITION IN 1D: GRAPHICAL METHOD (EXAMPLE PROBLEMS)

• Example problems provided to practice head-to-tail addition: multiple vectors with given magnitudes and directions (e.g., 5 km at 20° East of North; 6 km at 30° North of East; 7 km at 20° East of North; 4 km at 30° South of East).

VECTOR ADDITION IN 1D: COMPONENT METHOD

• The components form of the vector: \vec{A} = \langle Ax, Ay \rangle
• When adding two vectors perpendicular, we can use the Pythagorean theorem to find the resultant magnitude:
  • If the vector has components Ax$and$Ay$, then magnitude is$|\vec{A}| = \sqrt{Ax^2 + Ay^2}.
• General method: any vector in the plane can be written as the sum of an x-component vector and a y-component vector: \vec{A} = \langle Ax, Ay \rangle = Ax\hat{i} + Ay\hat{j}

COMPONENT METHOD: PRACTICAL EXAMPLES

• Example 1: Find the magnitude of a vector with an x-component pointing west and a y-component of 12 m north and 25 m east? (Note: from the material: “Find the magnitude of A with an x-component west and a y-component of 12 m, north of 25 m.”) The phrasing implies components: Ax = -25\text{ m}, \quad Ay = +12\text{ m}$; magnitude:$|\vec{A}| = \sqrt{(-25)^2 + 12^2}\,\text{m}.
• Example 2: A cross-country skier skis 1.00 km north and then 2.00 km east; how far and in what direction is she from the starting point? Use components: \vec{A} = \langle 2.00\text{ km}, 1.00\text{ km} \rangle; \quad |\,\vec{A}\,| = \sqrt{(2.00)^2 + (1.00)^2}\text{ km}$with direction arctangent:$\theta = \tan^{-1}\left( \frac{1.00}{2.00} \right) relative to east axis (or appropriate reference).

COMPONENT METHOD: ADDITION PROBLEMS

• Example: A = 5 km, 20° East of North; B = 6 km, 30° North of East; find resultant using components or graphical method (practice problems provided in the material).

ADDITIONAL PRACTICE PROBLEMS (MORE COMPLEX SCENARIOS)

• Examples given to practice vector addition and components include:
  • 7 km, 20° East of North and 4 km, 30° South of East; find resultant magnitude and direction.
  • The material emphasizes consistent use of components and proper angle conventions with the Cartesian plane.

SUMMARY OF KEY FORMULAS AND CONCEPTS

• Vector magnitude in components: |\vec{A}| = \sqrt{Ax^2 + Ay^2}
• Vector representation: \vec{A} = \langle Ax, Ay \rangle = Ax\hat{i} + Ay\hat{j}
• Direction angle relative to axes can be found by \theta = \tan^{-1}\left( \frac{Ay}{Ax} \right), with quadrant considered.
• For magnitude-angle form: |\vec{A}| = \text{magnitude},\quad \theta = \text{direction relative to reference axis}
• For scalar quantities: magnitude only; for vector quantities: magnitude and direction.
• For unit conversions, always multiply by a ratio of equivalent units to cancel units and preserve the numerical value; example conversions described above with given factors.

ASYNCHRONOUS TASK (STUDY QUESTIONS)

• Questions to prepare for the next class:
  1. What is the difference between speed and velocity?
  2. What is acceleration?
  3. What is the difference between distance and displacement?
  4. What are the kinematic equations?
  5. What is the relationship of speed, distance, and time mathematically?
• Use these prompts to test understanding of vectors, measurement, and kinematics.

CONNECTIONS TO FOUNDATIONAL PRINCIPLES AND REAL-WORLD RELEVANCE

• Measurement concepts underpin all experimental physics and engineering design, ensuring that data are comparable across experiments and over time.
• SI units and prefixes enable scientists worldwide to communicate quantities unambiguously, from lab measurements to large-scale simulations.
• Understanding uncertainty, accuracy, and precision helps in assessing data quality, informing experimental design, and interpreting results in both research and industry.
• Vectors are fundamental in describing physical quantities that have direction, such as force, velocity, and displacement; mastering vector addition and decomposition is essential in mechanics, navigation, physics simulations, and robotics.

ETHICAL, PHILOSOPHICAL, AND PRACTICAL IMPLICATIONS

• Ethical: accurate measurement and honest reporting of uncertainties are crucial for safety, reliability, and trust in science and engineering.
• Philosophical: measurement links perception to quantifiable reality; precision and accuracy reflect limits of instruments and methods, reminding us that knowledge is probabilistic and contingent on technique.
• Practical: unit consistency and proper conversion minimize errors in design, manufacturing, healthcare, and infrastructure projects.

NOTES ON FORMATTING AND EXPRESSIONS

• All mathematical expressions are presented in LaTeX format within double dollar signs, e.g., |\vec{A}| = \sqrt{Ax^2 + Ay^2}.
• Numbers and units are kept explicit to reflect the provided material (e.g., 57\text{ lb} \rightarrow 25.84\text{ kg}$using$2.204\text{ lb} = 1\text{ kg}).
• Where the material provides specific numbers or conversion factors, they are included exactly as given (e.g., 1 m = 39.370 in; 1 gal = 3.785 L).
• Some problems are stated as exercises or conversion challenges; these are included with the standard formulas to encourage practice and verification.

QUICK REFERENCE MATERIAL

• SI base units: m, kg, s, K, A, cd, mol.
• Derived units examples: N, Pa, Hz, C, Ω, Wb, S, T, H, Bq, J, Sv.
• Prefix scale (short form): T, G, M, k, da, d, c, m, μ, n, p with exponents 10^12 down to 10^-12.
• Conversion shortcuts used in the material:
  • 1\text{ gal} = 3.785\text{ L}
  • 39.370\text{ in} = 1\text{ m}
  • 2.204\text{ lb} = 1\text{ kg}
  • 1\text{ h} = 3600\text{ s}
  • 1\text{ m} = 100\text{ cm} (inferred from metric system context)
  • 1\text{ L} = 1000\text{ mL}$$ (inferred from metric system context)