Units, Physical Quantities and Vectors

What is Physics?

• Physics is defined as the study of matter and energy.
• It serves as the foundation for all sciences.
• Key principles include:
  • Motion of particles and waves.
  • Interactions between particles.
  • Properties of molecules, atoms, and larger-scale matter (gases, liquids, etc.).

Units

SI Units of Fundamental Quantities

• Second: Time Measurement

  • Defined as 9,192,631,770 times the period of vibration of radiation from cesium-133 atom.

• Metre: Length Measurement

  • Defined as the distance light travels in a vacuum in 1/299,792,458 seconds.

• Kilogram: Mass Measurement

  • Defined as the mass of a platinum-iridium alloy cylinder preserved in France.

Unit Prefixes

• Prefixes are used for identifying common multiples and submultiples of SI units.

Conversion of Units

• Conversion involves multiplying by a ratio that equals 1.

• Example:

  • To convert 80 km/h to m/s:
    $80 ext{ km/h} = 80 imes \frac{1000 ext{ m}}{1 ext{ km}} imes \frac{1 ext{ h}}{3600 ext{ s}} = 22.2 ext{ m/s}$

• Another Example:

  • Convert 3.5 ns into Ts:
    $3.5 ext{ ns} = 3.5 imes 10^{-9} ext{ s} = 3.5 imes 10^{-21} ext{ Ts}$

Significant Figures and Uncertainty

• Measurements always carry uncertainty.
  • Example of two measurements for the same pen:
  • Instrument 1: $0.8 ext{ cm}$ (1 significant figure)
  • Instrument 2: $0.806 ext{ cm}$ (3 significant figures)
• The last digit in a measurement indicates uncertainty.
  • E.g., first measurement: uncertainty of ±0.1 cm (range: 0.7 cm to 0.9 cm).

Examples of Significant Figures

1. $2.095 ext{ m} ext{ ± } 0.005 ext{ m}$ has 4 significant figures.
2. 220 ext{ Ω ± 10 ext{%}} has 2 significant figures.
3. $57.0 ext{ N ± 0.1 N}$ has 3 significant figures.

Rules for Calculations

• Multiplication/Division: Result has the same number of significant figures as the factor with the least significant figures.
• Addition/Subtraction: Result's decimal places should equal the term with the least decimal places.

Orders of Magnitude and Estimates

• An order of magnitude is a rough estimation of measurement in powers of 10.
• Example: Height of an adult human is between 1.5 m and 2.0 m or $10^0$.

Estimation Examples

• Estimating the number of breaths taken over a lifetime.
• Measuring the thickness of a sheet of paper.

Vectors

• A vector quantity has both magnitude and direction (e.g., displacement, velocity, force).
• A scalar quantity has only magnitude (e.g., time, mass, distance).

Vector Representation

• Represented graphically as arrows where:
  • Length indicates magnitude.
  • Direction of the arrow indicates direction.
• Example: Vectors A and B can be equal if both have the same magnitude and direction.

Vector Addition

• Vectors are not added by mere arithmetic; direction matters.
• For example, parallel vectors can be added algebraically, while antiparallel vectors are subtracted.

Adding Vectors Example

• An airplane flies 400 km south and then 300 km east. Resultant displacement is computed via vector addition.

Components of Vectors

• A vector can be divided into components: $ext{A} = ext{A}<em>x + ext{A}</em>y$ Where:
  • $ext{A}_x = ext{A} imes ext{cos}( heta)$
  • $ext{A}_y = ext{A} imes ext{sin}( heta)$
  • Magnitude of A:
    $| ext{A}| = \sqrt{ ext{A}<em>x^2 + ext{A}</em>y^2}$

Adding Multiple Vectors Using Components

• Total components:
  $R<em>x = A</em>x + B<em>x$$R</em>y = A<em>y + B</em>y$

Unit Vectors

• Unit vectors have a magnitude of 1 and are used to represent direction.
  • In Cartesian coordinates:
  • $extbf{i}, extbf{j}, extbf{k}$ represent x, y, z directions respectively.

Example of Resultant Calculation

• Given displacements:
  $ext{A} = (2, 5, -6) ext{ m}$
  $ext{B} = (-4, -4, 3) ext{ m}$
• Find the resultant vector by adding the respective components.

Scalar and Vector Products

Scalar Product (Dot Product)

• Defined as: $ext{A} \bullet ext{B} = | ext{A}|| ext{B}| ext{cos}( heta)$
  • Results in a scalar quantity.
  • Examples include calculating the dot product between two vectors.

Vector Product (Cross Product)

• Defined as: $ext{A} imes ext{B} = | ext{A}|| ext{B}| ext{sin}( heta)$
  • The result is a vector (direction determined by the right-hand rule).
• Properties of Vector Products:
  • $extbf{i} imes extbf{j} = extbf{k}$
  • $extbf{j} imes extbf{k} = extbf{i}$
  • Anticommutative property: $ext{A} imes ext{B} = -( ext{B} imes ext{A})$

Example of Vector Product Calculation

• Given:
  $ext{A} = (-3, 0, 1) ext{ units}$
  $ext{B} = (-5, 4, 3) ext{ units}$
• Find:
  • The vector product $ext{A} imes ext{B}$
  • The angle between vectors A and B.