Chapter 1 Notes: Physical Quantities, Units, and Vectors

1. Physical Quantities, Units, and Vectors

• Physical quantities (properties that can be measured and expressed numerically) are categorized as:

  • Basic (base) quantities
  • Derived quantities

• Basic (base) quantities and their SI units:

  • Length → unit: meters, symbol: mm? No — base unit is the meter (m)
  • Time → unit: second (s)
  • Mass → unit: kilogram (kg)
  • Electrical current → unit: ampere (A)
  • Temperature → unit: kelvin (K)
  • Amount of substance → unit: mole (mol)
  • (Note: This chapter focuses on length, time, mass, current, temperature, amount of substance as base quantities; other base quantities like luminous intensity are not listed in the transcript.)

• Derived quantities (expressed in terms of base quantities) and their typical units:

  • Velocity → symbol: $v$, unit: $ext{m s}^{-1}$
  • Acceleration → symbol: $a$, unit: $ext{m s}^{-2}$
  • Work → symbol: $W$, unit: $ext{J}$ (joule)
  • Force → symbol: $F$, unit: $ext{N}$ (newton)
  • Pressure → symbol: $P$, unit: $ext{Pa}$ (pascal) = $ext{N m}^{-2}$
  • Energy → symbol: $E$, unit: $ext{J}$
  • Power → symbol: (often $P$ in other contexts; unit: $ext{W}$) (watt)
  • Frequency → symbol: $f$, unit: $ext{s}^{-1}$ = $ext{Hz}$

• Unit prefixes (SI prefixes) to present large/small values (examples from the table in the slides):

  • $10^{-9}$ → nano, symbol: n
  • $10^{-6}$ → micro, symbol: μ
  • $10^{-3}$ → milli, symbol: m
  • $10^{-2}$ → centi, symbol: c
  • $10^{3}$ → kilo, symbol: k
  • $10^{6}$ → mega, symbol: M
  • $10^{9}$ → giga, symbol: G
  • $10^{12}$ → tera, symbol: T
  • (Note: The transcript lists prefixes in a compact form; this is the standard set used in SI practice.)

• Examples of unit conversions (practice problems from the transcript):

  • Area: $15~ ext{mm}^2 = ?~ ext{m}^2$
  • Volume: $2~ ext{cm}^3 = ?~ ext{m}^3$
  • Density: $450~ ext{g cm}^{-3} = ?~ ext{kg m}^{-3}$
  • Speed: $65~ ext{km h}^{-1} = ?~ ext{m s}^{-1}$

• Worked solutions (step-by-step):

  • Area: 1 mm$^2$ = $1 imes10^{-6}$ m$^2$, so
    $15~ ext{mm}^2 = 15 imes10^{-6}~ ext{m}^2 = 1.5 imes10^{-5}~ ext{m}^2.$
  • Volume: 1 cm$^3$ = $1 imes10^{-6}$ m$^3$, so
    $2~ ext{cm}^3 = 2 imes10^{-6}~ ext{m}^3.$
  • Density: $450~ ext{g cm}^{-3} = 450 imes(10^{-3}~ ext{kg}) imes(10^{6}~ ext{m}^{-3}) = 4.5 imes10^{5}~ ext{kg m}^{-3}.$
  • Speed: 65~ ext{km h}^{-1} = 65 imes rac{1000 ext{ m}}{3600 ext{ s}} \approx 18.1~ ext{m s}^{-1}.

• 1.1.1 Unit Prefixes (quick reference)

  • Nano: $10^{-9}$, symbol: $ext{n}$
  • Micro: $10^{-6}$, symbol: $ext{μ}$
  • Milli: $10^{-3}$, symbol: $ext{m}$
  • Centi: $10^{-2}$, symbol: $ext{c}$
  • Kilo: $10^{3}$, symbol: $ext{k}$
  • Mega: $10^{6}$, symbol: $ext{M}$
  • Giga: $10^{9}$, symbol: $ext{G}$
  • Tera: $10^{12}$, symbol: $ext{T}$

• 1. Scalars and Vectors (intro)

  • Scalar quantity: magnitude only; examples include mass, time, temperature, pressure, electric current, work, energy, etc.; mathematics: ordinary algebra.
  • Vector quantity: magnitude and direction; examples include displacement, velocity, acceleration, force, momentum, electric/magnetic field, etc.; mathematics: vector algebra.

• 1.2 Scalars and Vectors (overview)

  • Scalars: quantity with only magnitude.
  • Vectors: quantity with both magnitude and direction.
  • Vectors follow vector algebra rules (addition, subtraction, scalar multiplication).

• 1.2.1 Vectors

  • A⃗ represents a vector; length of the arrow corresponds to magnitude; direction corresponds to direction of the vector.
  • Displacement, velocity, acceleration can be depicted as vectors with bold notation (e.g., oldsymbol{v}, oldsymbol{a}, oldsymbol{s}).
  • Notation: magnitude sometimes written as |oldsymbol{A}|.
  • Vector equality: Two vectors are equal if both magnitude and direction are the same.
  • Notation conventions: boldface vectors; magnitude denoted separately, e.g. |oldsymbol{A}|

• 1.2.2 Direction of Vectors

  • Direction can be described via compass directions (east, west, north, south, NE, NW, SE, SW) or by an angle with a reference line.
  • Example: a velocity of 20 m s$^{-1}$ at 50° above the horizontal.
  • Cartesian coordinates for 2-D direction: if a vector makes an angle θ with +x axis, components are $R<em>x = R \, \cos\theta$ and $R</em>y = R \, \sin\theta$.

• 1.2.3 Addition of Vectors

  • Graphical methods to add two vectors: parallelogram method and triangle method.
  • Parallelogram rule: if you have vectors A⃗ and B⃗, the resultant is the diagonal of the parallelogram formed by A⃗ and B⃗.
  • Triangle method: place the tail of B⃗ at the head of A⃗; the resultant is from the tail of A⃗ to the head of B⃗.
  • Commutativity: A⃗ + B⃗ = B⃗ + A⃗.

• 1.2.4 Subtraction of Vectors

  • Subtraction can be treated as addition with the negative: A⃗ − B⃗ = A⃗ + (−B⃗).
  • Can be represented via parallelogram or triangle construction.

• 1.2.5 Unit Vectors

  • Unit vectors denote direction with magnitude 1: $\hat{a}, \hat{b}, \hat{c}$ or along Cartesian axes: $\hat{i}, \hat{j}, \hat{k}$.
  • A unit vector is dimensionless.
  • In 3-D, the standard basis is $\hat{i}, \hat{j}, \hat{k}$ corresponding to the x-, y-, and z-axes respectively.

• 1.2.6 Resolving a Vector

  • In 2-D: for a vector of magnitude $R$ at angle $\theta$ with +x axis:
  • components: $R<em>x = R \cos\theta, \quad R</em>y = R \sin\theta$
  • magnitude: $R = \sqrt{R<em>x^2 + R</em>y^2}$
  • direction: $\theta = \tan^{-1}\left(\frac{R<em>y}{R</em>x}\right)$ (in the appropriate quadrant)
  • In Cartesian form: a vector can be written as $\boldsymbol{R} = R<em>x \hat{i} + R</em>y \hat{j} + R_z \hat{k}$ in 3-D.
  • In 3-D, magnitude: $R = \sqrt{R<em>x^2 + R</em>y^2 + R<em>z^2}$; direction cosines if needed are $\cos\alpha = R</em>x/R, \ \cos\beta = R<em>y/R, \ \cos\gamma = R</em>z/R$.
  • Example: vector expressed in unit vectors: $\boldsymbol{r} = x \hat{i} + y \hat{j} + z \hat{k}$ (with appropriate units).

• 1.2.6 Resolving a Vector — Examples

  • Example 1.2: Given vectors \mathbf{a} and \mathbf{b}, compute sums and differences component-wise and/or magnitude.
  • Example 1.3: Car velocity example: a) component due north, b) component due east for a velocity of 50 m s$^{-1}$ at 30° north of east.
  • If direction is 30° north of east, then along x (east) and y (north):
    • $v_x = v \cos 60^{\circ} = 50 \times 0.5 = 25\ \text{m s}^{-1}$
    • $v_y = v \sin 60^{\circ} = 50 \times \frac{\sqrt{3}}{2} \approx 43.3\ \text{m s}^{-1}$
    • Interpretation: east component 25 m s$^{-1}$, north component 43.3 m s$^{-1}$.
  • Example 1.4: A force of 100 N at 150° from the +x axis.
  • Components:
    • $F_x = F \cos 150^{\circ} = 100 \cos 150^{\circ} = -86.6\ \text{N}$
    • $F_y = F \sin 150^{\circ} = 100 \sin 150^{\circ} = 50.0\ \text{N}$
  • Example 1.5: Three forces F1, F2, F3 acting on a particle O. Find the resultant force.
  • Given components (from the transcript):
    • F1 has x-component 0 N and y-component 10 N: $F1<em>x = 0, \quad F1</em>y = 10\,\text{N}$
    • F2 has y-component $F2_y = 30\sin 60^{\circ} \approx 26\,\text{N}$
    • F3 has x-component $F3_x = -40\cos 30^{\circ} \approx -34.6\,\text{N}$
  • Sum of components:
    • $F<em>x = F1</em>x + F2<em>x + F3</em>x = 0 + 0 + (-34.6) \approx -34.6\,\text{N}$
    • $F<em>y = F1</em>y + F2<em>y + F3</em>y = 10 + 26 + 0 \approx 36.0\,\text{N}$
  • Resultant magnitude and direction:
    • $F<em>r = \sqrt{F</em>x^2 + F_y^2} = \sqrt{(-34.6)^2 + (36.0)^2} \approx 49.9\,\text{N}$
    • Direction from +x axis: $\theta = \tan^{-1}\left(\frac{F<em>y}{F</em>x}\right) \approx \tan^{-1}\left(\frac{36}{-34.6}\right) \approx 133.7^{\circ}$
    • (Alternative: from the negative x-axis, it is about 46.3° above it.)

• 2-D and 3-D Cartesian coordinates

  • Cartesian coordinates in 2-D: position vector $\boldsymbol{s} = (x, y)$; sometimes written as $\boldsymbol{s} = x \hat{i} + y \hat{j}$ with units of length (e.g., meters).
  • Cartesian coordinates in 3-D: position vector $\boldsymbol{s} = (x, y, z)$; expressed as $\boldsymbol{s} = x \hat{i} + y \hat{j} + z \hat{k}$.
  • Examples from the slides:
  • 2-D: $\boldsymbol{s} = (1\,\text{m}, 5\,\text{m})$ with components along the x- and y-axes.
  • 3-D: $\boldsymbol{s} = (4, 3, 2)\,\text{m} = 4\hat{i} + 3\hat{j} + 2\hat{k}\,\text{m}$; magnitude $|\boldsymbol{s}| = \sqrt{4^2 + 3^2 + 2^2} = \sqrt{29} \approx 5.39\,\text{m}.$

• 2. Polar coordinates and directions

  • Polar coordinates describe a vector with magnitude and angle relative to a reference line; in 2-D, a vector can be written with magnitude and angle, e.g., $R, \theta$, where the components are $R<em>x = R \cos\theta$ and $R</em>y = R \sin\theta$.
  • The slides indicate a representation using plus/minus signs and an angle reference (e.g., 150° or 30° with respect to axes).

• 1.2.3 and 1.2.4: Graphical vector addition and subtraction (recap)

  • Parallelogram method: construct a parallelogram with the two vectors, the resultant is the diagonal from the tails.
  • Triangle method: place the tail of B⃗ at the head of A⃗; the resultant is from the tail of A⃗ to the head of B⃗.
  • Subtraction is equivalent to addition with a negated vector: $\boldsymbol{A} - \boldsymbol{B} = \boldsymbol{A} + (-\boldsymbol{B})$.

• 1.2.5 Unit Vectors (detailed)

  • Unit vectors denote directions with a magnitude of 1: $\hat{a}, \hat{b}, \hat{c}$, or the Cartesian basis $\hat{i}, \hat{j}, \hat{k}$.
  • Any vector can be written as a linear combination of unit vectors: $\boldsymbol{A} = A<em>x \hat{i} + A</em>y \hat{j} + A_z \hat{k}.$

• 1.2.6 Resolving a Vector (more details)

  • In 2-D, magnitude and direction can be obtained from components or from polar form:
  • Components: $A<em>x, A</em>y$; magnitude: $A = \sqrt{A<em>x^2 + A</em>y^2}$; direction: $\theta = \tan^{-1}(A<em>y/A</em>x)$.
  • In 3-D, magnitude and direction can be described with direction cosines (cosines of angles with axes) or with components along $\hat{i}, \hat{j}, \hat{k}$.

• 3 Example problems overview (summary from the transcript)

  • Example 1.2: Given two vectors, compute sums and differences, and the resulting magnitude and direction. Emphasizes component-wise calculation.
  • Example 1.3: Velocity components for a car moving at 50 m s$^{-1}$ in a direction described as north 30° east.
  • Components (one common convention):
    • East component: $v_x = v \cos(60^{\circ}) = 25\,\text{m s}^{-1}$
    • North component: $v_y = v \sin(60^{\circ}) \approx 43.3\,\text{m s}^{-1}$
  • Example 1.4: Force of 100 N at 150° from +x axis; components as shown above.
  • Example 1.5: Three-forces problem; compute the x- and y-components of each force, sum them, then obtain the resultant magnitude and direction. The partial numbers show typical steps (e.g., F1x, F1y, F2y, F3x) and the final resultant angle relative to +x axis.

• Summary of key formulas (concise reference)

  • Base quantities (examples): Length: $L = m$; Time: $t = s$; Mass: $m = kg$; Current: $I = A$; Temperature: $T = K$; Amount: $n = mol$.
  • Derived quantities (examples):
  • Velocity: $v = \frac{\Delta s}{\Delta t}, \ [v] = \text{m s}^{-1}$
  • Acceleration: $a = \frac{\Delta v}{\Delta t}, \ [a] = \text{m s}^{-2}$
  • Force: $F = m a \ [F] = \text{N}$
  • Work: $W = F \cdot s \ [W] = \text{N m} = \text{J}$
  • Momentum: $p = m v \ [p] = \text{kg m s}^{-1}$
  • Pressure: $P = \frac{F}{A} \ [P] = \text{N m}^{-2} = \text{Pa}$
  • Frequency: $f = \frac{1}{T} = s^{-1} = \text{Hz}$
  • Unit conversion: 1 cm$^3$ = 1e-6 m$^3$, 1 mm$^2$ = 1e-6 m$^2$, 1 g cm$^{-3}$ to kg m$^{-3}$: 1 g cm$^{-3}$ = 1000 kg m$^{-3}$, etc.
  • Vector basics:
  • Vector: $\boldsymbol{A} = A<em>x \hat{i} + A</em>y \hat{j} + A_z \hat{k}$
  • Magnitude: $|\boldsymbol{A}| = \sqrt{A<em>x^2 + A</em>y^2 + A_z^2}$
  • Unit vector: $\hat{u} = \frac{\boldsymbol{A}}{|\boldsymbol{A}|}$
  • Direction angle (2-D): $\theta = \tan^{-1}\left(\frac{A<em>y}{A</em>x}\right)$
  • Components (2-D): $A<em>x = A \cos\theta,\quad A</em>y = A \sin\theta$

• Notes on presentation and practice

  • Always check that vector components are in the same reference frame (same axes, same angle convention).
  • When converting units, track each factor carefully and verify the final unit matches the quantity being measured.
  • For multi-vector problems, decomposing into x- and y-components (and z, if needed) before combining is usually simplest.

• Quick reference figures mentioned in the transcript

  • Figure 1.1: Visual depiction of equal vectors and scalar vs. vector equality criteria (magnitude and direction both match).
  • Figure 1.2 to Figure 1.5: Illustrations of vector addition (parallelogram/triangle), vector subtraction, and component resolution (2-D and 3-D).

• Practical implications

  • Consistency in units ensures correct dimensional analysis and prevents errors in calculations for physics problems.
  • Understanding vector resolution is essential for solving problems in mechanics, projectile motion, forces, and dynamics.
  • The ability to decompose into components ties directly into Newton’s laws, dynamics, and work-energy analyses in real-world engineering and physics contexts.

• Connection to prior principles

  • Builds on the notion of measurement, accuracy, and SI units introduced earlier in physics education.
  • Uses foundational algebra and trigonometry (cosine, sine, tangent, Pythagoras) to manipulate vectors and coordinates.

• Ethical/philosophical/practical implications (brief)

  • Accurate measurement and careful unit handling reflect scientific integrity and reliability in modeling physical systems.
  • Misuse or sloppy handling of units can lead to unsafe designs in engineering, medicine, and technology; practice emphasizes discipline and attention to detail.

• Final quick recap

  • Distinguish base vs. derived quantities; know common SI units and prefixes.
  • Be fluent with vector notation, operations, and resolving vectors into components in 2-D and 3-D.
  • Practice area/volume-density-speed conversions to reinforce unit fluency.
  • Apply parallelogram/triangle methods for addition, and subtract by adding a negative.
  • Use unit vectors to express directions and magnitudes clearly in any dimension.