Notes: Units, Quantities, and Vectors

1.1 The Nature of Physics

• Physics is an experimental science based on measurements and observations.

• Physicists look for patterns in observations to construct theories of universal validity.

• Key assumptions:

  • The universe is comprehensible, logically consistent, rational, and explicable.

  • Physical objects, their motion, and their interactions are described by physical laws.

  • Experience shows that all known phenomena can be explained by physical laws that have been painstakingly deduced over centuries.

• Math is the language of physics and is used extensively in this course.

• Practice problem set assigned in MP (Math Practice).

1.2 Solving Physics Problems

• Problem-Solving Strategy (1.1 Solving Physics Problems)

  • IDENTIFY the relevant concepts:

  • Use the problem’s physical conditions to decide which physics concepts are relevant.

  • Identify the target variables: the quantities you are solving for (e.g., projectile speed at impact, siren intensity, image size by a lens).

  • Identify the known quantities (stated or implied).

  • SET UP the problem:

  • Choose equations that will be used; ensure variables match those in the equations.

  • If appropriate, draw a sketch of the situation.

  • Estimate results and predict physical behavior when possible; tips are given in worked examples.

  • EXECUTE the solution:

  • Perform the math; study the worked examples for guidance.

  • EVALUATE your answer:

  • Compare with estimates; reconsider if there is a discrepancy.

  • If an algebraic expression is involved, verify behavior when variables take extreme values.

  • Note quantities of particular significance and consider more general or harder versions of the problem.

1.3 Standards and Units

• Physical quantities can be expressed in terms of basic units:

  • Length: meter (m)

  • Time: second (s)

  • Mass: kilogram (kg) [Note: the pound is a unit of force, not mass]

• SI (Système International) (aka “metric” or “MKS”) is the primary system used in this course:

  • Length: meter (m)

  • Time: second (s)

  • Mass: kilogram (kg)

• Other unit systems:

  • CGS: centimeter (cm), gram (g), second (s)

  • Planck units: ħ = c = G = 1 (natural units) used in particle physics and astrophysics; momentum, position, time, and energy can be expressed in Planck units.

• Planck-scale context: physics of the very small, very fast, or very large is often described with these natural units.

• Momentum, position, time, and energy can be expressed in natural units when appropriate.

• Key note: math is essential to physics; quantitative rigor is central.

1.3 Prefixes and Power of Ten (Power of ten prefixes)

• Prefixes and their powers of ten:

  • atto- (a): 10^{-18}

  • femto- (f): 10^{-15}

  • pico- (p): 10^{-12}

  • nano- (n): 10^{-9}

  • micro- (μ): 10^{-6}

  • milli- (m): 10^{-3}

  • centi- (c): 10^{-2}

  • kilo- (k): 10^{3}

  • mega- (M): 10^{6}

  • giga- (G): 10^{9}

  • tera- (T): 10^{12}

  • peta- (P): 10^{15}

  • exa- (E): 10^{18}

• Note: prefixes help express large and small quantities succinctly in measurements and calculations.

1.3 References for Standards

• Time (second, s):

  • A standard reference can be determined by various methods (mean solar day, pendulum swing, quartz crystal vibrations, etc.).

  • Atomic clocks are the most accurate.

  • Current definition: 1 second = 9,192,631,770 oscillations of the cesium-133 atomic transition.

• Length (meter, m):

  • The standard uses the speed of light c and the second definition above.

  • The meter is defined as 1/299,792,458 of the distance traveled by light in vacuum in 1 second.

• Mass (kilogram, kg):

  • Defined by the International Prototype Kilogram (IPK) kept in Sèvres, France; tied to electrical constants and other fundamental constants like Planck’s constant.

  • Aims for long-term stability and reproducibility; modern references aim to be more fundamental.

• Note: 2018 context; references to institutions such as NIST provide high-precision realizations.

1.4 Unit Conversions

• Example: How fast in m/s is 70 mph?

  • 1 mile = 1600 meters

  • 1 hour = 3600 seconds

  • Calculation:
    $70.0 ext{ [mi hr}^{-1} ext{]} imes rac{1600 ext{ [m]}}{1 ext{ [mi]}} imes rac{1}{3600 ext{ [s]}} = 31.1 ext{ [m s}^{-1} ext{]}$

• Practical rule: convert to SI units early to avoid conversion errors when reporting results.

1.4 Unit Conversions and Derived Units

• In this course, questions can be posed in any unit system, but answers are typically reported in SI units: [m], [kg], [s].

• Many new quantities have their own derived units; e.g., Work has unit Joule (J).

• Work unit:

  • $ext{J} = ext{kg} \, ext{m}^2 \, ext{s}^{-2}$

• Summary: convert to SI units to avoid simple errors.

1.5 Uncertainty and Significant Figures

• All measured quantities have an associated uncertainty, e.g., a meter measure might be 135.3 ± 0.5 mm.

• Significant figures (sig figs): digits that carry meaning in a measurement value; rules:

  • All digits are significant except leading zeros, trailing zeros without explicit decimal significance, or digits propagated by calculations with higher precision than the original data.

• Arithmetic rules:

  • Multiplication and division: result cannot have more significant figures than the quantity with the fewest significant figures in the calculation.

  • Example:
    $34.4354 imes 3.5 = 120.5239 <br>ightarrow 120.$ or $1.2 imes 10^{2}$

  • Addition and subtraction: result is limited by the term with the fewest digits to the right of the decimal point.

  • Example:
    $34.4354 + 3.5 = 37.9354 <br>ightarrow 37.9$

• Example using circumference and diameter: if $C = frac{C}{D} D$, and given measurements, e.g., $C = 424 ext{ mm}, D = 135 ext{ mm}$ then
  $frac{C}{D} = frac{424}{135} ext{ (dimensionless)} \approx 3.14074… \text{Proper sig figs: } \pi ext{ (in context) } \approx 3.14.$

1.6 Estimates and Orders of Magnitude

• Estimation is a crucial skill for quick, rough calculations and plausibility checks.

• Back-of-the-envelope thinking can save time and guard against errors or misrepresentations.

• Common approach:

  • Step 1: Identify what you know.

  • Step 2: Use what you know to estimate what you don’t know.

  • Step 3: Do the arithmetic with reasonable rounding to maintain a plausible order of magnitude.

• Examples mentioned:

  • How long would it take to walk from Miami to Los Angeles? (order of magnitude estimate.)

  • How many dollars would completely cover the US mainland? (order of magnitude estimate.)

1.7 Vectors and Vector Addition

• Vectors: quantities with both magnitude and direction.

• Scalars: quantities with magnitude only.

• Notation: boldface with arrow (e.g., V), or letters like a, b, α, γ for scalars; bold with arrow for vectors.

• Vectors are position-independent; they are essential in physics and engineering.

• Representations:

  • Graphical and numerical representations.

  • Example representations:

  • $\boldsymbol{V} = 3 \boldsymbol{e}<em>x + 3 \boldsymbol{e}</em>y$

  • $\boldsymbol{V} = -3 \boldsymbol{i} + 3 \boldsymbol{j}$

  • $\boldsymbol{V} = 10 ext{ m}, heta = 150^ ext{°} ext{ (relative to horizontal)}$

• Vectors can be added graphically by placing them head-to-tail; order of addition does not affect the resultant vector.

  • For example:

  • If vectors are A, B, and C, then A + B = C, etc.

  • A − B = A + (−B)

1.8 Components of Vectors

• Any vector can be decomposed into components along coordinate axes (orthogonal, e.g., Cartesian axes).

• Two common notations:

  • $\boldsymbol{A} = A<em>x \boldsymbol{B x} \ \boldsymbol{A} = A</em>x \boldsymbol{B x} \, \oplus \, A_y \boldsymbol{B y}$

  • Magnitude-angle form: $|\boldsymbol{A}| = \sqrt{A<em>x^2 + A</em>y^2}, \tan \theta = \frac{A<em>y}{A</em>x}$ where θ is the angle relative to the x-axis.

• Components are convenient for algebraic manipulation and solving vector problems.

1.9 Unit Vectors

• Unit vectors have magnitude 1 and no units; they point along a coordinate axis.

• Examples: \hat{i}, \hat{j}, \hat{k} for x, y, z directions respectively.

• One unit vector per physical dimension; used to express vectors in components.

1.10 Products of Vectors

• Vectors can be multiplied to yield either a scalar or a vector:

• Dot product (scalar product):

  • $\boldsymbol{A} \cdot \boldsymbol{B} = |\boldsymbol{A}| |\boldsymbol{B}| \cos \theta$

  • It picks out the component of one vector along the other.

  • In component form: if \boldsymbol{A} = Ax \hat{i} + Ay \hat{j} + Az \hat{k} and \boldsymbol{B} = Bx \hat{i} + By \hat{j} + Bz \hat{k}, then
    $\boldsymbol{A} \cdot \boldsymbol{B} = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$

• Cross product (vector product):

  • $\boldsymbol{A} \times \boldsymbol{B} = |\boldsymbol{A}| |\boldsymbol{B}| \sin \theta \, \hat{n}$

  • The resulting vector is perpendicular to both input vectors; direction is given by the right-hand rule.

  • Magnitude is $|\boldsymbol{A} \times \boldsymbol{B}| = |\boldsymbol{A}| |\boldsymbol{B}| \sin \theta$.

  • Order matters: $\boldsymbol{A} \times \boldsymbol{B} \neq \boldsymbol{B} \times \boldsymbol{A}$ and $\boldsymbol{A} \times \boldsymbol{B} = -\boldsymbol{B} \times \boldsymbol{A}$.

• Cross product is an axial vector; its direction is perpendicular to both input vectors.

• Example rules for basis vectors:

  • $\hat{i} \times \hat{i} = 0, \quad \hat{j} \times \hat{j} = 0, \quad \hat{k} \times \hat{k} = 0$

  • $\hat{i} \times \hat{j} = \hat{k}, \quad \hat{j} \times \hat{k} = \hat{i}, \quad \hat{k} \times \hat{i} = \hat{j}$

• Component form for dot product:
  $\boldsymbol{A} \cdot \boldsymbol{B} = (A<em>x \hat{i} + A</em>y \hat{j} + A<em>z \hat{k}) \cdot (B</em>x \hat{i} + B<em>y \hat{j} + B</em>z \hat{k})$
  $= A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$

• Component form for cross product:
  $\boldsymbol{A} \times \boldsymbol{B} = (A<em>x \hat{i} + A</em>y \hat{j} + A<em>z \hat{k}) \times (B</em>x \hat{i} + B<em>y \hat{j} + B</em>z \hat{k})$
  $= (A<em>y B</em>z - A<em>z B</em>y) \hat{i} + (A<em>z B</em>x - A<em>x B</em>z) \hat{j} + (A<em>x B</em>y - A<em>y B</em>x) \hat{k}$

• Determinant method (determinant form) for cross product:

  • $\boldsymbol{A} \times \boldsymbol{B} = \begin{vmatrix} \hat{i} &amp; \hat{j} &amp; \hat{k} \\ A<em>x &amp; A</em>y &amp; A<em>z \\ B</em>x &amp; B<em>y &amp; B</em>z \end{vmatrix}$

  • This yields the component form above and works for any 3D vectors.

• Practical notes:

  • The determinant expression is a convenient computational alternative when not using the right-hand rule.

  • In many problems, you will choose the method (geometric intuition, component form, or determinant) that makes the solution simplest.

1.10 Determinant method: cross product (summary)

• The determinant method provides a compact, general approach to compute cross products in 3D:

  • $\boldsymbol{A} \times \boldsymbol{B} = \begin{vmatrix} \hat{i} &amp; \hat{j} &amp; \hat{k} \\ A<em>x &amp; A</em>y &amp; A<em>z \\ B</em>x &amp; B<em>y &amp; B</em>z \end{vmatrix}$

  • Result is a vector with components
    $(A<em>y B</em>z - A<em>z B</em>y) \hat{i} + (A<em>z B</em>x - A<em>x B</em>z) \hat{j} + (A<em>x B</em>y - A<em>y B</em>x) \hat{k}$

• This method reinforces the right-hand rule intuition and is widely used when dealing with multiple cross products or symbolic vectors.

Quick reference: common vector identities and conventions

• Dot product: $\boldsymbol{A} \cdot \boldsymbol{B} = |\boldsymbol{A}| |\boldsymbol{B}| \cos \theta$

• Cross product magnitude: $|\boldsymbol{A} \times \boldsymbol{B}| = |\boldsymbol{A}| |\boldsymbol{B}| \sin \theta$

• Cross product direction: perpendicular to both inputs, following the right-hand rule; sign changes with order.

• Unit vectors: $\hat{i}, \hat{j}, \hat{k}$ form the standard basis in 3D Cartesian coordinates.

• Angle relationships: $\tan \theta = \frac{A<em>y}{A</em>x}$ for components in the plane relative to the x-axis.

• Magnitude of a vector from components: $|\boldsymbol{A}| = \sqrt{A<em>x^2 + A</em>y^2 + A_z^2}$ (3D case; 2D uses two components).

Additional notes and conceptual connections

• The nature of physics relies on patterns and laws valid across scales; mathematics provides the precise language to describe these patterns.

• Standardization of units and careful attention to uncertainty are essential for credible measurements and comparisons.

• Vector algebra (addition, decomposition, and products) is foundational for describing physical quantities like force, velocity, momentum, and fields in multiple dimensions.

• Understanding SI base and derived units, as well as common prefixes, is critical for communicating physical quantities clearly and avoiding errors in calculations.

• Practical problem-solving approach (identify, set up, execute, evaluate) encourages systematic thinking and cross-checking results against physical intuition and estimates.