Chapter 1: Units, Physical Quantities, and Vectors – Study Notes

Units, Physical Quantities, and Vectors

Introduction to Physics

Physics is the study of the fundamental laws of nature, aiming to quantitatively explain physical phenomena using mathematical equations and to compare theoretical predictions with experimental results.

• Physical Theories: Patterns that relate natural phenomena.

• Physical Laws or Principles: Well-established or widely used theories.

The Scientific Method in Physics

The scientific method is a systematic approach to investigating natural phenomena.

• Conduct systematic, reproducible, and quantitative observations and measurements.

• Identify relevant parameters and observables.

• Define models (mathematical, geometrical, conceptual) to describe relationships.

• Develop general theories ("Laws") to organize and explain observations.

• Derive and test predictions.

Measurements and Observables

Measurements are essential for quantifying physical phenomena. Observables are quantities that can be measured and require operational definitions.

• Examples: Distance, time, mass.

• Most observables have a dimension and a unit.

Basic Physical Quantities and SI Units

The laws of physics are expressed in terms of basic quantities, each with a standard unit in the International System (SI):

• Length: meter (m)

• Mass: kilogram (kg)

• Time: second (s)

These units are internationally agreed upon to ensure consistency in scientific communication.

Examples of Basic Quantities

• Distance: Dimension is length; unit is meter (m). Defined by comparison with a standard meter.

• Duration: Dimension is time; unit is second (s). Defined by the period of oscillation of a cesium clock.

• Mass (Inertia): Dimension is mass; unit is kilogram (kg). Defined by comparison with a standard kilogram.

Standards and Units

• SI units are used for all measurements in physics.

• Other units (e.g., inches, pounds, quarts) are sometimes encountered, especially in the United States.

• Conversion between units is often necessary.

Scientific Notation and Prefixes

Scientific notation and prefixes are used to express very large or very small numbers efficiently.

• Scientific Notation: $3 \times 10^3 = 3000$

• Prefixes: centi ($10^{-2}$, c), kilo ($10^3$, k), micro ($10^{-6}$, \mu), etc.

Power of Ten	Prefix	Symbol
$10^{-12}$	pico	p
$10^{-9}$	nano	n
$10^{-6}$	micro	\mu
$10^{-3}$	milli	m
$10^{-2}$	centi	c
$10^3$	kilo	k
$10^6$	mega	M
$10^9$	giga	G

Unit Consistency and Conversion

Always ensure that units are consistent in calculations. Use conversion factors to change from one unit to another.

• Example: $1.25$ hours $\times (60 \ \text{min/hour}) \times (60 \ \text{s/min}) = 4.50 \times 10^3$ s

• Conversion factors are ratios equal to one, used to change units without altering the value.

Dimensional Analysis

Dimensional analysis checks that the dimensions on both sides of an equation are consistent.

• For example, velocity $v$ has dimensions $[L][T]^{-1}$, and area $A$ has dimensions $[L]^2$.

• Only quantities with the same dimensions can be added or equated.

Precision, Uncertainty, and Significant Figures

Measurements have limitations, and results must reflect these limitations using significant figures.

• Significant Figures: The digits in a measurement that are known with certainty plus one estimated digit.

• For multiplication/division: The result should have as many significant figures as the input with the least number.

• For addition/subtraction: The result should have as many decimal places as the input with the least decimal places.

• Express uncertainties as $x \pm \Delta x$.

Example: $L = 16.3$ cm (3 sig. figs.), $W = 4.5$ cm (2 sig. figs.) $\Rightarrow$ Area $= 73$ cm$^2$

Estimates and Order of Magnitude

Order-of-magnitude estimates are rough calculations, accurate within a factor of 10, useful for checking plausibility.

• Example: $110$students$\approx 100$ students

• Size of an atom $\approx 10^{-10}$ m

Notation in Physics

• $\propto$ : proportional to

• $<$ : less than

• $>$ : greater than

• $\ll$ : much less than

• $\approx$ : approximately equal

• $\Delta x$ : change in $x$

• $|x|$ : magnitude of $x$

• $\sum x_i$ : sum over $x_i$

Vectors and Scalars

Scalars

A scalar is a physical quantity described by magnitude only (with units).

• Examples: Temperature, mass, time, energy.

• Scalars add like ordinary numbers.

Vectors

A vector is a physical quantity described by both magnitude and direction.

• Examples: Displacement, velocity, force, acceleration.

• Vectors are represented by boldface letters or with an arrow above the symbol (e.g., $\vec{A}$).

• The length of the arrow represents magnitude; the direction of the arrow shows direction.

Specifying Vectors

• By magnitude and direction (e.g., "4 m at 30° above the x-axis").

• By components in a Cartesian coordinate system: $\vec{A} = (A_x, A_y, A_z)$.

• Use trigonometry to convert between forms: $A_x = A \cos \theta$, $A_y = A \sin \theta$.

Unit Vectors

Unit vectors have magnitude 1 and indicate direction along coordinate axes.

• Common unit vectors: $\hat{i}$ (x-axis), $\hat{j}$ (y-axis), $\hat{k}$ (z-axis).

• Any vector can be written as $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$.

Vector Addition and Subtraction

• Graphically: Use the "tip-to-tail" or parallelogram method.

• Algebraically: Add corresponding components: $\vec{A} + \vec{B} = (A_x + B_x, A_y + B_y, A_z + B_z)$.

• Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

• Vector subtraction: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

Multiplication of Vectors

There are two main ways to multiply vectors:

• Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = AB \cos \theta$ (results in a scalar)

• Vector (Cross) Product: $\vec{A} \times \vec{B} = AB \sin \theta \; \hat{n}$ (results in a vector perpendicular to both $\vec{A}$ and $\vec{B}$)

Where $\theta$ is the angle between the two vectors, and $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$ (right-hand rule).

Operation	Result	Formula
Dot Product	Scalar	$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$
Cross Product	Vector	$\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$

Properties of Vector Operations

• Dot product is commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$

• Cross product is anti-commutative: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$

• Cross product of parallel vectors is zero.

Problem Solving in Physics

Effective problem solving involves:

• Identifying relevant concepts and quantities.

• Setting up equations using symbols and units.

• Carrying out calculations with correct significant figures and units.

• Checking results for consistency and plausibility (order of magnitude).

Important: Avoid "plug and chug"; focus on understanding and logical reasoning.

Summary Table: SI Base Units

Quantity	Unit Name	Unit Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s

Example: To convert 8.5 inches to centimeters:

• $8.5 \; \text{in} \times \frac{2.54 \; \text{cm}}{1 \; \text{in}} = 22 \; \text{cm}$

Additional info: These notes provide foundational concepts for all subsequent topics in introductory physics, including the importance of units, dimensional analysis, and vector operations.