Units, Physical Quantities, and Vectors – Foundations of Physics

Units, Physical Quantities, and Vectors

Introduction to Physics and Its Branches

Physics is the experimental science that studies the nature and properties of matter and energy. It relies on fundamental observations, quantitative measurements, and mathematical language to describe natural phenomena. The field of physics is divided into several main areas:

• Classical Mechanics – studies motion and forces (Phys. 111)

• Electricity and Magnetism – studies electric and magnetic phenomena (Phys. 121)

• Waves and Optics (Phys. 234, 418)

• Relativity (Phys. 420, 421)

• Statistical Physics and Thermodynamics (Phys. 335)

• Quantum Mechanics (Phys. 442)

Classical mechanics focuses on the motion of physical bodies under the influence of forces, using laws of motion and conservation principles, and is valid when quantum and relativistic effects are negligible.

Standards and Units

Fundamental Physical Quantities and SI Units

Physics uses standard units to measure physical quantities. The International System of Units (SI) is the most widely used system. The three fundamental SI units are:

• Length – meter (m)

• Mass – kilogram (kg)

• Time – second (s)

Other SI base units include:

• Temperature – kelvin (K)

• Electric current – ampere (A)

• Luminous intensity – candela (cd)

• Amount of substance – mole (mol)

Prefixes for SI Units

Prefixes are used to denote multiples or submultiples of units:

• 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): $10^{9}$

Unit Systems and Conversions

Other unit systems include the U.S. customary system (foot, slug, second) and the cgs system (centimeter, gram, second). Conversion between units is essential in physics. For example:

• $1~ ext{mile} = 1609~ ext{m}$

• $1~ ext{ft} = 0.3048~ ext{m}$

• $1~ ext{in} = 2.54~ ext{cm}$

• $1~ ext{kg} = 2.205~ ext{lb}$

To convert units, use conversion factors as multipliers. For example, to convert 21.5 inches to centimeters:

• $21.5~ ext{in} imes \frac{2.54~ ext{cm}}{1~ ext{in}} = 54.6~ ext{cm}$

Uncertainty and Significant Figures

Measurement Uncertainty

All measurements have some degree of uncertainty, which is reflected in the number of significant figures (sig figs) reported.

• Significant Figures are the digits in a number that are known with certainty plus one estimated digit.

• Rules for significant figures:

  • All nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros are significant only if there is a decimal point.

• For multiplication/division: the result has as many significant figures as the factor with the fewest sig figs.

• For addition/subtraction: the result has as many decimal places as the term with the fewest decimal places.

Estimates and Orders of Magnitude

Order of Magnitude

An order of magnitude estimate is a rough approximation, usually to the nearest power of ten. It is useful for checking the plausibility of results and for making quick calculations.

• Example: $0.00635 = 6.35 \times 10^{-3}$ (order of magnitude: $10^{-3}$)

• Example: $4693~ ext{m} = 4.693 \times 10^{3}~ ext{m}$ (order of magnitude: $10^{3}$)

Dimensional Analysis

Principles of Dimensional Analysis

Dimensional analysis is a method to check the consistency of equations and to derive relationships between physical quantities. Each physical quantity can be expressed in terms of fundamental dimensions:

• Length (L)

• Mass (M)

• Time (T)

Examples:

• Velocity: $[v] = LT^{-1}$

• Acceleration: $[a] = LT^{-2}$

• Force: $[F] = MLT^{-2}$

• Energy: $[E] = ML^{2}T^{-2}$

• Power: $[P] = ML^{2}T^{-3}$

Equations must be dimensionally consistent: both sides must have the same dimensions.

Buckingham Pi Theorem

The Buckingham Pi theorem states that any physically meaningful equation involving a certain number of physical variables can be equivalently rewritten as an equation involving a set of dimensionless parameters (called Pi terms).

• If $N$ variables are involved and $k$ are independent dimensions, then there are $N-k$ dimensionless parameters.

• This theorem is useful for reducing the complexity of physical problems and for scaling analysis.

Vectors and Scalars

Definitions

• Scalar quantity: Specified by a single value with units (e.g., mass, temperature, energy).

• Vector quantity: Specified by both magnitude and direction (e.g., displacement, velocity, force).

Vectors are often denoted by boldface letters (e.g., v) or with an arrow above the letter (e.g., $\vec{v}$).

Graphical Representation and Addition of Vectors

• Vectors are represented as arrows; length indicates magnitude, arrowhead indicates direction.

• To add vectors graphically, use the head-to-tail method or the parallelogram method.

• Commutative Law: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$

• Associative Law: $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$

When adding more than two vectors, the order does not affect the resultant.

Vector Subtraction and Scalar Multiplication

• Subtracting $\vec{B}$ from $\vec{A}$ is equivalent to adding $-\vec{B}$ to $\vec{A}$: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

• Multiplying a vector by a positive scalar changes its magnitude but not its direction; multiplying by a negative scalar reverses its direction.

Components of Vectors

Vectors in two or three dimensions can be resolved into components along the coordinate axes:

• For a vector $\vec{A}$ at angle $\theta$ from the x-axis:

  • $A_x = A \cos \theta$

  • $A_y = A \sin \theta$

  • $A = \sqrt{A_x^2 + A_y^2}$

  • $\tan \theta = \frac{A_y}{A_x}$

• In three dimensions: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

Products of Vectors

Scalar (Dot) Product

The scalar product (dot product) of two vectors $\vec{A}$ and $\vec{B}$ is defined as:

• $\vec{A} \cdot \vec{B} = AB \cos \theta$

• Where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$

• In component form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

• The dot product is maximum when vectors are parallel ($\theta = 0^\circ$), zero when perpendicular ($\theta = 90^\circ$), and negative when antiparallel ($\theta = 180^\circ$).

Vector (Cross) Product

The vector product (cross product) of two vectors $\vec{A}$ and $\vec{B}$ is a vector $\vec{C}$ defined as:

• $\vec{C} = \vec{A} \times \vec{B}$

• $|\vec{C}| = AB \sin \theta$

• Direction is given by the right-hand rule.

• In component form:

• $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y) \hat{i} + (A_z B_x - A_x B_z) \hat{j} + (A_x B_y - A_y B_x) \hat{k}$

Properties:

• The cross product is anticommutative: $\vec{A} \times \vec{B} = - (\vec{B} \times \vec{A})$

• Unit vector cross products: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$

Applications of Vector Products

• Torque: $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector and $\vec{F}$ is the force.

• Angular momentum: $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{p}$ is the linear momentum.

Summary Table: SI Base Units

Physical Quantity	Unit Name	Unit Symbol
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

Problem-Solving Strategies in Physics

• Identify relevant concepts, target variables, and known quantities.

• Set up the problem: choose equations and draw a sketch.

• Execute the solution: perform calculations.

• Evaluate your answer: check for reasonableness and consistency.

Additional info:

• Some context and examples were inferred to clarify fragmented points and ensure completeness.

• Table entries and some formulae were reconstructed for clarity and academic completeness.