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): $10^{-3}$

• 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 mile = 1609 m = 1.609 km

• 1 inch = 2.54 cm

• 1 ft = 0.3048 m

To convert units, use conversion factors as multipliers. For example, to convert 3 minutes to seconds:

$3 \text{ min} \times \frac{60 \text{ s}}{1 \text{ min}} = 180 \text{ s}$

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 sig figs 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 within a factor of 10. Scientific notation is used to express large or small numbers conveniently.

• Example: $0.00635 = 6.35 \times 10^{-3}$

• Example: $4693 = 4.693 \times 10^{3}$

Dimensional Analysis

Principles of Dimensional Analysis

Dimensional analysis checks the consistency of equations and helps derive relationships between physical quantities. The fundamental dimensions are:

• Length (L)

• Mass (M)

• Time (T)

Examples of derived quantities:

• Velocity: $[v] = LT^{-1}$

• Acceleration: $[a] = LT^{-2}$

• Energy: $[E] = ML^{2}T^{-2}$

• Power: $[P] = ML^{2}T^{-3}$

• Force: $[F] = MLT^{-2}$

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 (Pi terms). This is useful for reducing the number of variables in a problem.

For example, if $F(X_1, X_2, ..., X_n) = 0$ and only $k$ of the $n$ parameters are independent, the equation can be rewritten in terms of $n-k$ dimensionless parameters.

Vectors and Scalars

Definitions

• Scalar quantity: Specified by a single value with units; has no direction (e.g., mass, temperature, energy).

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

Vectors are often denoted with arrows above the letter (handwritten) or boldface (printed): $\vec{v}$ or v.

Graphical Representation and Addition of Vectors

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

• Vectors can be added graphically using the head-to-tail method or parallelogram method.

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

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

Vector Subtraction and Scalar Multiplication

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

• 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

Any vector in two or three dimensions can be expressed in terms of its components along the coordinate axes:

• For a vector $\vec{A}$ in 2D: $A_x = A \cos \theta$, $A_y = A \sin \theta$

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$

• Direction: $\tan \theta = \frac{A_y}{A_x}$

In 3D, vectors are written as $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$, where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the x, y, and z axes.

Products of Vectors

Scalar (Dot) Product

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

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

• Result is a scalar (no direction).

• Can be positive, negative, or zero depending on the angle $\theta$ between the vectors.

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

Vector (Cross) Product

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

$\vec{C} = \vec{A} \times \vec{B}$

The magnitude is $C = AB \sin \theta$, where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$. The direction is given by the right-hand rule and is perpendicular to the plane containing $\vec{A}$ and $\vec{B}$.

• 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:

• Anticommutative: $\vec{A} \times \vec{B} = - (\vec{B} \times \vec{A})$

• Distributive over addition: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$

Right-Hand Rule

To determine the direction of the cross product, point the fingers of your right hand in the direction of $\vec{A}$, curl them toward $\vec{B}$, and your thumb points in the direction of $\vec{A} \times \vec{B}$.

Summary Table: SI Base Units

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

Example Applications

• Estimating the period of a pendulum using dimensional analysis.

• Calculating the resultant displacement of a person walking in multiple directions using vector components.

• Finding the torque produced by a force using the cross product.

Additional info: Some context and explanations have been expanded for clarity and completeness, including the summary table and example applications.