Physics is the quantitative study of matter, energy, space, and time. This chapter compiles fundamental physical constants, the International System of Units (SI), Newtonian mechanics, conservation principles, and the foundational equations of classical electromagnetism and wave dynamics.
1.1 The International System of Units (SI) — Base Units
| Physical Quantity | Unit Name | Symbol | Standard Definition Basis |
|---|
| Time | second | $\text{s}$ | Hyperfine transition frequency of Caesium-133 ($\Delta\nu_{\text{Cs}} = 9\,192\,631\,770\text{ Hz}$) |
| Length | meter | $\text{m}$ | Distance light travels in vacuum in $1/299\,792\,458$ seconds |
| Mass | kilogram | $\text{kg}$ | Fixed numerical value of the Planck constant ($h = 6.626\,070\,15 \times 10^{-34}\text{ J}\cdot\text{s}$) |
| Electric Current | ampere | $\text{A}$ | Elementary electric charge ($e = 1.602\,176\,634 \times 10^{-19}\text{ C}$) |
| Thermodynamic Temperature | kelvin | $\text{K}$ | Boltzmann constant ($k_B = 1.380\,649 \times 10^{-23}\text{ J/K}$) |
| Amount of Substance | mole | $\text{mol}$ | Avogadro constant ($N_A = 6.022\,140\,76 \times 10^{23}\text{ mol}^{-1}$) |
| Luminous Intensity | candela | $\text{cd}$ | Luminous efficacy of monochromatic radiation of frequency $540 \times 10^{12}\text{ Hz}$ |
1.2 Fundamental Physical Constants
| Constant | Symbol | Standard Value | SI Units |
|---|
| Speed of Light in Vacuum | $c$ | $299\,792\,458$ (exact) | $\text{m}\cdot\text{s}^{-1}$ |
| Gravitational Constant | $G$ | $6.674\,30(15) \times 10^{-11}$ | $\text{m}^3\cdot\text{kg}^{-1}\cdot\text{s}^{-2}$ ($\text{N}\cdot\text{m}^2/\text{kg}^2$) |
| Planck Constant | $h$ | $6.626\,070\,15 \times 10^{-34}$ (exact) | $\text{J}\cdot\text{s}$ |
| Reduced Planck Constant | $\hbar = h / (2\pi)$ | $1.054\,571\,817 \times 10^{-34}$ | $\text{J}\cdot\text{s}$ |
| Elementary Electric Charge | $e$ | $1.602\,176\,634 \times 10^{-19}$ (exact) | $\text{C}$ |
| Boltzmann Constant | $k_B$ | $1.380\,649 \times 10^{-23}$ (exact) | $\text{J}\cdot\text{K}^{-1}$ |
| Avogadro Constant | $N_A$ | $6.022\,140\,76 \times 10^{23}$ (exact) | $\text{mol}^{-1}$ |
| Universal Gas Constant | $R = N_A k_B$ | $8.314\,462\,618$ (exact) | $\text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$ |
| Permittivity of Free Space | $\varepsilon_0$ | $8.854\,187\,8128 \times 10^{-12}$ | $\text{F}\cdot\text{m}^{-1}$ ($\text{C}^2/\text{N}\cdot\text{m}^2$) |
| Permeability of Free Space | $\mu_0$ | $1.256\,637\,062 \times 10^{-6}$ | $\text{N}\cdot\text{A}^{-2}$ ($\text{T}\cdot\text{m}/\text{A}$) |
| Coulomb Constant | $k_e = \frac{1}{4\pi\varepsilon_0}$ | $8.987\,551\,792 \times 10^9$ | $\text{N}\cdot\text{m}^2\cdot\text{C}^{-2}$ |
| Standard Gravitational Acceleration | $g_0$ | $9.80665$ (exact) | $\text{m}\cdot\text{s}^{-2}$ |
| Electron Rest Mass | $m_e$ | $9.109\,383\,70 \times 10^{-31}$ | $\text{kg}$ |
| Proton Rest Mass | $m_p$ | $1.672\,621\,92 \times 10^{-27}$ | $\text{kg}$ |
| Neutron Rest Mass | $m_n$ | $1.674\,927\,49 \times 10^{-27}$ | $\text{kg}$ |
1.3 Newton's Three Laws of Motion
| Law | Formal Principle | Mathematical Statement | Physical Significance |
|---|
First Law
(Inertia) | An object remains at rest or in uniform motion at constant velocity unless acted upon by a net external force. | $\sum \vec{F} = 0 \implies \vec{v} = \text{constant}$ | Defines inertial reference frames; resistance of matter to acceleration. |
Second Law
(Force & Acceleration) | The rate of change of linear momentum equals the net external force applied to the body. | $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} = m\vec{a}$ (for constant $m$) | Quantifies dynamical motion; $1\text{ N} = 1\text{ kg}\cdot\text{m/s}^2$. |
Third Law
(Action-Reaction) | When one body exerts a force on a second body, the second body exerts an equal and opposite force on the first. | $\vec{F}_{A\rightarrow B} = -\vec{F}_{B\rightarrow A}$ | Forces occur exclusively in simultaneous pairs across distinct bodies. |
1.4 Kinematic Equations for Constant Acceleration
For one-dimensional rectilinear motion with initial velocity $v_0$, final velocity $v$, acceleration $a$, displacement $\Delta x$, and elapsed time $t$:
| Equation | Variables Present | Variable Omitted |
|---|
| $v = v_0 + at$ | $v, v_0, a, t$ | $\Delta x$ (Displacement) |
| $\Delta x = v_0 t + \frac{1}{2}at^2$ | $\Delta x, v_0, a, t$ | $v$ (Final velocity) |
| $v^2 = v_0^2 + 2a\Delta x$ | $v, v_0, a, \Delta x$ | $t$ (Elapsed time) |
| $\Delta x = \frac{v_0 + v}{2}t$ | $\Delta x, v_0, v, t$ | $a$ (Acceleration) |
| $\Delta x = vt - \frac{1}{2}at^2$ | $\Delta x, v, a, t$ | $v_0$ (Initial velocity) |
1.5 Work, Energy & Power
| Concept | Formula | Units | Notes |
|---|
| Work Done | $W = \vec{F} \cdot \vec{d} = F d \cos\theta = \int \vec{F}\cdot d\vec{r}$ | Joule ($\text{J}$) | Zero work if displacement is perpendicular to force. |
| Kinetic Energy (Translational) | $K = \frac{1}{2}mv^2 = \frac{p^2}{2m}$ | $\text{J}$ | Work-Energy Theorem: $W_{\text{net}} = \Delta K$. |
| Gravitational Potential Energy | $U_g = mgh$ (near surface)
$U_g = -\frac{G M m}{r}$ (general) | $\text{J}$ | Reference datum where $U=0$ is arbitrary locally or at $r\rightarrow\infty$ globally. |
| Elastic Potential Energy | $U_e = \frac{1}{2}kx^2$ | $\text{J}$ | Governed by Hooke's Law: $F_s = -kx$. |
| Conservation of Mechanical Energy | $E = K + U = \text{constant}$ | $\text{J}$ | Holds in isolated systems with conservative forces. |
| Power | $P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}$ | Watt ($\text{W} = \text{J/s}$) | $1\text{ horsepower} = 745.7\text{ W}$. |
1.6 Momentum, Collisions & Gravitation
| Law / Concept | Equation | Physical Meaning |
|---|
| Linear Momentum | $\vec{p} = m\vec{v}$ | Vector quantity conserved in isolated systems: $\sum \vec{p}_i = \sum \vec{p}_f$. |
| Impulse | $\vec{J} = \int \vec{F}\,dt = \vec{F}_{\text{avg}}\Delta t = \Delta \vec{p}$ | Change in momentum produced by a force over duration $\Delta t$. |
| Elastic Collision | $\Delta K = 0 \quad\text{and}\quad \Delta\vec{p} = 0$ | Both mechanical kinetic energy and linear momentum are conserved. |
| Inelastic Collision | $\Delta K \neq 0 \quad (\text{kinetic energy lost to heat/sound})$ | Momentum is strictly conserved; bodies stick together if perfectly inelastic. |
| Newton's Universal Gravitation | $F_g = G \frac{m_1 m_2}{r^2}$ | Inverse-square law of mutual attraction between masses. |
| Kepler's 3rd Law of Planetary Motion | $T^2 = \left(\frac{4\pi^2}{G M}\right) a^3$ | The square of orbital period is proportional to the cube of semi-major axis. |
1.7 Electromagnetism & Wave Mechanics
| Law | Equation | Significance |
|---|
| Coulomb's Law | $F_e = k_e \frac{|q_1 q_2|}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{|q_1 q_2|}{r^2}$ | Electrostatic force between point charges. |
| Ohm's Law | $V = I R$ | Voltage drop across an ideal ohmic resistor. |
| Electrical Power | $P = IV = I^2 R = \frac{V^2}{R}$ | Rate of electrical energy dissipation into heat or work. |
| Lorentz Force Law | $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$ | Total force on charged particle in electric and magnetic fields. |
| Wave Speed Relation | $v = f \lambda$ | Propagation speed equals frequency times wavelength. |
| Snell's Law of Refraction | $n_1 \sin\theta_1 = n_2 \sin\theta_2$ | Path of light crossing interface between refractive indices $n_1$ and $n_2$. |
| Planck-Einstein Relation | $E = hf = \frac{hc}{\lambda}$ | Energy of a photon proportional to electromagnetic frequency. |
| Mass-Energy Equivalence | $E = mc^2$ | Equivalence of rest mass and rest energy in special relativity. |