Halliday Physics Essentials

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### Kinematics #### 1D Motion - **Position:** $x(t)$ - **Velocity:** $v = \frac{dx}{dt}$ - **Acceleration:** $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ - **Constant Acceleration Formulas:** - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 2D/3D Motion (Vectors) - **Position Vector:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - **Velocity Vector:** $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ - **Acceleration Vector:** $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ - **Projectile Motion (ignoring air resistance):** - $v_x = v_{0x}$ (constant) - $x = x_0 + v_{0x}t$ - $v_y = v_{0y} - gt$ - $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ - **Uniform Circular Motion:** - **Speed:** $v = \frac{2\pi r}{T}$ - **Centripetal Acceleration:** $a_c = \frac{v^2}{r} = \omega^2 r$ (directed towards center) ### Newton's Laws of Motion - **Newton's First Law (Inertia):** An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. - **Newton's Second Law:** $\vec{F}_{net} = m\vec{a}$ - $\sum \vec{F} = m\vec{a}$ - **Weight:** $\vec{W} = m\vec{g}$ - **Newton's Third Law:** If object A exerts a force $\vec{F}_{AB}$ on object B, then object B exerts an equal and opposite force $\vec{F}_{BA}$ on object A: $\vec{F}_{AB} = -\vec{F}_{BA}$. #### Forces - **Friction:** - **Static Friction:** $f_s \le \mu_s N$ (prevents motion) - **Kinetic Friction:** $f_k = \mu_k N$ (opposes motion) - $\mu_s > \mu_k$ - **Tension:** Force transmitted through a string/cable. - **Normal Force:** Force perpendicular to a surface, preventing penetration. ### Work & Energy - **Work done by a constant force:** $W = \vec{F} \cdot \Delta\vec{r} = F\Delta r \cos\theta$ - **Work done by a variable force:** $W = \int \vec{F} \cdot d\vec{r}$ - **Kinetic Energy:** $K = \frac{1}{2}mv^2$ - **Work-Energy Theorem:** $W_{net} = \Delta K$ - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ - **Potential Energy:** - **Gravitational (near Earth):** $U_g = mgh$ - **Elastic (Spring):** $U_s = \frac{1}{2}kx^2$ - **Conservation of Mechanical Energy:** $E_{mech} = K + U$ - If only conservative forces do work: $E_{mech,i} = E_{mech,f}$ - If non-conservative forces do work: $W_{nc} = \Delta E_{mech}$ ### Momentum & Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta\vec{p}$ - **Impulse-Momentum Theorem:** $\vec{F}_{avg}\Delta t = \Delta\vec{p}$ - **Conservation of Linear Momentum:** If $\vec{F}_{net,ext} = 0$, then $\vec{P}_{total,i} = \vec{P}_{total,f}$. - **Collisions:** - **Elastic:** Both momentum and kinetic energy are conserved. - **Inelastic:** Momentum conserved, kinetic energy is NOT conserved. - **Completely Inelastic:** Objects stick together after collision. - **Center of Mass:** - $x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$ - $\vec{v}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$ - $\vec{F}_{net,ext} = M_{total}\vec{a}_{CM}$ ### Rotational Motion - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ - **Constant Angular Acceleration Formulas:** - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ - **Relating Linear & Angular Variables:** - $s = r\theta$ - $v_t = r\omega$ (tangential speed) - $a_t = r\alpha$ (tangential acceleration) - $a_c = \frac{v^2}{r} = r\omega^2$ (centripetal acceleration) - **Moment of Inertia:** $I = \sum m_i r_i^2 = \int r^2 dm$ - Parallel-Axis Theorem: $I = I_{CM} + Md^2$ - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ - $\tau = rF\sin\phi$ - **Newton's Second Law for Rotation:** $\tau_{net} = I\alpha$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Work & Power in Rotation:** - $W = \int \tau d\theta$ - $P = \tau\omega$ - **Angular Momentum:** $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ - **Conservation of Angular Momentum:** If $\vec{\tau}_{net,ext} = 0$, then $\vec{L}_{total,i} = \vec{L}_{total,f}$. ### Gravitation - **Newton's Law of Universal Gravitation:** $F = G\frac{m_1 m_2}{r^2}$ - $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ - **Gravitational Potential Energy:** $U = -G\frac{m_1 m_2}{r}$ - **Escape Speed:** $v_{esc} = \sqrt{\frac{2GM}{R}}$ - **Kepler's Laws:** 1. **Law of Orbits:** Planets move in elliptical orbits with the Sun at one focus. 2. **Law of Areas:** A line connecting a planet to the Sun sweeps out equal areas in equal times. 3. **Law of Periods:** $T^2 \propto a^3$ (for circular orbits, $T^2 = (\frac{4\pi^2}{GM})r^3$) ### Oscillations & Waves #### Simple Harmonic Motion (SHM) - **Displacement:** $x(t) = A\cos(\omega t + \phi)$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (mass-spring system) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Simple Pendulum (small angles):** $T = 2\pi\sqrt{\frac{L}{g}}$ - **Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgd}}$ #### Waves - **Wave Speed:** $v = \lambda f = \frac{\omega}{k}$ - **Transverse Wave on a String:** $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear density) - **Sound Wave Speed:** $v = \sqrt{\frac{B}{\rho}}$ (B = bulk modulus, $\rho$ = density) - **Intensity:** $I = \frac{P}{A}$ - **Sound Level (decibels):** $\beta = 10 \log_{10}(\frac{I}{I_0})$ ($I_0 = 10^{-12} \text{ W/m}^2$) - **Doppler Effect:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ (top signs for approaching, bottom for receding) - **Standing Waves:** - **String fixed at both ends:** $L = n\frac{\lambda}{2}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) - **Open-Open/Closed-Closed Pipe:** $L = n\frac{\lambda}{2}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) - **Open-Closed Pipe:** $L = (2n-1)\frac{\lambda}{4}$, $f_n = (2n-1)\frac{v}{4L}$ ($n=1,2,3...$) ### Thermodynamics - **Temperature Scales:** - $T_F = \frac{9}{5}T_C + 32^\circ$ - $T_K = T_C + 273.15$ - **Thermal Expansion:** - **Linear:** $\Delta L = L\alpha\Delta T$ - **Volume:** $\Delta V = V\beta\Delta T \approx 3V\alpha\Delta T$ - **Heat Capacity & Latent Heat:** - **Heat Transfer:** $Q = mc\Delta T$ (no phase change) - **Phase Change:** $Q = mL_F$ (fusion), $Q = mL_V$ (vaporization) - **First Law of Thermodynamics:** $\Delta E_{int} = Q - W$ - $Q$: Heat added to system - $W$: Work done BY system - **Work done by gas (constant pressure):** $W = P\Delta V$ - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.314 \text{ J/(mol}\cdot\text{K)}$ - $k = 1.38 \times 10^{-23} \text{ J/K}$ - **Kinetic Theory of Gases:** - **Average Kinetic Energy:** $K_{avg} = \frac{3}{2}kT$ (for monatomic gas) - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}}$ (M = molar mass in kg/mol) - **Second Law of Thermodynamics:** - **Heat Engines:** $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - **Carnot Engine (ideal):** $\epsilon_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** $K = \frac{|Q_C|}{|W|}$ - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ - For an isolated system, $\Delta S \ge 0$. ### Electromagnetism #### Electrostatics - **Coulomb's Law:** $F = k\frac{|q_1 q_2|}{r^2}$ - $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ - Point Charge: $E = k\frac{|q|}{r^2}$ - **Electric Potential:** $V = \frac{U}{q_0}$ - Point Charge: $V = k\frac{q}{r}$ - $\vec{E} = -\nabla V$ - **Capacitance:** $C = \frac{Q}{V}$ - Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ - Energy Stored: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ - **Dielectrics:** $C = \kappa C_0$ #### Current & Resistance - **Current:** $I = \frac{dQ}{dt}$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho\frac{L}{A}$ - **Power:** $P = IV = I^2R = \frac{V^2}{R}$ - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ - **Capacitors in Parallel:** $C_{eq} = C_1 + C_2 + ...$ - **RC Circuits:** - **Charging:** $Q(t) = Q_{max}(1 - e^{-t/\tau})$, $\tau = RC$ - **Discharging:** $Q(t) = Q_0 e^{-t/\tau}$ #### Magnetism - **Magnetic Force on a Moving Charge:** $\vec{F}_B = q(\vec{v} \times \vec{B})$ - **Magnetic Force on a Current-Carrying Wire:** $\vec{F}_B = I(\vec{L} \times \vec{B})$ - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ - **Magnetic Field from Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field at center of Loop:** $B = \frac{\mu_0 I}{2R}$ - **Magnetic Field of Solenoid:** $B = \mu_0 n I$ ($n=$ turns per unit length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - **Faraday's Law of Induction:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - **Lenz's Law:** Induced current/EMF opposes the change in magnetic flux. - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Solenoid Inductance:** $L = \mu_0 n^2 A l$ - **Energy Stored in Inductor:** $U = \frac{1}{2}LI^2$ - **RL Circuits:** Current growth: $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$, $\tau = L/R$ - **LC Oscillations:** $\omega = \frac{1}{\sqrt{LC}}$ - **Maxwell's Equations (integral form):** 1. $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ (Gauss' Law for Electricity) 2. $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss' Law for Magnetism) 3. $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (Faraday's Law) 4. $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (Ampere-Maxwell Law) ### Optics #### Electromagnetic Waves - **Speed of Light:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$ - **Relationship:** $c = \lambda f$ - **Poynting Vector:** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (direction of energy flow) - **Intensity:** $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{E_{max}B_{max}}{2\mu_0}$ #### Geometric Optics - **Law of Reflection:** $\theta_i = \theta_r$ - **Law of Refraction (Snell's Law):** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Critical Angle:** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) - **Thin Lens/Mirror Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - **Magnification:** $m = -\frac{i}{p} = \frac{h_i}{h_p}$ - **Sign Conventions:** - $p$: positive if object is real (in front of lens/mirror) - $i$: positive if image is real (behind lens, in front of mirror) - $f$: positive for converging lens/concave mirror - $h$: positive if upright #### Wave Optics - **Young's Double-Slit Experiment:** - **Bright Fringes:** $d\sin\theta = m\lambda$ ($m=0, \pm1, \pm2, ...$) - **Dark Fringes:** $d\sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm1, \pm2, ...$) - **Single-Slit Diffraction:** - **Dark Fringes:** $a\sin\theta = m\lambda$ ($m=\pm1, \pm2, ...$) - **Diffraction Grating:** - **Bright Fringes:** $d\sin\theta = m\lambda$ ($m=0, \pm1, \pm2, ...$) - **Thin Film Interference (for normal incidence):** - **Reflected Light:** - **Constructive:** $2n_{film}t = (m + \frac{1}{2})\lambda$ (if 1 or 3 phase shifts) - **Destructive:** $2n_{film}t = m\lambda$ (if 0 or 2 phase shifts) - Phase shift occurs upon reflection if $n_{incident} ### Modern Physics #### Relativity - **Lorentz Factor:** $\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$ - **Length Contraction:** $L = L_0/\gamma$ - **Time Dilation:** $\Delta t = \gamma \Delta t_0$ - **Relativistic Momentum:** $p = \gamma mv$ - **Relativistic Energy:** $E = \gamma mc^2$ - **Rest Energy:** $E_0 = mc^2$ - **Kinetic Energy:** $K = (\gamma - 1)mc^2$ #### Quantum Physics - **Planck's Quantum Hypothesis:** $E = hf$ - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($\Phi = $ work function) - **Photon Momentum:** $p = \frac{h}{\lambda}$ - **De Broglie Wavelength:** $\lambda = \frac{h}{p}$ - **Heisenberg Uncertainty Principle:** - $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - $\Delta E \Delta t \ge \frac{\hbar}{2}$ - $\hbar = \frac{h}{2\pi}$ - **Schrödinger Equation:** (Time-independent 1D) $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ - **Probability Density:** $|\psi(x)|^2$ - **Quantization of Energy (e.g., Hydrogen Atom):** $E_n = -\frac{13.6 \text{ eV}}{n^2}$ #### Nuclear Physics - **Mass Defect & Binding Energy:** $\Delta E_B = \Delta m c^2$ - $\Delta m = (Z m_p + N m_n) - M_{nucleus}$ - **Radioactive Decay:** $N(t) = N_0 e^{-\lambda t}$ - **Half-life:** $T_{1/2} = \frac{\ln 2}{\lambda}$