Halliday Physics Fundamentals

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1. Measurement & Vectors SI Base Units: Length (m), Mass (kg), Time (s), Current (A), Temp (K), Amount (mol), Luminous Intensity (cd) Prefixes: Giga ($10^9$), Mega ($10^6$), Kilo ($10^3$), Centi ($10^{-2}$), Milli ($10^{-3}$), Micro ($10^{-6}$), Nano ($10^{-9}$), Pico ($10^{-12}$) Vector Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\theta = \tan^{-1}(A_y/A_x)$ Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \phi = A_x B_x + A_y B_y + A_z B_z$ Cross Product Magnitude: $|\vec{A} \times \vec{B}| = AB \sin \phi$ Cross Product Vector: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ 2. Kinematics (1D & 2D) Position: $x(t)$ Displacement: $\Delta x = x_f - x_i$ Average Velocity: $v_{avg} = \Delta x / \Delta t$ Instantaneous Velocity: $v = dx/dt$ Average Acceleration: $a_{avg} = \Delta v / \Delta t$ Instantaneous Acceleration: $a = dv/dt = d^2x/dt^2$ Constant Acceleration Equations: $v = v_0 + at$ $x - x_0 = v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ $x - x_0 = vt - \frac{1}{2}at^2$ Free Fall: $a = -g = -9.8 \text{ m/s}^2$ (downward) Projectile Motion (Horizontal x, Vertical y): $v_x = v_{0x} = v_0 \cos \theta_0$ $x = x_0 + v_{0x} t$ $v_y = v_{0y} - gt = v_0 \sin \theta_0 - gt$ $y = y_0 + v_{0y} t - \frac{1}{2}gt^2$ $v_y^2 = v_{0y}^2 - 2g(y - y_0)$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for $y_0=0$) 3. Newton's Laws 1st Law: Object at rest stays at rest, object in motion stays in motion with constant velocity, unless acted on by a net force. 2nd Law: $\vec{F}_{net} = m\vec{a}$ Weight: $W = mg$ 3rd Law: If A exerts force on B, B exerts equal and opposite force on A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) Friction: Static: $f_s \le \mu_s N$ Kinetic: $f_k = \mu_k N$ Usually $\mu_s > \mu_k$ Tension: Force transmitted through a string/rope. Normal Force: Perpendicular force from a surface. Centripetal Force: $F_c = \frac{mv^2}{r}$ (directed towards center of circle) 4. Work, Energy & Power Work Done by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd \cos \phi$ Work Done by Variable Force (1D): $W = \int_{x_i}^{x_f} F(x) dx$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ (Hooke's Law: $F_s = -kx$) Conservative Forces: Work done is path independent (e.g., gravity, spring). $W_c = -\Delta U$. Non-Conservative Forces: Work done is path dependent (e.g., friction). Conservation of Mechanical Energy: $E_{mech} = K + U = \text{constant}$ (if only conservative forces do work) General Conservation of Energy: $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ Power (Average): $P_{avg} = W/\Delta t$ Power (Instantaneous): $P = dW/dt = \vec{F} \cdot \vec{v}$ 5. Momentum & Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Newton's 2nd Law (Momentum Form): $\vec{F}_{net} = d\vec{p}/dt$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum: If $\vec{F}_{net, ext} = 0$, then $\vec{P}_{total} = \text{constant}$ Collisions: Elastic: Both momentum and kinetic energy are conserved. Inelastic: Momentum conserved, kinetic energy NOT conserved. Perfectly Inelastic: Objects stick together; momentum conserved, max KE loss. Center of Mass (CM): Discrete Particles: $x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$ Continuous Body: $x_{CM} = \frac{1}{M} \int x \, dm$ $\vec{P}_{total} = M \vec{v}_{CM}$ $\vec{F}_{net, ext} = M \vec{a}_{CM}$ 6. Rotational Motion Angular Position: $\theta$ (radians) Angular Displacement: $\Delta \theta$ Average Angular Velocity: $\omega_{avg} = \Delta \theta / \Delta t$ Instantaneous Angular Velocity: $\omega = d\theta/dt$ Average Angular Acceleration: $\alpha_{avg} = \Delta \omega / \Delta t$ Instantaneous Angular Acceleration: $\alpha = d\omega/dt$ Rotational Kinematics (Constant $\alpha$): $\omega = \omega_0 + \alpha t$ $\theta - \theta_0 = \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Linear-Angular Relations: $s = r\theta$ (arc length) $v_t = r\omega$ (tangential speed) $a_t = r\alpha$ (tangential acceleration) $a_r = v_t^2/r = r\omega^2$ (radial/centripetal acceleration) Rotational Inertia (Moment of Inertia): $I = \sum m_i r_i^2 = \int r^2 dm$ Parallel-Axis Theorem: $I = I_{CM} + Md^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F} \implies \tau = rF \sin \phi$ Newton's 2nd Law for Rotation: $\tau_{net} = I\alpha$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Work in Rotation: $W = \int_{\theta_i}^{\theta_f} \tau d\theta$ Power in Rotation: $P = \tau \omega$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) Newton's 2nd Law (Angular Momentum Form): $\vec{\tau}_{net} = d\vec{L}/dt$ Conservation of Angular Momentum: If $\vec{\tau}_{net, ext} = 0$, then $\vec{L}_{total} = \text{constant}$ 7. Gravity Newton's Law of 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: 1st: Orbits are ellipses with sun at one focus. 2nd: Line connecting planet to sun sweeps equal areas in equal times. 3rd: $T^2 \propto a^3$ (for circular orbit $T^2 = (\frac{4\pi^2}{GM})r^3$) 8. Oscillations & Waves Simple Harmonic Motion (SHM): Position: $x(t) = x_m \cos(\omega t + \phi)$ Angular Frequency: $\omega = \sqrt{k/m}$ (spring) or $\omega = \sqrt{g/L}$ (pendulum) Frequency: $f = \omega / (2\pi)$ Period: $T = 1/f = 2\pi/\omega$ Velocity: $v(t) = -\omega x_m \sin(\omega t + \phi)$ Acceleration: $a(t) = -\omega^2 x_m \cos(\omega t + \phi) = -\omega^2 x(t)$ Energy: $E = \frac{1}{2}kx_m^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ Traveling Wave: $y(x,t) = y_m \sin(kx \pm \omega t + \phi)$ Wave Number: $k = 2\pi/\lambda$ Wave Speed: $v = \lambda f = \omega/k$ Speed on Stretched String: $v = \sqrt{\tau/\mu}$ ($\tau$ tension, $\mu$ linear density) Superposition: $y_{net} = y_1 + y_2$ Standing Waves: Formed by superposition of two identical waves traveling in opposite directions. Strings Fixed at Both Ends: $\lambda_n = 2L/n$, $f_n = n(v/2L) = n f_1$ ($n=1,2,3...$) 9. Thermodynamics Temperature Scales: $T_F = \frac{9}{5}T_C + 32^\circ$ $T_K = T_C + 273.15$ Thermal Expansion: Length: $\Delta L = L \alpha \Delta T$ Volume: $\Delta V = V \beta \Delta T$ ($\beta \approx 3\alpha$) Heat Capacity: $Q = C \Delta T$ Specific Heat: $Q = c m \Delta T$ Latent Heat (Phase Change): $Q = L m$ ($L_f$ fusion, $L_v$ vaporization) Heat Transfer: Conduction: $P_{cond} = \frac{Q}{t} = k A \frac{T_H - T_C}{L}$ Radiation: $P_{rad} = \sigma \epsilon A T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/(m}^2\text{K}^4)$) First Law of Thermodynamics: $\Delta E_{int} = Q - W$ ($Q$ heat added, $W$ work done by system) Work Done by Gas: $W = \int P dV$ 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: $K_{avg} = \frac{3}{2}kT$ (per molecule) RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ ($M$ molar mass in kg/mol) Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of an isolated system never decreases. $\Delta S \ge 0$. Entropy Change: $\Delta S = \int dQ/T$ (reversible) Heat Engine Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ Carnot Engine Efficiency (Ideal): $\epsilon_C = 1 - \frac{T_C}{T_H}$ Refrigerator Coefficient of Performance: $K = \frac{|Q_C|}{|W|}$ Carnot Refrigerator COP: $K_C = \frac{T_C}{T_H - T_C}$ 10. Electric Fields & Forces Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ ($k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) Electric Field: $\vec{E} = \vec{F}/q_0$ Field of Point Charge: $E = k \frac{|q|}{r^2}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on Dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ Gauss' Law: $\oint \vec{E} \cdot d\vec{A} = Q_{enc}/\epsilon_0$ 11. Electric Potential Potential Energy: $\Delta U = -W = - \int_i^f \vec{F} \cdot d\vec{s}$ Electric Potential: $V = U/q_0$ Potential Difference: $\Delta V = V_f - V_i = - \int_i^f \vec{E} \cdot d\vec{s}$ Potential of Point Charge: $V = k q/r$ Relation E and V: $E_x = -\partial V/\partial x$, $\vec{E} = -\nabla V$ Capacitance: $C = Q/V$ Parallel Plate Capacitor: $C = \epsilon_0 A/d$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Capacitors in Series: $1/C_{eq} = \sum 1/C_i$ Energy Stored in Capacitor: $U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ Dielectric Constant: $C = \kappa C_0$ ($C_0$ vacuum capacitance) 12. Current & Resistance Electric Current: $I = dQ/dt$ Current Density: $\vec{J} = n q \vec{v}_d$ ($n$ charge carriers/volume, $v_d$ drift speed) Ohm's Law: $V = IR$ Resistance: $R = \rho L/A$ ($\rho$ resistivity) Resistivity Temperature Dependence: $\rho - \rho_0 = \rho_0 \alpha (T - T_0)$ Power in Circuits: $P = IV = I^2R = V^2/R$ Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $1/R_{eq} = \sum 1/R_i$ Kirchhoff's Rules: Junction Rule: $\sum I_{in} = \sum I_{out}$ Loop Rule: $\sum \Delta V = 0$ around any closed loop RC Circuits (Charging Capacitor): $q(t) = C\mathcal{E}(1 - e^{-t/RC})$ RC Circuits (Discharging Capacitor): $q(t) = Q_0 e^{-t/RC}$ Time Constant: $\tau = RC$ 13. Magnetic Fields & Forces Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ ($F_B = |q|vB \sin \phi$) Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ ($F_B = ILB \sin \phi$) Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NIA\hat{n}$ Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = I B / (ne L)$ 14. Sources of Magnetic Fields Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ ($\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$) Magnetic Field of Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Force Between Two Parallel Wires: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ Magnetic Field at Center of Circular Loop: $B = \frac{\mu_0 I}{2R}$ Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Solenoid: $B = \mu_0 n I$ ($n$ turns/length) Toroid: $B = \frac{\mu_0 N I}{2\pi r}$ 15. Electromagnetic Induction Faraday's Law of Induction: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$ Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Motional EMF: $\mathcal{E} = BLv$ Lenz's Law: Induced current's magnetic field opposes the change in magnetic flux. Inductance: $L = N\Phi_B/I$ Solenoid Inductance: $L = \mu_0 n^2 A l$ Inductors in Series: $L_{eq} = \sum L_i$ Inductors in Parallel: $1/L_{eq} = \sum 1/L_i$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Energy Density of Magnetic Field: $u_B = B^2/(2\mu_0)$ RL Circuits (Current Rise): $I(t) = (\mathcal{E}/R)(1 - e^{-t/\tau_L})$ RL Circuits (Current Decay): $I(t) = I_0 e^{-t/\tau_L}$ Time Constant: $\tau_L = L/R$ Mutual Inductance: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ 16. Electromagnetic Waves Maxwell's Equations (Integral Form): Gauss' Law for E: $\oint \vec{E} \cdot d\vec{A} = Q_{enc}/\epsilon_0$ Gauss' Law for B: $\oint \vec{B} \cdot d\vec{A} = 0$ Faraday's Law: $\oint \vec{E} \cdot d\vec{s} = - \frac{d\Phi_B}{dt}$ Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ Speed of Light: $c = 1/\sqrt{\mu_0 \epsilon_0} = 3.00 \times 10^8 \text{ m/s}$ Relation E and B: $E = cB$ Poynting Vector (Intensity): $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ Average Intensity: $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{1}{\mu_0 c} (\frac{E_m}{\sqrt{2}})^2$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 17. Light & Optics Reflection: $\theta_i = \theta_r$ Refraction (Snell's Law): $n_1 \sin \theta_1 = n_2 \sin \theta_2$ Index of Refraction: $n = c/v$ Critical Angle: $\sin \theta_c = n_2/n_1$ (for total internal reflection, $n_1 > n_2$) Thin Lenses & Spherical Mirrors: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -i/p = h'/h$ Focal Length: $f = R/2$ (mirror), $1/f = (n-1)(1/r_1 - 1/r_2)$ (lensmaker's) Sign Conventions: $p$: + real object, - virtual object $i$: + real image, - virtual image $f$: + converging (concave mirror, convex lens), - diverging (convex mirror, concave lens) $R$: + center on side of outgoing light, - center on side of incident light $h'$: + upright, - inverted 18. Interference & Diffraction Young's Double Slit: Bright Fringes (Max): $d \sin \theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Dark Fringes (Min): $d \sin \theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) Fringe Spacing: $\Delta y = L\lambda/d$ Thin Film Interference: Consider phase changes upon reflection (air-film, film-substrate). Path difference $2L$. Phase change $0$ or $\pi$. Single Slit Diffraction: Minima: $a \sin \theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Diffraction Grating: Max: $d \sin \theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh's Criterion: $\theta_R = 1.22 \lambda/D$ (circular aperture) 19. Modern Physics Photoelectric Effect: $K_{max} = hf - \Phi$ ($h=6.626 \times 10^{-34} \text{ J}\cdot\text{s}$, $\Phi$ work function) Photon Energy: $E = hf = hc/\lambda$ Momentum of Photon: $p = h/\lambda = E/c$ Compton Effect: $\Delta \lambda = \lambda' - \lambda = \frac{h}{mc}(1 - \cos \phi)$ De Broglie Wavelength: $\lambda = h/p = h/(mv)$ Heisenberg Uncertainty Principle: $\Delta x \Delta p_x \ge \hbar/2$ $\Delta E \Delta t \ge \hbar/2$ ($\hbar = h/(2\pi)$) Schrödinger Equation (1D Time-Independent): $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ Particle in a Box (1D): Energy Levels: $E_n = \frac{h^2 n^2}{8mL^2}$ ($n=1,2,3...$) Wave Function: $\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L})$ Atomic Structure: Bohr Radius: $a_0 = 0.0529 \text{ nm}$ Hydrogen Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ Radioactive Decay: $N(t) = N_0 e^{-\lambda t}$ Half-Life: $T_{1/2} = \frac{\ln 2}{\lambda}$ Mass-Energy Equivalence: $E = mc^2$