Halliday

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1. Measurement SI Base Units: Length: meter (m) Mass: kilogram (kg) Time: second (s) Current: ampere (A) Temperature: kelvin (K) Amount of substance: mole (mol) Luminous intensity: candela (cd) Prefixes: Giga (G): $10^9$ Mega (M): $10^6$ Kilo (k): $10^3$ Centi (c): $10^{-2}$ Milli (m): $10^{-3}$ Micro ($\mu$): $10^{-6}$ Nano (n): $10^{-9}$ Pico (p): $10^{-12}$ 2. Kinematics 2.1. One-Dimensional Motion Displacement: $\Delta x = x_f - x_i$ Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Instantaneous Velocity: $v = \frac{dx}{dt}$ Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$ Instantaneous Acceleration: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ 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 Acceleration: $g = 9.8 \text{ m/s}^2$ (downward) 2.2. Two- and Three-Dimensional Motion 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: $v_{0x} = v_0 \cos\theta_0$ $v_{0y} = v_0 \sin\theta_0$ $x = (v_0 \cos\theta_0)t$ $y = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ $v_y = v_0 \sin\theta_0 - gt$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Uniform Circular Motion: Centripetal Acceleration: $a = \frac{v^2}{r}$ (directed towards center) Period: $T = \frac{2\pi r}{v}$ 3. Newton's Laws of Motion Newton's First Law: 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: $\sum \vec{F} = m\vec{a}$ Newton's Third Law: If object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on object A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) 3.1. Specific Forces Gravitational Force (Weight): $F_g = mg$ Normal Force: $F_N$ (perpendicular to surface) Tension: $F_T$ (along a cord/rope) Friction: Static Friction: $f_s \le \mu_s F_N$ Kinetic Friction: $f_k = \mu_k F_N$ Drag Force (Air Resistance): $D = \frac{1}{2}C\rho Av^2$ (at high speeds) 4. Work and Energy Work Done by a Constant Force: $W = \vec{F} \cdot \vec{d} = Fd\cos\phi$ Work Done by a Variable Force: $W = \int_{x_i}^{x_f} F(x) dx$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ (instantaneous), $P_{avg} = \frac{W}{\Delta t}$ 4.1. Potential Energy Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ Conservation of Mechanical Energy: $E_{mech} = K + U = \text{constant}$ (if only conservative forces do work) Conservation of Energy (General): $W_{ext} = \Delta E_{mech} + \Delta E_{therm} + \Delta E_{int}$ Non-Conservative Work: $W_{nc} = \Delta E_{mech}$ 5. Momentum and Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Newton's Second Law (Momentum Form): $\sum \vec{F} = \frac{d\vec{p}}{dt}$ Impulse: $\vec{J} = \int_{t_i}^{t_f} \vec{F}(t)dt = \Delta \vec{p}$ Conservation of Linear Momentum: If $\sum \vec{F}_{ext} = 0$, then $\vec{P}_{total} = \text{constant}$ Collisions: Elastic Collision: Momentum and kinetic energy are conserved. Inelastic Collision: Momentum conserved, kinetic energy not conserved. Perfectly Inelastic Collision: Objects stick together; momentum conserved, kinetic energy not conserved. Center of Mass: $x_{com} = \frac{1}{M}\sum m_ix_i$ $\vec{v}_{com} = \frac{1}{M}\sum m_i\vec{v}_i$ $\vec{P}_{total} = M\vec{v}_{com}$ 6. Rotation Angular Position: $\theta$ (radians) Angular Displacement: $\Delta\theta = \theta_f - \theta_i$ Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Constant Angular Acceleration: $\omega = \omega_0 + \alpha t$ $\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$ Relations to Linear Quantities: Arc Length: $s = r\theta$ Tangential Speed: $v_t = r\omega$ Tangential Acceleration: $a_t = r\alpha$ Centripetal Acceleration: $a_c = \frac{v_t^2}{r} = r\omega^2$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Moment of Inertia: $I = \sum m_ir_i^2$ (discrete), $I = \int r^2 dm$ (continuous) Parallel-Axis Theorem: $I = I_{com} + Mh^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F} = rF\sin\phi$ Newton's Second Law for Rotation: $\sum \tau = I\alpha$ Work Done by Torque: $W = \int_{\theta_i}^{\theta_f} \tau d\theta$ Power for Rotation: $P = \tau\omega$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) Conservation of Angular Momentum: If $\sum \vec{\tau}_{ext} = 0$, then $\vec{L}_{total} = \text{constant}$ 7. Gravity Newton's Law of Gravitation: $F = G\frac{m_1m_2}{r^2}$ ($G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$) Gravitational Potential Energy: $U = -\frac{GMm}{r}$ (relative to $U=0$ at $r=\infty$) Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's Laws: 1. Orbits are ellipses with the Sun at one focus. 2. A line joining a planet and the Sun sweeps out equal areas in equal times. 3. The square of the orbital period $T$ is proportional to the cube of the semi-major axis $a$: $T^2 = (\frac{4\pi^2}{GM})a^3$. 8. Oscillations Simple Harmonic Motion (SHM): Position: $x(t) = x_m \cos(\omega t + \phi)$ Velocity: $v(t) = -\omega x_m \sin(\omega t + \phi)$ Acceleration: $a(t) = -\omega^2 x_m \cos(\omega t + \phi) = -\omega^2 x(t)$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) Period: $T = \frac{2\pi}{\omega}$ Frequency: $f = \frac{1}{T} = \frac{\omega}{2\pi}$ Energy in SHM: $E = \frac{1}{2}kx_m^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ 9. Waves Wave Speed: $v = \lambda f$ Transverse Wave on a String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$: tension, $\mu$: linear mass density) Sound Speed in Fluid: $v = \sqrt{\frac{B}{\rho}}$ ($B$: bulk modulus, $\rho$: density) Sound Speed in Solid Rod: $v = \sqrt{\frac{Y}{\rho}}$ ($Y$: Young's modulus) Intensity: $I = \frac{P}{A}$ (Power per unit area) Intensity Level (dB): $\beta = (10 \text{ dB}) \log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D: detector, S: source) Standing Waves: String fixed at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) Open-open pipe: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) Open-closed pipe: $\lambda_n = \frac{4L}{n}$, $f_n = n\frac{v}{4L}$ ($n=1,3,5...$) 10. Thermodynamics Temperature Scales: $T_C = T_K - 273.15$ $T_F = \frac{9}{5}T_C + 32$ Thermal Expansion: Linear: $\Delta L = L\alpha\Delta T$ Volume: $\Delta V = V\beta\Delta T = V(3\alpha)\Delta T$ First Law of Thermodynamics: $\Delta E_{int} = Q - W$ ($Q$: heat added, $W$: work done by system) Heat Transfer: Conduction: $P_{cond} = kA\frac{dT}{dx}$ Convection: Involves fluid movement. Radiation: $P_{rad} = \epsilon\sigma AT^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$) Heat Capacity: $Q = C\Delta T$, $C = mc$ (specific heat) Phase Change: $Q = Lm$ ($L$: latent heat) Ideal Gas Law: $PV = nRT = NkT$ ($R = 8.31 \text{ J/mol}\cdot\text{K}$, $k = 1.38 \times 10^{-23} \text{ J/K}$) Kinetic Theory of Gases: Average Kinetic Energy: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ Entropy Change: $\Delta S = \int \frac{dQ}{T}$ (reversible), $\Delta S \ge 0$ (isolated system) Carnot Engine: Efficiency: $\epsilon = 1 - \frac{T_L}{T_H}$ $\frac{Q_L}{Q_H} = \frac{T_L}{T_H}$ 11. Electric Fields Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0}\frac{|q_1q_2|}{r^2}$ ($\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2$) Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ Electric Field of Point Charge: $\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on Dipole in E-field: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ Gauss' Law: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ 12. Electric Potential Potential Difference: $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ Potential Energy: $\Delta U = q\Delta V$ Potential of Point Charge: $V = \frac{1}{4\pi\epsilon_0}\frac{q}{r}$ Relation between E and V: $\vec{E} = -\nabla V$, for 1D: $E_x = -\frac{dV}{dx}$ 13. Capacitance Capacitance: $C = \frac{q}{V}$ (Unit: Farad, F) Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{q^2}{C} = \frac{1}{2}qV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Dielectric: $C = \kappa C_{air}$, $\kappa$: dielectric constant Capacitors in Parallel: $C_{eq} = \sum C_i$ Capacitors in Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ 14. Current and Resistance Electric Current: $I = \frac{dq}{dt}$ (Unit: Ampere, A) Current Density: $\vec{J} = nq\vec{v}_d$ ($n$: charge carrier density, $\vec{v}_d$: drift velocity) Ohm's Law: $V = IR$ Resistance: $R = \rho\frac{L}{A}$ ($\rho$: resistivity) Power in Circuits: $P = IV = I^2R = \frac{V^2}{R}$ Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ 15. DC Circuits Kirchhoff's Laws: Junction Rule: $\sum I_{in} = \sum I_{out}$ Loop Rule: $\sum \Delta V = 0$ around any closed loop RC Circuits: Charging Capacitor: $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ Discharging Capacitor: $q(t) = q_0 e^{-t/RC}$, $I(t) = -\frac{q_0}{RC}e^{-t/RC}$ Time Constant: $\tau = RC$ 16. Magnetic Fields Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ ($\vec{\mu}$: magnetic dipole moment) Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ 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: $B = \frac{\mu_0 I}{2\pi r}$ Magnetic Field of Loop Center: $B = \frac{\mu_0 I}{2R}$ Magnetic Field of Solenoid: $B = \mu_0 n I$ ($n$: turns per unit length) Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ 17. Induction and Inductance Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Lenz's Law: Induced current opposes the change in magnetic flux. Motional EMF: $\mathcal{E} = BLv$ Inductance: $L = \frac{N\Phi_B}{I}$ (Unit: Henry, H) Solenoid Inductance: $L = \mu_0 n^2 A l$ Self-Induced EMF: $\mathcal{E}_L = -L\frac{dI}{dt}$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Energy Density of B-field: $u_B = \frac{B^2}{2\mu_0}$ RL Circuits: Current build-up: $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ Current decay: $I(t) = I_0 e^{-t/\tau_L}$ Time Constant: $\tau_L = \frac{L}{R}$ 18. Electromagnetic Waves Speed of Light: $c = \frac{1}{\sqrt{\mu_0\epsilon_0}} = 3.00 \times 10^8 \text{ m/s}$ Wave Speed: $c = \lambda f$ Relation between E and B: $E = cB$ Intensity: $I = \frac{1}{c\mu_0}E_{rms}^2 = \frac{c}{\mu_0}B_{rms}^2 = \frac{1}{2c\mu_0}E_m^2$ Radiation Pressure: $P_r = \frac{I}{c}$ (absorbed), $P_r = \frac{2I}{c}$ (reflected) 19. Light: Reflection and Refraction Law of Reflection: $\theta_1 = \theta_1'$ Snell's Law (Refraction): $n_1\sin\theta_1 = n_2\sin\theta_2$ Index of Refraction: $n = \frac{c}{v}$ Critical Angle: $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) 20. Lenses and Mirrors Mirror Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Lensmaker's Equation: $\frac{1}{f} = (n-1)(\frac{1}{r_1} - \frac{1}{r_2})$ Magnification: $m = -\frac{i}{p} = \frac{h'}{h}$ Sign Conventions: $p$: + real object, - virtual object $i$: + real image, - virtual image $f$: + concave mirror/converging lens, - convex mirror/diverging lens $r$: + center on same side as outgoing light, - opposite side $h'$: + upright, - inverted 21. Interference Young's Double-Slit Experiment: Bright Fringes (maxima): $d\sin\theta = m\lambda$ ($m=0,\pm 1,\pm 2,...$) Dark Fringes (minima): $d\sin\theta = (m+\frac{1}{2})\lambda$ ($m=0,\pm 1,\pm 2,...$) Fringe Spacing (small $\theta$): $\Delta y = \frac{\lambda L}{d}$ Thin-Film Interference: Path difference $2L$ Reflected light phase change: $\frac{1}{2}\lambda$ if $n_{film} > n_{incident}$ Condition for constructive/destructive depends on phase changes 22. Diffraction Single-Slit Diffraction: Minima: $a\sin\theta = m\lambda$ ($m=\pm 1,\pm 2,...$) Diffraction Grating: Maxima: $d\sin\theta = m\lambda$ ($m=0,\pm 1,\pm 2,...$) Rayleigh's Criterion (Resolution): $\theta_R = 1.22\frac{\lambda}{D}$ (circular aperture) 23. Modern Physics Essentials Planck's Quantum Hypothesis: $E = hf$ ($h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$) 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: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ $\Delta E \Delta t \ge \frac{\hbar}{2}$ Schrödinger Equation (Time-Independent): $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U\psi = E\psi$ Energy-Mass Equivalence: $E = mc^2$