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: mole (mol) Luminous Intensity: candela (cd) Scientific Notation: $A \times 10^n$ Significant Figures: Non-zero digits are significant. Zeros between non-zero digits are significant. Trailing zeros after a decimal point are significant. 2. Kinematics (1D & 2D) 1D Motion Position: $x(t)$ 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: $a = \frac{dv}{dt}$ Constant Acceleration Equations: $v = v_0 + at$ $x = x_0 + v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ Free Fall: $a = -g = -9.8 \text{ m/s}^2$ (downwards) 2D Motion Position Vector: $\vec{r} = x\hat{i} + y\hat{j}$ Velocity Vector: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j}$ Acceleration Vector: $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j}$ Projectile Motion: $v_x = v_{0x} = v_0 \cos\theta_0$ (constant) $x = x_0 + (v_0 \cos\theta_0)t$ $v_y = v_{0y} - gt = v_0 \sin\theta_0 - gt$ $y = y_0 + (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ Uniform Circular Motion: Speed: $v = \frac{2\pi r}{T}$ Centripetal Acceleration: $a_c = \frac{v^2}{r}$ (directed towards center) 3. Newton's Laws of Motion Newton's 1st 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 2nd Law: $\sum \vec{F} = m\vec{a}$ Force: $\vec{F}$ (Newtons, N) Mass: $m$ (kg) Acceleration: $\vec{a}$ (m/s$^2$) Newton's 3rd Law: If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. $\vec{F}_{AB} = -\vec{F}_{BA}$ Weight: $W = mg$ (force due to gravity) Normal Force: $\vec{F}_N$ (perpendicular to surface) Friction: Static: $f_s \le \mu_s F_N$ Kinetic: $f_k = \mu_k F_N$ Tension: $\vec{T}$ (force through a string/rope) 4. Work and Energy Work Done by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ (Joules, J) Work Done by Variable Force: $W = \int \vec{F} \cdot d\vec{r}$ 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: $U_s = \frac{1}{2}kx^2$ (for a spring) Conservation of Mechanical Energy: $E = K + U = \text{constant}$ (if only conservative forces do work) $E_i = E_f \implies K_i + U_i = K_f + U_f$ Non-Conservative Forces: $W_{nc} = \Delta E = \Delta K + \Delta U$ Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ (Watts, W) 5. Momentum and Collisions Linear Momentum: $\vec{p} = m\vec{v}$ (kg m/s) Newton's 2nd Law (Momentum Form): $\sum \vec{F} = \frac{d\vec{p}}{dt}$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta\vec{p} = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum: If $\sum \vec{F}_{ext} = 0$, then $\vec{P}_{total} = \text{constant}$ $\vec{p}_{1i} + \vec{p}_{2i} = \vec{p}_{1f} + \vec{p}_{2f}$ Collisions: Elastic: Both momentum and kinetic energy are conserved. Inelastic: Momentum conserved, kinetic energy NOT conserved. Completely Inelastic: Objects stick together. Momentum conserved, kinetic energy NOT conserved. Center of Mass: $x_{CM} = \frac{1}{M}\sum m_i x_i$ $\vec{v}_{CM} = \frac{1}{M}\sum m_i \vec{v}_i$ 6. Rotational Motion Angular Position: $\theta$ (radians) Angular Displacement: $\Delta\theta$ Angular Velocity: $\omega = \frac{d\theta}{dt}$ (rad/s) Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ (rad/s$^2$) Rotational Kinematics (Constant $\alpha$): $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Tangential Speed: $v_t = r\omega$ Tangential Acceleration: $a_t = r\alpha$ Centripetal Acceleration: $a_c = \frac{v_t^2}{r} = r\omega^2$ Moment of Inertia: $I = \sum m_i r_i^2$ (kg m$^2$) (for point masses) For continuous body: $I = \int r^2 dm$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F} = rF\sin\theta$ (N m) Newton's 2nd Law for Rotation: $\sum \tau = I\alpha$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p}$ (for point particle) For rigid body: $L = I\omega$ Conservation of Angular Momentum: If $\sum \tau_{ext} = 0$, then $\vec{L}_{total} = \text{constant}$ $I_i\omega_i = I_f\omega_f$ 7. Gravity Newton's Law of Universal Gravitation: $F = G \frac{m_1 m_2}{r^2}$ $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$ Gravitational Acceleration: $g = G \frac{M}{r^2}$ 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 the Sun at one focus. 2nd: A line joining a planet and the Sun sweeps out equal areas in equal times. 3rd: $T^2 \propto a^3$ (period squared is proportional to semi-major axis cubed) 8. Oscillations Simple Harmonic Motion (SHM): Position: $x(t) = A \cos(\omega t + \phi)$ Velocity: $v(t) = -A\omega \sin(\omega t + \phi)$ Acceleration: $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass system) Angular Frequency: $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum for small angles) Period: $T = \frac{2\pi}{\omega}$ Frequency: $f = \frac{1}{T} = \frac{\omega}{2\pi}$ Total Energy in SHM: $E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$ 9. Waves Wave Speed: $v = \lambda f$ Transverse Wave on String: $v = \sqrt{\frac{T}{\mu}}$ (T = tension, $\mu$ = linear mass density) Sound Speed in Fluid: $v = \sqrt{\frac{B}{\rho}}$ (B = bulk modulus, $\rho$ = density) Intensity: $I = \frac{P}{A}$ (Power/Area, W/m$^2$) Intensity Level (dB): $\beta = (10 \text{ dB}) \log_{10}\left(\frac{I}{I_0}\right)$ ($I_0 = 10^{-12} \text{ W/m}^2$) Standing Waves: String fixed at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1,2,3...$) Open-open pipe: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1,2,3...$) Open-closed pipe: $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ ($n=1,3,5...$) Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D = detector, S = source; use top signs for "towards", bottom for "away") 10. Thermodynamics Temperature Scales: $T_F = \frac{9}{5}T_C + 32^\circ$ $T_K = T_C + 273.15$ Thermal Expansion: Linear: $\Delta L = L_0 \alpha \Delta T$ Volume: $\Delta V = V_0 \beta \Delta T$, where $\beta \approx 3\alpha$ Heat Transfer: Conduction: $P_{cond} = kA \frac{\Delta T}{L}$ Convection: Involves fluid movement. Radiation: $P_{rad} = \sigma e A T^4$ (Stefan-Boltzmann Law) Specific Heat: $Q = mc\Delta T$ Latent Heat (Phase Change): $Q = mL$ First Law of Thermodynamics: $\Delta E_{int} = Q - W$ $Q$: Heat added to system $W$: Work done BY system Work Done by Gas: $W = \int P dV$ Ideal Gas Law: $PV = nRT = Nk_B T$ $R = 8.314 \text{ J/(mol K)}$ $k_B = 1.38 \times 10^{-23} \text{ J/K}$ Kinetic Theory of Gases: Average Kinetic Energy: $K_{avg} = \frac{3}{2}k_B T$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of an isolated system never decreases. $\Delta S \ge 0$ Heat Engines: Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ Carnot Efficiency: $\epsilon_C = 1 - \frac{T_C}{T_H}$ Refrigerators/Heat Pumps: Coefficient of Performance (Cooling): $K = \frac{|Q_C|}{|W|}$ Coefficient of Performance (Heating): $K_{HP} = \frac{|Q_H|}{|W|}$ Entropy Change: $\Delta S = \int \frac{dQ}{T}$ (for reversible processes) 11. Electric Fields and Forces Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2$ $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N m}^2)$ $e = 1.602 \times 10^{-19} \text{ C}$ Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ (N/C or V/m) Electric Field from Point Charge: $E = k \frac{|q|}{r^2}$ 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 Electric Potential Energy: $\Delta U = -W = -q_0 \int \vec{E} \cdot d\vec{s}$ Electric Potential: $V = \frac{U}{q_0}$ (Volts, V) Potential Difference: $\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{s}$ Potential from Point Charge: $V = k \frac{q}{r}$ Relationship E and V: $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ Parallel Plate Capacitor: $E = \frac{\sigma}{\epsilon_0}$ 13. Capacitance Capacitance: $C = \frac{Q}{V}$ (Farads, F) Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Energy Stored in Capacitor: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ Energy Density of E-field: $u_E = \frac{1}{2}\epsilon_0 E^2$ Capacitors in Parallel: $C_{eq} = C_1 + C_2 + ...$ Capacitors in Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ Dielectrics: $C = \kappa C_0$, $E = E_0/\kappa$ 14. Current and Resistance Electric Current: $I = \frac{dQ}{dt}$ (Amperes, A) Current Density: $\vec{J} = nq\vec{v}_d$ Relationship to Current: $I = \int \vec{J} \cdot d\vec{A}$ Ohm's Law: $V = IR$ (Ohms, $\Omega$) Resistance: $R = \rho \frac{L}{A}$ ($\rho$ = resistivity) Power in Circuits: $P = IV = I^2 R = \frac{V^2}{R}$ Resistors in Series: $R_{eq} = R_1 + R_2 + ...$ Resistors in Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ RC Circuits (Charging): $Q(t) = Q_{max}(1 - e^{-t/RC})$ RC Circuits (Discharging): $Q(t) = Q_0 e^{-t/RC}$ Time Constant: $\tau = RC$ 15. Magnetic Fields and Forces Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ (Newtons, N) Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ Magnetic Moment of Current Loop: $\vec{\mu} = NI\vec{A}$ Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Potential Energy of Magnetic Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = \frac{IB}{net}$ 16. Sources of Magnetic Fields Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$ Magnetic Field of Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Force Between Parallel Wires: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ 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: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ 17. Faraday's Law and Induction Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ (Webers, Wb) Faraday's Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ (Volts, V) Lenz's Law: Induced current/EMF opposes the change in magnetic flux. Motional EMF: $\mathcal{E} = BLv$ Inductance: $L = \frac{N\Phi_B}{I}$ (Henrys, H) Energy Stored in Inductor: $U_L = \frac{1}{2}LI^2$ Energy Density of B-field: $u_B = \frac{B^2}{2\mu_0}$ RL Circuits (Current Growth): $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ RL Circuits (Current Decay): $I(t) = I_0 e^{-t/\tau_L}$ Inductive Time Constant: $\tau_L = L/R$ LC Oscillations: $\omega = \frac{1}{\sqrt{LC}}$ 18. AC Circuits RMS Values: $V_{rms} = \frac{V_{max}}{\sqrt{2}}$, $I_{rms} = \frac{I_{max}}{\sqrt{2}}$ Capacitive Reactance: $X_C = \frac{1}{\omega C}$ Inductive Reactance: $X_L = \omega L$ Impedance (RLC Series): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ Resonance Frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$ Average Power: $P_{avg} = I_{rms}V_{rms}\cos\phi$ (Power Factor: $\cos\phi$) 19. Electromagnetic Waves Speed of Light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$ Wave Speed: $c = \lambda f$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (Direction of propagation, W/m$^2$) Intensity (Average Poynting Vector): $I = S_{avg} = \frac{1}{2\mu_0}E_{max}B_{max} = \frac{E_{max}^2}{2\mu_0 c} = \frac{cB_{max}^2}{2\mu_0}$ Radiation Pressure: Perfectly absorbing: $P_{rad} = I/c$ Perfectly reflecting: $P_{rad} = 2I/c$ 20. Light, Reflection, Refraction Law of Reflection: $\theta_i = \theta_r$ Snell's Law (Refraction): $n_1 \sin\theta_1 = n_2 \sin\theta_2$ Index of Refraction: $n = c/v$ Critical Angle: $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) Thin Lens / Mirror Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ $p$: object distance (real +) $i$: image distance (real +, virtual -) $f$: focal length (converging +, diverging -) Magnification: $M = -\frac{i}{p} = \frac{h'}{h}$ (upright +, inverted -) Lensmaker's Equation: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ 21. Interference Young's Double-Slit Experiment: Constructive: $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Destructive: $d\sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) Fringe separation: $\Delta y = \frac{\lambda L}{d}$ Thin Films: Path difference $2t$ (plus phase shift at reflection, if any) Phase Change upon Reflection: $180^\circ$ if reflecting from higher $n$ medium. 22. Diffraction Single-Slit Diffraction: Minima: $a\sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Width of central maximum: $2\frac{\lambda L}{a}$ Diffraction Grating: Maxima: $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh's Criterion (Resolution): $\theta_{min} = 1.22 \frac{\lambda}{D}$ 23. 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: $\vec{p} = \gamma m\vec{v}$ Relativistic Energy: $E = \gamma mc^2 = K + mc^2$ Rest Energy: $E_0 = mc^2$ Kinetic Energy: $K = (\gamma - 1)mc^2$ Energy-Momentum Relation: $E^2 = (pc)^2 + (mc^2)^2$ 24. Quantum Physics Photon Energy: $E = hf = \frac{hc}{\lambda}$ $h = 6.626 \times 10^{-34} \text{ J s}$ (Planck's constant) Photoelectric Effect: $K_{max} = hf - \Phi$ ($\Phi$ = work function) Compton Effect: $\Delta\lambda = \frac{h}{mc}(1 - \cos\phi)$ De Broglie Wavelength: $\lambda = \frac{h}{p}$ Heisenberg Uncertainty Principle: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ $\Delta E \Delta t \ge \frac{\hbar}{2}$ $\hbar = h/(2\pi)$ Schrödinger Equation: (Time-independent for 1D) $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ 25. Atomic Physics Bohr Model (Hydrogen): Quantized Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ Radii: $r_n = n^2 a_0$, $a_0 = 0.0529 \text{ nm}$ (Bohr radius) Quantum Numbers: Principal ($n=1,2,3...$): Energy, size Orbital ($l=0,1,...,n-1$): Shape, angular momentum Magnetic ($m_l=-l,...,l$): Orientation of orbital ang. mom. Spin ($m_s=\pm 1/2$): Intrinsic spin Pauli Exclusion Principle: No two electrons can have the same set of four quantum numbers. 26. Nuclear Physics Atomic Number (Z): Number of protons Mass Number (A): Number of protons + neutrons Nuclear Radius: $R = R_0 A^{1/3}$ ($R_0 \approx 1.2 \text{ fm}$) Binding Energy: $\Delta E = \Delta m c^2$ ($\Delta m$ = mass defect) Radioactive Decay Law: $N(t) = N_0 e^{-\lambda t}$ Half-life: $T_{1/2} = \frac{\ln 2}{\lambda}$ Activity: $R = |\frac{dN}{dt}| = \lambda N$ (Becquerel, Bq)