Halliday

Cheatsheet Content

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): $10^{-12}$ Significant Figures: Rules for precision in calculations. 2. Motion in 1D Displacement: $\Delta x = x_f - x_i$ Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Average Speed: $s_{avg} = \frac{\text{total distance}}{\Delta t}$ 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: $v = v_0 + at$ $x - x_0 = v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ Free Fall: $a = -g = -9.8 \text{ m/s}^2$ (downwards) 3. Vectors Components: $A_x = A \cos\theta$, $A_y = A \sin\theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\tan\theta = \frac{A_y}{A_x}$ Unit Vector Notation: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ Vector Addition: $\vec{C} = \vec{A} + \vec{B} \Rightarrow C_x = A_x + B_x$, $C_y = A_y + B_y$ Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = AB \cos\phi = A_x B_x + A_y B_y + A_z B_z$ Vector (Cross) Product: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ Magnitude: $|\vec{A} \times \vec{B}| = AB \sin\phi$ Direction: Right-hand rule 4. Motion in 2D & 3D 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: $x = (v_0 \cos\theta_0)t$ $y = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ $v_x = v_0 \cos\theta_0$ $v_y = v_0 \sin\theta_0 - gt$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Max height: $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ Uniform Circular Motion: Speed: $v = \frac{2\pi r}{T}$ Centripetal Acceleration: $a_c = \frac{v^2}{r}$ (directed towards center) 5. Force & Motion (Newton's Laws) 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: $\vec{F}_{net} = m\vec{a}$ 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. ($\vec{F}_{AB} = -\vec{F}_{BA}$) Weight: $W = mg$ Normal Force ($F_N$): Perpendicular to surface. Tension ($T$): Force transmitted through a string/cable. Friction: Static: $f_s \le \mu_s F_N$ Kinetic: $f_k = \mu_k F_N$ $\mu_s \ge \mu_k$ Drag Force (high speed): $D = \frac{1}{2} C \rho A v^2$ Terminal Speed: Occurs when $D = mg$. 6. Work & Energy Work done by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd \cos\phi$ Work done by 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), $P_{avg} = \frac{W}{\Delta t}$ Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy: $U_s = \frac{1}{2}kx^2$ (Hooke's Law: $F_s = -kx$) Mechanical Energy: $E_{mech} = K + U$ Conservation of Mechanical Energy: $\Delta E_{mech} = 0 \Rightarrow K_i + U_i = K_f + U_f$ (for conservative forces only) Conservation of Energy (General): $W_{ext} = \Delta E_{mech} + \Delta E_{int} + \Delta E_{thermal}$ $W_{nc} = \Delta E_{mech}$ (work by non-conservative forces) 7. Center of Mass & Momentum Center of Mass: 1D: $x_{com} = \frac{1}{M} \sum m_i x_i = \frac{1}{M} \int x \, dm$ 3D: $\vec{r}_{com} = \frac{1}{M} \sum m_i \vec{r}_i$ Momentum: $\vec{p} = m\vec{v}$ Newton's Second Law (Momentum Form): $\vec{F}_{net} = \frac{d\vec{p}}{dt}$ Impulse: $\vec{J} = \int \vec{F}(t) dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ Conservation of Momentum: $\vec{P}_{total,i} = \vec{P}_{total,f}$ (if $\vec{F}_{net,ext} = 0$) Collisions: Elastic: Momentum and Kinetic Energy conserved. Inelastic: Momentum conserved, Kinetic Energy NOT conserved. Perfectly Inelastic: Objects stick together, momentum conserved, max KE loss. 8. Rotation Angular Position: $\theta$ (radians) Angular Displacement: $\Delta\theta = \theta_f - \theta_i$ Average Angular Velocity: $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$ Instantaneous Angular Velocity: $\omega = \frac{d\theta}{dt}$ Average Angular Acceleration: $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$ Instantaneous Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Constant Angular Acceleration Equations: (analogous to linear) $\omega = \omega_0 + \alpha t$ $\theta - \theta_0 = \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relating Linear & Angular Variables: $s = r\theta$ $v_t = r\omega$ (tangential speed) $a_t = r\alpha$ (tangential acceleration) $a_c = \frac{v_t^2}{r} = \omega^2 r$ (centripetal acceleration) Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Moment of Inertia: $I = \sum m_i r_i^2 = \int r^2 dm$ Parallel-Axis Theorem: $I = I_{com} + Md^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$, magnitude $\tau = rF\sin\phi$ Newton's Second Law for Rotation: $\tau_{net} = I\alpha$ Work done by Torque: $W = \int \tau d\theta$ Power: $P = \tau\omega$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) Newton's Second Law (Angular Momentum Form): $\vec{\tau}_{net} = \frac{d\vec{L}}{dt}$ Conservation of Angular Momentum: $L_{total,i} = L_{total,f}$ (if $\vec{\tau}_{net,ext} = 0$) 9. Equilibrium & Elasticity Static Equilibrium Conditions: $\sum \vec{F}_{ext} = 0$ (translational equilibrium) $\sum \vec{\tau}_{ext} = 0$ (rotational equilibrium) Stress & Strain: Stress: Force per unit area ($\sigma = F/A$) Strain: Fractional deformation ($\epsilon = \Delta L / L_0$) Young's Modulus (tension/compression): $E = \text{stress} / \text{strain} = (F/A) / (\Delta L/L_0)$ Shear Modulus (shearing): $G = (F/A) / (\Delta x/L_0)$ Bulk Modulus (volume compression): $B = -\Delta P / (\Delta V/V_0)$ 10. Gravitation Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$ (where $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$) Gravitational Acceleration: $g = G \frac{M_{Earth}}{R_{Earth}^2}$ Gravitational Potential Energy: $U = -G \frac{m_1 m_2}{r}$ (zero at infinity) Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's Laws: Orbits are ellipses with the Sun at one focus. Equal areas swept in equal times ($\frac{dA}{dt} = \text{constant}$). $T^2 \propto a^3$ (period squared proportional to semi-major axis cubed). For circular orbits: $T^2 = (\frac{4\pi^2}{GM}) r^3$. 11. Fluids Density: $\rho = m/V$ Pressure: $P = F/A$ Pressure in Fluid at Depth: $P = P_0 + \rho g h$ Pascal's Principle: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. Buoyant Force (Archimedes' Principle): $F_B = \rho_{fluid} V_{disp} g$ Equation of Continuity: $A_1 v_1 = A_2 v_2$ (for incompressible fluid) Bernoulli's Equation: $P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$ 12. 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: $\omega = \sqrt{k/m}$ (spring-mass) Period: $T = 2\pi/\omega$ Frequency: $f = 1/T$ Simple Pendulum: $T = 2\pi \sqrt{L/g}$ (for small angles) Physical Pendulum: $T = 2\pi \sqrt{I/mgd}$ Damped SHM: $x(t) = x_m e^{-bt/2m} \cos(\omega' t + \phi)$ Forced Oscillations & Resonance: Max amplitude when driving frequency equals natural frequency. 13. Waves Wave Speed: $v = \lambda f$ Transverse Wave on String: $v = \sqrt{\tau/\mu}$ ($\tau$ tension, $\mu$ linear mass density) Speed of Sound: $v = \sqrt{B/\rho}$ (B bulk modulus) Intensity: $I = \frac{P}{A}$ Sound Level: $\beta = (10 \text{ dB}) \log_{10}(I/I_0)$ Standing Waves: On string fixed at both ends: $\lambda_n = 2L/n$, $f_n = n(v/2L)$ ($n=1,2,3...$) Open-open pipe: $\lambda_n = 2L/n$, $f_n = n(v/2L)$ ($n=1,2,3...$) Open-closed pipe: $\lambda_n = 4L/n$, $f_n = n(v/4L)$ ($n=1,3,5...$) Beats: $f_{beat} = |f_1 - f_2|$ Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (top signs for approaching, bottom signs for receding) 14. Temperature & Heat 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: $\Delta V = V \beta \Delta T$, where $\beta = 3\alpha$ Heat Capacity: $Q = C \Delta T$ Specific Heat: $Q = mc \Delta T$ Latent Heat (Phase Change): $Q = mL$ Heat Transfer: Conduction: $P_{cond} = kA \frac{dT}{dx}$ Convection: Heat transfer via fluid movement. Radiation: $P_{rad} = \sigma \epsilon A T^4$ (Stefan-Boltzmann Law) 15. Thermodynamics 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: $PV = nRT = NkT$ ($R = 8.31 \text{ J/mol}\cdot\text{K}$, $k = 1.38 \times 10^{-23} \text{ J/K}$) Kinetic Theory of Gases: Average KE per molecule: $K_{avg} = \frac{3}{2}kT$ RMS speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ Internal Energy of Ideal Gas: $E_{int} = \frac{3}{2}nRT$ (monatomic) Molar Specific Heat: Constant Volume: $C_V = \frac{3}{2}R$ (monatomic), $C_V = \frac{5}{2}R$ (diatomic) Constant Pressure: $C_P = C_V + R$ Ratio: $\gamma = C_P/C_V$ Adiabatic Process: $PV^\gamma = \text{constant}$ Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of isolated system never decreases. 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: $K = \frac{|Q_C|}{|W|} = \frac{|Q_C|}{|Q_H| - |Q_C|}$ Entropy: $\Delta S = \int \frac{dQ}{T}$ (reversible), $\Delta S \ge 0$ (isolated systems) 16. Electric Charge & Field Quantization of Charge: $q = ne$ ($e = 1.602 \times 10^{-19} \text{ C}$) Conservation of Charge: Total charge in isolated system is constant. Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ (where $k = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2 = \frac{1}{4\pi\epsilon_0}$) Electric Field: $\vec{E} = \vec{F}/q_0$ Electric Field of 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: $U = -\vec{p} \cdot \vec{E}$ 17. Gauss' Law Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\epsilon_0 \Phi_E = q_{enc}$ Applications: Finding E-field for symmetric charge distributions (sphere, cylinder, infinite plane). 18. Electric Potential Potential Energy Change: $\Delta U = -W = -q_0 \int \vec{E} \cdot d\vec{s}$ Electric Potential: $V = U/q_0$ Potential Difference: $\Delta V = V_f - V_i = - \int_i^f \vec{E} \cdot d\vec{s}$ Potential due to Point Charge: $V = k \frac{q}{r}$ Potential due to Dipole: $V = \frac{k p \cos\theta}{r^2}$ Relating E-field and Potential: $\vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k} \right)$ 19. Capacitance Capacitance: $C = Q/V$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{1}{2}Q^2/C = \frac{1}{2}QV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Capacitors in Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Dielectrics: $C = \kappa C_{air}$, $E = E_{air}/\kappa$ 20. Current & Resistance Electric Current: $I = \frac{dQ}{dt}$ Current Density: $\vec{J} = n e \vec{v}_d$ Ohm's Law (Microscopic): $\vec{E} = \rho \vec{J}$ Resistance: $R = V/I$ Resistivity: $\rho = RA/L$ Resistance and Temperature: $\rho - \rho_0 = \rho_0 \alpha(T - T_0)$ Power in Circuits: $P = IV = I^2 R = V^2/R$ 21. Circuits EMF: $\mathcal{E}$ Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ 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: $\tau = RC$ 22. Magnetic Fields Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$, magnitude $F_B = |q|vB\sin\phi$ Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ Magnetic Field Units: Tesla (T), Gauss (G) ($1 \text{ T} = 10^4 \text{ G}$) Hall Effect: $V_H = \frac{IB}{net}$ Cyclotron Motion: Radius: $r = \frac{mv}{|q|B}$ Period: $T = \frac{2\pi m}{|q|B}$ Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NI\vec{A}$ Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ 23. Magnetic Fields from Currents 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 Parallel Currents: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ Magnetic Field at Center of Loop: $B = \frac{\mu_0 I}{2R}$ Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Magnetic Field of Solenoid: $B = \mu_0 n I$ (n = turns/length) Magnetic Field of Toroid: $B = \frac{\mu_0 I N}{2\pi r}$ 24. Induction & 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 direction opposes change in magnetic flux. Motional EMF: $\mathcal{E} = BLv$ Inductance: $L = \frac{N\Phi_B}{I}$ Self-Inductance of Solenoid: $L = \mu_0 n^2 A l$ RL Circuits (Current Growth): $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ RL Circuits (Current Decay): $I(t) = I_0 e^{-t/\tau_L}$ Time Constant: $\tau_L = L/R$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Magnetic Energy Density: $u_B = \frac{B^2}{2\mu_0}$ Mutual Inductance: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ 25. Electromagnetic Oscillations & AC LC Oscillations: Angular Frequency: $\omega = \frac{1}{\sqrt{LC}}$ Charge: $Q(t) = Q_{max} \cos(\omega t + \phi)$ Current: $I(t) = -\omega Q_{max} \sin(\omega t + \phi)$ Driven RLC Circuits: Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Inductive Reactance: $X_L = \omega L$ Capacitive Reactance: $X_C = \frac{1}{\omega C}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ Current Amplitude: $I = V/Z$ Resonance: $X_L = X_C \Rightarrow \omega_0 = \frac{1}{\sqrt{LC}}$ Power in AC Circuits: $P_{avg} = I_{rms} V_{rms} \cos\phi = I_{rms}^2 R$ (power factor $\cos\phi$) Transformers: $\frac{V_S}{V_P} = \frac{N_S}{N_P}$ (ideal) 26. Maxwell's Equations & EM Waves Gauss' Law for Electricity: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ Gauss' Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$ Faraday's Law: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (displacement current $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$) Speed of EM Waves: $c = \frac{1}{\sqrt{\mu_0\epsilon_0}} = 3 \times 10^8 \text{ m/s}$ Relationship E and B: $E = cB$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$ (direction of propagation, magnitude is intensity) Intensity: $I = S_{avg} = \frac{1}{c\mu_0} E_{rms}^2 = \frac{c}{\mu_0} B_{rms}^2$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 27. 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: $n = c/v$ Total Internal Reflection: Occurs when $\theta_1 > \theta_c$, where $\sin\theta_c = n_2/n_1$ ($n_1 > n_2$) 28. Lenses & Mirrors Mirror/Lens Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -\frac{i}{p} = \frac{h'}{h}$ Focal Length: Spherical Mirror: $f = R/2$ Lensmaker's Equation: $\frac{1}{f} = (n-1)\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$ Sign Conventions: $p$: + real object, - virtual object $i$: + real image, - virtual image $f$: + converging (concave mirror, converging lens), - diverging (convex mirror, diverging lens) $R$: + center on same side as outgoing light (mirror), + center on same side as outgoing light (lens) $h'$: + upright, - inverted 29. 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 on screen: $\Delta y = \frac{\lambda L}{d}$ Thin-Film Interference: Path difference is $2L$. Phase changes on reflection at higher $n$. 30. Diffraction Single Slit Diffraction: Dark Fringes (minima): $a \sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Diffraction Grating: Bright Fringes (maxima): $d \sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh's Criterion: $\theta_R = 1.22 \frac{\lambda}{D}$ (for circular aperture) 31. Relativity Postulates: The laws of physics are the same for all inertial reference frames. The speed of light in vacuum is the same for all inertial observers. 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: $E^2 = (pc)^2 + (mc^2)^2$ 32. Quantum Physics Photoelectric Effect: $K_{max} = hf - \Phi$ ($h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$) Photon Energy: $E = hf = hc/\lambda$ Compton Effect: $\Delta\lambda = \frac{h}{mc}(1 - \cos\phi)$ De Broglie Wavelength: $\lambda = h/p$ Heisenberg Uncertainty Principle: $\Delta x \Delta p_x \ge \hbar/2$ $\Delta E \Delta t \ge \hbar/2$ Schrödinger Equation (Time-Independent 1D): $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ 33. Atomic Physics Bohr Model (Hydrogen): Radii: $r_n = a_0 n^2$ ($a_0 = 0.0529 \text{ nm}$, Bohr radius) Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ Photon Energy: $\Delta E = hf = E_f - E_i$ Quantum Numbers: $n$: principal (energy, size) $1, 2, 3, ...$ $l$: orbital (shape) $0, 1, ..., n-1$ $m_l$: magnetic (orientation) $-l, ..., 0, ..., +l$ $m_s$: spin ($\pm 1/2$) Pauli Exclusion Principle: No two electrons can occupy the same quantum state. 34. Nuclear Physics Atomic Notation: $^A_Z X$ ($A$ mass number, $Z$ atomic number) Mass Defect: $\Delta m = (Z m_p + N m_n - M_{nucleus})$ Binding Energy: $E_B = \Delta m c^2$ Radioactive Decay: $N(t) = N_0 e^{-\lambda t}$ Half-Life: $T_{1/2} = \frac{\ln 2}{\lambda}$ Activity: $R = |\frac{dN}{dt}| = \lambda N$