Halliday

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1. Measurement and Vectors 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) Vector Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\theta = \tan^{-1} \left( \frac{A_y}{A_x} \right)$ Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \phi = A_x B_x + A_y B_y + A_z B_z$ 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 of Cross Product: $|\vec{A} \times \vec{B}| = AB \sin \phi$ 2. Kinematics (1D & 2D) 1D Motion Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$ Constant Acceleration Equations: $v = v_0 + at$ $x - x_0 = v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ 2D Motion (Projectile) Horizontal Motion: $x = x_0 + v_{0x}t$ ($a_x = 0$) Vertical Motion: $v_y = v_{0y} - gt$ $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ $v_y^2 = v_{0y}^2 - 2g(y - y_0)$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Circular Motion Centripetal Acceleration: $a_c = \frac{v^2}{r}$ 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: $\vec{F}_{net} = m\vec{a}$ 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: $W = mg$ Friction: Static: $f_s \le \mu_s N$ Kinetic: $f_k = \mu_k N$ 4. Work and Energy Work Done by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd \cos \phi$ 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: $W_{net} = \Delta K$ Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy: $U_s = \frac{1}{2}kx^2$ Conservation of Mechanical Energy: $E_{mech} = K + U = \text{constant}$ (if only conservative forces) Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ 5. Momentum and Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ Conservation of Linear Momentum: If $\vec{F}_{net,ext} = 0$, then $\vec{P}_{total} = \text{constant}$ Collisions: Elastic: Kinetic energy conserved ($K_{before} = K_{after}$) Inelastic: Kinetic energy not conserved ($K_{before} \ne K_{after}$) Perfectly Inelastic: Objects stick together after collision. Center of Mass: $\vec{r}_{com} = \frac{1}{M} \sum m_i \vec{r}_i$ 6. Rotation Angular Position: $\theta$ (radians) Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Rotational Kinematics (Constant $\alpha$): $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relationship to Linear: $s = r\theta$, $v = r\omega$, $a_t = r\alpha$, $a_c = r\omega^2$ Rotational Inertia: $I = \sum m_i r_i^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ Newton's Second Law for Rotation: $\tau_{net} = I\alpha$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ Conservation of Angular Momentum: If $\vec{\tau}_{net,ext} = 0$, then $\vec{L}_{total} = \text{constant}$ 7. Gravity Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$ Gravitational Potential Energy: $U = -G \frac{m_1 m_2}{r}$ 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. $T^2 \propto a^3$ (period squared proportional to semi-major axis cubed). $\frac{T^2}{a^3} = \frac{4\pi^2}{GM}$ 8. Oscillations Simple Harmonic Motion (SHM): $x(t) = x_m \cos(\omega t + \phi)$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) Period: $T = \frac{2\pi}{\omega}$ Frequency: $f = \frac{1}{T}$ Energy in SHM: $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kx_m^2$ 9. Waves Wave Speed: $v = \lambda f$ Transverse Wave on String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear density) Sound Speed in Fluid: $v = \sqrt{\frac{B}{\rho}}$ ($B$ = bulk modulus, $\rho$ = density) Sound Speed in Solid Rod: $v = \sqrt{\frac{E}{\rho}}$ ($E$ = Young's modulus) Intensity: $I = \frac{P}{A}$ Intensity Level (dB): $\beta = (10 \text{ dB}) \log_{10} \frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) 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; + top 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: $\Delta L = L\alpha\Delta T$ Volume: $\Delta V = V\beta\Delta T$ ($\beta = 3\alpha$) Heat Capacity: $Q = C\Delta T = mc\Delta T$ Latent Heat: $Q = mL$ (L: latent heat of fusion/vaporization) First Law of Thermodynamics: $\Delta E_{int} = Q - W$ Work Done by Gas: $W = \int P dV$ Ideal Gas Law: $PV = nRT = NkT$ Kinetic Theory of Gases: Average Kinetic Energy: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ Heat Transfer: Conduction: $P_{cond} = kA \frac{T_H - T_C}{L}$ Radiation: $P_{rad} = \sigma \epsilon A T^4$ (Stefan-Boltzmann Law) Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of an isolated system never decreases: $\Delta S \ge 0$. Carnot Engine Efficiency: $\epsilon = 1 - \frac{T_C}{T_H}$ Entropy Change: $\Delta S = \int \frac{dQ}{T}$ 11. Electric Fields Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ ($k = \frac{1}{4\pi\epsilon_0} \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ Field due to 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: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ 12. Electric Potential Electric 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 due to Point Charge: $V = k \frac{q}{r}$ Relationship between E and V: $\vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k} \right)$ 13. Capacitance Capacitance: $C = \frac{q}{V}$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{q^2}{2C} = \frac{1}{2}qV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Dielectrics: $C = \kappa C_{air}$ ($E = E_{air}/\kappa$) 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: $I = \frac{dq}{dt}$ Current Density: $\vec{J} = nq\vec{v}_d$ ($n$: charge carrier density, $v_d$: drift speed) Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ ($\rho$: resistivity) Power in Circuits: $P = IV = I^2 R = \frac{V^2}{R}$ Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ RC Circuits (Charging): $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R} e^{-t/RC}$ ($\tau = RC$) 15. 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: $\vec{\tau} = \vec{\mu} \times \vec{B}$ ($\vec{\mu} = NI\vec{A}$ is magnetic dipole moment) Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ 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 of Solenoid: $B = \mu_0 n I$ ($n$: turns per unit length) Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ 16. 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: $L = \frac{N\Phi_B}{I}$ Self-Induced EMF: $\mathcal{E}_L = -L \frac{dI}{dt}$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Energy Density: $u_B = \frac{B^2}{2\mu_0}$ RL Circuits (Current buildup): $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ ($\tau_L = L/R$) LC Oscillations: $\omega = \frac{1}{\sqrt{LC}}$ LRC Circuits: Damped oscillations. 17. Electromagnetic Waves Maxwell's Equations (Integral Form): $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ (Gauss' Law for E) $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss' Law for B) $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (Faraday's Law) $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (Ampere-Maxwell Law) Speed of EM Waves: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Wave Equation: $\frac{\partial^2 E}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 E}{\partial t^2}$ (similar for B) Relationship between E and B: $E = cB$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$ (direction of energy flow) Intensity (Average Poynting Vector): $I = S_{avg} = \frac{1}{c\mu_0} E_{rms}^2 = \frac{c}{\mu_0} B_{rms}^2$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 18. Light and Optics Reflection and 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: $\sin \theta_c = n_2/n_1$ (for $n_1 > n_2$) Mirrors and Lenses Mirror/Lens Equation: $\frac{1}{f} = \frac{1}{p} + \frac{1}{i}$ ($f$: focal length, $p$: object dist, $i$: image dist) Magnification: $m = -\frac{i}{p} = \frac{h_i}{h_p}$ Sign Conventions: $p$: + (real object), - (virtual object) $i$: + (real image), - (virtual image) $f$: + (converging/concave), - (diverging/convex) $h$: + (upright), - (inverted) Interference Path Difference: $\Delta L = d \sin \theta$ Young's Double Slit: Constructive: $d \sin \theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Destructive: $d \sin \theta = (m + \frac{1}{2})\lambda$ Fringe separation: $\Delta y = \frac{\lambda L}{d}$ Thin Films: Phase change of $\pi$ (or $1/2 \lambda$) occurs upon reflection if $n_{incident} Diffraction Single Slit: Destructive: $a \sin \theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Diffraction Grating: Constructive: $d \sin \theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh Criterion: $\theta_{res} = 1.22 \frac{\lambda}{D}$ (circular aperture) 19. Modern Physics Quantum Physics Planck's Constant: $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$ Photon Energy: $E = hf = \frac{hc}{\lambda}$ Photoelectric Effect: $K_{max} = hf - \Phi$ ($\Phi$: work function) 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 1D): $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ Atomic Physics Bohr Model Energy Levels (Hydrogen): $E_n = -\frac{13.6 \text{ eV}}{n^2}$ Hydrogen Spectral Lines: $\frac{1}{\lambda} = R (\frac{1}{n_f^2} - \frac{1}{n_i^2})$ ($R$: Rydberg constant) Quantum Numbers: $n$ (principal), $l$ (orbital), $m_l$ (magnetic), $m_s$ (spin) Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers. Nuclear Physics Mass-Energy Equivalence: $E = mc^2$ Binding Energy: Energy required to break a nucleus into its constituent nucleons. Calculated from mass defect. Radioactive Decay Law: $N(t) = N_0 e^{-\lambda t}$ ($\lambda$: decay constant) Half-Life: $T_{1/2} = \frac{\ln 2}{\lambda}$ Activity: $R = |\frac{dN}{dt}| = \lambda N$