Halliday Physics Summary

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### Units & Measurements - **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) - **Dimensional Analysis:** Check consistency of equations. Dimensions must match on both sides. - **Significant Figures:** Rules for addition/subtraction, multiplication/division. - **Scientific Notation:** $A \times 10^n$, where $1 \le A ### Kinematics - **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 - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### Projectile Motion - Horizontal: $v_x = v_{0x}$ (constant), $x = x_0 + v_{0x}t$ - Vertical: $v_y = v_{0y} - gt$, $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ - Use $g = 9.8 \, m/s^2$ (downwards) ### Newton's Laws of Motion - **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. - **2nd Law:** $\sum \vec{F} = m\vec{a}$ - **3rd Law:** For every action, there is an equal and opposite reaction ($\vec{F}_{AB} = -\vec{F}_{BA}$). #### Forces - **Weight:** $W = mg$ - **Normal Force:** Perpendicular to surface. - **Friction:** $f_s \le \mu_s N$ (static), $f_k = \mu_k N$ (kinetic) - **Tension:** Force transmitted through a string/cable. - **Spring Force (Hooke's Law):** $F_s = -kx$ ### Work & Energy - **Work by Constant Force:** $W = \vec{F} \cdot \Delta\vec{r} = F \Delta r \cos\theta$ - **Work by Variable Force:** $W = \int \vec{F} \cdot d\vec{r}$ - **Kinetic Energy:** $K = \frac{1}{2}mv^2$ - **Work-Kinetic Energy Theorem:** $W_{net} = \Delta K$ #### Potential Energy - **Gravitational:** $U_g = mgh$ - **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:** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ (if non-conservative forces do work) - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### Momentum & Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta\vec{p}$ - **Newton's 2nd Law (Momentum Form):** $\sum \vec{F} = \frac{d\vec{p}}{dt}$ - **Conservation of Linear Momentum:** $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$ (if net external force is zero) #### Collisions - **Elastic:** Both momentum and kinetic energy are conserved. - **Inelastic:** Momentum conserved, kinetic energy is NOT conserved ($K_{final} ### Rotational Motion - **Angular Displacement:** $\Delta\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ #### Constant Angular Acceleration - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### Relations to Linear Variables - Arc Length: $s = r\theta$ - Tangential Speed: $v_t = r\omega$ - Tangential Acceleration: $a_t = r\alpha$ - Centripetal Acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$ - Centripetal Force: $F_c = ma_c$ #### Torque & Moment of Inertia - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F} = rF\sin\phi$ - **Moment of Inertia:** $I = \sum m_i r_i^2 = \int r^2 dm$ - **Newton's 2nd Law for Rotation:** $\sum \vec{\tau} = I\vec{\alpha}$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Angular Momentum:** $\vec{L} = I\vec{\omega}$ (for rigid body) - **Conservation of Angular Momentum:** $\sum \vec{L}_{initial} = \sum \vec{L}_{final}$ (if net external torque is zero) ### Gravitation - **Newton's Law of Universal Gravitation:** $F = G \frac{m_1 m_2}{r^2}$ - $G = 6.67 \times 10^{-11} \, N \cdot m^2/kg^2$ - **Gravitational Potential Energy:** $U = -G \frac{m_1 m_2}{r}$ - **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. $T^2 \propto r^3$ (for circular orbits, $T^2 = (\frac{4\pi^2}{GM})r^3$) ### Oscillations & Waves #### Simple Harmonic Motion (SHM) - **Displacement:** $x(t) = A \cos(\omega t + \phi)$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (pendulum) - **Period:** $T = \frac{2\pi}{\omega}$ - **Frequency:** $f = \frac{1}{T}$ - **Velocity:** $v(t) = -A\omega \sin(\omega t + \phi)$ - **Acceleration:** $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$ - **Energy:** $E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ #### Waves - **Wave Speed:** $v = \lambda f$ - **Transverse Wave on String:** $v = \sqrt{\frac{T}{\mu}}$ ($\mu$ = mass/unit length) - **Sound Speed in Fluid:** $v = \sqrt{\frac{B}{\rho}}$ ($B$ = bulk modulus, $\rho$ = density) - **Intensity:** $I = \frac{P}{A}$ - **Interference:** Constructive (path diff = $m\lambda$), Destructive (path diff = $(m+\frac{1}{2})\lambda$) - **Standing Waves:** Nodes (zero displacement), Antinodes (max displacement) - String fixed at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1,2,3,...$) - Open-open/Closed-closed pipe: Same as string - Open-closed pipe: $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ ($n=1,3,5,...$) - **Doppler Effect:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ (Upper signs for approaching, lower for receding) ### 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: $\Delta V = V_0 \beta \Delta T$ ($\beta \approx 3\alpha$) - **Heat Capacity & Latent Heat:** - Heat: $Q = mc\Delta T$ (specific heat $c$) - Phase Change: $Q = mL$ (latent heat $L$) - **Heat Transfer:** - Conduction: $P_{cond} = kA \frac{dT}{dx}$ - Convection: Fluid motion - Radiation: $P_{rad} = \epsilon \sigma A T^4$ (Stefan-Boltzmann Law) - **First Law of Thermodynamics:** $\Delta E_{int} = Q - W$ - $E_{int}$ = internal energy, $Q$ = heat added to system, $W$ = work done BY system. - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.314 \, J/(mol \cdot K)$, $k = 1.38 \times 10^{-23} \, J/K$ - **Work done by gas (isobaric):** $W = P\Delta V$ - **Second Law of Thermodynamics:** - Heat flows spontaneously from hot to cold. - Entropy of isolated system never decreases ($\Delta S \ge 0$). - **Carnot Engine Efficiency:** $\epsilon = 1 - \frac{T_C}{T_H}$ ### Electric Forces & Fields - **Coulomb's Law:** $F = k \frac{|q_1 q_2|}{r^2}$ - $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \, N \cdot m^2/C^2$ - $\epsilon_0 = 8.85 \times 10^{-12} \, C^2/(N \cdot m^2)$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Field of Point Charge:** $E = k \frac{|q|}{r^2}$ - **Electric Dipole:** $\vec{p} = q\vec{d}$ - **Electric Flux:** $\Phi_E = \int \vec{E} \cdot d\vec{A}$ - **Gauss's Law:** $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ ### Electric Potential - **Potential Energy:** $\Delta U = -W = -q_0 \int \vec{E} \cdot d\vec{s}$ - **Electric Potential:** $V = \frac{U}{q_0}$ - **Potential Difference:** $\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{s}$ - **Potential from Point Charge:** $V = k \frac{q}{r}$ - **Relation between E and V:** $\vec{E} = -\vec{\nabla}V$ (in 1D, $E_x = -\frac{dV}{dx}$) ### Capacitance & Dielectrics - **Capacitance:** $C = \frac{Q}{V}$ - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Energy Stored:** $U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ - **Energy Density:** $u_E = \frac{1}{2}\epsilon_0 E^2$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ - **Capacitors in Parallel:** $C_{eq} = \sum C_i$ - **Dielectrics:** $C = \kappa C_0$, $\epsilon = \kappa \epsilon_0$ ### Current & Resistance - **Electric Current:** $I = \frac{dQ}{dt}$ - **Current Density:** $\vec{J} = nq\vec{v}_d$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ ($\rho$ = resistivity) - **Power Dissipation:** $P = IV = I^2R = \frac{V^2}{R}$ - **Resistors in Series:** $R_{eq} = \sum R_i$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ ### DC Circuits - **Kirchhoff's Rules:** 1. **Junction Rule:** $\sum I_{in} = \sum I_{out}$ 2. **Loop Rule:** $\sum \Delta V = 0$ - **RC Circuits (Charging Capacitor):** $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = I_{max}e^{-t/RC}$ - **RC Circuits (Discharging Capacitor):** $Q(t) = Q_0 e^{-t/RC}$, $I(t) = I_0 e^{-t/RC}$ - **Time Constant:** $\tau = RC$ ### Magnetic Forces & Fields - **Magnetic Force on Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ - Direction by RHR. - **Magnetic Force on Current:** $\vec{F}_B = I\vec{L} \times \vec{B}$ - **Magnetic Field from Current (Biot-Savart Law):** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ - $\mu_0 = 4\pi \times 10^{-7} \, T \cdot m/A$ - **Field of Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Field at Center of Loop:** $B = \frac{\mu_0 I}{2R}$ - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - **Field of Solenoid:** $B = \mu_0 n I$ ($n$ = turns/unit length) - **Magnetic Dipole Moment:** $\vec{\mu} = NIA\hat{n}$ - **Torque on Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ ### 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 opposes the change in magnetic flux. - **Motional EMF:** $\mathcal{E} = B L v$ - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Self-Inductance of Solenoid:** $L = \mu_0 n^2 A l$ - **Energy Stored in Inductor:** $U_B = \frac{1}{2}LI^2$ - **Energy Density of Magnetic Field:** $u_B = \frac{B^2}{2\mu_0}$ - **RL Circuits:** - Current increase: $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ - Current decrease: $I(t) = I_0 e^{-t/\tau_L}$ - Time Constant: $\tau_L = \frac{L}{R}$ ### AC Circuits - **RMS Values:** $V_{rms} = \frac{V_{max}}{\sqrt{2}}$, $I_{rms} = \frac{I_{max}}{\sqrt{2}}$ - **Reactance:** - Inductive: $X_L = \omega L$ - Capacitive: $X_C = \frac{1}{\omega C}$ - **Impedance (RLC Series):** $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - **Phase Angle:** $\tan\phi = \frac{X_L - X_C}{R}$ - **Resonance:** $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ - **Power Factor:** $\cos\phi = \frac{R}{Z}$ - **Average Power:** $P_{avg} = I_{rms}V_{rms}\cos\phi$ ### Electromagnetic Waves - **Speed of Light:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \, m/s$ - **Wave Speed:** $c = \lambda f$ - **Relationship between E and B:** $E = cB$ - **Poynting Vector:** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (energy flux) - **Intensity:** $I = S_{avg} = \frac{E_{max}B_{max}}{2\mu_0} = \frac{E_{rms}B_{rms}}{\mu_0} = \frac{E_{rms}^2}{c\mu_0}$ - **Radiation Pressure:** $P_{rad} = \frac{I}{c}$ (absorbed), $P_{rad} = \frac{2I}{c}$ (reflected) ### Light & Optics #### Reflection - **Law of Reflection:** $\theta_i = \theta_r$ - **Plane Mirrors:** Image is virtual, upright, same size, same distance behind mirror. #### Refraction - **Snell's Law:** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Index of Refraction:** $n = \frac{c}{v}$ - **Total Internal Reflection:** Occurs when $n_1 > n_2$ and $\theta_1 > \theta_c$, where $\sin\theta_c = \frac{n_2}{n_1}$. #### Lenses & Mirrors (Thin Lens/Mirror Equation) - $\frac{1}{f} = \frac{1}{p} + \frac{1}{i}$ - **Magnification:** $m = -\frac{i}{p} = \frac{h'}{h}$ - **Sign Conventions:** - $p$: + real object, - virtual object - $i$: + real image, - virtual image - $f$: + converging (concave mirror, convex lens), - diverging (convex mirror, concave lens) - $h'$: + upright, - inverted #### Interference - **Young's Double-Slit:** - Bright Fringes: $d \sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - Dark Fringes: $d \sin\theta = (m + \frac{1}{2})\lambda$ - **Thin Films:** - Phase change upon reflection if $n_{incident} ### Modern Physics #### 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:** $p = \gamma mv$ - **Relativistic Energy:** $E = \gamma mc^2 = K + mc^2$ - **Rest Energy:** $E_0 = mc^2$ #### Quantum Physics - **Planck's Quantum Hypothesis:** $E = hf$ - $h = 6.626 \times 10^{-34} \, J \cdot s$ - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($\Phi$ = work function) - **Photon Momentum:** $p = \frac{h}{\lambda}$ - **Compton Effect:** $\Delta\lambda = \frac{h}{mc}(1 - \cos\phi)$ - **De Broglie Wavelength:** $\lambda = \frac{h}{p}$ - **Heisenberg Uncertainty Principle:** - $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - $\Delta E \Delta t \ge \frac{\hbar}{2}$ - **Schrödinger Equation:** (Time-independent for 1D) $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ - $|\psi|^2$ is probability density. #### Atomic Physics - **Bohr Model (Hydrogen):** - Energy Levels: $E_n = -\frac{13.6 \, eV}{n^2}$ - Radius: $r_n = a_0 n^2$ ($a_0 = 0.0529 \, nm$) - **Quantum Numbers:** $n, l, m_l, m_s$ - **Pauli Exclusion Principle:** No two electrons in an atom can have the same set of four quantum numbers. #### Nuclear Physics - **Mass Defect & Binding Energy:** $E_b = \Delta m c^2$ - **Radioactive Decay:** $N(t) = N_0 e^{-\lambda t}$ - Half-life: $T_{1/2} = \frac{\ln 2}{\lambda}$ - **Fission:** Large nucleus splits. - **Fusion:** Small nuclei combine.