Halliday Physics Essentials

Cheatsheet Content

### Kinematics #### 1. Motion in 1D - **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:** $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ #### 2. Constant Acceleration Formulas (1D) - $v = v_0 + at$ - $x = x_0 + v_0 t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 3. 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:** $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ #### 4. Projectile Motion - **Horizontal (constant velocity):** $v_x = v_{0x}$, $x = x_0 + v_{0x}t$ - **Vertical (constant acceleration $g$):** $v_y = v_{0y} - gt$, $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ - **Max Height:** $H = \frac{v_{0y}^2}{2g}$ - **Range:** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for level ground) ### Newton's Laws of Motion #### 1. First 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. #### 2. Second Law (Force & Acceleration) - $\vec{F}_{net} = m\vec{a}$ - **Weight:** $\vec{W} = m\vec{g}$ #### 3. Third Law (Action-Reaction) - 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. $\vec{F}_{AB} = -\vec{F}_{BA}$ #### 4. Friction - **Static Friction:** $f_s \le \mu_s N$ (prevents motion) - **Kinetic Friction:** $f_k = \mu_k N$ (opposes motion) - $\mu_s > \mu_k$ ### Work & Energy #### 1. Work - **Constant Force:** $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ - **Variable Force:** $W = \int \vec{F} \cdot d\vec{r}$ #### 2. Kinetic Energy - $K = \frac{1}{2}mv^2$ #### 3. Work-Kinetic Energy Theorem - $W_{net} = \Delta K = K_f - K_i$ #### 4. Potential Energy - **Gravitational:** $U_g = mgh$ - **Elastic (Spring):** $U_s = \frac{1}{2}kx^2$ - **Relationship between F and U:** $F_x = -\frac{dU}{dx}$ #### 5. Conservation of Energy - **Mechanical Energy:** $E_{mech} = K + U$ - **Conservative Forces (frictionless):** $E_{mech, i} = E_{mech, f}$ - **Non-Conservative Forces (with friction):** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ #### 6. Power - **Average Power:** $P_{avg} = \frac{\Delta W}{\Delta t}$ - **Instantaneous Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### Momentum & Collisions #### 1. Linear Momentum - $\vec{p} = m\vec{v}$ #### 2. Impulse - $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ #### 3. Conservation of Momentum - If $\vec{F}_{net, ext} = 0$, then $\Delta \vec{P}_{sys} = 0$, so $\vec{P}_{sys, i} = \vec{P}_{sys, f}$ - $\sum m_i \vec{v}_{i,i} = \sum m_i \vec{v}_{i,f}$ #### 4. Collisions - **Elastic:** Kinetic energy is conserved. - **Inelastic:** Kinetic energy is NOT conserved. - **Perfectly Inelastic:** Objects stick together after collision. - $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f$ #### 5. Center of Mass - **Position:** $\vec{r}_{CM} = \frac{1}{M} \sum m_i \vec{r}_i$ or $\frac{1}{M} \int \vec{r} dm$ - **Velocity:** $\vec{v}_{CM} = \frac{1}{M} \sum m_i \vec{v}_i$ - **Newton's 2nd Law for CM:** $\vec{F}_{net, ext} = M\vec{a}_{CM}$ ### Rotational Motion #### 1. Rotational Kinematics - **Angular Position:** $\theta$ (rad) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ (rad/s) - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ (rad/s$^2$) - **Constant Angular Acceleration:** - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### 2. Relationship between Linear and Angular - $s = r\theta$ - $v_t = r\omega$ (tangential speed) - $a_t = r\alpha$ (tangential acceleration) - $a_c = \frac{v_t^2}{r} = r\omega^2$ (centripetal acceleration) #### 3. Moment of Inertia - $I = \sum m_i r_i^2$ or $\int r^2 dm$ - **Parallel-Axis Theorem:** $I = I_{CM} + Md^2$ #### 4. Torque - $\vec{\tau} = \vec{r} \times \vec{F}$ - Magnitude: $\tau = rF \sin\phi = r F_t$ - **Newton's 2nd Law for Rotation:** $\tau_{net} = I\alpha$ #### 5. Rotational Kinetic Energy - $K_{rot} = \frac{1}{2}I\omega^2$ - **Total Kinetic Energy (Rolling):** $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$ #### 6. Angular Momentum - $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ - **Conservation of Angular Momentum:** If $\tau_{net, ext} = 0$, then $\vec{L}_{sys, i} = \vec{L}_{sys, f}$ ### Gravitation #### 1. Newton's Law of Universal Gravitation - $F = G \frac{m_1 m_2}{r^2}$ (G = $6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$) #### 2. Gravitational Acceleration - $g = G \frac{M_E}{R_E^2}$ (on Earth's surface) - $g(r) = G \frac{M}{r^2}$ (at distance $r$ from center of mass M) #### 3. Gravitational Potential Energy - $U = -G \frac{m_1 m_2}{r}$ (relative to $U=0$ at $r=\infty$) #### 4. Escape Speed - $v_{esc} = \sqrt{\frac{2GM}{R}}$ #### 5. Kepler's Laws - **1st Law:** Orbits are ellipses with the Sun at one focus. - **2nd Law:** A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. (Conservation of Angular Momentum) - **3rd Law:** $T^2 \propto a^3$ for planets orbiting the Sun, where $T$ is orbital period and $a$ is semi-major axis. - For circular orbits: $T^2 = \left(\frac{4\pi^2}{GM}\right)r^3$ ### Oscillations & Waves #### 1. Simple Harmonic Motion (SHM) - **Displacement:** $x(t) = x_m \cos(\omega t + \phi)$ - **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) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Simple Pendulum:** $T = 2\pi\sqrt{\frac{L}{g}}$ (for small angles) #### 2. Waves - **Wave Speed:** $v = \lambda f$ - **Transverse Wave on String:** $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$=tension, $\mu$=linear density) - **Speed of Sound:** $v = \sqrt{\frac{B}{\rho}}$ ($B$=bulk modulus, $\rho$=density) - **Intensity:** $I = \frac{P}{A}$ - **Decibel Level:** $\beta = (10 \text{ dB}) \log_{10} \frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) #### 3. Superposition & Interference - **Principle of Superposition:** $y_{net}(x,t) = y_1(x,t) + y_2(x,t)$ - **Constructive Interference:** Path difference = $n\lambda$ (for waves in phase) - **Destructive Interference:** Path difference = $(n + \frac{1}{2})\lambda$ #### 4. Standing Waves - **Strings fixed at both ends:** - Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1,2,3,...$) - Frequencies: $f_n = \frac{nv}{2L} = nf_1$ (harmonics) - **Pipes open at both ends:** Same as strings. - **Pipes closed at one end:** - Wavelengths: $\lambda_n = \frac{4L}{n}$ ($n=1,3,5,...$) - Frequencies: $f_n = \frac{nv}{4L} = nf_1$ (odd harmonics only) #### 5. Doppler Effect - $f' = f \frac{v \pm v_D}{v \mp v_S}$ - (+$v_D$ if detector moves toward source, -$v_D$ if away) - (-$v_S$ if source moves toward detector, +$v_S$ if away) ### Thermodynamics #### 1. Temperature & Heat - **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:** $\Delta V = V\beta\Delta T$ ($\beta \approx 3\alpha$) - **Heat Transfer:** $Q = mc\Delta T$ (specific heat) - **Phase Changes:** $Q = mL_F$ (fusion), $Q = mL_V$ (vaporization) - **Heat Transfer Mechanisms:** - **Conduction:** $P_{cond} = kA\frac{T_H - T_C}{L}$ - **Convection:** Heat transfer by fluid motion. - **Radiation:** $P_{rad} = \sigma A e T^4$ (Stefan-Boltzmann Law) #### 2. Ideal Gas Law - $PV = nRT = NkT$ - $R = 8.31 \text{ J/(mol}\cdot\text{K)}$ (gas constant) - $k = 1.38 \times 10^{-23} \text{ J/K}$ (Boltzmann constant) - $N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$ (Avogadro's number) #### 3. Kinetic Theory of Gases - **Average Kinetic Energy per molecule:** $K_{avg} = \frac{3}{2}kT$ - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ #### 4. First Law of Thermodynamics - $\Delta E_{int} = Q - W$ - $\Delta E_{int}$: change in internal energy - $Q$: heat added to system - $W$: work done BY system - **Work done BY gas:** $W = \int P dV$ - **Isobaric (constant P):** $W = P\Delta V$ - **Isochoric (constant V):** $W = 0$ - **Isothermal (constant T):** $W = nRT \ln\left(\frac{V_f}{V_i}\right)$ - **Adiabatic ($Q=0$):** $PV^\gamma = \text{constant}$ - **Internal Energy of Ideal Gas:** $E_{int} = n C_V T$ - Monatomic: $C_V = \frac{3}{2}R$, $\gamma = 5/3$ - Diatomic: $C_V = \frac{5}{2}R$, $\gamma = 7/5$ - **Molar Specific Heats:** $C_P = C_V + R$ #### 5. Second Law of Thermodynamics - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ (reversible process) - **Isolated Systems:** $\Delta S \ge 0$ - **Heat Engines:** $e = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - **Carnot Engine (ideal):** $e_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** $K = \frac{|Q_C|}{|W|}$ (refrigerator), $K_{HP} = \frac{|Q_H|}{|W|}$ (heat pump) - **Carnot Refrigerator:** $K_C = \frac{T_C}{T_H - T_C}$ ### Electricity #### 1. Electric Charge & Force - **Quantization of Charge:** $q = ne$ ($e = 1.602 \times 10^{-19} \text{ C}$) - **Conservation of Charge:** Total charge in an isolated system is conserved. - **Coulomb's Law:** $F = k \frac{|q_1 q_2|}{r^2}$ ($k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) #### 2. Electric Field - **Definition:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Point Charge:** $E = k \frac{|q|}{r^2}$ (radially outward/inward) - **Electric Dipole:** $\vec{p} = q\vec{d}$ - **Torque on Dipole:** $\vec{\tau} = \vec{p} \times \vec{E}$ - **Potential Energy of Dipole:** $U = -\vec{p} \cdot \vec{E}$ #### 3. Gauss' Law - $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - **Electric Flux:** $\Phi_E = \int \vec{E} \cdot d\vec{A}$ #### 4. Electric Potential - **Definition:** $V = \frac{U}{q_0}$ - **Potential Difference:** $\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{s}$ - **Point Charge:** $V = k \frac{q}{r}$ - **Relationship E and V:** $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ #### 5. Capacitance - **Definition:** $C = \frac{Q}{V}$ - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Series Capacitors:** $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ - **Parallel Capacitors:** $C_{eq} = \sum C_i$ - **Energy Stored:** $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ - **Dielectrics:** $C' = \kappa C_0$, $E' = E_0/\kappa$ #### 6. Current & Resistance - **Current:** $I = \frac{dQ}{dt} = nAv_d e$ - **Current Density:** $\vec{J} = n e \vec{v}_d$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ ($\rho$=resistivity) - **Resistivity Temperature Dependence:** $\rho - \rho_0 = \rho_0 \alpha (T - T_0)$ - **Power in Circuits:** $P = IV = I^2R = \frac{V^2}{R}$ - **Series Resistors:** $R_{eq} = \sum R_i$ - **Parallel Resistors:** $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ #### 7. DC Circuits - **Kirchhoff's Junction Rule:** $\sum I_{in} = \sum I_{out}$ - **Kirchhoff's Loop Rule:** $\sum \Delta V = 0$ - **RC Circuits (Charging):** $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = \frac{V_0}{R}e^{-t/RC}$ - **RC Circuits (Discharging):** $Q(t) = Q_0 e^{-t/RC}$, $I(t) = -\frac{Q_0}{RC}e^{-t/RC}$ - **Time Constant:** $\tau = RC$ ### Magnetism #### 1. Magnetic Field & Force - **Magnetic Force on Moving Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ - Magnitude: $F_B = |q|vB \sin\phi$ - **Magnetic Force on Current:** $\vec{F}_B = I\vec{L} \times \vec{B}$ - **Hall Effect:** $V_H = \frac{IB}{net}$ #### 2. Sources of Magnetic Field - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ - **Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Center of Current Loop:** $B = \frac{\mu_0 I}{2R}$ - **Solenoid:** $B = \mu_0 n I$ (n = turns/length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ #### 3. Magnetic Dipole - **Magnetic Dipole Moment:** $\vec{\mu} = NIA\hat{n}$ - **Torque on Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ - **Potential Energy:** $U = -\vec{\mu} \cdot \vec{B}$ ### Electromagnetic Induction #### 1. Magnetic Flux - $\Phi_B = \int \vec{B} \cdot d\vec{A}$ #### 2. Faraday's Law of Induction - $\mathcal{E} = -\frac{d\Phi_B}{dt}$ #### 3. Lenz's Law - The induced current will flow in a direction that opposes the change in magnetic flux that produced it. #### 4. Motional EMF - $\mathcal{E} = BLv$ #### 5. Inductance - **Self-Inductance:** $L = \frac{N\Phi_B}{I}$ - Solenoid: $L = \mu_0 n^2 A l$ - **Inductor Energy:** $U_B = \frac{1}{2}LI^2$ - **RL Circuits (Current Growth):** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$, $\tau_L = L/R$ - **RL Circuits (Current Decay):** $I(t) = I_0 e^{-t/\tau_L}$ #### 6. LC Oscillations - **Angular Frequency:** $\omega = \frac{1}{\sqrt{LC}}$ - **Energy Conservation:** $U_E + U_B = \frac{Q^2}{2C} + \frac{1}{2}LI^2 = \text{constant}$ #### 7. Maxwell's Equations (Integral Form) - **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:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ #### 8. Electromagnetic Waves - **Speed of Light:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3.00 \times 10^8 \text{ m/s}$ - **Relationship E and B:** $E = cB$ - **Intensity:** $I = \frac{1}{c\mu_0}E_{rms}^2 = \frac{1}{2c\mu_0}E_m^2$ - **Radiation Pressure:** $P_{rad} = I/c$ (absorbed), $2I/c$ (reflected) ### Light & Optics #### 1. Reflection - **Law of Reflection:** $\theta_i = \theta_r$ - **Plane Mirrors:** Image is virtual, upright, same size, same distance behind mirror. #### 2. Refraction - **Snell's Law:** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Critical Angle:** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) - **Apparent Depth:** $d' = d \frac{n_2}{n_1}$ #### 3. Lenses & Mirrors - **Mirror/Lens Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - $p$: object distance, $i$: image distance, $f$: focal length - Concave mirror/converging lens: $f > 0$ - Convex mirror/diverging lens: $f 1$: magnified, $|m| 0$: upright, $m 0$ (real object) - $i > 0$ (real image, opposite side of lens/same side of mirror as light) - $i 0$ (converging), $f 0$ (upright object), $h_i > 0$ (upright image) #### 4. Interference - **Young's Double Slit:** - **Constructive (bright fringes):** $d \sin\theta = m\lambda$ - **Destructive (dark fringes):** $d \sin\theta = (m + \frac{1}{2})\lambda$ - Fringe spacing: $\Delta y = \frac{\lambda L}{d}$ - **Thin Film Interference:** - Phase change upon reflection if $n_{incident}