Halliday Physics Essentials

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### 1D & 2D Kinematics #### 1D Constant Acceleration - **Displacement:** $\Delta x = v_{avg} \Delta t = v_0 t + \frac{1}{2}at^2$ - **Velocity:** $v = v_0 + at$ - **Velocity-Displacement:** $v^2 = v_0^2 + 2a\Delta x$ - **Average Velocity:** $v_{avg} = \frac{v_0 + v}{2}$ #### 2D Projectile Motion - **Horizontal (x-axis):** $v_x = v_{0x}$, $\Delta x = v_{0x} t$ - **Vertical (y-axis):** $v_y = v_{0y} - gt$, $\Delta y = v_{0y} t - \frac{1}{2}gt^2$, $v_y^2 = v_{0y}^2 - 2g\Delta y$ - **Range:** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for $\Delta y = 0$) - **Maximum Height:** $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ #### Relative Motion - **Relative Position:** $\vec{r}_{PA} = \vec{r}_{PB} + \vec{r}_{BA}$ - **Relative Velocity:** $\vec{v}_{PA} = \vec{v}_{PB} + \vec{v}_{BA}$ ### 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:** $\sum \vec{F} = m\vec{a}$ - **Weight:** $W = mg$ - **Friction (Static):** $f_s \le \mu_s N$ - **Friction (Kinetic):** $f_k = \mu_k N$ - **3rd Law:** For every action, there is an equal and opposite reaction. $\vec{F}_{AB} = -\vec{F}_{BA}$ #### Free-Body Diagrams - Draw all forces acting *on* the object. - Resolve forces into components. - Apply $\sum F_x = ma_x$ and $\sum F_y = ma_y$. ### Work, Energy, & Power #### Work - **Constant Force:** $W = \vec{F} \cdot \Delta \vec{r} = F \Delta r \cos\theta$ - **Variable Force (1D):** $W = \int_{x_i}^{x_f} F(x) dx$ - **Work-Kinetic Energy Theorem:** $W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$ #### Potential Energy - **Gravitational:** $U_g = mgh$ - **Spring:** $U_s = \frac{1}{2}kx^2$ - **Conservative Force:** $F_x = -\frac{dU}{dx}$ #### Conservation of Energy - **Mechanical Energy:** $E_{mech} = K + U$ - **Isolated System (Conservative Forces):** $E_{mech,i} = E_{mech,f}$ - $\frac{1}{2}mv_i^2 + U_i = \frac{1}{2}mv_f^2 + U_f$ - **Non-Conservative Forces:** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ #### Power - **Average Power:** $P_{avg} = \frac{\Delta W}{\Delta t}$ - **Instantaneous Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### Momentum & Collisions #### Momentum - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ - **Impulse-Momentum Theorem:** $\vec{J}_{net} = \Delta \vec{p}$ #### Conservation of Momentum - **Isolated System:** $\sum \vec{p}_i = \sum \vec{p}_f$ - $\vec{p}_{1i} + \vec{p}_{2i} = \vec{p}_{1f} + \vec{p}_{2f}$ #### Collisions - **Elastic Collision:** Momentum AND Kinetic Energy are conserved. - 1D, object 2 at rest: $v_{1f} = \frac{m_1-m_2}{m_1+m_2}v_{1i}$, $v_{2f} = \frac{2m_1}{m_1+m_2}v_{1i}$ - **Inelastic Collision:** Momentum conserved, Kinetic Energy NOT conserved. - **Perfectly Inelastic Collision:** Objects stick together. Momentum conserved. - $m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = (m_1+m_2)\vec{v}_f$ #### Center of Mass - **Position:** $\vec{r}_{CM} = \frac{1}{M}\sum m_i \vec{r}_i$ - For continuous object: $\vec{r}_{CM} = \frac{1}{M}\int \vec{r} dm$ - **Velocity:** $\vec{v}_{CM} = \frac{1}{M}\sum m_i \vec{v}_i = \frac{\vec{P}_{tot}}{M}$ - **Newton's 2nd Law for System:** $\sum \vec{F}_{ext} = M\vec{a}_{CM}$ ### Rotational Motion #### Kinematics - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ - **Constant Angular Acceleration:** - $\omega = \omega_0 + \alpha t$ - $\Delta \theta = \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha \Delta \theta$ #### Relation to Linear Variables - **Arc Length:** $s = r\theta$ - **Tangential Speed:** $v_t = r\omega$ - **Tangential Acceleration:** $a_t = r\alpha$ - **Centripetal Acceleration:** $a_c = \frac{v^2}{r} = r\omega^2$ #### Rotational Dynamics - **Moment of Inertia:** $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous) - **Parallel-Axis Theorem:** $I = I_{CM} + Md^2$ - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$, $\tau = rF\sin\phi$ - **Newton's 2nd Law (Rotational):** $\sum \tau = I\alpha$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Work (Rotational):** $W = \int \tau d\theta$ - **Power (Rotational):** $P = \tau\omega$ #### Angular Momentum - **Angular Momentum (Particle):** $\vec{l} = \vec{r} \times \vec{p}$ - **Angular Momentum (Rigid Body):** $L = I\omega$ - **Newton's 2nd Law (Angular Momentum):** $\sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$ - **Conservation of Angular Momentum:** If $\sum \vec{\tau}_{ext} = 0$, then $\vec{L}_{initial} = \vec{L}_{final}$ - $I_i\omega_i = I_f\omega_f$ ### Gravitation - **Newton's Law of Universal Gravitation:** $F = G \frac{m_1 m_2}{r^2}$ - $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ - **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. Equal areas swept in equal times ($\frac{dA}{dt} = \text{constant}$). 3. Period squared is proportional to semi-major axis cubed ($T^2 \propto a^3$ for planets orbiting same star). - $T^2 = (\frac{4\pi^2}{GM})r^3$ (for circular orbits) ### Oscillations & Waves #### Simple Harmonic Motion (SHM) - **Displacement:** $x(t) = A\cos(\omega t + \phi)$ - **Velocity:** $v(t) = -\omega A\sin(\omega t + \phi)$ - **Acceleration:** $a(t) = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x(t)$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ (spring-mass), $T = 2\pi\sqrt{\frac{L}{g}}$ (simple pendulum) - **Energy in SHM:** $E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ #### Waves - **Wave Speed:** $v = \lambda f$ - **Transverse Wave on String:** $v = \sqrt{\frac{T}{\mu}}$ ($\mu$ = mass per unit length) - **Sound Speed in Fluid:** $v = \sqrt{\frac{B}{\rho}}$ ($B$ = bulk modulus, $\rho$ = density) - **Sound Speed in Solid Rod:** $v = \sqrt{\frac{Y}{\rho}}$ ($Y$ = Young's modulus) #### Superposition & Interference - **Principle of Superposition:** $y_{net}(x,t) = y_1(x,t) + y_2(x,t)$ - **Constructive Interference:** Path difference $\Delta L = n\lambda$ - **Destructive Interference:** Path difference $\Delta L = (n + \frac{1}{2})\lambda$ #### Standing Waves - **On a String (fixed ends):** - Wavelengths: $\lambda_n = \frac{2L}{n}$, $n=1, 2, 3, ...$ - Frequencies: $f_n = \frac{nv}{2L} = nf_1$ - **In an Open-Open Pipe:** - Wavelengths: $\lambda_n = \frac{2L}{n}$, $n=1, 2, 3, ...$ - Frequencies: $f_n = \frac{nv}{2L} = nf_1$ - **In an Open-Closed Pipe:** - Wavelengths: $\lambda_n = \frac{4L}{n}$, $n=1, 3, 5, ...$ - Frequencies: $f_n = \frac{nv}{4L} = nf_1$ #### Doppler Effect - **Moving Source, Moving Observer:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ - Top sign for "towards", bottom for "away". - $v_D$: speed of detector, $v_S$: speed of source, $v$: speed of sound. ### Thermodynamics #### Temperature & Heat - **Temperature Scales:** - $T_F = \frac{9}{5}T_C + 32^\circ$ - $T_K = T_C + 273.15$ - **Thermal Expansion (Linear):** $\Delta L = L_0 \alpha \Delta T$ - **Thermal Expansion (Volume):** $\Delta V = V_0 \beta \Delta T$, where $\beta \approx 3\alpha$ - **Heat Transfer:** $Q = mc\Delta T$ (specific heat), $Q = mL$ (latent heat) - **Conduction:** $P_{cond} = \frac{Q}{t} = kA\frac{T_H - T_C}{L}$ - **Radiation:** $P_{rad} = \sigma \epsilon A T^4$ (Stefan-Boltzmann Law) #### Ideal Gases - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.314 \text{ J/mol}\cdot\text{K}$ - $k = 1.38 \times 10^{-23} \text{ J/K}$ (Boltzmann constant) - **Kinetic Theory of Gases:** - Average Kinetic Energy: $K_{avg} = \frac{3}{2}kT$ - RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ #### First Law of Thermodynamics - **Statement:** $\Delta E_{int} = Q - W$ - $Q$: heat added to system, $W$: work done BY system. - **Work done by gas (constant pressure):** $W = P\Delta V$ - **Work done by gas (variable pressure):** $W = \int P dV$ - **Internal Energy (Ideal Gas):** $E_{int} = n C_V T$ - Monatomic: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$ - Diatomic: $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$ - $\gamma = C_P/C_V$ #### Thermodynamic Processes - **Adiabatic:** $Q=0$, $PV^\gamma = \text{constant}$ - **Isothermal:** $\Delta T=0$, $\Delta E_{int}=0$, $W=Q$, $PV = \text{constant}$ - **Isobaric:** $\Delta P=0$, $W=P\Delta V$ - **Isochoric:** $\Delta V=0$, $W=0$, $\Delta E_{int}=Q$ #### Second Law of Thermodynamics - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ (reversible) - $\Delta S \ge 0$ for isolated system (entropy always increases or stays same). - **Heat Engines:** $e = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - Carnot Efficiency: $e_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** $K = \frac{|Q_C|}{|W|}$ (refrigerator), $K_{HP} = \frac{|Q_H|}{|W|}$ (heat pump) - Carnot Coefficient of Performance: $K_C = \frac{T_C}{T_H - T_C}$ (refrigerator) ### Electricity #### Electric Force & Field - **Coulomb's Law:** $F = k \frac{|q_1 q_2|}{r^2}$ - $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ (test charge) - Point Charge: $E = k \frac{|q|}{r^2}$ - **Electric Dipole Moment:** $\vec{p} = q\vec{d}$ #### Gauss's Law - **Statement:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - Used for highly symmetric charge distributions. #### Electric Potential - **Potential Energy:** $\Delta U = -W = -q_0 \int \vec{E} \cdot d\vec{s}$ - **Electric Potential:** $V = \frac{U}{q_0}$ - Point Charge: $V = k \frac{q}{r}$ - Relation to Field: $\vec{E} = -\nabla V = -(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k})$ - **Capacitance:** $C = \frac{Q}{V}$ - Parallel Plate: $C = \frac{\epsilon_0 A}{d}$ - Energy Stored: $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$ - Dielectric: $C = \kappa C_0$ #### Current & Resistance - **Current:** $I = \frac{dQ}{dt} = nqv_d A$ - **Current Density:** $\vec{J} = nq\vec{v}_d$ - **Resistance:** $R = \frac{V}{I}$ (Ohm's Law) - Resistivity: $R = \rho \frac{L}{A}$ - Temperature Dependence: $\rho = \rho_0[1 + \alpha(T-T_0)]$ - **Power Dissipation:** $P = IV = I^2 R = \frac{V^2}{R}$ #### DC Circuits - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ - **Capacitors in Parallel:** $C_{eq} = C_1 + C_2 + ...$ - **Kirchhoff's Rules:** 1. **Junction Rule:** $\sum I_{in} = \sum I_{out}$ 2. **Loop Rule:** $\sum \Delta V = 0$ - **RC Circuits (Charging Capacitor):** $Q(t) = Q_f(1 - e^{-t/RC})$, $I(t) = I_0 e^{-t/RC}$ - Time Constant: $\tau = RC$ ### Magnetism #### Magnetic Force - **Force on Moving Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$, $F_B = |q|vB\sin\phi$ - **Force on Current-Carrying Wire:** $\vec{F}_B = I\vec{L} \times \vec{B}$ - **Torque on Current Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ - Magnetic Dipole Moment: $\vec{\mu} = NI\vec{A}$ #### Magnetic Fields - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ - **Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Current Loop (center):** $B = \frac{\mu_0 I}{2R}$ - **Solenoid:** $B = \mu_0 n I$ ($n$ = turns per unit length) - $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ #### Ampere's Law - **Statement:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - Used for highly symmetric current distributions. #### Faraday's Law of Induction - **Statement:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - Lenz's Law: Induced current opposes the change in magnetic flux. - **Motional EMF:** $\mathcal{E} = BLv$ #### Inductance - **Self-Inductance:** $L = \frac{N\Phi_B}{I}$ - Solenoid: $L = \mu_0 n^2 A l$ - **Induced EMF:** $\mathcal{E}_L = -L\frac{dI}{dt}$ - **Energy Stored in Inductor:** $U_B = \frac{1}{2}LI^2$ - **LR Circuits (Current Growth):** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ - Time Constant: $\tau_L = L/R$ #### LC & RLC Circuits - **LC Oscillations (Angular Frequency):** $\omega = \frac{1}{\sqrt{LC}}$ - **RLC (Damped):** $\omega' = \sqrt{\frac{1}{LC} - (\frac{R}{2L})^2}$ ### Light & Optics #### Electromagnetic Waves - **Speed of Light:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$ - **Wave Speed:** $c = \lambda f$ - **Energy Density (E-field):** $u_E = \frac{1}{2}\epsilon_0 E^2$ - **Energy Density (B-field):** $u_B = \frac{1}{2\mu_0} B^2$ - **Poynting Vector (Intensity):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{1}{2c\mu_0}E_{max}^2$ #### Reflection & Refraction - **Law of Reflection:** $\theta_i = \theta_r$ - **Snell's Law:** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - Index of Refraction: $n = c/v$ - **Total Internal Reflection:** Occurs when $n_1 > n_2$ and $\theta_1 > \theta_c$, where $\sin\theta_c = n_2/n_1$ #### Mirrors & Lenses - **Mirror/Lens Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - $p$: object distance, $i$: image distance, $f$: focal length - **Magnification:** $m = -\frac{i}{p} = \frac{h_i}{h_p}$ - **Sign Conventions:** - $p$ positive if real object (in front of mirror/lens) - $i$ positive if real image (in front of mirror, behind lens) - $f$ positive for concave mirror/converging lens - $f$ negative for convex mirror/diverging lens - $h_i$ positive if upright, negative if inverted #### Interference - **Young's Double-Slit Experiment:** - Constructive: $d\sin\theta = m\lambda$, $m=0, \pm 1, \pm 2, ...$ - Destructive: $d\sin\theta = (m + \frac{1}{2})\lambda$, $m=0, \pm 1, \pm 2, ...$ - **Thin Films:** Phase changes upon reflection ($n_1