Halliday Physics Essentials

Cheatsheet Content

### Kinematics: 1D Motion - **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:** $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ #### Constant Acceleration Equations - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ ### Kinematics: 2D & 3D Motion - **Position Vector:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - **Displacement Vector:** $\Delta\vec{r} = \vec{r}_f - \vec{r}_i$ - **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 (Constant $g$, $a_x=0$) - $v_x = v_{0x}$ - $x = x_0 + v_{0x}t$ - $v_y = v_{0y} - gt$ - $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ - $v_y^2 = v_{0y}^2 - 2g(y - y_0)$ #### Uniform Circular Motion - **Speed:** $v = \frac{2\pi R}{T}$ - **Centripetal Acceleration:** $a_c = \frac{v^2}{R}$ (directed towards center) ### Newton's Laws of Motion - **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. - **Second Law:** $\sum \vec{F} = m\vec{a}$ (Net force equals mass times acceleration) - **Third Law:** If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) #### Common Forces - **Weight:** $F_g = mg$ (directed downwards) - **Normal Force:** $F_N$ (perpendicular to surface) - **Tension:** $T$ (along a string/rope) - **Friction:** - **Static:** $f_s \le \mu_s F_N$ - **Kinetic:** $f_k = \mu_k F_N$ ### Work and Energy - **Work done by constant force:** $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$ - **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:** $U_g = mgh$ (near Earth's surface) - **Elastic Potential Energy:** $U_s = \frac{1}{2}kx^2$ (for a spring) - **Conservative Force:** Work done is path-independent ($W_c = -\Delta U$) - **Non-Conservative Force:** Work done is path-dependent (e.g., friction) - **Conservation of Mechanical Energy:** $E_{mech} = K + U$ (if only conservative forces do work, $\Delta E_{mech} = 0$) - **General Conservation of Energy (with non-conservative work):** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### Momentum and Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Newton's Second Law (Momentum Form):** $\sum \vec{F} = \frac{d\vec{p}}{dt}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta\vec{p}$ - **Conservation of Linear Momentum:** If $\sum \vec{F}_{ext} = 0$, then $\Delta\vec{P}_{sys} = 0$, so $\vec{P}_i = \vec{P}_f$. - **Collisions:** - **Elastic:** Kinetic energy is conserved ($\Delta K = 0$) - **Inelastic:** Kinetic energy is NOT conserved ($\Delta K \ne 0$) - **Perfectly Inelastic:** Objects stick together after collision. - **Center of Mass:** $\vec{r}_{CM} = \frac{\sum m_i\vec{r}_i}{\sum m_i} = \frac{1}{M_{tot}}\int \vec{r} dm$ ### Rotation - **Angular Position:** $\theta$ (radians) - **Angular Displacement:** $\Delta\theta = \theta_f - \theta_i$ - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ #### Constant Angular Acceleration Equations - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### Rotational-Translational Correspondence | Translational | Rotational | |---------------|------------| | $x$ | $\theta$ | | $v$ | $\omega$ | | $a$ | $\alpha$ | | $m$ | $I$ (Moment of Inertia) | | $\vec{F}$ | $\vec{\tau}$ (Torque) | | $K = \frac{1}{2}mv^2$ | $K = \frac{1}{2}I\omega^2$ | | $\vec{p} = m\vec{v}$ | $\vec{L} = I\vec{\omega}$ | - **Moment of Inertia:** $I = \sum m_i r_i^2 = \int r^2 dm$ - **Parallel-Axis Theorem:** $I = I_{CM} + Md^2$ - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F} = rF\sin\phi$ - **Newton's Second Law for Rotation:** $\sum \vec{\tau} = I\vec{\alpha}$ - **Angular Momentum:** $\vec{L} = I\vec{\omega}$ - **Conservation of Angular Momentum:** If $\sum \vec{\tau}_{ext} = 0$, then $\Delta\vec{L}_{sys} = 0$, so $\vec{L}_i = \vec{L}_f$. ### Gravitation - **Newton's Law of Universal Gravitation:** $F = G \frac{m_1 m_2}{r^2}$ (attractive force) - **Gravitational Potential Energy:** $U = -G \frac{m_1 m_2}{r}$ (for two point masses) - **Escape Speed:** $v_{esc} = \sqrt{\frac{2GM}{R}}$ - **Kepler's Laws:** 1. Law of Orbits: Planets move in elliptical orbits with the Sun at one focus. 2. Law of Areas: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. (Consequence of angular momentum conservation) 3. Law of Periods: $T^2 \propto a^3$ (for circular orbits, $T^2 = (\frac{4\pi^2}{GM})r^3$) ### Oscillations - **Simple Harmonic Motion (SHM):** - **Position:** $x(t) = A\cos(\omega t + \phi)$ - **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:** $\omega = \sqrt{\frac{k}{m}}$ (mass-spring system) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Frequency:** $f = \frac{1}{T}$ - **Simple Pendulum (small angles):** $T = 2\pi\sqrt{\frac{L}{g}}$ - **Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgd}}$ ### Waves - **Wave Speed:** $v = \lambda f$ - **Transverse Wave on a String:** $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear mass density) - **Sound Wave Speed in Fluid:** $v = \sqrt{\frac{B}{\rho}}$ ($B$ = bulk modulus, $\rho$ = density) - **Sound Wave Speed in Solid Rod:** $v = \sqrt{\frac{Y}{\rho}}$ ($Y$ = Young's modulus) - **Intensity:** $I = \frac{P}{A}$ - **Sound Level (dB):** $\beta = (10 \text{ dB})\log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) - **Standing Waves on a String (fixed ends):** - Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1,2,3,...$) - Frequencies: $f_n = n\frac{v}{2L}$ - **Standing Waves in Open-Open Pipe:** Same as string. - **Standing Waves in Open-Closed Pipe:** - Wavelengths: $\lambda_n = \frac{4L}{n}$ ($n=1,3,5,...$) - Frequencies: $f_n = n\frac{v}{4L}$ - **Doppler Effect:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ ($v_D$ is detector, $v_S$ is source; use top signs for "towards", bottom for "away") ### 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:** $\Delta V = V\beta\Delta T$ ($\beta \approx 3\alpha$) - **Heat Capacity:** $Q = C\Delta T$ - **Specific Heat:** $Q = mc\Delta T$ - **Latent Heat (Phase Change):** $Q = mL$ - **Heat Transfer:** - **Conduction:** $P_{cond} = kA\frac{T_H - T_C}{L}$ - **Radiation:** $P_{rad} = \sigma\epsilon A T^4$ (Stefan-Boltzmann Law) - **Ideal Gas Law:** $PV = nRT = NkT$ - **Kinetic Theory of Gases:** - Average Kinetic Energy: $K_{avg} = \frac{3}{2}kT$ - RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ - **Internal Energy of Ideal Gas:** $E_{int} = nC_V T$ - Monatomic: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$ - Diatomic: $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$ (at moderate temps) - **First Law of Thermodynamics:** $\Delta E_{int} = Q - W$ - $W = \int P dV$ (work done BY gas) - **Thermodynamic Processes:** - **Isothermal ($T$=const):** $\Delta E_{int} = 0$, $Q = W = nRT\ln(\frac{V_f}{V_i})$ - **Adiabatic ($Q$=0):** $\Delta E_{int} = -W$, $PV^\gamma$=const, $TV^{\gamma-1}$=const ($\gamma = C_P/C_V$) - **Isobaric ($P$=const):** $W = P\Delta V$ - **Isochoric ($V$=const):** $W = 0$, $\Delta E_{int} = Q$ - **Heat Engines:** - Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - Carnot Efficiency: $\epsilon_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** - Coefficient of Performance: $K = \frac{|Q_C|}{|W|}$ (refrigerator); $K_{HP} = \frac{|Q_H|}{|W|}$ (heat pump) - Carnot COP: $K_C = \frac{T_C}{T_H - T_C}$ - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ (reversible) - Second Law of Thermodynamics: $\Delta S_{sys} + \Delta S_{env} \ge 0$ ### Electric Forces and Fields - **Coulomb's Law:** $\vec{F} = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}\hat{r}$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ (force per unit test charge) - **Electric Field of Point Charge:** $E = \frac{1}{4\pi\epsilon_0}\frac{|q|}{r^2}$ - **Electric Dipole Moment:** $\vec{p} = q\vec{d}$ (from -q to +q) - **Torque on Dipole in E-field:** $\vec{\tau} = \vec{p} \times \vec{E}$ - **Potential Energy of Dipole:** $U = -\vec{p} \cdot \vec{E}$ - **Gauss' Law:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ ### Electric Potential - **Potential Difference:** $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ - **Electric Potential Energy:** $\Delta U = q\Delta V$ - **Potential of Point Charge:** $V = \frac{1}{4\pi\epsilon_0}\frac{q}{r}$ - **Relation between E and V:** $\vec{E} = -\nabla V = -(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k})$ ### Capacitance and Dielectrics - **Capacitance:** $C = \frac{Q}{V}$ - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Energy Stored in Capacitor:** $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$ - **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:** $C = \kappa C_0$ (where $\kappa$ is dielectric constant) ### Current and Resistance - **Electric Current:** $I = \frac{dQ}{dt}$ - **Current Density:** $J = \frac{I}{A} = nqv_d$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ (where $\rho$ is resistivity) - **Power in Circuit:** $P = IV = I^2R = \frac{V^2}{R}$ - **Resistors in Series:** $R_{eq} = \sum R_i$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ ### DC Circuits - **Kirchhoff's Junction Rule:** $\sum I_{in} = \sum I_{out}$ - **Kirchhoff's Loop Rule:** $\sum \Delta V = 0$ (around any closed loop) - **RC Circuits:** - **Charging:** $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = I_{max}e^{-t/RC}$ - **Discharging:** $Q(t) = Q_0 e^{-t/RC}$, $I(t) = I_0 e^{-t/RC}$ - **Time Constant:** $\tau = RC$ ### Magnetic Forces and Fields - **Magnetic Force on Moving Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ - Magnitude: $F_B = |q|vB\sin\theta$ - **Magnetic Force on Current-Carrying Wire:** $\vec{F}_B = I\vec{L} \times \vec{B}$ - **Torque on Current Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ (where $\vec{\mu}$ is magnetic dipole moment, $\vec{\mu} = NIA\hat{n}$) - **Potential Energy of Magnetic 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:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field at Center of Circular 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 per unit length) ### Electromagnetic Induction - **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 flows in a direction that opposes the change in magnetic flux that caused it. - **Motional EMF:** $\mathcal{E} = BLv$ - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Self-Inductance of Solenoid:** $L = \mu_0 n^2 A l$ - **Energy Stored in Inductor:** $U_B = \frac{1}{2}LI^2$ - **RL Circuits:** - **Current build-up:** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ - **Current decay:** $I(t) = I_0 e^{-t/\tau_L}$ - **Time Constant:** $\tau_L = \frac{L}{R}$ - **LC Oscillations:** $\omega = \frac{1}{\sqrt{LC}}$ - **LRC Circuits:** Damped oscillations ### 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 term) - **Speed of EM Waves in Vacuum:** $c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ - **Relation between E and B in EM Wave:** $E = cB$ - **Poynting Vector (Energy Flow):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - **Intensity of EM Wave:** $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{c}{\mu_0}B_{rms}^2$ - **Radiation Pressure:** $P_{rad} = \frac{I}{c}$ (absorbed), $P_{rad} = \frac{2I}{c}$ (reflected) ### Light & Optics - **Speed of Light:** $c = f\lambda$ - **Index of Refraction:** $n = \frac{c}{v}$ - **Snell's Law (Refraction):** $n_1\sin\theta_1 = n_2\sin\theta_2$ - **Total Internal Reflection:** Occurs when $n_1 > n_2$ and $\theta_1 > \theta_c$, where $\sin\theta_c = \frac{n_2}{n_1}$ - **Mirror/Lens Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - $p$: object distance (positive if real object) - $i$: image distance (positive if real image, negative if virtual) - $f$: focal length (positive for converging lens/concave mirror, negative for diverging/convex) - **Magnification:** $m = -\frac{i}{p} = \frac{h_i}{h_p}$ - **Sign Conventions (General):** - $p>0$ for real object (always true for single lens/mirror) - $i>0$ for real image (same side as reflected light for mirror, opposite for lens) - $i 0$ for concave mirror / converging lens - $f 0$ for upright, $h_i ### Interference & Diffraction - **Young's Double-Slit Experiment:** - **Constructive Interference (Bright Fringes):** $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Destructive Interference (Dark Fringes):** $d\sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Fringe Spacing (small angles):** $y_m = \frac{m\lambda L}{d}$ - **Thin Film Interference:** - Phase change upon reflection: occurs if light reflects from a boundary where $n_2 > n_1$. - Conditions vary based on number of phase changes (0, 1, or 2) - **Single-Slit Diffraction:** - **Minima (Dark Fringes):** $a\sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) - **Diffraction Grating:** - **Maxima (Bright Fringes):** $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Rayleigh's Criterion (Resolution):** $\theta_R = 1.22\frac{\lambda}{D}$ (for circular aperture) ### Physical Constants - **Gravitational Constant:** $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ - **Acceleration due to Gravity:** $g = 9.8 \text{ m/s}^2$ - **Speed of Light:** $c = 3.00 \times 10^8 \text{ m/s}$ - **Elementary Charge:** $e = 1.60 \times 10^{-19} \text{ C}$ - **Permittivity of Free Space:** $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ - **Coulomb Constant:** $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - **Permeability of Free Space:** $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ - **Boltzmann Constant:** $k = 1.38 \times 10^{-23} \text{ J/K}$ - **Universal Gas Constant:** $R = 8.314 \text{ J}/(\text{mol}\cdot\text{K})$ - **Avogadro's Number:** $N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$