Halliday Physics Essentials

Cheatsheet Content

### 1. Kinematics #### 1.1. One-Dimensional Motion - **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:** $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ #### 1.2. Constant Acceleration Equations - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 1.3. Two-Dimensional Motion (Projectile Motion) - **Horizontal:** $v_x = v_{0x}$, $x = x_0 + v_{0x}t$ - **Vertical:** $v_y = v_{0y} - gt$, $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$, $v_y^2 = v_{0y}^2 - 2g(y - y_0)$ - **Range:** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for $y_0=0$) ### 2. Newton's Laws of Motion - **Newton's 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. - **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 ($\vec{F}_{AB} = -\vec{F}_{BA}$). #### 2.1. Forces - **Weight:** $W = mg$ - **Friction:** - Static: $f_s \le \mu_s N$ - Kinetic: $f_k = \mu_k N$ - **Tension:** Force transmitted through a string, rope, cable. - **Normal Force:** Perpendicular force exerted by a surface. ### 3. Work & Energy - **Work done by a constant force:** $W = \vec{F} \cdot \Delta\vec{r} = F \Delta r \cos\theta$ - **Work done by a variable force:** $W = \int_{x_i}^{x_f} F(x) dx$ - **Kinetic Energy:** $K = \frac{1}{2}mv^2$ - **Work-Kinetic Energy Theorem:** $W_{net} = \Delta K = K_f - K_i$ #### 3.1. Potential Energy - **Gravitational Potential Energy:** $U_g = mgh$ - **Elastic Potential Energy (Spring):** $U_s = \frac{1}{2}kx^2$ - **Work done by conservative force:** $W_c = -\Delta U$ #### 3.2. Conservation of Energy - **Mechanical Energy:** $E_{mech} = K + U$ - **Conservation of Mechanical Energy:** $K_i + U_i = K_f + U_f$ (if only conservative forces do work) - **Generalized Conservation of Energy:** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ (if non-conservative forces do work) #### 3.3. Power - **Average Power:** $P_{avg} = \frac{\Delta W}{\Delta t}$ - **Instantaneous Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### 4. Momentum & Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Newton's Second Law (Momentum Form):** $\vec{F}_{net} = \frac{d\vec{p}}{dt}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta\vec{p}$ #### 4.1. Conservation of Momentum - **Conservation of Linear Momentum:** If $\vec{F}_{net, ext} = 0$, then $\vec{P}_{total, i} = \vec{P}_{total, f}$ - For a system of particles: $\sum \vec{p}_i = \text{constant}$ #### 4.2. Collisions - **Elastic Collision:** Both momentum and kinetic energy are conserved. - **Inelastic Collision:** Momentum is conserved, but kinetic energy is NOT conserved ($K_f ### 5. Rotational Motion - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ #### 5.1. Rotational Kinematics (Constant Angular Acceleration) - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### 5.2. Relationship between Linear and Angular Quantities - **Arc Length:** $s = r\theta$ - **Tangential Speed:** $v_t = r\omega$ - **Tangential Acceleration:** $a_t = r\alpha$ - **Centripetal Acceleration:** $a_c = \frac{v^2}{r} = r\omega^2$ #### 5.3. Rotational Dynamics - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ or $\tau = rF\sin\phi$ - **Newton's Second Law for Rotation:** $\tau_{net} = I\alpha$ - **Moment of Inertia:** $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous) - **Parallel-Axis Theorem:** $I = I_{CM} + Md^2$ - **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 $\tau_{net, ext} = 0$, then $\vec{L}_{total, i} = \vec{L}_{total, f}$ ### 6. Gravitation - **Newton's Law of 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. A line from the Sun to a planet sweeps out equal areas in equal times. 3. $T^2 \propto r^3$ (where $r$ is semi-major axis) ### 7. Oscillations (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:** $\omega = \sqrt{\frac{k}{m}}$ (mass-spring system) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Frequency:** $f = \frac{1}{T}$ - **Period of a Simple Pendulum:** $T = 2\pi\sqrt{\frac{L}{g}}$ (for small angles) - **Period of a Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgd}}$ ### 8. Waves - **Wave Speed:** $v = \lambda f$ - **Transverse Wave on a String:** $v = \sqrt{\frac{T}{\mu}}$ ($\mu$ = mass/unit length) - **Sound Wave Speed:** $v = \sqrt{\frac{B}{\rho}}$ (fluids), $v = \sqrt{\frac{Y}{\rho}}$ (solids) - **Intensity:** $I = \frac{P}{A}$ - **Intensity Level (Decibels):** $\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)$ where $I_0 = 10^{-12} \text{ W/m}^2$ #### 8.1. Superposition & Interference - **Principle of Superposition:** When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements. - **Constructive Interference:** Path difference = $n\lambda$ - **Destructive Interference:** Path difference = $(n + \frac{1}{2})\lambda$ #### 8.2. Standing Waves - **On a String (fixed ends):** - Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1, 2, 3, ...$) - Frequencies: $f_n = n\frac{v}{2L}$ - **In an Open-Open Pipe:** - Wavelengths: $\lambda_n = \frac{2L}{n}$ - **In an Open-Closed Pipe:** - Wavelengths: $\lambda_n = \frac{4L}{n}$ ($n=1, 3, 5, ...$) #### 8.3. Doppler Effect - **Moving Source / Stationary Observer:** - Approaching: $f' = f \frac{v}{v - v_s}$ - Receding: $f' = f \frac{v}{v + v_s}$ - **Moving Observer / Stationary Source:** - Approaching: $f' = f \frac{v + v_o}{v}$ - Receding: $f' = f \frac{v - v_o}{v}$ - **General (Source and Observer Moving):** $f' = f \frac{v \pm v_o}{v \mp v_s}$ (top signs for approaching, bottom for receding) ### 9. Thermodynamics - **Temperature Scales:** - $T_F = \frac{9}{5}T_C + 32^\circ$ - $T_K = T_C + 273.15$ - **Thermal Expansion:** - Linear: $\Delta L = \alpha L_0 \Delta T$ - Volume: $\Delta V = \beta V_0 \Delta T$ ($\beta \approx 3\alpha$) - **Heat Capacity:** $Q = C\Delta T = mc\Delta T$ - **Latent Heat:** $Q = mL$ (L = latent heat of fusion or vaporization) #### 9.1. First Law of Thermodynamics - $\Delta E_{int} = Q - W$ - $Q$ = heat added to system - $W$ = work done BY system ($W = \int P dV$) #### 9.2. Heat Transfer - **Conduction:** $P_{cond} = kA\frac{T_H - T_C}{L}$ - **Convection:** Heat transfer via fluid motion. - **Radiation (Stefan-Boltzmann Law):** $P_{rad} = \sigma A e T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$) #### 9.3. Ideal Gases - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.314 \text{ J/(mol}\cdot\text{K})$ - $k = 1.38 \times 10^{-23} \text{ J/K}$ - **Kinetic Theory of Gases:** - Average Kinetic Energy per molecule: $K_{avg} = \frac{3}{2}kT$ - RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ - **Internal Energy of Ideal Gas:** - Monatomic: $E_{int} = \frac{3}{2}nRT$ - Diatomic: $E_{int} = \frac{5}{2}nRT$ (at moderate temps) #### 9.4. Second Law of Thermodynamics - **Heat Engines:** $e = \frac{W_{eng}}{Q_H} = 1 - \frac{|Q_C|}{Q_H}$ - **Carnot Efficiency (Ideal Engine):** $e_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** - COP (Refrigerator): $K = \frac{|Q_C|}{|W|}$ - COP (Heat Pump): $K = \frac{|Q_H|}{|W|}$ - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ - **Entropy Statement:** $\Delta S_{total} \ge 0$ (for any process) ### 10. Electric Forces & Fields - **Coulomb's Law:** $F = k \frac{|q_1 q_2|}{r^2}$ - $k = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2 = \frac{1}{4\pi\epsilon_0}$ - $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Field of a Point Charge:** $E = k \frac{|q|}{r^2}$ - **Electric Dipole Moment:** $\vec{p} = q\vec{d}$ #### 10.1. Gauss's Law - $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ #### 10.2. Electric Potential - **Potential Energy:** $\Delta U = -W = -q_0 \int \vec{E} \cdot d\vec{s}$ - **Electric Potential:** $V = \frac{U}{q_0}$ or $\Delta V = -\int \vec{E} \cdot d\vec{s}$ - **Potential of a Point Charge:** $V = k \frac{q}{r}$ - **Relationship between E and V:** $\vec{E} = -\nabla V$ (in 1D, $E_x = -\frac{dV}{dx}$) #### 10.3. Capacitance - **Capacitance:** $C = \frac{Q}{V}$ - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Energy Stored:** $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ - **Dielectrics:** $C = \kappa C_0$ ### 11. Current & Resistance - **Electric Current:** $I = \frac{dQ}{dt} = nqv_d A$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ - Resistivity: $\rho = \rho_0[1 + \alpha(T - T_0)]$ - **Power Dissipation:** $P = IV = I^2R = \frac{V^2}{R}$ #### 11.1. DC Circuits - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Kirchhoff's Rules:** - **Junction Rule:** $\sum I_{in} = \sum I_{out}$ - **Loop Rule:** $\sum \Delta V = 0$ - **RC Circuits (Charging Capacitor):** $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = I_{max}e^{-t/RC}$ - Time Constant: $\tau = RC$ ### 12. Magnetic Forces & Fields - **Magnetic Force on a Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ - Magnitude: $F_B = |q|vB\sin\theta$ - **Magnetic Force on a Current-Carrying Wire:** $\vec{F}_B = I\vec{L} \times \vec{B}$ - Magnitude: $F_B = ILB\sin\theta$ - **Torque on a Current Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ - Magnetic Dipole Moment: $\vec{\mu} = NIA\hat{n}$ #### 12.1. Sources of Magnetic Field - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ - $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ - **Magnetic Field of a Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field at Center of Current Loop:** $B = \frac{\mu_0 I}{2R}$ - **Magnetic Field of a Solenoid:** $B = \mu_0 n I$ ($n$ = turns/unit length) #### 12.2. Ampere's Law - $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ ### 13. Electromagnetic Induction - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A} = BA\cos\theta$ - **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 produced it. - **Motional EMF:** $\mathcal{E} = BLv$ #### 13.1. Inductance - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Solenoid Inductance:** $L = \mu_0 n^2 A l$ - **Energy Stored in Inductor:** $U_L = \frac{1}{2}LI^2$ - **RL Circuits (Charging Inductor):** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$, $\tau = L/R$ ### 14. 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:** $\tan\phi = \frac{X_L - X_C}{R}$ - **Power Factor:** $\cos\phi = R/Z$ - **Resonance:** $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ ### 15. Electromagnetic Waves - **Speed of Light:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3.00 \times 10^8 \text{ m/s}$ - **Relationship between E and B:** $E = cB$ - **Wave Speed:** $c = \lambda f$ - **Poynting Vector (Intensity):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - Average Intensity: $I = S_{avg} = \frac{E_{max}B_{max}}{2\mu_0} = \frac{E_{max}^2}{2\mu_0 c} = \frac{cB_{max}^2}{2\mu_0}$ - **Radiation Pressure:** $P_{rad} = I/c$ (perfect absorption), $2I/c$ (perfect reflection) ### 16. Light & Optics #### 16.1. Reflection & 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: $n = \frac{c}{v}$ - **Critical Angle:** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) #### 16.2. Mirrors & Lenses - **Mirror/Lens Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - $p$ = object distance, $i$ = image distance, $f$ = focal length - **Magnification:** $M = -\frac{i}{p} = \frac{h'}{h}$ - **Sign Conventions:** - $p$ always positive (real object) - $i$ positive for real image (same side as object for mirrors, opposite for lenses) - $f$ positive for concave mirror/converging lens; negative for convex mirror/diverging lens - $h'$ positive for upright image #### 16.3. Interference - **Young's Double-Slit Experiment:** - Constructive: $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - Destructive: $d\sin\theta = (m + \frac{1}{2})\lambda$ - Fringe spacing: $y = L \tan\theta \approx L\frac{m\lambda}{d}$ - **Thin Films:** - Condition for constructive/destructive depends on phase shifts at boundaries. (0 or $\pi$ shift) #### 16.4. Diffraction - **Single Slit Diffraction (Minima):** $a\sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) - **Diffraction Grating (Maxima):** $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Rayleigh's Criterion (Resolution):** $\theta_{min} = 1.22 \frac{\lambda}{D}$ ### 17. Modern Physics #### 17.1. Quantum Physics - **Planck's Hypothesis:** $E = hf$ - $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$ - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($\Phi$ = work function) - **Photon Momentum:** $p = \frac{h}{\lambda}$ - **Compton Effect:** $\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$ - **De Broglie Wavelength:** $\lambda = \frac{h}{p}$ - **Heisenberg Uncertainty Principle:** $\Delta x \Delta p_x \ge \frac{\hbar}{2}$, $\Delta E \Delta t \ge \frac{\hbar}{2}$ ($\hbar = h/2\pi$) #### 17.2. Atomic Physics - **Bohr Model (Hydrogen):** - Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ - Radius: $r_n = a_0 n^2$ ($a_0 = 0.0529 \text{ nm}$) - **Energy of Emitted/Absorbed Photon:** $\Delta E = |E_f - E_i| = hf$ #### 17.3. Nuclear Physics - **Mass-Energy Equivalence:** $E = mc^2$ - **Binding Energy:** (Mass defect) $\times c^2$ - **Radioactive Decay Law:** $N(t) = N_0 e^{-\lambda t}$ - Half-Life: $T_{1/2} = \frac{\ln 2}{\lambda}$ - **Activity:** $R = |\frac{dN}{dt}| = \lambda N$