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:** $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ ##### Constant Acceleration Equations: - $v = v_0 + at$ - $x = x_0 + v_0 t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 1.2. Two- and Three-Dimensional Motion - **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:** $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ ##### Projectile Motion (Horizontal $a_x = 0$, Vertical $a_y = -g$): - $x = (v_0 \cos\theta_0)t$ - $y = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ - $v_y = v_0 \sin\theta_0 - gt$ - **Range:** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ - **Max Height:** $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ ##### Uniform Circular Motion: - **Centripetal Acceleration:** $a_c = \frac{v^2}{r}$ (directed towards center) - **Period:** $T = \frac{2\pi r}{v}$ ### 2. Newton's Laws of Motion - **Newton's First Law:** 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:** $\sum \vec{F} = m\vec{a}$ - **Newton's Third Law:** If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. #### 2.1. Forces - **Weight:** $W = mg$ (gravitational force) - **Normal Force:** $N$ (perpendicular to surface) - **Friction Force:** - Static: $f_s \le \mu_s N$ - Kinetic: $f_k = \mu_k N$ - **Tension:** $T$ (force transmitted through a string/cable) - **Spring Force (Hooke's Law):** $F_s = -kx$ (where $k$ is spring constant, $x$ is displacement from equilibrium) ### 3. Work, Energy, and Power #### 3.1. Work - **Work done by constant force:** $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$ - **Work done by variable force:** $W = \int \vec{F} \cdot d\vec{r}$ - **Work-Kinetic Energy Theorem:** $W_{net} = \Delta K = K_f - K_i$ #### 3.2. Kinetic Energy - **Translational Kinetic Energy:** $K = \frac{1}{2}mv^2$ #### 3.3. Potential Energy - **Gravitational Potential Energy:** $U_g = mgh$ - **Elastic Potential Energy (Spring):** $U_s = \frac{1}{2}kx^2$ - **Relationship between Force and Potential Energy:** $F_x = -\frac{dU}{dx}$ #### 3.4. Conservation of Energy - **Mechanical Energy:** $E_{mech} = K + U$ - **Conservation of Mechanical Energy (only conservative forces):** $E_{mech,i} = E_{mech,f}$ or $K_i + U_i = K_f + U_f$ - **Conservation of Energy (with non-conservative forces):** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ #### 3.5. Power - **Average Power:** $P_{avg} = \frac{\Delta W}{\Delta t}$ - **Instantaneous Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### 4. Momentum and Collisions #### 4.1. Linear Momentum - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Newton's Second Law (in terms of momentum):** $\sum \vec{F}_{net} = \frac{d\vec{p}}{dt}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ #### 4.2. Conservation of Linear Momentum - **If $\sum \vec{F}_{ext} = 0$:** $\vec{P}_{total,i} = \vec{P}_{total,f}$ - **For a system of particles:** $\vec{P}_{CM} = M\vec{V}_{CM}$ #### 4.3. Collisions - **Elastic Collision:** Both momentum and kinetic energy are conserved. - **Inelastic Collision:** Momentum is conserved, but kinetic energy is NOT conserved. - **Perfectly Inelastic Collision:** Objects stick together after collision; momentum is conserved, kinetic energy is NOT conserved. ### 5. Rotational Motion #### 5.1. Rotational Kinematics - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ ##### Constant Angular Acceleration Equations: - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ ##### Relations between Linear and Angular Variables: - $s = r\theta$ - $v_t = r\omega$ (tangential speed) - $a_t = r\alpha$ (tangential acceleration) - $a_c = \frac{v_t^2}{r} = r\omega^2$ (centripetal acceleration) #### 5.2. Torque and Moment of Inertia - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ (magnitude $\tau = rF\sin\phi$) - **Newton's Second Law for Rotation:** $\sum \tau = I\alpha$ - **Moment of Inertia:** $I = \sum m_i r_i^2$ (discrete particles) or $I = \int r^2 dm$ (continuous object) - Parallel-Axis Theorem: $I = I_{CM} + Md^2$ #### 5.3. Rotational Kinetic Energy and Work - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Work done by Torque:** $W = \int \tau d\theta$ - **Power:** $P = \tau\omega$ #### 5.4. Angular Momentum - **Angular Momentum of a particle:** $\vec{L} = \vec{r} \times \vec{p}$ (magnitude $L = rp\sin\phi$) - **Angular Momentum of a rigid body:** $L = I\omega$ - **Newton's Second Law (in terms of angular momentum):** $\sum \vec{\tau}_{net} = \frac{d\vec{L}}{dt}$ - **Conservation of Angular Momentum:** If $\sum \vec{\tau}_{ext} = 0$, then $\vec{L}_{total,i} = \vec{L}_{total,f}$ ### 6. Gravitation - **Newton's Law of Universal Gravitation:** $F = G\frac{m_1 m_2}{r^2}$ (where $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$) - **Gravitational Potential Energy:** $U = -G\frac{m_1 m_2}{r}$ (relative to $U=0$ at $r=\infty$) - **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. 3. Law of Periods: $T^2 \propto r^3$ (where $r$ is the semi-major axis) - For circular orbits: $T^2 = \left(\frac{4\pi^2}{GM}\right)r^3$ ### 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:** $f = \frac{1}{T} = \frac{\omega}{2\pi}$ - **Period of Simple Pendulum (small angles):** $T = 2\pi\sqrt{\frac{L}{g}}$ - **Period of Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgd}}$ - **Energy in SHM:** $E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ (conserved) ### 8. Waves #### 8.1. Transverse and Longitudinal Waves - **Wave Speed:** $v = \lambda f$ - **Speed on a stretched string:** $v = \sqrt{\frac{\tau}{\mu}}$ (where $\tau$ is tension, $\mu$ is linear mass density) #### 8.2. Superposition and 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 = $m\lambda$ (where $m=0, 1, 2, ...$) - **Destructive Interference:** Path difference = $(m + \frac{1}{2})\lambda$ #### 8.3. Standing Waves - **Standing Waves on a string fixed at both ends:** - Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1, 2, 3, ...$) - Frequencies: $f_n = \frac{nv}{2L} = nf_1$ (harmonics) - $f_1 = \frac{v}{2L}$ (fundamental frequency) - **Standing Waves in pipes:** - Open at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1, 2, 3, ...$) - Open at one end, closed at other: $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ ($n=1, 3, 5, ...$ only odd harmonics) #### 8.4. Sound Waves - **Speed of sound in ideal gas:** $v = \sqrt{\frac{\gamma P}{\rho}}$ - **Sound Intensity:** $I = \frac{P}{A}$ - **Intensity Level (Decibels):** $\beta = (10 \text{ dB})\log_{10}\left(\frac{I}{I_0}\right)$ (where $I_0 = 10^{-12} \text{ W/m}^2$) - **Doppler Effect:** $f' = f \left(\frac{v \pm v_D}{v \mp v_S}\right)$ - Top signs for "towards", bottom for "away" ($v_D$ is detector speed, $v_S$ is source speed) ### 9. Thermodynamics #### 9.1. Temperature and 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: $\Delta V = V\beta\Delta T$ (where $\beta = 3\alpha$) - **Heat Capacity:** $Q = C\Delta T$ - **Specific Heat:** $Q = mc\Delta T$ - **Latent Heat (Phase Change):** $Q = mL$ (where $L_f$ is latent heat of fusion, $L_v$ is latent heat of vaporization) - **Heat Transfer:** - Conduction: $P_{cond} = kA\frac{T_H - T_C}{L}$ - Convection: Involves fluid movement (complex) - Radiation: $P_{rad} = \sigma A e T^4$ (Stefan-Boltzmann Law, $\sigma = 5.67 \times 10^{-8} \text{ W/(m}^2\text{K}^4)$) #### 9.2. First Law of Thermodynamics - **Internal Energy:** $\Delta E_{int} = Q - W$ - $Q$ is heat added to system, $W$ is work done BY system. - **Work done by a gas:** $W = \int P dV$ - Isobaric ($P$ constant): $W = P\Delta V$ - Isothermal ($T$ constant): $W = nRT\ln\left(\frac{V_f}{V_i}\right)$ - Adiabatic ($Q=0$): $PV^\gamma = \text{constant}$ - Isochoric ($V$ constant): $W=0$ - **Molar Specific Heat:** - Constant Volume: $C_V = \frac{\Delta E_{int}}{n\Delta T}$ - Constant Pressure: $C_P = C_V + R$ - For monatomic ideal gas: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$, $\gamma = \frac{C_P}{C_V} = \frac{5}{3}$ #### 9.3. Second Law of Thermodynamics - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ - **For reversible process:** $\Delta S = \frac{Q}{T}$ - **Entropy always increases for an isolated system:** $\Delta S \ge 0$ - **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 (Refrigerator): $K = \frac{|Q_C|}{|W|}$ - Coefficient of Performance (Heat Pump): $K_{HP} = \frac{|Q_H|}{|W|}$ ### 10. Electric Forces and Fields #### 10.1. Electric Charge - **Quantization of Charge:** $q = ne$ (where $e = 1.602 \times 10^{-19} \text{ C}$) - **Conservation of Charge:** Total charge in an isolated system is conserved. #### 10.2. Coulomb's Law - **Force between two point charges:** $F = k\frac{|q_1 q_2|}{r^2}$ (where $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) - $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ (permittivity of free space) #### 10.3. Electric Field - **Electric Field due to a point charge:** $\vec{E} = \frac{\vec{F}}{q_0} = k\frac{q}{r^2}\hat{r}$ - **Electric Field due to continuous charge distribution:** $\vec{E} = \int d\vec{E} = \int k\frac{dq}{r^2}\hat{r}$ #### 10.4. Gauss' Law - **Electric Flux:** $\Phi_E = \oint \vec{E} \cdot d\vec{A}$ - **Gauss' Law:** $\Phi_E = \frac{q_{enc}}{\epsilon_0}$ ### 11. Electric Potential - **Electric Potential Energy:** $\Delta U = -W = - \int \vec{F} \cdot d\vec{s}$ - **Electric Potential:** $V = \frac{U}{q_0}$ or $\Delta V = \frac{\Delta U}{q_0}$ - **Potential difference (voltage):** $\Delta V = -\int_A^B \vec{E} \cdot d\vec{s}$ - **Potential due to a point charge:** $V = k\frac{q}{r}$ - **Potential due to continuous charge distribution:** $V = \int dV = \int k\frac{dq}{r}$ - **Relationship between E-field and Potential:** $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ - **Capacitance:** $C = \frac{Q}{V}$ - Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ - Capacitors in Parallel: $C_{eq} = C_1 + C_2 + ...$ - Capacitors in Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ - **Energy stored in a capacitor:** $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ - **Energy Density of E-field:** $u_E = \frac{1}{2}\epsilon_0 E^2$ ### 12. Current, Resistance, and DC Circuits #### 12.1. Electric Current - **Current:** $I = \frac{dQ}{dt} = nAv_d e$ (where $n$ is charge carrier density, $v_d$ is drift speed) - **Current Density:** $\vec{J} = n e \vec{v}_d$ - **Ohm's Law (Microscopic):** $\vec{J} = \sigma \vec{E}$ or $\vec{E} = \rho \vec{J}$ (where $\sigma$ is conductivity, $\rho$ is resistivity) #### 12.2. Resistance - **Resistance:** $R = \frac{V}{I}$ - **Resistance of a wire:** $R = \rho \frac{L}{A}$ - **Temperature dependence of resistivity:** $\rho = \rho_0(1 + \alpha(T - T_0))$ #### 12.3. DC Circuits - **Power in a circuit:** $P = IV = I^2 R = \frac{V^2}{R}$ - **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:** 1. Junction Rule: Sum of currents entering a junction equals sum of currents leaving it ($\sum I_{in} = \sum I_{out}$). 2. Loop Rule: Sum of potential differences around any closed loop is zero ($\sum \Delta V = 0$). - **RC Circuits (Charging Capacitor):** - $Q(t) = Q_{max}(1 - e^{-t/RC})$ - $I(t) = I_{max}e^{-t/RC}$ - **Time Constant:** $\tau = RC$ - **RC Circuits (Discharging Capacitor):** - $Q(t) = Q_0 e^{-t/RC}$ - $I(t) = I_0 e^{-t/RC}$ ### 13. Magnetic Fields and Forces #### 13.1. Magnetic Force - **Force on a moving charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ (magnitude $F_B = |q|vB\sin\phi$) - Direction by right-hand rule (for positive charge) - **Force on a current-carrying wire:** $\vec{F}_B = I\vec{L} \times \vec{B}$ (magnitude $F_B = ILB\sin\phi$) - **Magnetic Torque on a current loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ (where $\vec{\mu} = IA\hat{n}$ is magnetic dipole moment) - Potential energy of a magnetic dipole: $U = -\vec{\mu} \cdot \vec{B}$ #### 13.2. Sources of Magnetic Field - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{s} \times \hat{r}}{r^2}$ (where $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$) - **Magnetic Field of a long straight wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field at center of a circular loop:** $B = \frac{\mu_0 I}{2R}$ - **Magnetic Field of a solenoid:** $B = \mu_0 n I$ (where $n$ is turns per unit length) #### 13.3. Ampere's Law - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ ### 14. Electromagnetic Induction #### 14.1. Faraday's Law of Induction - **Motional EMF:** $\mathcal{E} = BLv$ (for a conductor of length L moving perp. to B) - **Faraday's Law:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ (induced EMF) - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ #### 14.2. Lenz's Law - The direction of the induced current is such that it opposes the change in magnetic flux that produced it. #### 14.3. Inductance - **Self-Inductance:** $L = \frac{N\Phi_B}{I}$ - Inductor EMF: $\mathcal{E}_L = -L\frac{dI}{dt}$ - **Inductance of a Solenoid:** $L = \frac{\mu_0 N^2 A}{l}$ - **Energy stored in an inductor:** $U_L = \frac{1}{2}LI^2$ - **Energy Density of B-field:** $u_B = \frac{B^2}{2\mu_0}$ #### 14.4. LR Circuits (Growth of Current):** - $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ - **Time Constant:** $\tau_L = \frac{L}{R}$ ### 15. Alternating Current (AC) Circuits - **RMS Values (for sinusoidal AC):** - $V_{rms} = \frac{V_{max}}{\sqrt{2}}$ - $I_{rms} = \frac{I_{max}}{\sqrt{2}}$ - **Reactance:** - Inductive Reactance: $X_L = \omega L$ - Capacitive Reactance: $X_C = \frac{1}{\omega C}$ - **Impedance (RLC Series Circuit):** $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - **Phase Angle:** $\tan\phi = \frac{X_L - X_C}{R}$ - **Power in AC Circuits:** - Average Power: $P_{avg} = I_{rms}V_{rms}\cos\phi$ (where $\cos\phi$ is power factor) - For a resistor: $P_{avg} = I_{rms}^2 R$ - **Resonance in RLC Circuit:** $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ ### 16. Electromagnetic Waves - **Speed of Light in Vacuum:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3.00 \times 10^8 \text{ m/s}$ - **Wave Speed:** $c = \lambda f$ - **Relationship between E and B field amplitudes:** $E_{max} = c B_{max}$ - **Poynting Vector (Direction of energy flow):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - **Intensity:** $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{c}{2\mu_0}B_{max}^2 = \frac{E_{max}B_{max}}{2\mu_0}$ - **Radiation Pressure:** - Total absorption: $P_r = \frac{I}{c}$ - Total reflection: $P_r = \frac{2I}{c}$ ### 17. Light and Optics #### 17.1. Reflection and Refraction - **Law of Reflection:** $\theta_i = \theta_r$ - **Snell's Law (Law of Refraction):** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - Index of Refraction: $n = \frac{c}{v}$ - **Critical Angle (for total internal reflection):** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) #### 17.2. Mirrors and Lenses - **Mirror/Thin Lens Equation:** $\frac{1}{f} = \frac{1}{p} + \frac{1}{i}$ - $f$: focal length (+ for concave mirror/converging lens, - for convex mirror/diverging lens) - $p$: object distance (+ if in front of mirror/lens) - $i$: image distance (+ for real image, - for virtual image) - **Magnification:** $m = -\frac{i}{p} = \frac{h'}{h}$ - $|m| > 1$ (magnified), $|m| 0$ (upright), $m ### 18. Modern Physics (Brief Overview) #### 18.1. 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: $E_0 = mc^2$ - Kinetic Energy: $K = (\gamma - 1)mc^2$ #### 18.2. Quantum Physics - **Planck's Quantum Hypothesis:** $E = hf$ (where $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$) - **Photoelectric Effect:** $K_{max} = hf - \Phi$ (where $\Phi$ is work function) - **Photon Momentum:** $p = \frac{h}{\lambda}$ - **De Broglie Wavelength:** $\lambda = \frac{h}{p}$ - **Heisenberg Uncertainty Principle:** - Position and Momentum: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - Energy and Time: $\Delta E \Delta t \ge \frac{\hbar}{2}$ (where $\hbar = \frac{h}{2\pi}$) - **Schrödinger Equation (Time-Independent 1D):** $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ - **Probability Density:** $|\psi|^2$