Halliday

Cheatsheet Content

1. Measurement SI Base Units: Length: meter (m) Mass: kilogram (kg) Time: second (s) Current: ampere (A) Temperature: kelvin (K) Amount: mole (mol) Luminous Intensity: candela (cd) Prefixes: Giga (G): $10^9$ Mega (M): $10^6$ Kilo (k): $10^3$ Centi (c): $10^{-2}$ Milli (m): $10^{-3}$ Micro ($\mu$): $10^{-6}$ Nano (n): $10^{-9}$ Pico (p): $10^{-12}$ 2. Kinematics 2.1. One-Dimensional Motion Displacement: $\Delta x = x_f - x_i$ Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Average Speed: $s_{avg} = \frac{\text{total distance}}{\Delta t}$ 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: $v = v_0 + at$ $x = x_0 + v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ Free Fall: $a = -g = -9.8 \text{ m/s}^2$ (downward) 2.2. Two- and Three-Dimensional Motion Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Displacement Vector: $\Delta \vec{r} = \vec{r}_f - \vec{r}_i$ Average Velocity Vector: $\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$ Instantaneous Velocity Vector: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ Average Acceleration Vector: $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$ Instantaneous Acceleration Vector: $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ Projectile Motion: $v_{0x} = v_0 \cos \theta_0$ $v_{0y} = v_0 \sin \theta_0$ $x = (v_0 \cos \theta_0)t$ $y = (v_0 \sin \theta_0)t - \frac{1}{2}gt^2$ $v_y = v_0 \sin \theta_0 - gt$ $v_y^2 = (v_0 \sin \theta_0)^2 - 2gy$ Range $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Uniform Circular Motion: Speed: $v = \frac{2\pi r}{T}$ Centripetal Acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$ (directed towards center) Angular Speed: $\omega = \frac{2\pi}{T}$ 3. 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: $\Sigma \vec{F} = m\vec{a}$ Newton's Third Law: For every action, there is an equal and opposite reaction ($\vec{F}_{AB} = -\vec{F}_{BA}$). Gravitational Force (Weight): $\vec{F}_g = m\vec{g}$ Normal Force: $\vec{F}_N$ (perpendicular to surface) Friction Force: Static: $f_s \le \mu_s F_N$ Kinetic: $f_k = \mu_k F_N$ Tension: $\vec{T}$ (pulling force via string/cable) Drag Force (approximate): $F_D = \frac{1}{2}C\rho Av^2$ Centripetal Force: $F_c = m\frac{v^2}{r}$ (not a new type of force, but the net force causing circular motion) 4. Work and Energy Work Done by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ Work Done by Variable Force (1D): $W = \int_{x_i}^{x_f} F(x) dx$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy: $U_s = \frac{1}{2}kx^2$ (for spring) Mechanical Energy: $E_{mech} = K + U$ Conservation of Mechanical Energy: $E_{mech,i} = E_{mech,f}$ (if only conservative forces do work) Conservative Force Work: $W_c = -\Delta U$ Non-Conservative Force Work: $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ Total Energy Conservation: $W_{ext} = \Delta E_{sys} = \Delta K + \Delta U + \Delta E_{th} + \Delta E_{int}$ Power: $P_{avg} = \frac{W}{\Delta t}$, $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ 5. Momentum and Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Newton's Second Law (Momentum Form): $\Sigma \vec{F} = \frac{d\vec{p}}{dt}$ Impulse: $\vec{J} = \int_{t_i}^{t_f} \vec{F} dt = \Delta \vec{p}$ Impulse-Momentum Theorem: $\vec{J} = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum: If $\Sigma \vec{F}_{ext} = 0$, then $\Sigma \vec{p}_i = \Sigma \vec{p}_f$ Collisions: Elastic: Both momentum and kinetic energy are conserved. Inelastic: Momentum conserved, kinetic energy NOT conserved. Perfectly Inelastic: Objects stick together; momentum conserved, maximum KE loss. Center of Mass: 1D: $x_{com} = \frac{1}{M}\sum m_i x_i$ 3D: $\vec{r}_{com} = \frac{1}{M}\sum m_i \vec{r}_i$ Velocity: $\vec{v}_{com} = \frac{1}{M}\sum m_i \vec{v}_i$ Acceleration: $\vec{a}_{com} = \frac{1}{M}\sum m_i \vec{a}_i$ $\Sigma \vec{F}_{ext} = M\vec{a}_{com}$ 6. Rotation Angular Position: $\theta$ (radians) Angular Displacement: $\Delta \theta = \theta_f - \theta_i$ Average Angular Velocity: $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$ Instantaneous Angular Velocity: $\omega = \frac{d\theta}{dt}$ Average Angular Acceleration: $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$ Instantaneous Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Constant Angular Acceleration Equations: $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relationship between Linear and Angular: $s = r\theta$ $v_t = r\omega$ (tangential speed) $a_t = r\alpha$ (tangential acceleration) $a_c = r\omega^2 = \frac{v_t^2}{r}$ (centripetal acceleration) Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Moment of Inertia: $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous) Parallel-Axis Theorem: $I = I_{com} + Md^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F} = rF\sin\phi$ Newton's Second Law for Rotation: $\Sigma \tau = I\alpha$ Work in Rotation: $W = \int_{\theta_i}^{\theta_f} \tau d\theta$ Power in Rotation: $P = \tau \omega$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) Newton's Second Law (Angular Momentum Form): $\Sigma \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$ Conservation of Angular Momentum: If $\Sigma \vec{\tau}_{ext} = 0$, then $\vec{L}_{initial} = \vec{L}_{final}$ 7. Gravity 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 Acceleration: $g = G\frac{M}{r^2}$ Gravitational Potential Energy: $U = -G\frac{m_1 m_2}{r}$ (zero at $r=\infty$) Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's Laws: 1. Orbits are ellipses with the Sun at one focus. 2. A line joining a planet and the Sun sweeps out equal areas in equal times. 3. $T^2 \propto r^3$ (where $r$ is semi-major axis); for circular orbits, $T^2 = (\frac{4\pi^2}{GM})r^3$ 8. Oscillations Simple Harmonic Motion (SHM): Position: $x(t) = x_m \cos(\omega t + \phi)$ 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: $\omega = \sqrt{\frac{k}{m}}$ (mass-spring) Period: $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ Frequency: $f = \frac{1}{T}$ Simple Pendulum: (for small angles) Period: $T = 2\pi\sqrt{\frac{L}{g}}$ Physical Pendulum: Period: $T = 2\pi\sqrt{\frac{I}{mgd}}$ (d is distance from pivot to COM) Damped SHM: $x(t) = x_m e^{-bt/2m} \cos(\omega' t + \phi)$ Forced Oscillations and Resonance: Amplitude peaks when driving frequency matches natural frequency. 9. Waves Wave Equation: $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Wave Speed: $v = \lambda f = \frac{\omega}{k}$ Angular Wave Number: $k = \frac{2\pi}{\lambda}$ Transverse Wave Speed on String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear density) Power Transmitted by Wave: $P = \frac{1}{2}\mu v \omega^2 y_m^2$ Sound Speed: $v = \sqrt{\frac{B}{\rho}}$ (B = bulk modulus, $\rho$ = density) Intensity: $I = \frac{P}{A}$ Intensity Level (dB): $\beta = (10 \text{ dB}) \log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) Standing Waves: For string fixed at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) For open-open or closed-closed pipe: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) For open-closed pipe: $\lambda_n = \frac{4L}{n}$, $f_n = n\frac{v}{4L}$ ($n=1,3,5...$) Beats: $f_{beat} = |f_1 - f_2|$ Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D = detector, S = source; top signs for approaching, bottom for receding) 10. Thermodynamics Temperature Conversion: $T_C = T_K - 273.15$ $T_F = \frac{9}{5}T_C + 32$ 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: $Q = cm\Delta T$ Latent Heat (Phase Change): $Q = Lm$ Heat Transfer: Conduction: $P_{cond} = \frac{kA(T_H - T_C)}{L}$ Radiation: $P_{rad} = \sigma A \epsilon T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$) Ideal Gas Law: $PV = nRT = NkT$ $R = 8.314 \text{ J/(mol}\cdot\text{K)}$ $k = 1.38 \times 10^{-23} \text{ J/K}$ $N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$ Kinetic Theory of Gases: Average KE per molecule: $K_{avg} = \frac{3}{2}kT$ RMS speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ First Law of Thermodynamics: $\Delta E_{int} = Q - W$ Work done by gas: $W = \int P dV$ For ideal gas: $\Delta E_{int} = n C_V \Delta T$ Molar Specific Heats: $C_P - C_V = R$ Monatomic gas: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$ Diatomic gas: $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$ (at moderate temps) Adiabatic Process: $PV^\gamma = \text{constant}$, $T V^{\gamma-1} = \text{constant}$, where $\gamma = C_P/C_V$ Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of isolated system never decreases. Heat Engines: Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ Carnot Engine: $\epsilon_C = 1 - \frac{T_C}{T_H}$ Refrigerators/Heat Pumps: Coefficient of Performance (COP): $K = \frac{|Q_C|}{|W|}$ (refrigerator) Carnot COP: $K_C = \frac{T_C}{T_H - T_C}$ Entropy: $\Delta S = \int \frac{dQ_{rev}}{T}$ For reversible process: $\Delta S = \frac{Q}{T}$ $\Delta S_{total} \ge 0$ 11. Electric Fields 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: $\vec{E} = \frac{\vec{F}}{q_0}$ Field of Point Charge: $E = k\frac{|q|}{r^2}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on Dipole in E-field: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ 12. Electric Potential Potential Energy Change: $\Delta U = -W = -q_0 \int \vec{E} \cdot d\vec{s}$ Electric Potential: $V = \frac{U}{q_0}$ Potential Difference: $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ Potential of Point Charge: $V = k\frac{q}{r}$ Potential due to Dipole (far field): $V = k\frac{p \cos\theta}{r^2}$ Relating E and V: $\vec{E} = -\nabla V = -(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k})$ Electric Potential Energy of System of Charges: $U = \sum_{i 13. Capacitance Capacitance: $C = \frac{q}{V}$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Cylindrical Capacitor: $C = 2\pi\epsilon_0 \frac{L}{\ln(b/a)}$ Spherical Capacitor: $C = 4\pi\epsilon_0 \frac{ab}{b-a}$ Isolated Sphere: $C = 4\pi\epsilon_0 R$ 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 Capacitor: $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{q^2}{C} = \frac{1}{2}qV$ Energy Density of E-field: $u_E = \frac{1}{2}\epsilon_0 E^2$ Dielectrics: $C = \kappa C_{air}$, $E = E_{air}/\kappa$ 14. Current and Resistance Electric Current: $I = \frac{dq}{dt}$ Current Density: $\vec{J} = nq\vec{v}_d$ Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ Resistivity: $\rho = \rho_0[1 + \alpha(T - T_0)]$ Power in Circuits: $P = IV = I^2 R = \frac{V^2}{R}$ 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: Junction Rule: $\Sigma I_{in} = \Sigma I_{out}$ Loop Rule: $\Sigma \Delta V = 0$ RC Circuits (Charging): $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ RC Circuits (Discharging): $q(t) = q_0 e^{-t/RC}$, $I(t) = -\frac{q_0}{RC}e^{-t/RC}$ Time Constant: $\tau = RC$ 15. Magnetic Fields Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B} = qvB\sin\phi$ Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NI\vec{A}$ Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = \frac{IB}{net}$ ($n$ = charge carrier density) 16. Sources of Magnetic Fields Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{s} \times \hat{r}}{r^2}$ $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ Field of Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Force Between Parallel Wires: $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$ Field at Center of Circular Loop: $B = \frac{\mu_0 I}{2R}$ Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Field of Solenoid: $B = \mu_0 nI$ ($n$ = turns per unit length) Field of Toroid: $B = \frac{\mu_0 NI}{2\pi r}$ 17. 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/EMF opposes the change in magnetic flux that produced it. Motional EMF: $\mathcal{E} = BLv$ Induced Electric Field: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ Inductance: $L = \frac{N\Phi_B}{I}$ Solenoid Inductance: $L = \mu_0 n^2 A L_{sol}$ RL Circuits (Current Increase): $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$, $\tau_L = L/R$ RL Circuits (Current Decrease): $I(t) = I_0 e^{-t/\tau_L}$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Energy Density of B-field: $u_B = \frac{B^2}{2\mu_0}$ Mutual Inductance: $\mathcal{E}_2 = -M\frac{dI_1}{dt}$ 18. Alternating Current (AC) RMS Values: $V_{rms} = V_m/\sqrt{2}$, $I_{rms} = I_m/\sqrt{2}$ Capacitive Reactance: $X_C = \frac{1}{\omega C}$ Inductive Reactance: $X_L = \omega L$ Impedance (RLC Series): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ Ohm's Law for AC: $V_{rms} = I_{rms} Z$ Power Factor: $\cos\phi$ Average Power: $P_{avg} = I_{rms}V_{rms}\cos\phi = I_{rms}^2 R$ Resonance in RLC: $X_L = X_C \Rightarrow \omega_0 = \frac{1}{\sqrt{LC}}$ Transformers: $\frac{V_S}{V_P} = \frac{N_S}{N_P} = \frac{I_P}{I_S}$ 19. Electromagnetic Waves Maxwell's Equations (Integral Form): $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ (Gauss' Law for E) $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss' Law for B) $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (Faraday's Law) $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (Ampere-Maxwell Law) Speed of EM Waves in Vacuum: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Relationship between E and B: $E = cB$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (direction of energy flow) Intensity (Average Poynting Vector): $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{1}{2c\mu_0}E_m^2$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 20. Optics 20.1. Reflection and 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: $n = c/v$ Critical Angle: $\sin\theta_c = n_2/n_1$ (for $n_1 > n_2$) 20.2. Mirrors and Lenses Mirror/Lens Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -\frac{i}{p} = \frac{h_i}{h_p}$ Focal Length: Spherical mirror: $f = R/2$ Lensmaker's Equation: $\frac{1}{f} = (n-1)(\frac{1}{r_1} - \frac{1}{r_2})$ Sign Conventions: (usually real positive, virtual negative) $p$: + if object is real (in front of mirror/lens) $i$: + if image is real (in front of mirror, behind lens) $f$: + for concave mirror/converging lens; - for convex mirror/diverging lens $h$: + if upright, - if inverted 20.3. Interference Young's Double Slit: Bright Fringes (maxima): $d \sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Dark Fringes (minima): $d \sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) Approximate positions on screen: $y = L \tan\theta \approx L\sin\theta$ Thin Film Interference: (for normal incidence) Phase change of $\pi$ (or $\lambda/2$) upon reflection if $n_1 Constructive: $2L = (m + \frac{1}{2})\lambda/n_{film}$ (if one phase shift) or $2L = m\lambda/n_{film}$ (if zero or two phase shifts) Destructive: $2L = m\lambda/n_{film}$ (if one phase shift) or $2L = (m + \frac{1}{2})\lambda/n_{film}$ (if zero or two phase shifts) 20.4. Diffraction Single Slit Diffraction: Dark Fringes (minima): $a \sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Diffraction Grating: Bright Fringes (maxima): $d \sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh's Criterion: $\theta_R = 1.22 \frac{\lambda}{D}$ (circular aperture)