Halliday

Cheatsheet Content

1. Measurement & Vectors SI Base Units: Length (m), Mass (kg), Time (s), Current (A), Temp (K), Amount (mol), Luminous Intensity (cd) Vector Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\tan \theta = \frac{A_y}{A_x}$ (mind quadrants) Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \theta = A_x B_x + A_y B_y + A_z B_z$ Cross Product (Magnitude): $|\vec{A} \times \vec{B}| = AB \sin \theta$ Cross Product (Components): $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ 2. Kinematics 2.1. 1D Motion Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$ Instantaneous Velocity: $v = \frac{dx}{dt}$ 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: $a = -g = -9.8 \text{ m/s}^2$ (downwards) 2.2. 2D & 3D 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: $v_{x0} = v_0 \cos \theta_0$ (constant) $v_{y0} = v_0 \sin \theta_0$ $y = y_0 + (v_0 \sin \theta_0) t - \frac{1}{2}gt^2$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Uniform Circular Motion: Speed: $v = \frac{2\pi r}{T}$ Centripetal Acceleration: $a_c = \frac{v^2}{r}$ (directed inward) 3. Newton's Laws of Motion Newton's 1st 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 2nd Law: $\sum \vec{F} = m\vec{a}$ Newton's 3rd Law: For every action, there is an equal and opposite reaction. $\vec{F}_{AB} = -\vec{F}_{BA}$ Weight: $W = mg$ Friction: Static: $f_s \le \mu_s N$ Kinetic: $f_k = \mu_k N$ Drag Force (High Speed): $D = \frac{1}{2}C\rho Av^2$ 4. Work & Energy Work (Constant Force): $W = \vec{F} \cdot \vec{d} = Fd \cos \theta$ Work (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: $U_g = mgh$ Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ Mechanical Energy: $E_{mech} = K + U$ Conservation of Mechanical Energy (Conservative Forces Only): $E_{mech,i} = E_{mech,f}$ Work by Non-Conservative Forces: $W_{nc} = \Delta E_{mech}$ Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ 5. Momentum & Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ Impulse-Momentum Theorem: $\vec{J} = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum (Closed, Isolated System): $\sum \vec{p}_i = \sum \vec{p}_f$ Collisions: Elastic: Kinetic energy conserved. $K_i = K_f$ Inelastic: Kinetic energy not conserved. Perfectly Inelastic: Objects stick together. Center of Mass (CM): $\vec{r}_{CM} = \frac{1}{M} \sum m_i \vec{r}_i$ Velocity of CM: $\vec{v}_{CM} = \frac{1}{M} \sum m_i \vec{v}_i$ 6. Rotation Angular Position: $\theta$ (radians) Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Rotational Kinematics (Constant $\alpha$): $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relating Linear & Angular: $s = r\theta$ $v_t = r\omega$ (tangential speed) $a_t = r\alpha$ (tangential acceleration) $a_c = \frac{v_t^2}{r} = r\omega^2$ (centripetal acceleration) Moment of Inertia: $I = \sum m_i r_i^2 = \int r^2 dm$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ (Magnitude: $\tau = rF \sin \phi$) Newton's 2nd Law for Rotation: $\sum \tau = I\alpha$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) Conservation of Angular Momentum: If $\sum \vec{\tau}_{ext} = 0$, then $L_i = L_f$ 7. Gravity Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$ Gravitational Potential Energy: $U = -G \frac{m_1 m_2}{r}$ Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's 3rd Law: $T^2 = \left(\frac{4\pi^2}{GM}\right) r^3$ (for circular orbits) 8. Oscillations Simple Harmonic Motion (SHM) Equation: $x(t) = A \cos(\omega t + \phi)$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) Period: $T = \frac{2\pi}{\omega}$ Frequency: $f = \frac{1}{T}$ Velocity in SHM: $v(t) = -A\omega \sin(\omega t + \phi)$ Acceleration in SHM: $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$ Energy in SHM: $E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ 9. Waves Wave Speed: $v = \lambda f$ Speed on String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$=tension, $\mu$=linear density) Wave Equation: $y(x,t) = A \sin(kx - \omega t + \phi)$ Wave Number: $k = \frac{2\pi}{\lambda}$ Angular Frequency: $\omega = 2\pi f$ Power of a Wave: $P = \frac{1}{2}\mu v \omega^2 A^2$ Interference: Constructive: $\Delta L = m\lambda$ Destructive: $\Delta L = (m + \frac{1}{2})\lambda$ Standing Waves on String (fixed ends): $\lambda_n = \frac{2L}{n}$ $f_n = n \frac{v}{2L}$ (harmonics) 10. Sound Waves Speed of Sound: $v = \sqrt{\frac{B}{\rho}}$ (fluids), $v = \sqrt{\frac{Y}{\rho}}$ (solids) Intensity: $I = \frac{P}{A}$ Sound Level: $\beta = (10 \text{ dB}) \log_{10}\left(\frac{I}{I_0}\right)$ ($I_0 = 10^{-12} \text{ W/m}^2$) Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (top signs for approaching) Standing Waves in Pipes: Open-Open: $\lambda_n = \frac{2L}{n}$, $f_n = n \frac{v}{2L}$ ($n=1,2,3...$) Open-Closed: $\lambda_n = \frac{4L}{n}$, $f_n = n \frac{v}{4L}$ ($n=1,3,5...$) Beats: $f_{beat} = |f_1 - f_2|$ 11. Thermodynamics Temperature Conversion: $T_F = \frac{9}{5}T_C + 32^\circ$ $T_K = T_C + 273.15$ Linear Thermal Expansion: $\Delta L = L\alpha\Delta T$ Volume Thermal Expansion: $\Delta V = V\beta\Delta T$ ($\beta = 3\alpha$) Heat Capacity: $Q = C\Delta T$ Specific Heat: $Q = mc\Delta T$ Latent Heat: $Q = mL$ First Law of Thermodynamics: $\Delta E_{int} = Q - W$ Work Done by Gas: $W = \int P dV$ Ideal Gas Law: $PV = nRT = NkT$ Kinetic Energy of Gas Molecule: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ Heat Transfer: Conduction: $P_{cond} = kA \frac{dT}{dx}$ Radiation: $P_{rad} = \sigma A e T^4$ Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of isolated system never decreases. Entropy Change: $\Delta S = \int \frac{dQ}{T}$ Carnot Efficiency: $\epsilon_C = 1 - \frac{T_C}{T_H}$ 12. Electric Fields & Forces Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ ($k = \frac{1}{4\pi\epsilon_0} \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) Electric Field (Point Charge): $\vec{E} = \frac{\vec{F}}{q_0} = k \frac{q}{r^2}\hat{r}$ Electric Field (Continuous Charge): $\vec{E} = \int d\vec{E} = \int k \frac{dq}{r^2}\hat{r}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on Dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ 13. Electric Potential Electric Potential Difference: $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ Potential (Point Charge): $V = k \frac{q}{r}$ Potential (Continuous Charge): $V = \int \frac{k dq}{r}$ Relation E and V: $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ Potential Energy (Point Charges): $U = k \frac{q_1 q_2}{r}$ Potential Energy (Charge in Field): $U = qV$ 14. Capacitance & Dielectrics Capacitance: $C = \frac{Q}{V}$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Capacitors in Parallel: $C_{eq} = C_1 + C_2 + ...$ Capacitors in Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ Dielectrics: $C = \kappa C_0$, $E = E_0/\kappa$ 15. Current & Resistance Current: $I = \frac{dQ}{dt}$ Current Density: $\vec{J} = nq\vec{v}_d$ ($v_d$=drift speed) Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ Resistivity: $\rho = \rho_0 [1 + \alpha(T - T_0)]$ Power Dissipated: $P = IV = I^2R = \frac{V^2}{R}$ Resistors in Series: $R_{eq} = R_1 + R_2 + ...$ Resistors in Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ RC Circuits: Charging: $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ Discharging: $Q(t) = Q_0 e^{-t/RC}$, $I(t) = -\frac{Q_0}{RC}e^{-t/RC}$ Time Constant: $\tau = RC$ 16. Magnetic Fields & Forces Magnetic Force on Charge: $\vec{F}_B = q(\vec{v} \times \vec{B})$ Magnetic Force on Current: $\vec{F}_B = I(\vec{L} \times \vec{B})$ Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ ($\vec{\mu}=IA\hat{n}$) Potential Energy of 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 (Long Straight Wire): $B = \frac{\mu_0 I}{2\pi r}$ Magnetic Field (Loop Center): $B = \frac{\mu_0 I}{2R}$ Magnetic Field (Solenoid): $B = \mu_0 n I$ ($n$=turns/length) Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Gauss's Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$ 17. Induction & Inductance Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Motional EMF: $\mathcal{E} = BLv$ Lenz's Law: Induced current opposes the change in magnetic flux. Inductance: $L = \frac{N\Phi_B}{I}$ Solenoid Inductance: $L = \mu_0 n^2 A l$ Induced EMF in Inductor: $\mathcal{E}_L = -L \frac{dI}{dt}$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Energy Density (Magnetic Field): $u_B = \frac{B^2}{2\mu_0}$ 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: $\tau_L = L/R$ 18. AC Circuits AC Voltage/Current: $V = V_m \sin(\omega t)$, $I = I_m \sin(\omega t - \phi)$ Reactance: Capacitive: $X_C = \frac{1}{\omega C}$ Inductive: $X_L = \omega L$ Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan \phi = \frac{X_L - X_C}{R}$ RMS Values: $V_{rms} = \frac{V_m}{\sqrt{2}}$, $I_{rms} = \frac{I_m}{\sqrt{2}}$ Average Power: $P_{avg} = I_{rms}V_{rms}\cos\phi = I_{rms}^2 R$ Resonance: $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ 19. Electromagnetic Waves Speed of Light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Wave Equation: $E = E_m \sin(kx - \omega t)$, $B = B_m \sin(kx - \omega t)$ Relation E and B: $E = cB$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (Direction of propagation, Power/Area) Intensity (Avg Poynting): $I = S_{avg} = \frac{1}{2c\mu_0}E_m^2 = \frac{1}{2}\epsilon_0 c E_m^2$ Radiation Pressure: $P_r = \frac{I}{c}$ (absorbed), $P_r = \frac{2I}{c}$ (reflected) 20. Light & Optics 20.1. 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: $\sin \theta_c = \frac{n_2}{n_1}$ ($n_1 > n_2$) 20.2. Mirrors & Lenses Mirror/Lens Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -\frac{i}{p} = \frac{h_i}{h_p}$ Sign Conventions: Real image: $i > 0$, Virtual image: $i Converging lens/concave mirror: $f > 0$ Diverging lens/convex mirror: $f Object in front: $p > 0$ 20.3. Interference & Diffraction Young's Double Slit: Bright Fringes: $d \sin \theta = m\lambda$ Dark Fringes: $d \sin \theta = (m + \frac{1}{2})\lambda$ Thin Films: (Consider phase shifts on reflection) Constructive: $2L = (m + \frac{1}{2})\lambda_n$ (one phase shift) or $2L = m\lambda_n$ (zero/two phase shifts) Destructive: $2L = m\lambda_n$ (one phase shift) or $2L = (m + \frac{1}{2})\lambda_n$ (zero/two phase shifts) Single Slit Diffraction: Dark Fringes: $a \sin \theta = m\lambda$ ($m = \pm 1, \pm 2, ...$) Diffraction Grating: Bright Fringes: $d \sin \theta = m\lambda$ ($m = 0, \pm 1, \pm 2, ...$) Rayleigh Criterion: $\theta_R = 1.22 \frac{\lambda}{D}$ (circular aperture)