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 (1D & 2D) 1D 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: $v = v_0 + at$ $x = x_0 + v_0t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ Free Fall: $a = -g = -9.8 \text{ m/s}^2$ (downwards) 2D Motion (Projectile Motion) Components: $x(t) = (v_0 \cos\theta_0)t$, $y(t) = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ Velocity Components: $v_x = v_0 \cos\theta_0$, $v_y = v_0 \sin\theta_0 - gt$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Maximum Height: $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ 3. Newton's Laws of Motion 1st 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. 2nd Law: $\vec{F}_{net} = m\vec{a}$ (Vector sum of forces equals mass times acceleration) 3rd Law (Action-Reaction): If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A. $\vec{F}_{AB} = -\vec{F}_{BA}$ Gravitational Force: $F_g = mg$ (near Earth's surface) Normal Force: $F_N$ (perpendicular to surface) Friction: Static: $f_s \le \mu_s F_N$ Kinetic: $f_k = \mu_k F_N$ Tension: $T$ (force transmitted through a string/rope) 4. Work, Energy, Power Work: $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ (constant force) Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Potential Energy: Gravitational: $U_g = mgh$ Elastic (Spring): $U_s = \frac{1}{2}kx^2$ Conservation of Mechanical Energy: $E_{mech} = K + U$ (if only conservative forces do work) $E_{mech,i} = E_{mech,f} \implies K_i + U_i = K_f + U_f$ Work done by non-conservative forces: $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ (instantaneous), $P_{avg} = \frac{W}{\Delta t}$ 5. Momentum & Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum: If $\vec{F}_{net, ext} = 0$, then $\vec{P}_{total,i} = \vec{P}_{total,f}$ Collisions: Elastic: Both momentum and kinetic energy are conserved. Inelastic: Momentum conserved, kinetic energy NOT conserved. Perfectly Inelastic: Objects stick together; momentum conserved, max KE loss. Center of Mass: $\vec{r}_{CM} = \frac{1}{M} \sum m_i \vec{r}_i$ or $\vec{r}_{CM} = \frac{1}{M} \int \vec{r} dm$ 6. Rotation Angular Position: $\theta$ (rad) Angular Velocity: $\omega = \frac{d\theta}{dt}$ (rad/s) Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ (rad/s$^2$) Constant Angular Acceleration Equations: $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ 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) Newton's 2nd Law for Rotation: $\tau_{net} = I\alpha$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ or $\tau = rF \sin\phi$ Moment of Inertia: $I = \sum m_i r_i^2$ or $I = \int r^2 dm$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Angular Momentum: $\vec{L} = I\vec{\omega}$ (for rigid body), $\vec{L} = \vec{r} \times \vec{p}$ Conservation of Angular Momentum: If $\vec{\tau}_{net, ext} = 0$, then $\vec{L}_{total,i} = \vec{L}_{total,f}$ 7. Gravity 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: $U = -G \frac{m_1 m_2}{r}$ (zero at $r = \infty$) Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's Laws: Orbits are ellipses with the Sun at one focus. A line from the Sun to a planet sweeps out equal areas in equal times. $T^2 \propto a^3$ (period squared proportional to semi-major axis cubed) 8. Oscillations Simple Harmonic Motion (SHM): $x(t) = x_m \cos(\omega t + \phi)$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (mass-spring), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) Period: $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ (mass-spring), $T = 2\pi\sqrt{\frac{L}{g}}$ (simple pendulum) Frequency: $f = \frac{1}{T}$ 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 a String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear density) 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: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) Standing Waves in Pipes: Open at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) Open at one, closed at one: $\lambda_n = \frac{4L}{n}$, $f_n = n\frac{v}{4L}$ ($n=1,3,5...$) Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (top signs for approaching, bottom for receding) 10. 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: $\Delta V = V\beta\Delta T$, where $\beta = 3\alpha$ Heat Transfer: Specific Heat: $Q = mc\Delta T$ Latent Heat (Phase Change): $Q = mL$ Conduction: $P_{cond} = \frac{Q}{t} = kA\frac{T_H - T_C}{L}$ Radiation: $P_{rad} = \sigma \epsilon A T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$) Ideal Gas Law: $PV = nRT = NkT$ ($R = 8.31 \text{ J/mol}\cdot\text{K}$, $k = 1.38 \times 10^{-23} \text{ J/K}$) 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: $\Delta E_{int} = Q - W$ $W = \int P dV$ (work done BY gas) Heat Engines & Refrigerators: Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ COP (Refrigerator): $K = \frac{|Q_C|}{|W|}$ COP (Heat Pump): $K_{hp} = \frac{|Q_H|}{|W|}$ Carnot Efficiency: $\epsilon_C = 1 - \frac{T_C}{T_H}$ Entropy: $\Delta S = \int \frac{dQ}{T}$ (reversible), $\Delta S \ge 0$ (Second Law) 11. Electric Fields & Potentials Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ ($k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ (test charge), $\vec{E} = k \frac{q}{r^2}\hat{r}$ (point charge) Electric Flux: $\Phi_E = \oint \vec{E} \cdot d\vec{A}$ Gauss' Law: $\epsilon_0 \Phi_E = q_{enc}$ Electric Potential Energy: $\Delta U = -W = -q_0 \int \vec{E} \cdot d\vec{s}$ Electric Potential: $V = \frac{U}{q_0}$, $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ Potential from Point Charge: $V = k \frac{q}{r}$ Relating E and V: $\vec{E} = -\vec{\nabla}V$ (in 1D, $E_x = -\frac{dV}{dx}$) 12. 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 Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Dielectrics: $C = \kappa C_{air}$, $E = E_{air}/\kappa$ 13. Current & Resistance Current: $I = \frac{dQ}{dt}$ Current Density: $\vec{J} = nq\vec{v}_d$ Ohm's Law: $V = IR$, $\vec{J} = \sigma \vec{E}$ Resistance: $R = \rho \frac{L}{A}$ ($\rho$ = resistivity) Power Dissipation: $P = IV = I^2R = \frac{V^2}{R}$ Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ RC Circuits: Charging: $Q(t) = Q_{max}(1 - e^{-t/\tau})$, $I(t) = I_{max}e^{-t/\tau}$ Discharging: $Q(t) = Q_0 e^{-t/\tau}$, $I(t) = I_0 e^{-t/\tau}$ Time Constant: $\tau = RC$ 14. Magnetic Fields Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ ($F_B = |q|vB\sin\phi$) Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ ($F_B = ILB\sin\phi$) Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ ($\vec{\mu}$ = magnetic dipole moment) Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ Magnetic Field from Wires: Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Center of Loop (N turns): $B = \frac{\mu_0 NI}{2R}$ Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Magnetic Field in Solenoid: $B = \mu_0 nI$ ($n$ = turns per unit length) 15. Induction & Inductance Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Lenz's Law: Induced current opposes the change in magnetic flux. Motional EMF: $\mathcal{E} = BLv$ (for conductor of length L moving perp. to B) Inductance: $L = \frac{N\Phi_B}{I}$ Solenoid Inductance: $L = \mu_0 n^2 A l$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Energy Density in 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: $\tau_L = L/R$ Mutual Inductance: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ 16. Electromagnetic Waves Maxwell's Equations: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ (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 Light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$ Wave Properties: $c = \lambda f$ Energy Density: $u_E = \frac{1}{2}\epsilon_0 E^2$, $u_B = \frac{1}{2\mu_0} B^2$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (direction of propagation) Intensity: $I = S_{avg} = \frac{P}{A} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{c}{\mu_0}B_{rms}^2$ Radiation Pressure: $P_{rad} = \frac{I}{c}$ (absorbed), $P_{rad} = \frac{2I}{c}$ (reflected) 17. Optics Reflection Law of Reflection: $\theta_i = \theta_r$ Plane Mirrors: Virtual, upright, same size, same distance behind mirror. Spherical Mirrors (Concave/Convex): Mirror Equation: $\frac{1}{f} = \frac{1}{p} + \frac{1}{i}$ Magnification: $m = -\frac{i}{p}$ Focal Length: $f = R/2$ Refraction Snell's Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$ Index of Refraction: $n = c/v$ Critical Angle: $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) Thin Lenses (Converging/Diverging): Lensmaker's Equation: $\frac{1}{f} = (n-1)\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$ Thin Lens Equation: $\frac{1}{f} = \frac{1}{p} + \frac{1}{i}$ Magnification: $m = -\frac{i}{p}$ 18. Interference & Diffraction Constructive Interference: $\Delta L = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Destructive Interference: $\Delta L = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) Young's Double Slit: Bright Fringes: $d \sin\theta = m\lambda$ Dark Fringes: $d \sin\theta = (m + \frac{1}{2})\lambda$ Thin Films: Phase change of $\pi$ (or $\lambda/2$) upon reflection from higher '$n$' medium. 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): $\theta_R = 1.22 \frac{\lambda}{D}$