Fundamentals of Physics Cheatsheet

Cheatsheet Content

1. Measurement & Vectors SI Base Units: Length (meter, m), Mass (kilogram, kg), Time (second, s) Scientific Notation: $N \times 10^n$ Vector Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\theta = \tan^{-1}(A_y/A_x)$ Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \phi = A_x B_x + A_y B_y + A_z B_z$ Cross Product Magnitude: $|\vec{A} \times \vec{B}| = AB \sin \phi$ Cross Product Components: $(\vec{A} \times \vec{B})_x = A_y B_z - A_z B_y$ $(\vec{A} \times \vec{B})_y = A_z B_x - A_x B_z$ $(\vec{A} \times \vec{B})_z = A_x B_y - A_y B_x$ 2. Kinematics (1D & 2D) 1D Motion Average Velocity: $v_{avg} = \Delta x / \Delta t$ Average Acceleration: $a_{avg} = \Delta v / \Delta t$ Constant Acceleration Equations: $v = v_0 + at$ $x = x_0 + v_0 t + \frac{1}{2} a t^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ 2D Motion (Projectile) Horizontal: $v_x = v_{0x}$, $x = x_0 + v_{0x} t$ Vertical: $v_y = v_{0y} - gt$, $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$, $v_y^2 = v_{0y}^2 - 2g(y - y_0)$ Relative Motion $\vec{v}_{PA} = \vec{v}_{PB} + \vec{v}_{BA}$ (P relative to A, P relative to B, B relative to A) 3. Newton's Laws 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. 2nd Law: $\vec{F}_{net} = m\vec{a}$ 3rd Law: If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) Gravitational Force: $F_g = mg$ (near Earth's surface) Friction: Static: $f_s \le \mu_s N$ Kinetic: $f_k = \mu_k N$ Centripetal Force: $F_c = mv^2/r$ (directed towards center of circle) 4. Work & Energy Work by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd \cos \phi$ Work 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: $U_s = \frac{1}{2} k x^2$ (for a spring) Conservation of Mechanical Energy: $E_{mech} = K + U$ (if only conservative forces do work) General Conservation of Energy: $\Delta E_{mech} = W_{nc}$ (non-conservative work) Power: $P_{avg} = W/\Delta t$, $P = \vec{F} \cdot \vec{v}$ 5. Momentum & Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ Impulse-Momentum Theorem: $\vec{F}_{avg} \Delta t = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum: $\vec{P}_{total} = constant$ (if $\vec{F}_{net, ext} = 0$) Elastic Collisions (1D): Both momentum and kinetic energy are conserved. $v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_{1i} + \frac{2m_2}{m_1 + m_2} v_{2i}$ $v_{2f} = \frac{2m_1}{m_1 + m_2} v_{1i} + \frac{m_2 - m_1}{m_1 + m_2} v_{2i}$ Inelastic Collisions: Momentum conserved, kinetic energy not conserved. Completely Inelastic: Objects stick together. Center of Mass (CM): $\vec{r}_{CM} = \frac{1}{M_{total}} \sum m_i \vec{r}_i$ 6. Rotation Angular Position: $\theta$ (radians) Angular Velocity: $\omega = d\theta/dt$ Angular Acceleration: $\alpha = d\omega/dt$ Constant Angular Acceleration: $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relating Linear & Angular: $s = r\theta$, $v = r\omega$, $a_t = r\alpha$, $a_c = v^2/r = r\omega^2$ 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: $I = I_{CM} + Mh^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$, $\tau = rF \sin \phi$ Newton's 2nd Law for Rotation: $\tau_{net} = I\alpha$ Work in Rotation: $W = \int \tau d\theta$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p}$, $L = I\omega$ (for rigid body) Conservation of Angular Momentum: $\vec{L}_{total} = constant$ (if $\vec{\tau}_{net, ext} = 0$) 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 infinity) Escape Speed: $v_{esc} = \sqrt{2GM/R}$ Kepler's Laws: Orbits are ellipses with the Sun at one focus. A line connecting a planet to the Sun sweeps out equal areas in equal times. $T^2 \propto a^3$ (period squared proportional to semi-major axis cubed). $T^2 = (\frac{4\pi^2}{GM})a^3$ 8. Oscillations Simple Harmonic Motion (SHM): $x(t) = x_m \cos(\omega t + \phi)$ Angular Frequency: $\omega = \sqrt{k/m}$ (mass-spring), $\omega = \sqrt{g/L}$ (simple pendulum) Period: $T = 2\pi/\omega$ Frequency: $f = 1/T = \omega/(2\pi)$ Energy in SHM: $E = \frac{1}{2} k x_m^2 = \frac{1}{2} m v_m^2$ Damped SHM: $x(t) = x_m e^{-bt/2m} \cos(\omega' t + \phi)$ Resonance: Max amplitude when driving frequency equals natural frequency. 9. Waves Wave Speed: $v = \lambda f$ Wave Equation: $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Wave Number: $k = 2\pi/\lambda$ Speed on String: $v = \sqrt{\tau/\mu}$ ($\tau$ = tension, $\mu$ = linear density) Power Transmitted: $P = \frac{1}{2} \mu v \omega^2 y_m^2$ Interference: Superposition of waves. Constructive: $\Delta L = n\lambda$ Destructive: $\Delta L = (n + \frac{1}{2})\lambda$ Standing Waves (String fixed at both ends): $\lambda_n = 2L/n$, $f_n = n(v/2L)$ Beat Frequency: $f_{beat} = |f_1 - f_2|$ Doppler Effect (Sound): $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D = detector, S = source; use top signs for "towards", bottom for "away") 10. Temperature, Heat & 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 Capacity: $Q = C \Delta T$ Specific Heat: $Q = cm\Delta T$ Latent Heat (Phase Change): $Q = L m$ ($L_f$ for fusion, $L_v$ for vaporization) Heat Transfer: Conduction: $P_{cond} = kA \frac{T_H - T_C}{L}$ Radiation: $P_{rad} = \sigma \epsilon A T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W}/\text{m}^2\text{K}^4$) Ideal Gas Law: $PV = nRT = NkT$ ($R = 8.31 \text{ J}/\text{mol}\cdot\text{K}$, $k = 1.38 \times 10^{-23} \text{ J}/\text{K}$) Kinetic Theory of Gases: Average KE per molecule: $K_{avg} = \frac{3}{2} kT$ RMS Speed: $v_{rms} = \sqrt{3RT/M} = \sqrt{3kT/m}$ First Law of Thermodynamics: $\Delta E_{int} = Q - W$ Work Done by Gas: $W = \int P dV$ Molar Specific Heats: $C_P = C_V + R$ Monatomic: $C_V = \frac{3}{2} R$, $C_P = \frac{5}{2} R$, $\gamma = 5/3$ Diatomic: $C_V = \frac{5}{2} R$, $C_P = \frac{7}{2} R$, $\gamma = 7/5$ (at moderate temps) Adiabatic Process: $PV^\gamma = constant$, $TV^{\gamma-1} = constant$ Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of an isolated system never decreases. Entropy Change: $\Delta S = \int dQ/T$ (reversible), $\Delta S = Q/T$ (isothermal) Efficiency of Heat Engine: $\epsilon = |W|/|Q_H| = 1 - |Q_C|/|Q_H|$ Carnot Engine Efficiency: $\epsilon_C = 1 - T_C/T_H$ Coefficient of Performance (Refrigerator): $K = |Q_C|/|W|$ 11. Electric Charge & Field Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ ($k = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) Electric Field: $\vec{E} = \vec{F}/q_0$ Field of Point Charge: $E = k |q|/r^2$ 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' Law: $\oint \vec{E} \cdot d\vec{A} = Q_{enc}/\epsilon_0$ ($\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2$) 12. Electric Potential Potential Difference: $\Delta V = V_f - V_i = -W_{ab}/q_0 = -\int_i^f \vec{E} \cdot d\vec{s}$ Potential due to Point Charge: $V = kq/r$ Potential due to Dipole: $V = \frac{kp \cos \theta}{r^2}$ Electric Field from Potential: $E_x = -\partial V/\partial x$, $\vec{E} = -\nabla V$ Potential Energy of System of Charges: $U = k \frac{q_1 q_2}{r}$ 13. Capacitance & Dielectrics Capacitance: $C = Q/V$ Parallel Plate Capacitor: $C = \epsilon_0 A/d$ Energy Stored: $U = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{1}{2} Q^2/C$ Energy Density: $u = \frac{1}{2} \epsilon_0 E^2$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Capacitors in Series: $1/C_{eq} = \sum (1/C_i)$ Dielectric: $C = \kappa C_{air}$, $\epsilon = \kappa \epsilon_0$ 14. Current & Resistance Electric Current: $I = dq/dt$ Current Density: $J = I/A = nev_d$ Resistance: $R = V/I$ (Ohm's Law) Resistivity: $\rho = E/J$, $R = \rho L/A$ Temperature Dependence of $\rho$: $\rho - \rho_0 = \rho_0 \alpha(T - T_0)$ Power Dissipation: $P = IV = I^2 R = V^2/R$ 15. DC Circuits EMF: $\mathcal{E}$ Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $1/R_{eq} = \sum (1/R_i)$ Kirchhoff's Rules: Junction Rule: $\sum I_{in} = \sum I_{out}$ Loop Rule: $\sum \Delta V = 0$ around any closed loop. RC Circuits: Charging: $Q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = (\mathcal{E}/R)e^{-t/RC}$ Discharging: $Q(t) = Q_0 e^{-t/RC}$, $I(t) = (Q_0/RC)e^{-t/RC}$ Time Constant: $\tau = RC$ 16. 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: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NI\vec{A}$ Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = I B / (neA)$ 17. Sources of Magnetic Field Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ ($\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m}/\text{A}$) Magnetic Field of Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Force Between Parallel Wires: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ Magnetic Field of Current Loop (center): $B = \frac{\mu_0 I}{2R}$ Magnetic Field of Solenoid: $B = \mu_0 n I$ ($n$ = turns per unit length) Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ 18. Induction & Inductance Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law of Induction: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$ Motional EMF: $\mathcal{E} = BLv$ Lenz's Law: Induced current opposes the change in magnetic flux. Inductance: $L = N\Phi_B/I$ Solenoid Inductance: $L = \mu_0 n^2 A l$ Energy Stored in Inductor: $U_B = \frac{1}{2} L I^2$ Energy Density of Magnetic Field: $u_B = \frac{1}{2\mu_0} B^2$ RL Circuits: Current build-up: $I(t) = (\mathcal{E}/R)(1 - e^{-t/\tau_L})$ Current decay: $I(t) = I_0 e^{-t/\tau_L}$ Time Constant: $\tau_L = L/R$ Mutual Inductance: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$, $M = \frac{N_2 \Phi_{B2}}{I_1}$ 19. Electromagnetic Oscillations & AC Circuits LC Oscillations: $Q(t) = Q_m \cos(\omega t + \phi)$, $I(t) = - \omega Q_m \sin(\omega t + \phi)$ Angular Frequency: $\omega = 1/\sqrt{LC}$ Energy in LC Circuit: $U = U_E + U_B = \frac{1}{2} Q^2/C + \frac{1}{2} L I^2 = constant$ RLC Series Circuit: Resonance Frequency: $\omega_d = \omega = 1/\sqrt{LC}$ Reactance: $X_L = \omega L$, $X_C = 1/(\omega C)$ Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan \phi = (X_L - X_C)/R$ RMS Current: $I_{rms} = V_{rms}/Z$ Average Power: $P_{avg} = I_{rms} V_{rms} \cos \phi = I_{rms}^2 R$ 20. Electromagnetic Waves Speed of Light: $c = 1/\sqrt{\mu_0 \epsilon_0} = 3 \times 10^8 \text{ m/s}$ Wave Speed: $c = \lambda f$ 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: $I = S_{avg} = \frac{1}{c\mu_0} E_{rms}^2 = \frac{1}{2c\mu_0} E_m^2$ Radiation Pressure: $P_r = I/c$ (total absorption), $P_r = 2I/c$ (total reflection) 21. Light: 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: $n = c/v$ Critical Angle for Total Internal Reflection: $\sin \theta_c = n_2/n_1$ (for $n_1 > n_2$) 22. Lenses & Mirrors Mirror/Lens Equation: $1/p + 1/i = 1/f$ Magnification: $m = -i/p$ Spherical Mirror Focal Length: $f = r/2$ Thin Lens Maker's Formula: $1/f = (n-1)(1/r_1 - 1/r_2)$ Sign Conventions: Real objects $p > 0$, virtual objects $p Real images $i > 0$, virtual images $i Concave mirrors $f > 0$, convex mirrors $f Converging lenses $f > 0$, diverging lenses $f Object/image height $h > 0$ (upright), $h 23. Interference Young's Double Slit: Bright Fringes: $d \sin \theta = m\lambda$ Dark Fringes: $d \sin \theta = (m + \frac{1}{2})\lambda$ Fringe Spacing: $\Delta y = \lambda L/d$ Thin Film Interference (normal incidence): Constructive: $2L = (m + \frac{1}{2})\lambda_n$ Destructive: $2L = m\lambda_n$ $\lambda_n = \lambda/n_{film}$ Phase changes at interface: $n_{medium} n_{film}$ (180 deg phase change) 24. Diffraction Single Slit Diffraction: Dark Fringes: $a \sin \theta = m\lambda$ ($m = \pm 1, \pm 2, ...$) Width of Central Max: $2\Delta y = 2L\lambda/a$ Diffraction Grating: Bright Fringes: $d \sin \theta = m\lambda$ ($m = 0, \pm 1, \pm 2, ...$) Rayleigh Criterion (Resolution): $\theta_R = 1.22 \lambda/D$ (circular aperture) 25. Special Relativity Postulates: The laws of physics are the same for all inertial reference frames. The speed of light in vacuum is the same for all inertial observers. Time Dilation: $\Delta t = \gamma \Delta t_0$ ($\gamma = 1/\sqrt{1 - (v/c)^2}$) Length Contraction: $L = L_0/\gamma$ Relativistic Momentum: $p = \gamma mv$ Relativistic Energy: $E = \gamma mc^2 = K + mc^2$ Rest Energy: $E_0 = mc^2$ Kinetic Energy: $K = (\gamma - 1)mc^2$ Energy-Momentum Relation: $E^2 = (pc)^2 + (mc^2)^2$ 26. Quantum Physics Planck's Quantum 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: $p = h/\lambda$ Compton Effect (wavelength shift): $\Delta \lambda = \frac{h}{mc}(1 - \cos \phi)$ De Broglie Wavelength: $\lambda = h/p$ Heisenberg Uncertainty Principle: $\Delta x \Delta p_x \ge \hbar/2$ $\Delta E \Delta t \ge \hbar/2$ ($\hbar = h/(2\pi)$) Schrödinger Equation (Time-Independent 1D): $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ 27. Atomic Physics Bohr Model Energy Levels (Hydrogen): $E_n = -13.6 \text{ eV}/n^2$ Hydrogen Spectral Series: $1/\lambda = R_H (\frac{1}{n_f^2} - \frac{1}{n_i^2})$ ($R_H = 1.097 \times 10^7 \text{ m}^{-1}$) Quantum Numbers: Principal ($n$): $1, 2, 3, ...$ (energy, size) Orbital ($l$): $0, 1, ..., n-1$ (shape) Magnetic ($m_l$): $-l, ..., 0, ..., +l$ (orientation) Spin ($m_s$): $\pm 1/2$ (intrinsic angular momentum) Pauli Exclusion Principle: No two electrons can have the same set of four quantum numbers. 28. Nuclear Physics Atomic Mass Unit: $1 \text{ u} = 1.66 \times 10^{-27} \text{ kg} \approx 931.5 \text{ MeV}/c^2$ Nuclear Radius: $R = R_0 A^{1/3}$ ($R_0 \approx 1.2 \text{ fm}$) Binding Energy: $E_B = (\sum m_{nucleons} - m_{nucleus})c^2$ Radioactive Decay Law: $N(t) = N_0 e^{-\lambda t}$ Half-life: $T_{1/2} = \ln(2)/\lambda$ Activity: $R = |\frac{dN}{dt}| = \lambda N$ Types of Decay: $\alpha$-decay: $^A_Z X \to ^{A-4}_{Z-2} Y + ^4_2 He$ $\beta^-$-decay: $^A_Z X \to ^A_{Z+1} Y + e^- + \bar{\nu}_e$ $\beta^+$-decay: $^A_Z X \to ^A_{Z-1} Y + e^+ + \nu_e$