Halliday Physics Essentials

Cheatsheet Content

### Kinematics: 1D - **Position:** $x(t)$ - **Displacement:** $\Delta x = x_f - x_i$ - **Average Velocity:** $v_{avg} = \frac{\Delta x}{\Delta t}$ - **Instantaneous Velocity:** $v(t) = \frac{dx}{dt}$ - **Average Acceleration:** $a_{avg} = \frac{\Delta v}{\Delta t}$ - **Instantaneous Acceleration:** $a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ #### Constant Acceleration Equations - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ ### Kinematics: 2D & 3D - **Position Vector:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - **Displacement Vector:** $\Delta\vec{r} = \vec{r}_f - \vec{r}_i$ - **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 (Constant $g$) - $v_{0x} = v_0 \cos\theta_0$, $v_{0y} = v_0 \sin\theta_0$ - $x = x_0 + v_{0x}t$ - $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ - $v_x = v_{0x}$ - $v_y = v_{0y} - gt$ ### Newton's Laws of Motion - **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:** $\Sigma\vec{F} = m\vec{a}$ - **3rd Law:** If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) #### Forces - **Weight:** $F_g = mg$ (directed downwards) - **Normal Force:** $F_N$ (perpendicular to surface) - **Friction:** - Static: $f_s \le \mu_s F_N$ - Kinetic: $f_k = \mu_k F_N$ - **Tension:** $T$ (along a cord/rope) ### Work & Energy - **Work done by a constant force:** $W = \vec{F} \cdot \Delta\vec{r} = F \Delta r \cos\theta$ - **Work done by a 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:** $U_s = \frac{1}{2}kx^2$ - **Conservation of Mechanical Energy:** $E_{mech} = K + U = \text{constant}$ (if only conservative forces do work) - **Conservation of Energy:** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ (where $W_{nc}$ is work done by non-conservative forces) - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### Momentum & Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta\vec{p}$ - **Conservation of Linear Momentum:** $\Sigma\vec{p}_{initial} = \Sigma\vec{p}_{final}$ (if no external forces) - **Collisions:** - **Elastic:** Momentum and Kinetic Energy conserved. - **Inelastic:** Momentum conserved, Kinetic Energy NOT conserved. - **Perfectly Inelastic:** Objects stick together, momentum conserved, max KE lost. - **Center of Mass:** $\vec{r}_{CM} = \frac{1}{M}\Sigma m_i\vec{r}_i$ ### Rotation - **Angular Position:** $\theta$ (radians) - **Angular Displacement:** $\Delta\theta$ - **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 - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### Relationships between 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) #### Rotational Dynamics - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F} = rF\sin\phi$ - **Newton's 2nd Law for Rotation:** $\Sigma\tau = I\alpha$ - **Moment of Inertia:** $I = \Sigma m_i r_i^2 = \int r^2 dm$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Angular Momentum:** $\vec{L} = I\vec{\omega}$ (for rigid body) - **Conservation of Angular Momentum:** $\Sigma\vec{L}_{initial} = \Sigma\vec{L}_{final}$ (if no external torque) ### Gravity - **Newton's Law of Universal Gravitation:** $F = G\frac{m_1m_2}{r^2}$ - **Gravitational Potential Energy:** $U = -G\frac{m_1m_2}{r}$ - **Escape Speed:** $v_{esc} = \sqrt{\frac{2GM}{R}}$ - **Kepler's Laws:** 1. Orbits are ellipses with the Sun at one focus. 2. Equal areas swept in equal times. 3. $T^2 \propto R^3$ (for circular orbits, $T^2 = (\frac{4\pi^2}{GM})R^3$) ### Oscillations & Waves #### Simple Harmonic Motion (SHM) - **Position:** $x(t) = A\cos(\omega t + \phi)$ - **Velocity:** $v(t) = -A\omega\sin(\omega t + \phi)$ - **Acceleration:** $a(t) = -A\omega^2\cos(\omega t + \phi) = -\omega^2x$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (pendulum) - **Period:** $T = \frac{2\pi}{\omega}$ - **Frequency:** $f = \frac{1}{T}$ #### Waves - **Wave Speed:** $v = \lambda f$ - **Speed on a string:** $v = \sqrt{\frac{T}{\mu}}$ (T=tension, $\mu$=linear density) - **Sound Speed in ideal gas:** $v = \sqrt{\frac{\gamma RT}{M}}$ - **Intensity:** $I = \frac{P}{A}$ - **Intensity Level (dB):** $\beta = (10 \text{ dB})\log_{10}\frac{I}{I_0}$ - **Doppler Effect:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D=detector, S=source; + for moving towards, - for moving away) ### 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: $\Delta V = V\beta\Delta T$ ($\beta \approx 3\alpha$) - **Ideal Gas Law:** $PV = nRT = NkT$ - **First Law of Thermodynamics:** $\Delta E_{int} = Q - W$ - $Q$: Heat added to system (+), removed (-) - $W$: Work done BY system (+), ON system (-) - **Work done by gas (isobaric):** $W = P\Delta V$ - **Heat Transfer:** - Conduction: $P_{cond} = kA\frac{T_H - T_C}{L}$ - Convection: Fluid motion - Radiation: $P_{rad} = \sigma A e T^4$ - **Specific Heat:** $Q = mc\Delta T$ - **Latent Heat:** $Q = mL$ - **Internal Energy of Monatomic Ideal Gas:** $E_{int} = \frac{3}{2}nRT$ - **Second Law of Thermodynamics:** - Heat flows spontaneously from hot to cold. - Entropy of isolated system never decreases. - **Efficiency of Heat Engine:** $e = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - **Carnot Engine Efficiency:** $e_C = 1 - \frac{T_C}{T_H}$ - **Entropy Change:** $\Delta S = \int \frac{dQ}{T}$ ### Electrostatics - **Coulomb's Law:** $F = k\frac{|q_1q_2|}{r^2}$ (where $k = \frac{1}{4\pi\epsilon_0}$) - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Electric Field from Point Charge:** $E = k\frac{|q|}{r^2}$ - **Electric Potential Energy:** $U = k\frac{q_1q_2}{r}$ - **Electric Potential:** $V = \frac{U}{q_0} = k\frac{q}{r}$ - **Relationship between E and V:** $\vec{E} = -\nabla V$ (for 1D, $E_x = -\frac{dV}{dx}$) - **Capacitance:** $C = \frac{Q}{V}$ - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Energy Stored in Capacitor:** $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ - **Dielectrics:** $C = \kappa C_0$ ### DC Circuits - **Current:** $I = \frac{dQ}{dt}$ - **Resistance (Ohm's Law):** $V = IR$ - **Resistivity:** $R = \rho\frac{L}{A}$ - **Power in Circuit:** $P = IV = I^2R = \frac{V^2}{R}$ - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ - **Capacitors in Parallel:** $C_{eq} = C_1 + C_2 + ...$ - **Kirchhoff's Laws:** - **Junction Rule:** $\Sigma I_{in} = \Sigma I_{out}$ - **Loop Rule:** $\Sigma \Delta V = 0$ - **RC Circuits (Charging):** $q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = I_{max}e^{-t/RC}$ - **Time Constant:** $\tau = RC$ ### Magnetism - **Magnetic Force on Moving Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ - **Magnetic Force on Current-Carrying Wire:** $\vec{F}_B = I\vec{L} \times \vec{B}$ - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{s} \times \hat{r}}{r^2}$ - **Magnetic Field of Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field of Solenoid:** $B = \mu_0 n I$ (n = turns per unit length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - **Faraday's Law of Induction:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - **Lenz's Law:** Induced current opposes the change in magnetic flux that created it. - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Energy Stored in Inductor:** $U_B = \frac{1}{2}LI^2$ - **RL Circuits (Current buildup):** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$, where $\tau = L/R$ ### Electromagnetic Waves - **Speed of EM waves in vacuum:** $c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3.00 \times 10^8 \text{ m/s}$ - **Wave Equation:** $\frac{\partial^2 E}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}$ - **Relationship between E and B:** $E = cB$ - **Poynting Vector (Intensity):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - **Intensity (average):** $I = S_{avg} = \frac{E_{max}B_{max}}{2\mu_0} = \frac{E_{max}^2}{2\mu_0 c}$ ### Optics - **Reflection:** Angle of incidence = Angle of reflection ($\theta_i = \theta_r$) - **Refraction (Snell's Law):** $n_1\sin\theta_1 = n_2\sin\theta_2$ - **Index of Refraction:** $n = \frac{c}{v}$ - **Thin Lens / Spherical Mirror Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - **Magnification:** $M = -\frac{i}{p} = \frac{h_i}{h_p}$ - **Lensmaker's Equation:** $\frac{1}{f} = (n-1)(\frac{1}{r_1} - \frac{1}{r_2})$ #### Interference & Diffraction - **Double Slit Interference (Bright Fringes):** $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Double Slit Interference (Dark Fringes):** $d\sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Single Slit Diffraction (Minima):** $a\sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) - **Rayleigh Criterion:** $\theta_{min} = 1.22\frac{\lambda}{D}$ (circular aperture) ### Modern Physics - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($h$=Planck's constant, $\Phi$=work function) - **Photon Energy:** $E = hf = \frac{hc}{\lambda}$ - **De Broglie Wavelength:** $\lambda = \frac{h}{p}$ - **Heisenberg Uncertainty Principle:** - $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - $\Delta E \Delta t \ge \frac{\hbar}{2}$ - **Schrödinger Equation (Time-Independent 1D):** $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ - **Bohr Model (Hydrogen Energy Levels):** $E_n = -\frac{13.6 \text{ eV}}{n^2}$ - **Mass-Energy Equivalence:** $E = mc^2$ #### Nuclear Physics - **Atomic Mass Unit (u):** $1 \text{ u} = 1.66 \times 10^{-27} \text{ kg} \approx 931.5 \text{ MeV}/c^2$ - **Binding Energy:** $E_b = (\Sigma m_{nucleons} - m_{nucleus})c^2$ - **Radioactive Decay Law:** $N(t) = N_0 e^{-\lambda t}$ - **Half-life:** $T_{1/2} = \frac{\ln 2}{\lambda}$