Halliday Physics Essentials

Cheatsheet Content

### 1. Kinematics #### 1.1. One-Dimensional 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 - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 1.2. Two- and Three-Dimensional 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 (Constant $g$, no air resistance) - $v_x = v_{0x}$ - $x = x_0 + v_{0x}t$ - $v_y = v_{0y} - gt$ - $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ - **Range:** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for $y_0=0$) ##### 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 Velocity:** $\omega = \frac{d\theta}{dt}$ ### 2. 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:** $\sum \vec{F} = m\vec{a}$ - **Weight:** $W = mg$ (force of gravity) - **Friction:** - Static: $f_s \le \mu_s N$ - Kinetic: $f_k = \mu_k N$ - **Newton's Third Law:** If object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on object A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) ### 3. Work and Energy - **Work done by constant force:** $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ - **Work done by 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:** $U_g = mgh$ (near Earth's surface) - **Elastic Potential Energy:** $U_s = \frac{1}{2}kx^2$ (for a spring) - **Conservation of Mechanical Energy:** $E_{mech} = K + U = \text{constant}$ (if only conservative forces do work) - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ - **Conservation of Energy:** $W_{nc} = \Delta E_{mech} + \Delta E_{int}$ (non-conservative forces) ### 4. Linear Momentum & Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Newton's Second Law (momentum form):** $\sum \vec{F} = \frac{d\vec{p}}{dt}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ - **Conservation of Linear Momentum:** $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$ (for isolated systems) - **Collisions:** - **Elastic:** Kinetic energy conserved. - **Inelastic:** Kinetic energy not conserved. - **Perfectly Inelastic:** Objects stick together. - **Center of Mass:** - $x_{CM} = \frac{1}{M}\sum m_i x_i$ - $\vec{v}_{CM} = \frac{1}{M}\sum m_i \vec{v}_i$ - $\vec{P}_{total} = M_{total}\vec{v}_{CM}$ ### 5. Rotation - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ ##### Constant Angular Acceleration Equations - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ ##### Relations between Linear and Angular Variables - $s = r\theta$ - $v_t = r\omega$ (tangential speed) - $a_t = r\alpha$ (tangential acceleration) - $a_c = \frac{v_t^2}{r} = r\omega^2$ (centripetal acceleration) ##### Torque and Moment of Inertia - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ or $\tau = rF\sin\phi$ - **Moment of Inertia:** $I = \sum m_i r_i^2 = \int r^2 dm$ - **Newton's Second Law for Rotation:** $\sum \tau = I\alpha$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Work done by Torque:** $W = \int \tau d\theta$ - **Power:** $P = \tau\omega$ ##### Angular Momentum - **Angular Momentum of particle:** $\vec{l} = \vec{r} \times \vec{p} = I\vec{\omega}$ - **Angular Momentum of rigid body:** $\vec{L} = I\vec{\omega}$ - **Conservation of Angular Momentum:** $\vec{L}_{initial} = \vec{L}_{final}$ (if net external torque is zero) - **Torque and Angular Momentum:** $\sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$ ### 6. Gravitation - **Newton's Law of Universal 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 Laws:** 1. Orbits are ellipses with Sun at one focus. 2. Equal areas swept in equal times. 3. $T^2 \propto r^3$ (for circular orbits: $T^2 = (\frac{4\pi^2}{GM})r^3$) ### 7. Oscillations (Simple Harmonic Motion) - **Displacement:** $x(t) = A \cos(\omega t + \phi)$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (mass-spring system) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Velocity:** $v(t) = -A\omega \sin(\omega t + \phi)$ - **Acceleration:** $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$ - **Simple Pendulum (small angles):** $T = 2\pi\sqrt{\frac{L}{g}}$ - **Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgd}}$ ### 8. Waves - **Wave Speed:** $v = \lambda f$ - **Wave on a String:** $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear density) - **Sound Speed in Fluid:** $v = \sqrt{\frac{B}{\rho}}$ ($B$ = bulk modulus, $\rho$ = density) - **Sound Speed in Solid Rod:** $v = \sqrt{\frac{Y}{\rho}}$ ($Y$ = Young's modulus) - **Intensity:** $I = \frac{P}{A}$ - **Sound Level (dB):** $\beta = 10 \log_{10}(\frac{I}{I_0})$ where $I_0 = 10^{-12} \text{ W/m}^2$ - **Doppler Effect:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D: detector, S: source; top sign for approach, bottom for recession) - **Standing Waves:** - **String (fixed ends):** $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) - **Open-Open Pipe:** $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ ($n=1,2,3...$) - **Open-Closed Pipe:** $\lambda_n = \frac{4L}{n}$, $f_n = n\frac{v}{4L}$ ($n=1,3,5...$) ### 9. 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: $\Delta V = V\beta\Delta T$ ($\beta \approx 3\alpha$) - **Heat Capacity:** $Q = C\Delta T = mc\Delta T$ - **Latent Heat:** $Q = mL$ (L = latent heat of fusion/vaporization) - **First Law of Thermodynamics:** $\Delta E_{int} = Q - W$ - $Q$: heat added to system - $W$: work done BY system ($W = \int P dV$) - **Ideal Gas Law:** $PV = nRT = Nk_BT$ - **Kinetic Theory of Gases:** - **Average Kinetic Energy:** $K_{avg} = \frac{3}{2}k_BT$ - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}}$ - **Work for Ideal Gas:** - Isobaric ($P=$ const): $W = P\Delta V$ - Isothermal ($T=$ const): $W = nRT \ln(\frac{V_f}{V_i})$ - Adiabatic ($Q=0$): $PV^\gamma = \text{constant}$ - **Heat Engines:** - **Efficiency:** $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - **Carnot Efficiency:** $\epsilon_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** - **Coefficient of Performance (Cooling):** $K = \frac{|Q_C|}{|W|}$ - **Coefficient of Performance (Heating):** $K_{HP} = \frac{|Q_H|}{|W|} = K+1$ - **Second Law of Thermodynamics:** - Heat flows spontaneously from hot to cold. - Entropy of an isolated system never decreases. - **Entropy Change:** $\Delta S = \int \frac{dQ}{T}$ (reversible processes) - **Entropy for Ideal Gas:** $\Delta S = nR \ln(\frac{V_f}{V_i}) + nC_V \ln(\frac{T_f}{T_i})$ ### 10. Electricity & Magnetism #### 10.1. Electric Fields - **Coulomb's Law:** $F = k \frac{|q_1 q_2|}{r^2}$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0} = k \frac{q}{r^2}\hat{r}$ - **Electric Potential Energy:** $U = k \frac{q_1 q_2}{r}$ - **Electric Potential:** $V = \frac{U}{q_0} = k \frac{q}{r}$ - **Relation between E and V:** $\vec{E} = -\nabla V$ (for uniform field: $E = -\frac{\Delta V}{\Delta s}$) - **Capacitance:** $C = \frac{Q}{V}$ - Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ - Energy stored: $U_E = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ - **Dielectrics:** $C = \kappa C_0$ #### 10.2. Current & Resistance - **Current:** $I = \frac{dQ}{dt} = nqv_dA$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ - **Resistivity (temperature dependent):** $\rho = \rho_0[1 + \alpha(T - T_0)]$ - **Power in Circuits:** $P = IV = I^2R = \frac{V^2}{R}$ - **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):** $Q(t) = Q_0(1 - e^{-t/RC})$, $I(t) = I_0 e^{-t/RC}$ - Time constant: $\tau = RC$ #### 10.3. Magnetic Fields - **Magnetic Force on a Charge:** $\vec{F}_B = q(\vec{v} \times \vec{B})$ - **Magnetic Force on a Current:** $\vec{F}_B = I(\vec{L} \times \vec{B})$ - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ - **Magnetic Field from Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field of Solenoid:** $B = \mu_0 n I$ ($n$ = turns/length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - **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. - **Inductance:** $L = \frac{N\Phi_B}{I}$ - Solenoid: $L = \mu_0 n^2 A l$ - Energy stored: $U_B = \frac{1}{2}LI^2$ - **RL Circuits (Charging):** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$, $\tau = L/R$ - **LC Oscillations:** $\omega = \frac{1}{\sqrt{LC}}$ - **Maxwell's Equations (Integral Form, in vacuum):** 1. $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ (Gauss' Law for E-field) 2. $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss' Law for B-field) 3. $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (Faraday's Law) 4. $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (Ampere-Maxwell Law) ### 11. Optics #### 11.1. Geometrical Optics - **Law of Reflection:** $\theta_i = \theta_r$ - **Snell's Law (Refraction):** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Critical Angle:** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) - **Thin Lens Equation / Spherical Mirrors:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - **Magnification:** $M = -\frac{i}{p}$ - **Lensmaker's Equation:** $\frac{1}{f} = (n-1)(\frac{1}{r_1} - \frac{1}{r_2})$ - **Sign Conventions (for mirrors/lenses):** - $p$: + if object real (front), - if virtual (back) - $i$: + if image real (front for mirror, back for lens), - if virtual (back for mirror, front for lens) - $f$: + for converging (concave mirror, convex lens), - for diverging (convex mirror, concave lens) - $r$: + if center of curvature in front of mirror / back of lens - $M$: + if upright, - if inverted #### 11.2. Wave Optics - **Young's Double-Slit Experiment:** - **Constructive Interference:** $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Destructive Interference:** $d\sin\theta = (m + \frac{1}{2})\lambda$ - **Thin Film Interference:** - **Optical Path Difference (OPD):** $2n_2d$ - Consider phase shifts upon reflection (180° if $n_1 ### 12. Modern Physics #### 12.1. Relativity - **Lorentz Factor:** $\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$ - **Length Contraction:** $L = L_0/\gamma$ - **Time Dilation:** $\Delta t = \gamma \Delta t_0$ - **Relativistic Momentum:** $p = \gamma mv$ - **Relativistic Energy:** $E = \gamma mc^2 = K + mc^2$ - **Mass-Energy Equivalence:** $E_0 = mc^2$ - **Relativistic Kinetic Energy:** $K = (\gamma - 1)mc^2$ #### 12.2. Quantum Physics - **Planck's Quantum Hypothesis:** $E = hf$ - **Photon Energy:** $E = \frac{hc}{\lambda}$ - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($\Phi$ = work function) - **De Broglie Wavelength:** $\lambda = \frac{h}{p} = \frac{h}{mv}$ - **Heisenberg Uncertainty Principle:** - Position-Momentum: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - Energy-Time: $\Delta E \Delta t \ge \frac{\hbar}{2}$ - **Schrödinger Equation (Time-Independent 1D):** $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi(x) = E\psi(x)$ - $\hbar = \frac{h}{2\pi}$ - **Quantum Numbers (for hydrogen atom):** - **Principal (n):** $1, 2, 3, ...$ (energy level) - **Orbital (l):** $0, 1, ..., n-1$ (shape of orbital) - **Magnetic ($m_l$):** $-l, ..., 0, ..., +l$ (orientation of orbital) - **Spin ($m_s$):** $\pm \frac{1}{2}$ (electron spin) - **Pauli Exclusion Principle:** No two electrons in an atom can have the same set of four quantum numbers. #### 12.3. Nuclear Physics - **Mass Defect and Binding Energy:** $E_B = \Delta m c^2$ - $\Delta m = (Z m_p + N m_n) - m_{nucleus}$ - **Radioactive Decay Law:** $N(t) = N_0 e^{-\lambda t}$ - **Half-life:** $T_{1/2} = \frac{\ln 2}{\lambda}$ ### 13. Fundamental Constants - **Speed of Light (c):** $3.00 \times 10^8 \text{ m/s}$ - **Gravitational Constant (G):** $6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ - **Elementary Charge (e):** $1.60 \times 10^{-19} \text{ C}$ - **Coulomb's Constant (k):** $8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - **Permittivity of Free Space ($\epsilon_0$):** $8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ - **Permeability of Free Space ($\mu_0$):** $4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ - **Planck's Constant (h):** $6.63 \times 10^{-34} \text{ J}\cdot\text{s}$ - **Reduced Planck's Constant ($\hbar$):** $1.05 \times 10^{-34} \text{ J}\cdot\text{s}$ - **Boltzmann Constant ($k_B$):** $1.38 \times 10^{-23} \text{ J/K}$ - **Avogadro's Number ($N_A$):** $6.02 \times 10^{23} \text{ mol}^{-1}$ - **Universal Gas Constant (R):** $8.31 \text{ J}/(\text{mol}\cdot\text{K})$ - **Mass of Electron ($m_e$):** $9.11 \times 10^{-31} \text{ kg}$ - **Mass of Proton ($m_p$):** $1.67 \times 10^{-27} \text{ kg}$ - **Acceleration due to Gravity (g):** $9.80 \text{ m/s}^2$ (Earth's surface)