Halliday Physics Essentials

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### 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 Formulas - $v = v_0 + at$ - $x = x_0 + v_0 t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 1.2 Two-Dimensional Motion - **Position Vector:** $\vec{r} = x\hat{i} + y\hat{j}$ - **Velocity Vector:** $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j}$ - **Acceleration Vector:** $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j}$ ##### Projectile Motion (Constant $a_x=0$, $a_y=-g$) - $v_x = v_{0x}$ - $x = x_0 + v_{0x}t$ - $v_y = v_{0y} - gt$ - $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ ### 2. Newton's Laws of Motion - **Newton's First Law:** An object remains at rest or in uniform motion in a straight line unless acted upon by an external force. - **Newton's Second Law:** $\sum \vec{F} = m\vec{a}$ - **Newton's Third Law:** If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. $\vec{F}_{AB} = -\vec{F}_{BA}$ #### Common Forces - **Weight:** $W = mg$ - **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 rope/cable) - **Spring Force (Hooke's Law):** $\vec{F}_s = -k\Delta\vec{x}$ ### 3. Work and Energy - **Work Done by Constant Force:** $W = \vec{F} \cdot \Delta\vec{r} = F \Delta r \cos\theta$ - **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:** $W_{net} = \Delta K = K_f - K_i$ #### Potential Energy - **Gravitational Potential Energy:** $U_g = mgh$ - **Elastic Potential Energy (Spring):** $U_s = \frac{1}{2}kx^2$ - **Relationship between Force and Potential Energy:** $F_x = -\frac{dU}{dx}$ #### Conservation of Energy - **Mechanical Energy:** $E_{mech} = K + U$ - **Conservation of Mechanical Energy (Conservative Forces Only):** $E_{mech,i} = E_{mech,f}$ or $\Delta E_{mech} = 0$ - **Conservation of Energy (Non-Conservative Forces):** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### 4. Momentum and Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Newton's Second Law (in terms of momentum):** $\sum \vec{F} = \frac{d\vec{p}}{dt}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta\vec{p} = \vec{p}_f - \vec{p}_i$ - **Conservation of Linear Momentum:** If $\sum \vec{F}_{ext} = 0$, then $\Delta\vec{P}_{total} = 0$ or $\vec{P}_{total,i} = \vec{P}_{total,f}$ #### Collisions - **Elastic Collision:** Both momentum and kinetic energy are conserved. - **Inelastic Collision:** Momentum is conserved, but kinetic energy is NOT conserved ($K_{initial} \ne K_{final}$). - **Perfectly Inelastic Collision:** Objects stick together after collision. Momentum is conserved, kinetic energy is NOT conserved. - **Center of Mass:** $x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$, $\vec{v}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$ ### 5. Rotational Motion - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ #### Rotational Kinematics (Constant Angular Acceleration) - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### Relationship between Linear and Angular Variables - **Arc Length:** $s = r\theta$ - **Tangential Speed:** $v_t = r\omega$ - **Tangential Acceleration:** $a_t = r\alpha$ - **Centripetal Acceleration:** $a_c = \frac{v_t^2}{r} = r\omega^2$ - **Centripetal Force:** $F_c = ma_c = \frac{mv_t^2}{r} = mr\omega^2$ #### Torque and Angular Momentum - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ or $\tau = rF\sin\phi$ - **Newton's Second Law for Rotation:** $\sum \tau = I\alpha$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Moment of Inertia:** $I = \sum m_i r_i^2$ (for point masses); $I = \int r^2 dm$ (for continuous bodies) - **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 $\Delta\vec{L}_{total} = 0$ or $\vec{L}_{total,i} = \vec{L}_{total,f}$ ### 6. Gravitation - **Newton's Law of Universal Gravitation:** $F = G\frac{m_1 m_2}{r^2}$ - **Gravitational Potential Energy:** $U = -G\frac{m_1 m_2}{r}$ (for $r \to \infty$, $U=0$) - **Escape Speed:** $v_{esc} = \sqrt{\frac{2GM}{R}}$ - **Kepler's Laws:** 1. Orbits are ellipses with the Sun at one focus. 2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3. $T^2 \propto a^3$ (Period squared is proportional to semimajor axis cubed) ### 7. Oscillations (Simple Harmonic Motion - SHM) - **Displacement:** $x(t) = A\cos(\omega t + \phi)$ - **Velocity:** $v(t) = -\omega A\sin(\omega t + \phi)$ - **Acceleration:** $a(t) = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x(t)$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (mass-spring system) - **Angular Frequency (Simple Pendulum):** $\omega = \sqrt{\frac{g}{L}}$ (for small angles) - **Period:** $T = \frac{2\pi}{\omega}$ - **Frequency:** $f = \frac{1}{T} = \frac{\omega}{2\pi}$ - **Energy of SHM:** $E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ ### 8. Waves - **Wave Speed:** $v = \lambda f$ - **Speed of Wave on a String:** $v = \sqrt{\frac{T}{\mu}}$ (T = tension, $\mu$ = linear mass density) - **Wave Equation:** $y(x,t) = A\sin(kx - \omega t + \phi)$ - **Wave Number:** $k = \frac{2\pi}{\lambda}$ - **Angular Frequency:** $\omega = 2\pi f$ - **Intensity:** $I = \frac{P}{A}$ (Power per unit area) - **Standing Waves:** Occur in confined media due to superposition of incident and reflected waves. - **Nodes:** Points of zero displacement. - **Antinodes:** Points of maximum displacement. - **String fixed at both ends:** $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ for $n=1,2,3,...$ (harmonics) - **Pipe open at both ends:** Same as string fixed at both ends. - **Pipe closed at one end:** $\lambda_n = \frac{4L}{n}$, $f_n = n\frac{v}{4L}$ for $n=1,3,5,...$ (odd harmonics) ### 9. Sound - **Speed of Sound in Air:** $v \approx 343 \text{ m/s}$ (at 20°C) - **Intensity Level (Decibels):** $\beta = (10 \text{ dB})\log_{10}\left(\frac{I}{I_0}\right)$ where $I_0 = 10^{-12} \text{ W/m}^2$ - **Doppler Effect:** $f' = f \left(\frac{v \pm v_D}{v \mp v_S}\right)$ - Use + for detector moving towards source, - for detector moving away. - Use - for source moving towards detector, + for source moving away. - $v_D$: detector speed, $v_S$: source speed, $v$: speed of sound. ### 10. Fluid Mechanics - **Density:** $\rho = \frac{m}{V}$ - **Pressure:** $P = \frac{F}{A}$ - **Pressure in a Fluid at Depth h:** $P = P_0 + \rho gh$ - **Pascal's Principle:** A change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. - **Archimedes' Principle:** Buoyant Force $F_B = \rho_{fluid} V_{displaced} g$ - **Equation of Continuity:** $A_1 v_1 = A_2 v_2$ (for incompressible fluid) - **Bernoulli's Equation:** $P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$ ### 11. Thermodynamics #### 11.1 Temperature and Heat - **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:** $Q = C\Delta T$ - **Specific Heat:** $Q = mc\Delta T$ - **Latent Heat (Phase Changes):** $Q = mL$ ($L_f$ for fusion, $L_v$ for vaporization) - **Heat Transfer:** - **Conduction:** $P_{cond} = \frac{Q}{\Delta t} = kA\frac{dT}{dx}$ - **Convection:** Heat transfer via fluid movement. - **Radiation (Stefan-Boltzmann Law):** $P_{rad} = \sigma A e T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/(m}^2\text{K}^4)$) #### 11.2 Ideal Gases - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.314 \text{ J/(mol K)}$ (gas constant) - $k = 1.38 \times 10^{-23} \text{ J/K}$ (Boltzmann constant) - $N$: number of molecules, $n$: number of moles - **Kinetic Theory of Gases:** - **Average Kinetic Energy per molecule:** $K_{avg} = \frac{3}{2}kT$ - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ #### 11.3 First Law of Thermodynamics - **Statement:** $\Delta E_{int} = Q - W$ - $\Delta E_{int}$: change in internal energy - $Q$: heat added to the system - $W$: work done BY the system ($W = \int P dV$) - **Internal Energy of Ideal Gas:** $E_{int} = \frac{3}{2}nRT$ (monatomic) - **Processes:** - **Isobaric:** Constant pressure, $W = P\Delta V$ - **Isochoric:** Constant volume, $W = 0$, $\Delta E_{int} = Q$ - **Isothermal:** Constant temperature, $\Delta E_{int} = 0$, $Q = W = nRT \ln\left(\frac{V_f}{V_i}\right)$ - **Adiabatic:** No heat exchange, $Q = 0$, $\Delta E_{int} = -W$, $PV^\gamma = \text{constant}$ ($\gamma = C_p/C_v$) #### 11.4 Second Law of Thermodynamics - **Statement:** Heat flows spontaneously from hot to cold, not the other way. - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ - **Entropy Statement:** The entropy of an isolated system never decreases. $\Delta S_{total} \ge 0$. - **Heat Engines:** - **Efficiency:** $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - **Carnot Engine (ideal):** $\epsilon_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** - **Coefficient of Performance (COP) Fridge:** $K = \frac{|Q_C|}{|W|}$ - **Coefficient of Performance (COP) Heat Pump:** $K = \frac{|Q_H|}{|W|}$ - **Carnot COP Fridge:** $K_C = \frac{T_C}{T_H - T_C}$ - **Carnot COP Heat Pump:** $K_C = \frac{T_H}{T_H - T_C}$ ### 12. Electrostatics - **Coulomb's Law:** $F = k\frac{|q_1 q_2|}{r^2}$ where $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ or $\vec{E} = k\frac{q}{r^2}\hat{r}$ - **Electric Flux (Gauss's Law):** $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - **Electric Potential Energy:** $U = k\frac{q_1 q_2}{r}$ - **Electric Potential:** $V = \frac{U}{q_0}$ or $V = k\frac{q}{r}$ - **Relationship between E and V:** $E_x = -\frac{dV}{dx}$, $\vec{E} = -\nabla V$ - **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:** $C = \kappa C_0$ ($C_0$ is capacitance without dielectric) ### 13. Current and Circuits - **Electric Current:** $I = \frac{dQ}{dt}$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ ($\rho$ is resistivity) - **Power in Circuits:** $P = IV = I^2 R = \frac{V^2}{R}$ #### Resistors in Series and Parallel - **Series:** $R_{eq} = R_1 + R_2 + ...$ - **Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ #### Kirchhoff's Rules - **Junction Rule:** $\sum I_{in} = \sum I_{out}$ (conservation of charge) - **Loop Rule:** $\sum \Delta V = 0$ around any closed loop (conservation of energy) - **RC Circuits:** - **Charging Capacitor:** $Q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ - **Discharging Capacitor:** $Q(t) = Q_0 e^{-t/RC}$, $I(t) = -\frac{Q_0}{RC}e^{-t/RC}$ - **Time Constant:** $\tau = RC$ ### 14. Magnetism - **Magnetic Force on a Moving Charge:** $\vec{F}_B = q(\vec{v} \times \vec{B})$ - **Magnetic Force on a Current-Carrying Wire:** $\vec{F}_B = I(\vec{L} \times \vec{B})$ - **Torque on a Current Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ where $\vec{\mu} = NIA\hat{n}$ (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 of a Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field of a Solenoid:** $B = \mu_0 n I$ ($n$ is turns per unit length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ ### 15. Electromagnetic Induction - **Faraday's Law of Induction:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ (magnitude of induced EMF) - **Lenz's Law:** The induced current or EMF opposes the change in magnetic flux that produced it. - **Motional EMF:** $\mathcal{E} = BLv$ (for a conductor moving perpendicular to B field) - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Self-Inductance of a Solenoid:** $L = \mu_0 n^2 A l$ - **Energy Stored in an Inductor:** $U_L = \frac{1}{2}LI^2$ - **RL Circuits:** - **Current build-up:** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$, $\tau = L/R$ - **Current decay:** $I(t) = I_0 e^{-t/\tau}$, $\tau = L/R$ - **LC Circuits:** - **Angular Frequency of Oscillation:** $\omega = \frac{1}{\sqrt{LC}}$ - **Charge:** $Q(t) = Q_{max}\cos(\omega t + \phi)$ - **RLC Circuits (Series):** - **Resonance Angular Frequency:** $\omega_0 = \frac{1}{\sqrt{LC}}$ ### 16. Electromagnetic Waves - **Speed of Light in Vacuum:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ - **Wave Speed:** $c = \lambda f$ - **Relationship between E and B field amplitudes:** $E_{max} = c B_{max}$ - **Poynting Vector (Energy Flow):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - **Intensity:** $I = S_{avg} = \frac{E_{max}B_{max}}{2\mu_0} = \frac{E_{max}^2}{2\mu_0 c} = \frac{c B_{max}^2}{2\mu_0}$ - **Radiation Pressure:** - **Perfectly Absorbing Surface:** $P_{rad} = \frac{I}{c}$ - **Perfectly Reflecting Surface:** $P_{rad} = \frac{2I}{c}$ ### 17. Optics #### 17.1 Geometric Optics - **Law of Reflection:** $\theta_i = \theta_r$ - **Law of Refraction (Snell's Law):** $n_1\sin\theta_1 = n_2\sin\theta_2$ - **Index of Refraction:** $n = \frac{c}{v}$ - **Critical Angle for Total Internal Reflection:** $\sin\theta_c = \frac{n_2}{n_1}$ (when $n_1 > n_2$) - **Thin Lens / Mirror Equation:** $\frac{1}{f} = \frac{1}{p} + \frac{1}{i}$ - $f$: focal length (+ for converging lens/concave mirror, - for diverging lens/convex mirror) - $p$: object distance (+ if real object) - $i$: image distance (+ if real image, - if virtual image) - **Magnification:** $m = -\frac{i}{p} = \frac{h_i}{h_p}$ #### 17.2 Wave Optics - **Young's Double-Slit Experiment:** - **Constructive Interference (bright fringes):** $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Destructive Interference (dark fringes):** $d\sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Fringe Spacing:** $\Delta y = \frac{\lambda L}{d}$ - **Single-Slit Diffraction:** - **Minima (dark fringes):** $a\sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) - **Diffraction Grating:** - **Maxima (bright fringes):** $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Thin Film Interference:** Depends on thickness, indices of refraction, and phase shifts upon reflection. - **Rayleigh Criterion:** $\theta_{min} = 1.22\frac{\lambda}{D}$ (for circular aperture) ### 18. Modern Physics (Brief Overview) #### 18.1 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:** $E = \gamma mc^2 = K + mc^2$ - **Rest Energy:** $E_0 = mc^2$ - **Kinetic Energy:** $K = (\gamma - 1)mc^2$ #### 18.2 Quantum Physics - **Planck's Hypothesis:** $E = hf$ (energy of a photon) - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($\Phi$ is work function) - **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-dependent) $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + U\Psi$ - **Quantized Energy Levels:** For bound systems (e.g., hydrogen atom). #### 18.3 Nuclear Physics - **Mass Defect and Binding Energy:** $E_b = \Delta m c^2$ - **Radioactive Decay Law:** $N(t) = N_0 e^{-\lambda t}$ - **Half-life:** $T_{1/2} = \frac{\ln 2}{\lambda}$ - **Nuclear Reactions:** Fission (splitting), Fusion (combining)