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 Formulas - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 1.2 Two-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 $a_x=0$, $a_y=-g$) - $v_x = v_{0x}$ - $x = x_0 + v_{0x}t$ - $v_y = v_{0y} - gt$ - $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ - **Range (R):** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for $y_0=0$) - **Maximum Height (H):** $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ (for $y_0=0$) ##### Uniform Circular Motion - **Speed:** $v = \frac{2\pi r}{T}$ - **Centripetal Acceleration:** $a_c = \frac{v^2}{r} = \omega^2 r$ (towards center) - **Period:** $T$ (time for one revolution) ### 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:** $\sum \vec{F} = m\vec{a}$ - In components: $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ - **Newton's Third Law:** If object A exerts a force $\vec{F}_{AB}$ on object B, then object B exerts a force $\vec{F}_{BA} = -\vec{F}_{AB}$ on object A. #### 2.1 Forces - **Weight:** $\vec{W} = m\vec{g}$ (magnitude $mg$) - **Normal Force:** $\vec{N}$ (perpendicular to surface) - **Tension:** $\vec{T}$ (along a rope/cable) - **Friction:** - **Static Friction:** $f_s \le \mu_s N$ (opposes impending motion) - **Kinetic Friction:** $f_k = \mu_k N$ (opposes actual motion) - **Drag Force (Approximate):** $D = \frac{1}{2}C\rho A v^2$ (for high speeds) ### 3. Work and Energy #### 3.1 Work - **Work done by constant force:** $W = \vec{F} \cdot \Delta\vec{r} = F \Delta r \cos\theta$ - **Work done by variable force:** $W = \int \vec{F} \cdot d\vec{r}$ - **Work-Kinetic Energy Theorem:** $W_{net} = \Delta K = K_f - K_i$ #### 3.2 Kinetic Energy - **Translational Kinetic Energy:** $K = \frac{1}{2}mv^2$ #### 3.3 Potential Energy - **Gravitational Potential Energy:** $U_g = mgy$ - **Elastic Potential Energy (Spring):** $U_s = \frac{1}{2}kx^2$ #### 3.4 Conservation of Energy - **Mechanical Energy:** $E_{mech} = K + U$ - **Conservation of Mechanical Energy:** If only conservative forces do work, $\Delta E_{mech} = 0 \implies E_{mech,f} = E_{mech,i}$ - **Conservation of Energy (General):** $W_{ext} = \Delta E_{mech} + \Delta E_{th} + \Delta E_{int}$ - $\Delta E_{th} = f_k d$ (thermal energy from friction) - $W_{ext}$ is work done by external non-conservative forces. #### 3.5 Power - **Average Power:** $P_{avg} = \frac{\Delta W}{\Delta t}$ - **Instantaneous 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 (Momentum Form):** $\sum \vec{F} = \frac{d\vec{p}}{dt}$ - **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 $\Delta\vec{P}_{sys} = 0 \implies \vec{P}_{sys,f} = \vec{P}_{sys,i}$ #### 4.1 Collisions - **Elastic Collision:** Both momentum and kinetic energy are conserved. - **Inelastic Collision:** Momentum is conserved, but kinetic energy is NOT conserved. - **Perfectly Inelastic Collision:** Objects stick together after collision. Momentum is conserved, but kinetic energy is NOT conserved. #### 4.2 Center of Mass - **Position of CM:** $\vec{r}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$ - **Velocity of CM:** $\vec{v}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i} = \frac{\vec{P}_{sys}}{M_{total}}$ - **Acceleration of CM:** $\vec{a}_{CM} = \frac{\sum m_i \vec{a}_i}{\sum m_i} = \frac{\sum \vec{F}_{ext}}{M_{total}}$ ### 5. Rotational Motion - **Angular Position:** $\theta$ (radians) - **Angular Displacement:** $\Delta\theta = \theta_f - \theta_i$ - **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:** $\alpha = \frac{d\omega}{dt}$ #### 5.1 Constant Angular Acceleration Formulas - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### 5.2 Relations Between Linear and Angular Variables - **Arc Length:** $s = r\theta$ - **Tangential Speed:** $v_t = r\omega$ - **Tangential Acceleration:** $a_t = r\alpha$ - **Centripetal Acceleration:** $a_c = \frac{v_t^2}{r} = \omega^2 r$ #### 5.3 Torque and Moment of Inertia - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ (magnitude $\tau = rF\sin\phi$) - **Newton's Second Law for Rotation:** $\sum \tau = I\alpha$ - **Moment of Inertia:** $I = \sum m_i r_i^2$ (for point masses) - For continuous objects, $I = \int r^2 dm$ - **Parallel-Axis Theorem:** $I = I_{CM} + Mh^2$ #### 5.4 Rotational Kinetic Energy - $K_{rot} = \frac{1}{2}I\omega^2$ - **Total Kinetic Energy (Rolling):** $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$ #### 5.5 Angular Momentum - **Angular Momentum of a particle:** $\vec{l} = \vec{r} \times \vec{p}$ - **Angular Momentum of a rigid body:** $\vec{L} = I\vec{\omega}$ - **Newton's Second Law (Angular Form):** $\sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$ - **Conservation of Angular Momentum:** If $\sum \vec{\tau}_{ext} = 0$, then $\Delta \vec{L}_{sys} = 0 \implies \vec{L}_{sys,f} = \vec{L}_{sys,i}$ ### 6. Gravitation - **Newton's Law of Universal Gravitation:** $F = G\frac{m_1 m_2}{r^2}$ - $G \approx 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ - **Gravitational Potential Energy:** $U = -G\frac{m_1 m_2}{r}$ - **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. The square of the orbital period $T$ is proportional to the cube of the semimajor axis $a$: $T^2 = (\frac{4\pi^2}{GM})a^3$ ### 7. Oscillations #### 7.1 Simple Harmonic Motion (SHM) - **Position:** $x(t) = x_m \cos(\omega t + \phi)$ - **Velocity:** $v(t) = -x_m\omega \sin(\omega t + \phi)$ - **Acceleration:** $a(t) = -x_m\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (for spring-mass system) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Frequency:** $f = \frac{1}{T} = \frac{\omega}{2\pi}$ - **Energy of SHM:** $E = \frac{1}{2}kx_m^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ #### 7.2 Pendulums - **Simple Pendulum (small angles):** $T = 2\pi\sqrt{\frac{L}{g}}$ - **Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgL_{CM}}}$ - **Torsional Pendulum:** $T = 2\pi\sqrt{\frac{I}{\kappa}}$ (where $\kappa$ is torsional constant) ### 8. Waves #### 8.1 Transverse and Longitudinal Waves - **Wave Speed:** $v = \lambda f$ - **Wave on a String:** $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear density) - **General Wave Equation:** $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}$ - **Sinusoidal Wave:** $y(x,t) = y_m \sin(kx - \omega t + \phi)$ - **Wave Number:** $k = \frac{2\pi}{\lambda}$ - **Angular Frequency:** $\omega = 2\pi f$ #### 8.2 Sound Waves - **Speed of Sound:** $v = \sqrt{\frac{B}{\rho}}$ ($B$ = bulk modulus, $\rho$ = density) - For ideal gas: $v = \sqrt{\frac{\gamma RT}{M}}$ - **Intensity:** $I = \frac{P}{A}$ - **Sound Level (decibels):** $\beta = (10 \text{ dB}) \log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) - **Doppler Effect:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ - Top signs for detector/source moving TOWARDS each other. - Bottom signs for detector/source moving AWAY from each other. #### 8.3 Standing Waves - **Standing Waves on a String (fixed ends):** - Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1, 2, 3, ...$) - Frequencies: $f_n = \frac{nv}{2L} = n f_1$ - **Standing Waves in Pipes:** - **Open at both ends:** $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1, 2, 3, ...$) - **Open at one end, closed at other:** $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ ($n=1, 3, 5, ...$) ### 9. Thermodynamics #### 9.1 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:** $Q = cm\Delta T$ - **Latent Heat (Phase Change):** $Q = Lm$ - **Heat Transfer:** - Conduction: $P_{cond} = kA\frac{T_H - T_C}{L}$ - Radiation: $P_{rad} = \sigma \epsilon A T^4$ (Stefan-Boltzmann Law) #### 9.2 First Law of Thermodynamics - **Internal Energy:** $\Delta E_{int} = Q - W$ - $Q$: Heat added to system (positive) - $W$: Work done BY system (positive) - **Work done by gas (constant pressure):** $W = P\Delta V$ - **Work done by gas (variable pressure):** $W = \int P dV$ #### 9.3 Kinetic Theory of Gases - **Ideal Gas Law:** $PV = nRT = NkT$ - $R \approx 8.31 \text{ J/(mol}\cdot\text{K)}$ - $k = R/N_A \approx 1.38 \times 10^{-23} \text{ J/K}$ - **Average Kinetic Energy (per molecule):** $K_{avg} = \frac{3}{2}kT$ - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}}$ - **Internal Energy of Monatomic Ideal Gas:** $E_{int} = \frac{3}{2}nRT$ #### 9.4 Second Law of Thermodynamics - **Entropy Change:** $\Delta S = \int \frac{dQ}{T}$ - For reversible process: $\Delta S = \frac{Q}{T}$ - **Second Law (Entropy Statement):** $\Delta S_{universe} \ge 0$ - **Heat Engines:** $e = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - **Carnot Engine (Ideal):** $e_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps (Coefficient of Performance):** - Refrigerator: $K = \frac{|Q_C|}{|W|}$ - Heat Pump: $K_{HP} = \frac{|Q_H|}{|W|}$ ### 10. Electricity #### 10.1 Electric Fields and Forces - **Coulomb's Law:** $F = k\frac{|q_1 q_2|}{r^2}$ - $k = \frac{1}{4\pi\epsilon_0} \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - $\epsilon_0 \approx 8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Electric Field of Point Charge:** $E = k\frac{|q|}{r^2}$ - **Electric Dipole Moment:** $\vec{p} = q\vec{d}$ - **Torque on Dipole in E-field:** $\vec{\tau} = \vec{p} \times \vec{E}$ - **Potential Energy of Dipole:** $U = -\vec{p} \cdot \vec{E}$ #### 10.2 Gauss' Law - **Electric Flux:** $\Phi_E = \int \vec{E} \cdot d\vec{A}$ - **Gauss' Law:** $\epsilon_0 \Phi_E = q_{enc}$ #### 10.3 Electric Potential - **Electric Potential Energy:** $\Delta U = -W_{field}$ - **Electric Potential:** $V = \frac{U}{q_0}$ - **Potential Difference:** $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ - **Potential due to Point Charge:** $V = k\frac{q}{r}$ - **Electric Field from Potential:** $\vec{E} = -\nabla V = -(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k})$ #### 10.4 Capacitance - **Capacitance:** $C = \frac{q}{V}$ - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Energy Stored:** $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{q^2}{C} = \frac{1}{2}qV$ - **Energy Density:** $u = \frac{1}{2}\epsilon_0 E^2$ - **Capacitors in Parallel:** $C_{eq} = \sum C_i$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ - **Dielectrics:** $C = \kappa C_{air}$ ### 11. DC Circuits - **Electric Current:** $I = \frac{dq}{dt} = nAv_d q$ - **Resistance:** $R = \rho \frac{L}{A}$ - **Ohm's Law:** $V = IR$ - **Power:** $P = IV = I^2R = \frac{V^2}{R}$ - **Resistors in Series:** $R_{eq} = \sum R_i$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ - **Kirchhoff's Laws:** - **Junction Rule:** $\sum I_{in} = \sum I_{out}$ - **Loop Rule:** $\sum \Delta V = 0$ - **RC Circuits (Charging Capacitor):** $q(t) = C\mathcal{E}(1 - e^{-t/RC})$ - **Time Constant:** $\tau = RC$ ### 12. Magnetism #### 12.1 Magnetic Fields and Forces - **Magnetic Force on Moving Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ (magnitude $qvB\sin\theta$) - **Magnetic Force on Current-Carrying Wire:** $\vec{F}_B = I\vec{L} \times \vec{B}$ (magnitude $ILB\sin\theta$) - **Torque on Current Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ - **Magnetic Dipole Moment:** $\vec{\mu} = NIA\hat{n}$ ($N$ turns, $I$ current, $A$ area) - **Potential Energy of Magnetic Dipole:** $U = -\vec{\mu} \cdot \vec{B}$ #### 12.2 Sources of Magnetic Field - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ - $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ - **Magnetic Field of Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field at Center of Circular Loop:** $B = \frac{\mu_0 I}{2R}$ - **Magnetic Field of Solenoid:** $B = \mu_0 n I$ ($n$ = turns per unit length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ #### 12.3 Faraday's Law and Induction - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - **Faraday's Law of Induction:** $\mathcal{E} = -N\frac{d\Phi_B}{dt}$ - **Lenz's Law:** Induced current/EMF opposes the change in magnetic flux that produced it. - **Motional EMF:** $\mathcal{E} = BLv$ (for conductor moving perpendicular to B) #### 12.4 Inductance - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Self-Inductance of Solenoid:** $L = \mu_0 n^2 A l$ - **Inductor EMF:** $\mathcal{E}_L = -L\frac{dI}{dt}$ - **Energy Stored in Inductor:** $U_B = \frac{1}{2}LI^2$ - **Energy Density of Magnetic Field:** $u_B = \frac{1}{2\mu_0} B^2$ - **RL Circuits (Current buildup):** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$ - **Time Constant:** $\tau = L/R$ ### 13. AC Circuits - **RLC Series Circuit:** - **Impedance:** $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - **Inductive Reactance:** $X_L = \omega L$ - **Capacitive Reactance:** $X_C = \frac{1}{\omega C}$ - **Phase Angle:** $\tan\phi = \frac{X_L - X_C}{R}$ - **RMS Current:** $I_{rms} = \frac{V_{rms}}{Z}$ - **Resonance:** $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ - **Power in AC Circuits:** $P_{avg} = I_{rms}V_{rms}\cos\phi$ - **Power Factor:** $\cos\phi$ ### 14. Electromagnetic Waves and Light - **Speed of Light in Vacuum:** $c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ - **Speed of Light in Medium:** $v = c/n$ ($n$ = refractive index) - **Relationship:** $c = \lambda f$ - **Energy of Photon:** $E = hf = \frac{hc}{\lambda}$ ($h \approx 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$) - **Intensity:** $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{1}{c\mu_0}B_{rms}^2 = \frac{E_{rms}B_{rms}}{\mu_0}$ - **Poynting Vector:** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ #### 14.1 Reflection and Refraction - **Law of Reflection:** $\theta_1 = \theta_1'$ - **Snell's Law (Law of Refraction):** $n_1\sin\theta_1 = n_2\sin\theta_2$ - **Critical Angle (Total Internal Reflection):** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) #### 14.2 Mirrors and Lenses - **Mirror/Lens Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - $p$: object distance (always +) - $i$: image distance (+ for real, - for virtual) - $f$: focal length (+ for concave mirror/converging lens, - for convex mirror/diverging lens) - **Magnification:** $m = -\frac{i}{p} = \frac{h_i}{h_p}$ - $|m|>1$: larger image, $|m| 0$: upright image, $m ### 15. 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:** $\vec{p} = \gamma m\vec{v}$ - **Relativistic Kinetic Energy:** $K = (\gamma - 1)mc^2$ - **Total Energy:** $E = \gamma mc^2 = K + mc^2$ - **Mass-Energy Equivalence:** $E_0 = mc^2$ (Rest Energy) - **Energy-Momentum Relation:** $E^2 = (pc)^2 + (mc^2)^2$ ### 16. Quantum Physics - **Photoelectric Effect:** $K_{max} = hf - \Phi$ - $\Phi$: work function - **De Broglie Wavelength:** $\lambda = \frac{h}{p} = \frac{h}{mv}$ - **Heisenberg Uncertainty Principle:** - $\Delta x \Delta p_x \ge \hbar/2$ - $\Delta E \Delta t \ge \hbar/2$ - **Schrödinger Equation (Time-Independent):** $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ - **Probability Density:** $|\psi(x)|^2$ ### 17. Nuclear Physics - **Atomic Mass Unit:** $1 \text{ u} \approx 1.66 \times 10^{-27} \text{ kg} \approx 931.5 \text{ MeV}/c^2$ - **Mass Defect and Binding Energy:** $E_B = \Delta m c^2$ - **Radioactive Decay:** $N(t) = N_0 e^{-\lambda t}$ - **Decay Constant:** $\lambda = \frac{\ln 2}{T_{1/2}}$ - **Half-life:** $T_{1/2}$ - **Activity:** $R = |\frac{dN}{dt}| = \lambda N$