Halliday Physics Essentials

Cheatsheet Content

### 1. Kinematics (Motion) #### 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:** $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ #### 1.2. Constant Acceleration Equations - $v = v_0 + at$ - $x = x_0 + v_0 t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 1.3. Two-Dimensional Motion (Projectile Motion) - **Position:** $\vec{r} = x\hat{i} + y\hat{j}$ - **Velocity:** $\vec{v} = v_x\hat{i} + v_y\hat{j}$ - $v_x = v_0 \cos\theta_0$ (constant) - $v_y = v_0 \sin\theta_0 - gt$ - **Acceleration:** $\vec{a} = -g\hat{j}$ - **Range:** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ - **Maximum Height:** $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ #### 1.4. Uniform Circular Motion - **Centripetal Acceleration:** $a_c = \frac{v^2}{r}$ (directed towards center) - **Period:** $T = \frac{2\pi r}{v}$ ### 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. ($\sum \vec{F} = 0 \Rightarrow \vec{a} = 0$) - **Newton's Second Law:** $\sum \vec{F} = m\vec{a}$ (Net force equals mass times acceleration) - **Newton's Third Law:** For every action, there is an equal and opposite reaction. ($\vec{F}_{AB} = -\vec{F}_{BA}$) #### 2.1. Forces - **Weight:** $W = mg$ - **Normal Force:** $F_N$ (perpendicular to surface) - **Tension:** $T$ (along a string/rope) - **Friction:** - **Static:** $f_s \le \mu_s F_N$ - **Kinetic:** $f_k = \mu_k F_N$ - **Drag Force (at high speed):** $D = \frac{1}{2}C\rho Av^2$ ### 3. Work & 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:** $W_{net} = \Delta K = K_f - K_i$ - **Gravitational Potential Energy:** $U_g = mgh$ (near Earth's surface) - **Elastic Potential Energy:** $U_s = \frac{1}{2}kx^2$ (for a spring) - **Conservative Forces:** Work done is path-independent ($W_c = -\Delta U$) - **Non-Conservative Forces:** Work done is path-dependent (e.g., friction) - **Conservation of Mechanical Energy:** $E_{mech} = K + U = \text{constant}$ (if only conservative forces do work) - **General Conservation of Energy (with non-conservative work):** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### 4. Momentum & Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ - **Impulse-Momentum Theorem:** $\vec{J} = \vec{p}_f - \vec{p}_i$ - **Conservation of Linear Momentum:** $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$ (if net external force is zero) - **Collisions:** - **Elastic:** Both momentum and kinetic energy are conserved. - **Inelastic:** Momentum conserved, kinetic energy not conserved. - **Perfectly Inelastic:** Objects stick together after collision. - **Center of Mass:** - $x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$ - $\vec{v}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$ - $\vec{a}_{CM} = \frac{\sum m_i \vec{a}_i}{\sum m_i} = \frac{\sum \vec{F}_{ext}}{M_{total}}$ ### 5. Rotational Motion - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ - **Rotational Kinematics (constant $\alpha$):** - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ - **Relating Linear & Angular:** - $s = r\theta$ - $v = r\omega$ - $a_t = r\alpha$ (tangential acceleration) - $a_c = r\omega^2 = v^2/r$ (centripetal acceleration) - **Moment of Inertia:** $I = \sum m_i r_i^2 = \int r^2 dm$ - **Parallel-Axis Theorem:** $I = I_{CM} + Mh^2$ - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ (magnitude $\tau = rF\sin\phi$) - **Newton's Second Law for Rotation:** $\sum \tau = I\alpha$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Work done by Torque:** $W = \int \tau d\theta$ - **Angular Momentum:** $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ - **Conservation of Angular Momentum:** $\sum \vec{L}_{initial} = \sum \vec{L}_{final}$ (if net external torque is zero) ### 6. Gravitation - **Newton's Law of Universal 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:** $U = -G\frac{m_1 m_2}{r}$ - **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. $T^2 \propto a^3$ (where $a$ is semi-major axis) ### 7. Oscillations & Waves #### 7.1. 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:** $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum, small angles) - **Period:** $T = \frac{2\pi}{\omega}$ - **Frequency:** $f = \frac{1}{T} = \frac{\omega}{2\pi}$ - **Energy in SHM:** $E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ #### 7.2. Waves - **Wave Speed:** $v = \lambda f$ - **Speed on a String:** $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear density) - **Intensity:** $I = P/A$ - **Sound Intensity Level (dB):** $\beta = 10 \log_{10}(I/I_0)$, where $I_0 = 10^{-12} \text{ W/m}^2$ - **Doppler Effect:** $f' = f \frac{v \pm v_D}{v \mp v_S}$ (top sign for approaching, bottom for receding) - **Standing Waves on a String (fixed ends):** - Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1, 2, 3, ...$) - Frequencies: $f_n = \frac{nv}{2L}$ - **Standing Waves in Open-Open Pipe:** Same as string. - **Standing Waves in Open-Closed Pipe:** - Wavelengths: $\lambda_n = \frac{4L}{n}$ ($n=1, 3, 5, ...$) - Frequencies: $f_n = \frac{nv}{4L}$ ### 8. Thermodynamics - **Temperature Scales:** - $T_C = (T_F - 32) \times 5/9$ - $T_K = T_C + 273.15$ - **Thermal Expansion:** - Linear: $\Delta L = \alpha L_0 \Delta T$ - Volume: $\Delta V = \beta V_0 \Delta T$, where $\beta \approx 3\alpha$ - **Heat Capacity:** $Q = C\Delta T = mc\Delta T$ - **Latent Heat:** $Q = mL$ (L = latent heat of fusion or vaporization) - **Heat Transfer:** - **Conduction:** $P_{cond} = kA\frac{T_H - T_C}{L}$ - **Convection:** Heat transfer via fluid motion. - **Radiation:** $P_{rad} = \sigma \epsilon A T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/(m}^2\text{K}^4)$) - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.314 \text{ J/(mol}\cdot\text{K})$ - $k = 1.38 \times 10^{-23} \text{ J/K}$ - **First Law of Thermodynamics:** $\Delta E_{int} = Q - W$ - $W = \int P dV$ (work done BY the system) - **Internal Energy of Ideal Gas:** $E_{int} = \frac{3}{2}nRT$ (monatomic) - **Processes:** - **Isothermal:** $\Delta T = 0 \Rightarrow \Delta E_{int} = 0 \Rightarrow Q = W$ - **Adiabatic:** $Q = 0 \Rightarrow \Delta E_{int} = -W$ - **Isobaric:** $\Delta P = 0 \Rightarrow W = P\Delta V$ - **Isovolumetric:** $\Delta V = 0 \Rightarrow W = 0 \Rightarrow \Delta E_{int} = Q$ - **Second Law of Thermodynamics:** - Heat flows spontaneously from hot to cold. - Entropy of an isolated system never decreases. - **Heat Engines:** - Efficiency: $e = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - Carnot Efficiency: $e_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** - Coefficient of Performance (COP): $K = \frac{|Q_C|}{|W|}$ (refrigerator); $K_{HP} = \frac{|Q_H|}{|W|}$ (heat pump) - Carnot COP: $K_C = \frac{T_C}{T_H - T_C}$ - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ - For reversible process: $\Delta S = \frac{Q}{T}$ ### 9. Electrostatics - **Coulomb's Law:** $\vec{F} = k\frac{|q_1 q_2|}{r^2}\hat{r}$ - $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0} = k\frac{q}{r^2}\hat{r}$ - **Electric Dipole Moment:** $\vec{p} = q\vec{d}$ - **Electric Potential Energy:** $\Delta U = -W = -q\Delta V$ - **Electric Potential:** $V = \frac{U}{q_0} = k\frac{q}{r}$ (for point charge) - **Relation between E and V:** $\vec{E} = -\nabla V$ (for 1D, $E_x = -\frac{dV}{dx}$) - **Gauss's Law:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ ### Capacitance & Dielectrics - **Capacitance:** $C = \frac{Q}{V}$ - **Parallel Plate Capacitor:** $C = \frac{\epsilon_0 A}{d}$ - **Capacitors in Parallel:** $C_{eq} = C_1 + C_2 + ...$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ - **Energy Stored in Capacitor:** $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$ - **Energy Density of Electric Field:** $u_E = \frac{1}{2}\epsilon_0 E^2$ - **Dielectrics:** $C = \kappa C_0$ ($C_0$ is capacitance without dielectric) ### 11. Current & Resistance - **Electric Current:** $I = \frac{dQ}{dt} = nqv_d A$ - **Current Density:** $\vec{J} = nq\vec{v}_d$ - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ - **Resistivity:** $\rho = \rho_0[1 + \alpha(T - T_0)]$ - **Power in Circuits:** $P = IV = I^2R = V^2/R$ - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Kirchhoff's Rules:** - **Junction Rule:** $\sum I_{in} = \sum I_{out}$ - **Loop Rule:** $\sum \Delta V = 0$ - **RC Circuits:** - Charging: $Q(t) = Q_{max}(1 - e^{-t/RC})$ - Discharging: $Q(t) = Q_0 e^{-t/RC}$ - Time Constant: $\tau = RC$ ### 12. Magnetism - **Magnetic Force on a Charge:** $\vec{F}_B = q(\vec{v} \times \vec{B})$ - **Magnetic Force on a Current-Carrying Wire:** $\vec{F}_B = I(\vec{L} \times \vec{B})$ - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - **Gauss's Law for Magnetism:** $\oint \vec{B} \cdot d\vec{A} = 0$ - **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 a Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Magnetic Field at Center of Loop:** $B = \frac{\mu_0 I}{2R}$ - **Magnetic Field of a Solenoid:** $B = \mu_0 n I$ (n = turns per unit length) - **Ampere's Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - **Torque on a Current Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ - Magnetic Dipole Moment: $\vec{\mu} = NIA\hat{n}$ ### 13. Electromagnetic Induction - **Faraday's Law of Induction:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - **Lenz's Law:** The induced current creates a magnetic field that opposes the change in magnetic flux. - **Motional EMF:** $\mathcal{E} = BLv$ - **Inductance:** $L = \frac{N\Phi_B}{I}$ - **Solenoid Inductance:** $L = \mu_0 n^2 A l$ - **Energy Stored in an Inductor:** $U_L = \frac{1}{2}LI^2$ - **Energy Density of Magnetic Field:** $u_B = \frac{B^2}{2\mu_0}$ - **RL Circuits:** - Current build-up: $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ - Current decay: $I(t) = I_0 e^{-t/\tau_L}$ - Time Constant: $\tau_L = L/R$ - **LC Oscillations:** $\omega = \frac{1}{\sqrt{LC}}$ - **RLC Series Circuit:** - Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - Reactances: $X_L = \omega L$, $X_C = \frac{1}{\omega C}$ - Resonant Frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$ ### 14. Electromagnetic Waves - **Speed of Light:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$ - **Wave Equation:** $E = E_m \sin(kx - \omega t)$, $B = B_m \sin(kx - \omega t)$ - **Relation between E and B:** $E = cB$ - **Poynting Vector (Intensity):** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ - Average Intensity: $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{1}{2c\mu_0}E_m^2$ - **Radiation Pressure:** $P_{rad} = I/c$ (total absorption); $P_{rad} = 2I/c$ (total reflection) ### 15. Optics #### 15.1. 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:** $\sin\theta_c = n_2/n_1$ (for $n_1 > n_2$) #### 15.2. Mirrors & Lenses - **Mirror/Lens Equation:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - $p$: object distance (positive if real object) - $i$: image distance (positive if real image) - $f$: focal length (positive for concave mirror/converging lens; negative for convex mirror/diverging lens) - **Magnification:** $m = -\frac{i}{p} = \frac{h'}{h}$ - $|m| > 1$: magnified; $|m| 0$: upright; $m n_f > n_2$ or similar reflection phase shifts) - **Single-Slit Diffraction:** - Dark Fringes (minima): $a\sin\theta = m\lambda$ ($m= \pm 1, \pm 2, ...$) - **Diffraction Grating:** - Bright Fringes (maxima): $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Rayleigh's Criterion (Resolution):** $\theta_{min} = 1.22\frac{\lambda}{D}$ (circular aperture)