Halliday Physics Essentials

Cheatsheet Content

### Kinematics - **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 - $v = v_0 + at$ - $x = x_0 + v_0t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### Free Fall - $a = -g = -9.8 \text{ m/s}^2$ (downwards) #### Projectile Motion - **Horizontal:** $v_x = v_{0x}$, $x = x_0 + v_{0x}t$ - **Vertical:** $v_y = v_{0y} - gt$, $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$, $v_y^2 = v_{0y}^2 - 2g(y - y_0)$ ### Newton's Laws of Motion - **First Law (Inertia):** 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. - **Second Law:** $\Sigma \vec{F} = m\vec{a}$ - **Weight:** $W = mg$ - **Third Law:** For every action, there is an equal and opposite reaction ($\vec{F}_{AB} = -\vec{F}_{BA}$) #### Types of Forces - **Normal Force:** $N$ (perpendicular to surface) - **Friction Force:** - **Static:** $f_s \le \mu_s N$ - **Kinetic:** $f_k = \mu_k N$ - **Tension Force:** $T$ (along a rope/string) - **Spring Force (Hooke's Law):** $F_s = -kx$ ### Work & Energy - **Work done by constant force:** $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$ - **Work done by variable force:** $W = \int \vec{F} \cdot d\vec{x}$ - **Kinetic Energy:** $K = \frac{1}{2}mv^2$ - **Work-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) - $\Delta E_{mech} = \Delta K + \Delta U = 0$ - **Conservation of Energy (General):** $W_{ext} = \Delta E_{mech} + \Delta E_{thermal} + \Delta E_{internal}$ - Or, $W_{nc} = \Delta E_{mech}$ (work by non-conservative forces) - **Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### Momentum & Collisions - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ - **Impulse-Momentum Theorem:** $\vec{J} = \vec{p}_f - \vec{p}_i$ - **Conservation of Linear Momentum:** $\Sigma \vec{p}_{initial} = \Sigma \vec{p}_{final}$ (if net external force is zero) - **Center of Mass:** - $x_{com} = \frac{1}{M}\sum m_i x_i$ - $\vec{v}_{com} = \frac{1}{M}\sum m_i \vec{v}_i$ - $\vec{P}_{com} = M\vec{v}_{com}$ - **Collisions:** - **Elastic:** Kinetic energy conserved. $K_{initial} = K_{final}$ - **Inelastic:** Kinetic energy not conserved. - **Completely Inelastic:** Objects stick together. ### Rotational Motion - **Angular Displacement:** $\Delta \theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ #### Constant Angular Acceleration - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### Relation to Linear Quantities (for a point at radius r) - Arc Length: $s = r\theta$ - Tangential Speed: $v_t = r\omega$ - Tangential Acceleration: $a_t = r\alpha$ - Centripetal Acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$ - Centripetal Force: $F_c = m a_c = \frac{mv^2}{r}$ #### Rotational Dynamics - **Moment of Inertia:** $I = \sum m_i r_i^2$ (for point masses) or $I = \int r^2 dm$ (for continuous bodies) - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F} = rF\sin\phi$ - **Newton's Second Law for Rotation:** $\Sigma \vec{\tau} = I\vec{\alpha}$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Work done by Torque:** $W = \int \tau d\theta$ - **Angular Momentum:** $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) - **Conservation of Angular Momentum:** $\Sigma \vec{L}_{initial} = \Sigma \vec{L}_{final}$ (if net external torque is zero) ### Oscillations & Waves - **Simple Harmonic Motion (SHM):** - Position: $x(t) = A\cos(\omega t + \phi)$ - 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: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum for small angles) - Period: $T = \frac{2\pi}{\omega}$ - Frequency: $f = \frac{1}{T}$ - **Mechanical Waves:** - Wave Speed: $v = \lambda f$ - Wave on a String: $v = \sqrt{\frac{T}{\mu}}$ (T = tension, $\mu$ = linear mass density) - Standing Waves: - Fixed ends (string): $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ for $n=1,2,3...$ - Open-open or closed-closed pipes: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ for $n=1,2,3...$ - Open-closed pipes: $\lambda_n = \frac{4L}{n}$, $f_n = n\frac{v}{4L}$ for $n=1,3,5...$ - **Sound Waves:** - Speed of Sound in Air: $v \approx 343 \text{ m/s}$ (at $20^\circ \text{C}$) - Intensity: $I = \frac{P}{A}$ - Intensity Level (decibels): $\beta = (10 \text{ dB})\log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) - Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D = detector, S = source; use top signs for "towards", bottom for "away") ### Thermodynamics - **Temperature Scales:** - $T_C = T_K - 273.15$ - $T_F = \frac{9}{5}T_C + 32$ - **Thermal Expansion:** - Linear: $\Delta L = L_0 \alpha \Delta T$ - Volume: $\Delta V = V_0 \beta \Delta T$ ($\beta = 3\alpha$) - **Heat and Internal Energy:** - Heat Capacity: $Q = C\Delta T$ - Specific Heat: $Q = mc\Delta T$ - Latent Heat (Phase Change): $Q = mL$ - **First Law of Thermodynamics:** $\Delta E_{int} = Q - W$ - $Q$: heat added to system, $W$: work done BY system - Work for gas: $W = \int P dV$ - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.31 \text{ J/(mol}\cdot\text{K)}$ (gas constant) - $k = 1.38 \times 10^{-23} \text{ J/K}$ (Boltzmann constant) - **Kinetic Theory of Gases:** - Average Kinetic Energy: $K_{avg} = \frac{3}{2}kT$ - RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ (M = molar mass in kg/mol) - **Thermodynamic Processes:** - **Isobaric:** Constant pressure ($W = P\Delta V$) - **Isochoric:** Constant volume ($W = 0$) - **Isothermal:** Constant temperature ($\Delta E_{int} = 0$, $Q = W = nRT\ln\frac{V_f}{V_i}$) - **Adiabatic:** No heat exchange ($Q = 0$, $\Delta E_{int} = -W$, $PV^\gamma = \text{constant}$) - **Heat Engines and Refrigerators:** - **Efficiency (Engine):** $e = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - **Coefficient of Performance (Refrigerator):** $K = \frac{|Q_C|}{|W|} = \frac{|Q_C|}{|Q_H| - |Q_C|}$ - **Carnot Cycle (Ideal):** $e_C = 1 - \frac{T_C}{T_H}$, $K_C = \frac{T_C}{T_H - T_C}$ - **Entropy:** $\Delta S = \int \frac{dQ}{T}$ - Second Law of Thermodynamics: $\Delta S_{universe} \ge 0$ ### Electricity & Magnetism #### Electrostatics - **Coulomb's Law:** $F = k\frac{|q_1 q_2|}{r^2}$ ($k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ or for point charge $\vec{E} = k\frac{q}{r^2}\hat{r}$ - **Electric Flux:** $\Phi_E = \int \vec{E} \cdot d\vec{A}$ - **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 for point charge $V = k\frac{q}{r}$ - **Relation 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: $U_C = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ - In series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ - In parallel: $C_{eq} = \sum C_i$ ### Circuits - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ - **Power in Circuits:** $P = IV = I^2R = \frac{V^2}{R}$ - **Resistors:** - In series: $R_{eq} = \sum R_i$ - In parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ - **Kirchhoff's Rules:** - **Junction Rule:** $\sum I_{in} = \sum I_{out}$ - **Loop Rule:** $\sum \Delta V = 0$ - **RC Circuits:** - Charging: $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = I_{max}e^{-t/RC}$ - Discharging: $Q(t) = Q_0 e^{-t/RC}$, $I(t) = I_0 e^{-t/RC}$ - Time Constant: $\tau = RC$ ### Magnetism - **Magnetic Force on a Moving Charge:** $\vec{F}_B = q(\vec{v} \times \vec{B})$ - Magnitude: $F_B = |q|vB\sin\theta$ - **Magnetic Force on a Current-Carrying Wire:** $\vec{F}_B = I(\vec{L} \times \vec{B})$ - Magnitude: $F_B = ILB\sin\theta$ - **Magnetic Field from a Current (Biot-Savart Law):** $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{l} \times \hat{r}}{r^2}$ - For long straight wire: $B = \frac{\mu_0 I}{2\pi r}$ - For 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 in Inductor: $U_L = \frac{1}{2}LI^2$ - **RL Circuits:** - Current build-up: $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$ - Current decay: $I(t) = I_0 e^{-t/\tau}$ - Time Constant: $\tau = L/R$ - **LC Oscillations:** $\omega = \frac{1}{\sqrt{LC}}$ ### Light & Optics - **Speed of Light:** $c = 3.00 \times 10^8 \text{ m/s}$ - **Electromagnetic Waves:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ - **Index of Refraction:** $n = \frac{c}{v}$ - **Snell's Law (Refraction):** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Total Internal Reflection:** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) - **Thin Lens Equation / Spherical Mirrors:** $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ - $f = R/2$ (for mirrors) - Magnification: $m = -\frac{i}{p}$ - **Sign Conventions:** - $p$: + real object, - virtual object - $i$: + real image, - virtual image - $f$: + converging (convex lens, concave mirror), - diverging (concave lens, convex mirror) - $h'$: + upright, - inverted - **Interference (Young's Double Slit):** - Bright Fringes (maxima): $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) - Dark Fringes (minima): $d\sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) - **Diffraction (Single Slit):** - Dark Fringes (minima): $a\sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) - **Diffraction Grating:** - Bright Fringes (maxima): $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) ### Modern Physics (Brief) - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$, Planck's constant) - **Photon Energy:** $E = hf = \frac{hc}{\lambda}$ - **De Broglie Wavelength:** $\lambda = \frac{h}{p}$ - **Heisenberg Uncertainty Principle:** - $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - $\Delta E \Delta t \ge \frac{\hbar}{2}$ ($\hbar = h/2\pi$) - **Energy-Mass Equivalence:** $E = mc^2$