physics / final.texon commit update physics cheatsheet (20fa557)
   1\documentclass[a4paper]{article}
   2\usepackage{multicol}
   3\usepackage[cm]{fullpage}
   4\usepackage{amsmath}
   5\usepackage{amssymb}
   6\setlength{\parindent}{0cm}
   7\usepackage[nodisplayskipstretch]{setspace}
   8\setstretch{1.3}
   9\usepackage{graphicx}
  10\usepackage{wrapfig}
  11\usepackage{enumitem}
  12\usepackage{supertabular}
  13\usepackage{tabularx}
  14\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt}
  15
  16
  17\begin{document}
  18
  19\pagenumbering{gobble}
  20\begin{multicols}{3}
  21
  22% +++++++++++++++++++++++
  23
  24{\huge Physics}\hfill Andrew Lorimer\hspace{2em}
  25
  26% +++++++++++++++++++++++
  27\section{Motion}
  28
  29  $\operatorname{m/s} \, \times \, 3.6 = \operatorname{km/h}$
  30
  31  \subsection*{Inclined planes}
  32    $F = m g \sin\theta - F_{\text{frict}} = m a$
  33
  34% -----------------------
  35  \subsection*{Banked tracks}
  36
  37    \includegraphics[height=4cm]{graphics/banked-track.png}
  38
  39    $\theta = \tan^{-1} {{v^2} \over rg}$
  40
  41    $\Sigma F$ always acts towards centre (horizontally)
  42
  43    $\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
  44
  45    Design speed $v = \sqrt{gr\tan\theta}$
  46
  47    $n\sin \theta = {mv^2 \div r}, \quad n\cos \theta = mg$
  48
  49% -----------------------
  50  \subsection*{Work and energy}
  51
  52    $W=Fx=\Delta \Sigma E$ (work)
  53
  54    $E_K = {1 \over 2}mv^2$ (kinetic)
  55
  56    $E_G = mgh$ (potential)
  57
  58    $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
  59
  60% -----------------------
  61  \subsection*{Horizontal circular motion}
  62
  63    $v = {{2 \pi r} \over T}$
  64
  65    $f = {1 \over T}, \quad T = {1 \over f}$
  66
  67    $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
  68
  69    $\Sigma F, a$ towards centre, $v$ tangential
  70
  71    $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
  72
  73    \includegraphics[height=4cm]{graphics/circ-forces.png}
  74
  75% -----------------------
  76  \subsection*{Vertical circular motion}
  77
  78    $T =$ tension, e.g. circular pendulum
  79
  80    $T+mg = {{mv^2}\over r}$ at highest point
  81
  82    $T-mg = {{mv^2} \over r}$ at lowest point
  83
  84% -----------------------
  85  \subsection*{Projectile motion}
  86    \begin{itemize}
  87      \item $v_x$ is constant: $v_x = {s \over t}$
  88      \item use suvat to find $t$ from $y$-component
  89      \item vertical component gravity: $a_y = -g$
  90    \end{itemize}
  91
  92    \begin{align*}
  93      v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\
  94      h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\
  95      x=ut\cos\theta \tag{$\Delta x$ at $t$} \\
  96      y=ut \sin \theta-{1 \over 2}gt^2 \tag{height at $t$} \\
  97      t={{2u\sin\theta}\over g} \tag{time of flight}\\
  98      d={v^2 \over g}\sin \theta \tag{horiz. range} \\
  99    \end{align*}
 100
 101    \includegraphics[height=3.2cm]{graphics/projectile-motion.png}
 102
 103% -----------------------
 104  \subsection*{Pulley-mass system}
 105
 106    $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
 107
 108    $\Sigma F = m_2g-m_1g=\Sigma ma$ (solve)
 109
 110% -----------------------
 111  \subsection*{Graphs}
 112    \begin{itemize}
 113      \item{Force-time: $A=\Delta \rho$}
 114      \item{Force-disp: $A=W$}
 115      \item{Force-ext: $m=k,\quad A=E_{spr}$}
 116      \item{Force-dist: $A=\Delta \operatorname{gpe}$}
 117      \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
 118    \end{itemize}
 119
 120% -----------------------
 121  \subsection*{Hooke's law}
 122
 123  $F=-kx$
 124
 125  $\text{elastic potential energy} = {1 \over 2}kx^2$
 126
 127  $x={2mg \over k}$
 128
 129% -----------------------
 130  \subsection*{Motion equations}
 131
 132    \begin{tabular}{ l r }
 133      & no \\
 134      $v=u+at$ & $x$ \\
 135      $x = {1 \over 2}(v+u)t$ & $a$ \\
 136      $x=ut+{1 \over 2}at^2$ & $v$ \\
 137      $x=vt-{1 \over 2}at^2$ & $u$ \\
 138      $v^2=u^2+2ax$ & $t$ \\
 139    \end{tabular}
 140
 141% -----------------------
 142  \subsection*{Momentum}
 143
 144    $\rho = mv$
 145
 146    $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
 147
 148    $\Sigma mv_0=\Sigma mv_1$ (conservation)
 149
 150    $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
 151
 152    $n$-body collisions: $\rho$ of each body is independent
 153
 154% ++++++++++++++++++++++
 155\section{Relativity}
 156
 157  \subsection*{Postulates}
 158    1. Laws of physics are constant in all intertial reference frames
 159
 160    2. Speed of light $c$ is the same to all observers (Michelson-Morley)
 161
 162    $\therefore \, t$ must dilate as speed changes
 163
 164    {\bf Inertial reference frame} $a=0$
 165
 166    {\bf Proper time $t_0$ $\vert$ length $l_0$} measured by observer in same frame as events
 167
 168% -----------------------
 169  \subsection*{Lorentz factor}
 170
 171    $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$
 172
 173    $t=t_0 \gamma$ ($t$ longer in moving frame)
 174
 175    $l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame)
 176
 177    $m=m_0 \gamma$ (mass dilation)
 178
 179    $$v = c\sqrt{1-{1 \over \gamma^2}}$$
 180
 181% -----------------------
 182  \subsection*{Energy and work}
 183
 184    $E_0 = mc^2$ (rest)
 185
 186    $E_{total} = E_K + E_{rest} = \gamma mc^2$
 187
 188    $E_K = (\gamma 1)mc^2$
 189
 190    $W = \Delta E = \Delta mc^2$
 191
 192% -----------------------
 193  \subsection*{Relativistic momentum}
 194
 195    $$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$
 196
 197    $\rho \rightarrow \infty$ as $v \rightarrow c$
 198
 199    $v=c$ is impossible (requires $E=\infty$)
 200
 201    $$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
 202
 203% -----------------------
 204  \subsection*{High-altitude muons}
 205    \begin{itemize}
 206      {\item $t$ dilation more muons reach Earth than expected}
 207      {\item normal half-life $2.2 \operatorname{\mu s}$ in stationary frame, $> 2.2 \operatorname{\mu s}$ observed from Earth}
 208    \end{itemize}
 209
 210% +++++++++++++++++++++++
 211\section{Fields and power}
 212
 213  \subsection*{Non-contact forces}
 214    \begin{itemize}
 215      {\item electric fields (dipoles \& monopoles)}
 216      {\item magnetic fields (dipoles only)}
 217      {\item gravitational fields (monopoles only)}
 218    \end{itemize}
 219
 220    \vspace{1em}
 221
 222    \begin{itemize}
 223      \item monopoles: lines towards centre
 224      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (or perpendicular to wire)
 225      \item closer field lines means larger force
 226      \item dot: out of page, cross: into page
 227      \item +ve corresponds to N pole
 228    \end{itemize}
 229
 230    \includegraphics[height=2cm]{graphics/field-lines.png}
 231    % \includegraphics[height=2cm]{graphics/bar-magnet-fields-rotated.png}
 232
 233% -----------------------
 234  \subsection*{Gravity}
 235
 236    \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]
 237    \[g={F_g \over m_2}=G{m_{1} \over r^2}\tag{field of $m_1$}\]
 238    \[E_g = mg \Delta h\tag{gpe}\]
 239    \[W = \Delta E_g = Fx\tag{work}\]
 240    \[w=m(g-a) \tag{app. weight}\]
 241
 242    % \columnbreak
 243
 244% -----------------------
 245  \subsection*{Satellites}
 246
 247    \[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
 248
 249    \[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\]
 250
 251    \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
 252
 253% -----------------------
 254  \subsection*{Magnetic fields}
 255    \begin{itemize}
 256      \item field strength $B$ measured in tesla
 257      \item magnetic flux $\Phi$ measured in weber
 258      \item charge $q$ measured in coulombs
 259      \item emf $\mathcal{E}$ measured in volts
 260    \end{itemize}
 261
 262    % \[{E_1 \over E_2}={r_1 \over r_2}^2\]
 263
 264    \[F=qvB\tag{$F$ on moving $q$}\]
 265    \[F=IlB\tag{$F$ of $B$ on $I$}\]
 266    \[B={mv \over qr}\tag{field strength on e-}\]
 267    \[r={mv \over qB} \tag{radius of $q$ in $B$}\]
 268
 269    if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
 270
 271% -----------------------
 272  \subsection*{Electric fields}
 273
 274    \[F=qE \tag{$E$ = strength} \]
 275    \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
 276    \[E=k{q \over r^2} \tag{field on point charge} \]
 277    \[E={V \over d} \tag{field between plates}\]
 278    \[F=BInl \tag{force on a coil} \]
 279    \[\Phi = B_{\perp}A\tag{magnetic flux} \]
 280    \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
 281    \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
 282
 283    \textbf{Lenz's law:}  $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
 284    (emf creates $I$ with associated field that opposes $\Delta \phi$)
 285
 286    \textbf{Eddy currents:} counter movement within a field
 287
 288    \textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
 289
 290    \includegraphics[height=2cm]{graphics/slap-2.jpeg}
 291    \includegraphics[height=3cm]{graphics/grip.png}
 292
 293    % \textbf{Right hand slap:} $B \perp I \perp F$ \\
 294    % ($I$ = thumb)
 295
 296    \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
 297
 298    \textbf{Transformers:} core strengthens \& focuses $\Phi$
 299
 300% -----------------------
 301  \subsection*{Particle acceleration}
 302
 303    $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
 304
 305    e- accelerated with $x$ V is given $x$ eV
 306
 307    \[W={1\over2}mv^2=qV \tag{field or points}\]
 308    \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
 309
 310
 311% -----------------------
 312  \subsection*{Power transmission}
 313
 314    % \begin{align*}
 315      \[V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \]
 316      \[P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \]
 317      \[V_{\operatorname{loss}}=IR \]
 318    % \end{align*}
 319
 320    Use high-$V$ side for correct $|V_{drop}|$
 321
 322    \begin{itemize}
 323      {\item Parallel $V$ is constant}
 324      {\item Series $V$ shared within branch}
 325    \end{itemize}
 326
 327    \includegraphics[height=4cm]{graphics/ac-generator.png}
 328
 329% -----------------------
 330  \subsection*{Motors}
 331% \begin{wrapfigure}{r}{-0.1\textwidth}
 332
 333    \includegraphics[height=4cm]{graphics/dc-motor-2.png}
 334    \includegraphics[height=3cm]{graphics/ac-motor.png} \\
 335
 336    Force on current-carying wire, not copper \\
 337    $F=0$ for front & back of coil (parallel) \\
 338    Any angle $> 0$ will produce force \\
 339% \end{wrapfigure}
 340    \textbf{DC:} split ring (two halves)
 341
 342% \begin{wrapfigure}{r}{0.3\textwidth}
 343
 344% \end{wrapfigure}
 345    \textbf{AC:} slip ring (separate rings with constant contact)
 346
 347% \pagebreak
 348
 349% +++++++++++++++++++++++
 350\section{Waves}
 351
 352  \textbf{nodes:} fixed on graph \\
 353  \textbf{amplitude:} max disp. from $y=0$ \\
 354  \textbf{rarefactions} and \textbf{compressions} \\
 355  \textbf{mechanical:} transfer of energy without net transfer of matter \\
 356
 357
 358  \textbf{Longitudinal (motion $||$ wave)}
 359  \includegraphics[width=6cm]{graphics/longitudinal-waves.png}
 360
 361  \textbf{Transverse (motion $\perp$ wave)}
 362  \includegraphics[width=6cm]{graphics/transverse-waves.png}
 363
 364  % -----------------------
 365  $T={1 \over f}\quad$(period: time for one cycle)
 366  $v=f \lambda \quad$(speed: displacement / sec)
 367
 368  % -----------------------
 369  \subsection*{Doppler effect}
 370
 371  When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to travel than $w_{n-1}$. $w_n$ reaches observer sooner than $w_{n-1}$ ("apparent" $\lambda$).
 372
 373  % -----------------------
 374  \subsection*{Interference}
 375
 376  \includegraphics[width=4.5cm]{graphics/possons-spot.png}
 377  Poissons's spot supports wave theory (circular diffraction)
 378
 379  \textbf{Standing waves} - constructive int. at resonant freq
 380
 381  \textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
 382
 383  \textbf{Incoherent} - e.g. incandescent bulb
 384
 385
 386
 387
 388
 389  % -----------------------
 390  \subsection*{Harmonics}
 391
 392  \(\lambda = {{al} \div n}\quad\) (\(\lambda\) for \(n^{th}\) harmonic)\\
 393  \(f = {nv \div al}\quad\) (\(f\) for \(n_{th}\) harmonic at length
 394  \(l\) and speed \(v\)) \\
 395  where \(a=2\) for antinodes at both ends, \(a=4\) for antinodes at one end
 396
 397  % -----------------------
 398  \subsection*{Polarisation}
 399  \includegraphics[height=3.5cm]{graphics/polarisation.png}
 400
 401  % -----------------------
 402  \subsection*{Diffraction}
 403  \includegraphics[width=6cm]{graphics/diffraction.jpg}
 404  \includegraphics[width=6cm]{graphics/diffraction-2.png}
 405  \begin{itemize}
 406    % \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
 407    \item Constructive: \(pd = n\lambda, n \in \mathbb{Z}\)
 408    \item Destructive: \(pd = (n-{1 \over 2})\lambda, n \in \mathbb{Z}\)
 409    \item Path difference: \(\Delta x = {{\lambda l }\over d}\) where \\
 410    % \(\Delta x\) = fringe spacing \\
 411    \(l\) = distance from source to observer\\
 412    \(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
 413    \item diffraction $\propto {\lambda \over d}$
 414    \item significant diffraction when ${\lambda \over \Delta x} \ge 1$
 415    \item diffraction creates distortion (electron $>$ optical microscopes)
 416  \end{itemize}
 417
 418
 419  % -----------------------
 420  \subsection*{Refraction}
 421  \includegraphics[height=3.5cm]{graphics/refraction.png}
 422
 423  When a medium changes character, energy is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}
 424
 425  angle of incidence $\theta_i =$ angle of reflection $\theta_r$
 426
 427  Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$
 428
 429  Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
 430
 431
 432% +++++++++++++++++++++++
 433\section{Light and Matter}
 434
 435  % -----------------------
 436  \subsection*{Planck's equation}
 437
 438  \[ f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c \]
 439  \[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
 440  \[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]
 441
 442  \subsection*{Force of electrons}
 443  \[ F={2P_{\text{in}}\over c} \]
 444  % \begin{align*}
 445    \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
 446    \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
 447    % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
 448  % \end{align*}
 449
 450  \subsection*{De Broglie's theory}
 451
 452  \[ \lambda = {h \over \rho} = {h \over mv} \]
 453  \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
 454  \begin{itemize}
 455    \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
 456    \item confirmed by e- and x-ray patterns
 457  \end{itemize}
 458
 459  \subsection*{X-ray electron interaction}
 460
 461  \begin{itemize}
 462    \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
 463    \item $\therefore 2\pi r = n{h \over mv} = n \lambda$ (circumference)
 464    \item if $2\pi r \ne n{h \over mv}$, no standing wave
 465    \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
 466    % \item calculating $h$: $\lambda = {h \over \rho}$
 467  \end{itemize}
 468
 469  \subsection*{Photoelectric effect}
 470
 471  \begin{itemize}
 472    \item $V_{\operatorname{supply}}$ does not affect photocurrent
 473    \item $V_{\operatorname{sup}} > 0$: attracted to +ve
 474    \item $V_{\operatorname{sup}} < 0$: attracted to -ve, $I\rightarrow 0$
 475    \item $v$ of e- depends on shell
 476    \item max current depends on intensity
 477  \end{itemize}
 478
 479  \subsubsection*{Threshold frequency $f_0$}
 480
 481  min $f$ for photoelectron release. if $f < f_0$, no photoelectrons.
 482
 483  \subsubsection*{Work function $\phi=hf_0$}
 484
 485  min $E$ for photoelectron release. determined by strength of bonding. Units: eV or J.
 486
 487  \subsubsection*{Kinetic energy E_K=hf - \phi = qV_0}
 488
 489
 490  $V_0 = E_K$ in eV \\
 491  % $E_K = x$-int of $V\cdot I$ graph (in eV) \\
 492  dashed line below $E_K=0$
 493
 494
 495  \subsubsection*{Stopping potential $V_0$ for min $I$}
 496
 497  $$V_0=h_{\text{eV}}(f-f_0)$$
 498
 499  \subsubsection*{Graph features}
 500
 501  \newcolumntype{b}{>{\hsize=.75\hsize}X}
 502\newcolumntype{s}{>{\hsize=.3\hsize}X}
 503
 504  \begin{tabularx}{\columnwidth}{bbbb}
 505\hline
 506&$m$&$x$-int&$y$-int \\
 507\hline
 508\hline
 509$f \cdot E_K$ & $h$ & $f_0$ & $-\phi$ \\
 510$V \cdot I$ &  & $V_0$ & intensity\\
 511$f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ &
 512\hline
 513\end{tabularx}
 514
 515
 516
 517  \subsection*{Spectral analysis}
 518
 519  \begin{itemize}
 520    \item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
 521    \item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
 522    \item Ionisation energy - min $E$ required to remove e-
 523    \item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
 524    \item No. of lines - include all possible states
 525  \end{itemize}
 526
 527  \subsection*{Uncertainty principle}
 528
 529  measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
 530
 531  \subsection*{Wave-particle duaity}
 532
 533  \subsubsection*{wave model}
 534  \begin{itemize}
 535    \item cannot explain photoelectric effect
 536    \item $f$ is irrelevant to photocurrent
 537    \item predicts delay between incidence and ejection
 538    \item speed depends on medium
 539    \item supported by bright spot in centre
 540  \end{itemize}
 541
 542  \subsubsection*{particle model}
 543
 544  \begin{itemize}
 545    \item explains photoelectric effect
 546    \item rate of photoelectron release $\propto$ intensity
 547    \item no time delay - one photon releases one electron
 548    \item double slit: photons interact. interference pattern still appears when a dim light source is used so that only one photon can pass at a time
 549    \item light exerts force
 550    \item light bent by gravity
 551    \item quantised energy
 552  \end{itemize}
 553
 554  % +++++++++++++++++++++++
 555  \section{Experimental \\ design}
 556
 557  \textbf{Absolute uncertainty} $\Delta$ \\
 558  (same units as quantity)
 559  \[ \Delta(m) = {{\mathcal{E}(m)} \over 100} \cdot m \]
 560  \[ (A \pm \Delta A) + (B \pm \Delta A) = (A+B) \pm (\Delta A + \Delta B) \]
 561  \[ (A \pm \Delta A) - (B \pm \Delta A) = (A-B) \pm (\Delta A + \Delta B) \]
 562  \[ c(A \pm \Delta A) = cA \pm c \Delta A \]
 563
 564  \textbf{Relative uncertainty} $\mathcal{E}$ (unitless)
 565  \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100 \]
 566  \[ (A \pm \mathcal{E} A) \cdot (B \pm \mathcal{E} B) = (A \cdot B) \pm (\mathcal{E} A + \mathcal{E} B) \]
 567  \[ (A \pm \mathcal{E} A) \div (B \pm \mathcal{E} B) = (A \div B) \pm (\mathcal{E} A + \mathcal{E} B) \]
 568  \[ (A \pm \mathcal{E} A)^n = (A^n \pm n \mathcal{E} A) \]
 569  \[ c(A \pm \mathcal{E} A)=cA \pm \mathcal{E} A \]
 570
 571  Uncertainty of a measurement is $1 \over 2$ the smallest division
 572
 573  \textbf{Precision} - concordance of values \\
 574  \textbf{Accuracy} - closeness to actual value\\
 575  \textbf{Random errors} - unpredictable, reduced by more tests \\
 576  \textbf{Systematic errors} - not reduced by more tests \\
 577  \textbf{Uncertainty} - margin of potential error \\
 578  \textbf{Error} - actual difference \\
 579  \textbf{Hypothesis} - can be tested experimentally \\
 580  \textbf{Model} - evidence-based but indirect representation
 581
 582\end{multicols}
 583
 584\end{document}