physics / final.texon commit add to cheatsheet (9ff87d7)
   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\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt}
  13
  14
  15\begin{document}
  16
  17\pagenumbering{gobble}
  18\begin{multicols}{3}
  19
  20% +++++++++++++++++++++++
  21
  22{\huge Physics}\hfill Andrew Lorimer\hspace{2em}
  23
  24% +++++++++++++++++++++++
  25\section{Motion}
  26
  27  $\operatorname{m/s} \, \times \, 3.6 = \operatorname{km/h}$
  28
  29  \subsection*{Inclined planes}
  30    $F = m g \sin\theta - F_{\text{frict}} = m a$
  31
  32% -----------------------
  33  \subsection*{Banked tracks}
  34
  35    \includegraphics[height=4cm]{graphics/banked-track.png}
  36
  37    $\theta = \tan^{-1} {{v^2} \over rg}$
  38
  39    $\Sigma F$ always acts towards centre (horizontally)
  40
  41    $\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
  42
  43    Design speed $v = \sqrt{gr\tan\theta}$
  44
  45    $n\sin \theta = {mv^2 \div r}, \quad n\cos \theta = mg$
  46
  47% -----------------------
  48  \subsection*{Work and energy}
  49
  50    $W=Fx=\Delta \Sigma E$ (work)
  51
  52    $E_K = {1 \over 2}mv^2$ (kinetic)
  53
  54    $E_G = mgh$ (potential)
  55
  56    $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
  57
  58% -----------------------
  59  \subsection*{Horizontal circular motion}
  60
  61    $v = {{2 \pi r} \over T}$
  62
  63    $f = {1 \over T}, \quad T = {1 \over f}$
  64
  65    $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
  66
  67    $\Sigma F, a$ towards centre, $v$ tangential
  68
  69    $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
  70
  71    \includegraphics[height=4cm]{graphics/circ-forces.png}
  72
  73% -----------------------
  74  \subsection*{Vertical circular motion}
  75
  76    $T =$ tension, e.g. circular pendulum
  77
  78    $T+mg = {{mv^2}\over r}$ at highest point
  79
  80    $T-mg = {{mv^2} \over r}$ at lowest point
  81
  82% -----------------------
  83  \subsection*{Projectile motion}
  84    \begin{itemize}
  85      \item{horizontal component of velocity is constant if no air resistance}
  86      \item{vertical component affected by gravity: $a_y = -g$}
  87    \end{itemize}
  88
  89    \begin{align*}
  90      v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\
  91      h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\
  92      x=ut\cos\theta \tag{$\Delta x$ at $t$} \\
  93      y=ut \sin \theta-{1 \over 2}gt^2 \tag{height at $t$} \\
  94      t={{2u\sin\theta}\over g} \tag{time of flight}\\
  95      d={v^2 \over g}\sin \theta \tag{horiz. range} \\
  96    \end{align*}
  97
  98    \includegraphics[height=3.2cm]{graphics/projectile-motion.png}
  99
 100% -----------------------
 101  \subsection*{Pulley-mass system}
 102
 103    $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
 104
 105    $\Sigma F = m_2g-m_1g=\Sigma ma$ (solve)
 106
 107% -----------------------
 108  \subsection*{Graphs}
 109    \begin{itemize}
 110      \item{Force-time: $A=\Delta \rho$}
 111      \item{Force-disp: $A=W$}
 112      \item{Force-ext: $m=k,\quad A=E_{spr}$}
 113      \item{Force-dist: $A=\Delta \operatorname{gpe}$}
 114      \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
 115    \end{itemize}
 116
 117% -----------------------
 118  \subsection*{Hooke's law}
 119
 120  $F=-kx$
 121
 122  $\text{elastic potential energy} = {1 \over 2}kx^2$
 123
 124  $x={2mg \over k}$
 125
 126% -----------------------
 127  \subsection*{Motion equations}
 128
 129    \begin{tabular}{ l r }
 130      & no \\
 131      $v=u+at$ & $x$ \\
 132      $x = {1 \over 2}(v+u)t$ & $a$ \\
 133      $x=ut+{1 \over 2}at^2$ & $v$ \\
 134      $x=vt-{1 \over 2}at^2$ & $u$ \\
 135      $v^2=u^2+2ax$ & $t$ \\
 136    \end{tabular}
 137
 138% -----------------------
 139  \subsection*{Momentum}
 140
 141    $\rho = mv$
 142
 143    $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
 144
 145    $\Sigma mv_0=\Sigma mv_1$ (conservation)
 146
 147    $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
 148
 149    $n$-body collisions: $\rho$ of each body is independent
 150
 151% ++++++++++++++++++++++
 152\section{Relativity}
 153
 154  \subsection*{Postulates}
 155    1. Laws of physics are constant in all intertial reference frames
 156
 157    2. Speed of light $c$ is the same to all observers (Michelson-Morley)
 158
 159    $\therefore \, t$ must dilate as speed changes
 160
 161    {\bf Inertial reference frame} $a=0$
 162
 163    {\bf Proper time $t_0$ $\vert$ length $l_0$} measured by observer in same frame as events
 164
 165% -----------------------
 166  \subsection*{Lorentz factor}
 167
 168    $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$
 169
 170    $t=t_0 \gamma$ ($t$ longer in moving frame)
 171
 172    $l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame)
 173
 174    $m=m_0 \gamma$ (mass dilation)
 175
 176    $$v = c\sqrt{1-{1 \over \gamma^2}}$$
 177
 178% -----------------------
 179  \subsection*{Energy and work}
 180
 181    $E_0 = mc^2$ (rest)
 182
 183    $E_{total} = E_K + E_{rest} = \gamma mc^2$
 184
 185    $E_K = (\gamma 1)mc^2$
 186
 187    $W = \Delta E = \Delta mc^2$
 188
 189% -----------------------
 190  \subsection*{Relativistic momentum}
 191
 192    $$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$
 193
 194    $\rho \rightarrow \infty$ as $v \rightarrow c$
 195
 196    $v=c$ is impossible (requires $E=\infty$)
 197
 198    $$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
 199
 200% -----------------------
 201  \subsection*{High-altitude muons}
 202    \begin{itemize}
 203      {\item $t$ dilation more muons reach Earth than expected}
 204      {\item normal half-life $2.2 \operatorname{\mu s}$ in stationary frame, $> 2.2 \operatorname{\mu s}$ observed from Earth}
 205    \end{itemize}
 206
 207% +++++++++++++++++++++++
 208\section{Fields and power}
 209
 210  \subsection*{Non-contact forces}
 211    \begin{itemize}
 212      {\item electric fields (dipoles \& monopoles)}
 213      {\item magnetic fields (dipoles only)}
 214      {\item gravitational fields (monopoles only)}
 215    \end{itemize}
 216
 217    \vspace{1em}
 218
 219    \begin{itemize}
 220      \item monopoles: lines towards centre
 221      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (or perpendicular to wire)
 222      \item closer field lines means larger force
 223      \item dot: out of page, cross: into page
 224      \item +ve corresponds to N pole
 225    \end{itemize}
 226
 227    \includegraphics[height=2cm]{graphics/field-lines.png}
 228    % \includegraphics[height=2cm]{graphics/bar-magnet-fields-rotated.png}
 229
 230% -----------------------
 231  \subsection*{Gravity}
 232
 233    \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]
 234    \[g={F_g \over m_2}=G{m_{1} \over r^2}\tag{field of $m_1$}\]
 235    \[E_g = mg \Delta h\tag{gpe}\]
 236    \[W = \Delta E_g = Fx\tag{work}\]
 237    \[w=m(g-a) \tag{app. weight}\]
 238
 239    % \columnbreak
 240
 241% -----------------------
 242  \subsection*{Satellites}
 243
 244    \[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
 245
 246    \[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\]
 247
 248    \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
 249
 250% -----------------------
 251  \subsection*{Magnetic fields}
 252    \begin{itemize}
 253      \item field strength $B$ measured in tesla
 254      \item magnetic flux $\Phi$ measured in weber
 255      \item charge $q$ measured in coulombs
 256      \item emf $\mathcal{E}$ measured in volts
 257    \end{itemize}
 258
 259    % \[{E_1 \over E_2}={r_1 \over r_2}^2\]
 260
 261    \[F=qvB\tag{$F$ on moving $q$}\]
 262    \[F=IlB\tag{$F$ of $B$ on $I$}\]
 263    \[B={mv \over qr}\tag{field strength on e-}\]
 264    \[r={mv \over qB} \tag{radius of $q$ in $B$}\]
 265
 266    if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
 267
 268% -----------------------
 269  \subsection*{Electric fields}
 270
 271    \[F=qE \tag{$E$ = strength} \]
 272    \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
 273    \[E=k{q \over r^2} \tag{field on point charge} \]
 274    \[E={V \over d} \tag{field between plates}\]
 275    \[F=BInl \tag{force on a coil} \]
 276    \[\Phi = B_{\perp}A\tag{magnetic flux} \]
 277    \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
 278    \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
 279
 280    \textbf{Lenz's law:}  $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
 281    (emf creates $I$ with associated field that opposes $\Delta \phi$)
 282
 283    \textbf{Eddy currents:} counter movement within a field
 284
 285    \textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
 286
 287    \includegraphics[height=2cm]{graphics/slap-2.jpeg}
 288    \includegraphics[height=3cm]{graphics/grip.png}
 289
 290    % \textbf{Right hand slap:} $B \perp I \perp F$ \\
 291    % ($I$ = thumb)
 292
 293    \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
 294
 295    \textbf{Transformers:} core strengthens \& focuses $\Phi$
 296
 297% -----------------------
 298  \subsection*{Particle acceleration}
 299
 300    $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
 301
 302    e- accelerated with $x$ V is given $x$ eV
 303
 304    \[W={1\over2}mv^2=qV \tag{field or points}\]
 305    \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
 306
 307
 308% -----------------------
 309  \subsection*{Power transmission}
 310
 311    % \begin{align*}
 312      \[V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \]
 313      \[P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \]
 314      \[V_{\operatorname{loss}}=IR \]
 315    % \end{align*}
 316
 317    Use high-$V$ side for correct $|V_{drop}|$
 318
 319    \begin{itemize}
 320      {\item Parallel $V$ is constant}
 321      {\item Series $V$ shared within branch}
 322    \end{itemize}
 323
 324    \includegraphics[height=4cm]{graphics/ac-generator.png}
 325
 326% -----------------------
 327  \subsection*{Motors}
 328% \begin{wrapfigure}{r}{-0.1\textwidth}
 329
 330    \includegraphics[height=4cm]{graphics/dc-motor-2.png}
 331    \includegraphics[height=3cm]{graphics/ac-motor.png} \\
 332
 333    Force on current-carying wire, not copper \\
 334    $F=0$ for front & back of coil (parallel) \\
 335    Any angle $> 0$ will produce force \\
 336% \end{wrapfigure}
 337    \textbf{DC:} split ring (two halves)
 338
 339% \begin{wrapfigure}{r}{0.3\textwidth}
 340
 341% \end{wrapfigure}
 342    \textbf{AC:} slip ring (separate rings with constant contact)
 343
 344% \pagebreak
 345
 346% +++++++++++++++++++++++
 347\section{Waves}
 348
 349  \textbf{nodes:} fixed on graph \\
 350  \textbf{amplitude:} max disp. from $y=0$ \\
 351  \textbf{rarefactions} and \textbf{compressions} \\
 352  \textbf{mechanical:} transfer of energy without net transfer of matter \\
 353
 354
 355  \textbf{Longitudinal (motion $||$ wave)}
 356  \includegraphics[width=6cm]{graphics/longitudinal-waves.png}
 357
 358  \textbf{Transverse (motion $\perp$ wave)}
 359  \includegraphics[width=6cm]{graphics/transverse-waves.png}
 360
 361  % -----------------------
 362  $T={1 \over f}\quad$(period: time for one cycle)
 363  $v=f \lambda \quad$(speed: displacement / sec)
 364
 365  % -----------------------
 366  \subsection*{Doppler effect}
 367
 368  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" $\lambda$).
 369
 370  % -----------------------
 371  \subsection*{Interference}
 372
 373  \textbf{Standing waves} - constructive int. at resonant freq
 374
 375  \textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
 376
 377  \textbf{Incoherent} - e.g. incandescent bulb
 378
 379
 380  % -----------------------
 381  \subsection*{Harmonics}
 382
 383  \(\lambda = {{al} \div n}\quad\) (\(\lambda\) for \(n^{th}\) harmonic)\\
 384  \(f = {nv \div al}\quad\) (\(f\) for \(n_{th}\) harmonic at length
 385  \(l\) and speed \(v\)) \\
 386  where \(a=2\) for antinodes at both ends, \(a=4\) for antinodes at one end
 387
 388  % -----------------------
 389  \subsection*{Polarisation}
 390  \includegraphics[height=3.5cm]{graphics/polarisation.png}
 391
 392  % -----------------------
 393  \subsection*{Diffraction}
 394  \includegraphics[width=6cm]{graphics/diffraction.jpg}
 395  \includegraphics[width=6cm]{graphics/diffraction-2.png}
 396  \begin{itemize}
 397    % \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
 398    \item Constructive: \(pd = n\lambda, n \in \mathbb{Z}\)
 399    \item Destructive: \(pd = (n-{1 \over 2})\lambda, n \in \mathbb{Z}\)
 400    \item Path difference: \(\Delta x = {{\lambda l }\over d}\) where \\
 401    % \(\Delta x\) = fringe spacing \\
 402    \(l\) = distance from source to observer\\
 403    \(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
 404    \item significant diffraction when ${\lambda \over \Delta x} \ge 1$
 405    \item diffraction creates distortion (electron $>$ optical microscopes)
 406  \end{itemize}
 407
 408
 409
 410  % -----------------------
 411  \subsection*{Refraction}
 412  \includegraphics[height=3.5cm]{graphics/refraction.png}
 413
 414  When a medium changes character, energy is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}
 415
 416  angle of incidence $\theta_i =$ angle of reflection $\theta_r$
 417
 418  Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$
 419
 420  Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
 421
 422
 423% +++++++++++++++++++++++
 424\section{Light and Matter}
 425
 426  % -----------------------
 427  \subsection*{Planck's equation}
 428
 429  \[ f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c \]
 430  \[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
 431  \[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]
 432
 433  \subsection*{Force of electrons}
 434  \[ F={2P_{\text{in}}\over c} \]
 435  % \begin{align*}
 436    \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
 437    \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
 438    % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
 439  % \end{align*}
 440
 441  \subsection*{Photoelectric effect}
 442
 443  \begin{itemize}
 444    \item $V_{\operatorname{supply}}$ does not affect photocurrent
 445    \item $V_{\operatorname{sup}} > 0$: e- attracted to collector anode
 446    \item $V_{\operatorname{sup}} < 0$: attracted to illuminated cathode, $I\rightarrow 0$
 447    \item $v$ of depends on ionisation energy (shell)
 448    \item max current depends on intensity
 449  \end{itemize}
 450
 451  \textbf{Threshold frequency $f_0$}
 452
 453  Minimum $f$ for photoelectrons to be ejected. $x$-intercept of frequency vs $E_K$ graph. if $f < f_0$, no photoelectrons are detected.
 454
 455  \textbf{Work function $\phi$}
 456
 457  Minimum $E$ required to release photoelectrons. Magnitude of $y$-intercept of $f$ vs $E_K$ graph. $\phi$ is determined by strength of bonding. Units: eV or J.
 458
 459  $\phi=hf_0$
 460
 461  \textbf{Kinetic energy}
 462
 463  E_{\operatorname{k-max}}=hf - \phi
 464
 465  voltage in circuit or stopping voltage = max $E_K$ in eV \\
 466  $E_K = x$-int of $V\cdot I$ graph (in eV) \\
 467  dashed line below $E_K=0$ ($E_K < 0$ is impossible)
 468
 469
 470  \textbf{Stopping potential $V$ for min $I$}
 471
 472  $V=h_{\text{eV}}(f-f_0)$
 473
 474  \subsection*{De Broglie's theory}
 475
 476  \[ \lambda = {h \over \rho} = {h \over mv} \]
 477  \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
 478  \begin{itemize}
 479    \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
 480    \item confirmed by similar e- and x-ray diff patterns
 481  \end{itemize}
 482
 483  \subsection*{X-ray electron interaction}
 484
 485  \begin{itemize}
 486    \item e- is only stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
 487    \item rearranging this, $2\pi r = n{h \over mv} = n \lambda$ (circumference)
 488    \item if $2\pi r \ne n{h \over mv}$, no standing wave
 489    \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
 490    \item calculating $h$: $\lambda = {h \over \rho}$
 491  \end{itemize}
 492
 493  \subsection*{Spectral analysis}
 494
 495  \begin{itemize}
 496    \item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
 497    \item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
 498    \item Ionisation energy - min $E$ required to remove e-
 499    \item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
 500    \item No. of lines - include all possible states
 501  \end{itemize}
 502
 503  \subsection*{Uncertainty principle}
 504
 505  measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
 506
 507  \subsection*{Wave-particle duaity}
 508
 509  \subsubsection*{wave model}
 510  \begin{itemize}
 511    \item cannot explain photoelectric effect
 512    \item $f$ is irrelevant to photocurrent
 513    \item predicts delay between incidence and ejection
 514    \item speed depends on medium
 515  \end{itemize}
 516
 517  \subsubsection*{particle model}
 518
 519  \begin{itemize}
 520    \item explains photoelectric effect
 521    \item rate of photoelectron release $\propto$ intensity
 522    \item no time delay - one photon releases one electron
 523    \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
 524    \item light exerts force
 525    \item light bent by gravity
 526    \item quantised energy
 527  \end{itemize}
 528
 529  % +++++++++++++++++++++++
 530  \section{Experimental \\ design}
 531
 532  \textbf{Absolute uncertainty} $\Delta$ \\
 533  (same units as quantity)
 534  \[ \Delta(m) = {{\mathcal{E}(m)} \over 100} \cdot m \]
 535  \[ (A \pm \Delta A) + (B \pm \Delta A) = (A+B) \pm (\Delta A + \Delta B) \]
 536  \[ (A \pm \Delta A) - (B \pm \Delta A) = (A-B) \pm (\Delta A + \Delta B) \]
 537  \[ c(A \pm \Delta A) = cA \pm c \Delta A \]
 538
 539  \textbf{Relative uncertainty} $\mathcal{E}$ (unitless)
 540  \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100 \]
 541  \[ (A \pm \mathcal{E} A) \cdot (B \pm \mathcal{E} B) = (A \cdot B) \pm (\mathcal{E} A + \mathcal{E} B) \]
 542  \[ (A \pm \mathcal{E} A) \div (B \pm \mathcal{E} B) = (A \div B) \pm (\mathcal{E} A + \mathcal{E} B) \]
 543  \[ (A \pm \mathcal{E} A)^n = (A^n \pm n \mathcal{E} A) \]
 544  \[ c(A \pm \mathcal{E} A)=cA \pm \mathcal{E} A \]
 545
 546  Uncertainty of a measurement is $1 \over 2$ the smallest division
 547
 548  \textbf{Precision} - concordance of values \\
 549  \textbf{Accuracy} - closeness to actual value\\
 550  \textbf{Random errors} - unpredictable, reduced by more tests \\
 551  \textbf{Systematic errors} - not reduced by more tests \\
 552  \textbf{Uncertainty} - margin of potential error \\
 553  \textbf{Error} - actual difference \\
 554  \textbf{Hypothesis} - can be tested experimentally \\
 555  \textbf{Model} - evidence-based but indirect representation
 556
 557\end{multicols}
 558
 559\end{document}