physics / final.texon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (3394f15)
   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    $F = m g \sin\theta F_{frict} = m a$
  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, but not necessarily 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% -----------------------
  46  \subsection*{Work and energy}
  47
  48    $W=Fx=\Delta \Sigma E$ (work)
  49
  50    $E_K = {1 \over 2}mv^2$ (kinetic)
  51
  52    $E_G = mgh$ (potential)
  53
  54    $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
  55
  56% -----------------------
  57  \subsection*{Horizontal circular motion}
  58
  59    $v = {{2 \pi r} \over T}$
  60
  61    $f = {1 \over T}, \quad T = {1 \over f}$
  62
  63    $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
  64
  65    $\Sigma F, a$ towards centre, $v$ tangential
  66
  67    $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
  68
  69    \includegraphics[height=4cm]{graphics/circ-forces.png}
  70
  71% -----------------------
  72  \subsection*{Vertical circular motion}
  73
  74    $T =$ tension, e.g. circular pendulum
  75
  76    $T+mg = {{mv^2}\over r}$ at highest point
  77
  78    $T-mg = {{mv^2} \over r}$ at lowest point
  79
  80% -----------------------
  81  \subsection*{Projectile motion}
  82    \begin{itemize}
  83      \item{horizontal component of velocity is constant if no air resistance}
  84      \item{vertical component affected by gravity: $a_y = -g$}
  85    \end{itemize}
  86
  87    \begin{align*}
  88      v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\
  89      h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\
  90      x=ut\cos\theta \tag{$\Delta x$ at $t$} \\
  91      y=ut \sin \theta-{1 \over 2}gt^2 \tag{height at $t$} \\
  92      t={{2u\sin\theta}\over g} \tag{time of flight}\\
  93      d={v^2 \over g}\sin \theta \tag{horiz. range} \\
  94    \end{align*}
  95
  96    \includegraphics[height=3.2cm]{graphics/projectile-motion.png}
  97
  98% -----------------------
  99  \subsection*{Pulley-mass system}
 100
 101    $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
 102
 103    $\Sigma F = m_2g-m_1g=\Sigma ma$ (solve)
 104
 105% -----------------------
 106  \subsection*{Graphs}
 107    \begin{itemize}
 108      \item{Force-time: $A=\Delta \rho$}
 109      \item{Force-disp: $A=W$}
 110      \item{Force-ext: $m=k,\quad A=E_{spr}$}
 111      \item{Force-dist: $A=\Delta \operatorname{gpe}$}
 112      \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
 113    \end{itemize}
 114
 115% -----------------------
 116  \subsection*{Hooke's law}
 117
 118  $F=-kx$
 119
 120  $E_{elastic} = {1 \over 2}kx^2$
 121
 122% -----------------------
 123  \subsection*{Motion equations}
 124
 125    \begin{tabular}{ l r }
 126      $v=u+at$ & $x$ \\
 127      $x = {1 \over 2}(v+u)t$ & $a$ \\
 128      $x=ut+{1 \over 2}at^2$ & $v$ \\
 129      $x=vt-{1 \over 2}at^2$ & $u$ \\
 130      $v^2=u^2+2ax$ & $t$ \\
 131    \end{tabular}
 132
 133% -----------------------
 134  \subsection*{Momentum}
 135
 136    $\rho = mv$
 137
 138    $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
 139
 140    $\Sigma mv_0=\Sigma mv_1$ (conservation)
 141
 142    $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
 143
 144    $n$-body collisions: $\rho$ of each body is independent
 145
 146% ++++++++++++++++++++++
 147\section{Relativity}
 148
 149  \subsection*{Postulates}
 150    1. Laws of physics are constant in all intertial reference frames
 151
 152    2. Speed of light $c$ is the same to all observers (Michelson-Morley)
 153
 154    $\therefore , t$ must dilate as speed changes
 155
 156    {\bf Inertial reference frame} $a=0$
 157
 158    {\bf Proper time $t_0$ $\vert$ length $l_0$} measured by observer in same frame as events
 159
 160% -----------------------
 161  \subsection*{Lorentz factor}
 162
 163    $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$
 164
 165    $t=t_0 \gamma$ ($t$ longer in moving frame)
 166
 167    $l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame)
 168
 169    $m=m_0 \gamma$ (mass dilation)
 170
 171    $$v = c\sqrt{1-{1 \over \gamma^2}}$$
 172
 173% -----------------------
 174  \subsection*{Energy and work}
 175
 176    $E_0 = mc^2$ (rest)
 177
 178    $E_{total} = E_K + E_{rest} = \gamma mc^2$
 179
 180    $E_K = (\gamma 1)mc^2$
 181
 182    $W = \Delta E = \Delta mc^2$
 183
 184% -----------------------
 185  \subsection*{Relativistic momentum}
 186
 187    $$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$
 188
 189    $\rho \rightarrow \infty$ as $v \rightarrow c$
 190
 191    $v=c$ is impossible (requires $E=\infty$)
 192
 193    $$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
 194
 195% -----------------------
 196  \subsection*{High-altitude muons}
 197    \begin{itemize}
 198      {\item $t$ dilation more muons reach Earth than expected}
 199      {\item normal half-life $2.2 \operatorname{\mu s}$ in stationary frame, $> 2.2 \operatorname{\mu s}$ observed from Earth}
 200    \end{itemize}
 201
 202% +++++++++++++++++++++++
 203\section{Fields and power}
 204
 205  \subsection*{Non-contact forces}
 206    \begin{itemize}
 207      {\item electric fields (dipoles \& monopoles)}
 208      {\item magnetic fields (dipoles only)}
 209      {\item gravitational fields (monopoles only)}
 210    \end{itemize}
 211
 212    \vspace{1em}
 213
 214    \begin{itemize}
 215      \item monopoles: lines towards centre
 216      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (or perpendicular to wire)
 217      \item closer field lines means larger force
 218      \item dot: out of page, cross: into page
 219      \item +ve corresponds to N pole
 220    \end{itemize}
 221
 222    \includegraphics[height=2cm]{graphics/field-lines.png}
 223
 224% -----------------------
 225  \subsection*{Gravity}
 226
 227    \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]
 228    \[g={F_g \over m_2}=G{m_{1} \over r^2}\tag{field of $m_1$}\]
 229    \[E_g = mg \Delta h\tag{gpe}\]
 230    \[W = \Delta E_g = Fx\tag{work}\]
 231    \[w=m(g-a) \tag{app. weight}\]
 232
 233    % \columnbreak
 234
 235% -----------------------
 236  \subsection*{Satellites}
 237
 238    \[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
 239
 240    \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
 241
 242    \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
 243
 244% -----------------------
 245  \subsection*{Magnetic fields}
 246    \begin{itemize}
 247      \item field strength $B$ measured in tesla
 248      \item magnetic flux $\Phi$ measured in weber
 249      \item charge $q$ measured in coulombs
 250      \item emf $\mathcal{E}$ measured in volts
 251    \end{itemize}
 252
 253    % \[{E_1 \over E_2}={r_1 \over r_2}^2\]
 254
 255    \[F=qvB\tag{$F$ on moving $q$}\]
 256    \[F=IlB\tag{$F$ of $B$ on $I$}\]
 257    \[r={mv \over qB} \tag{radius of $q$ in $B$}\]
 258
 259    if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
 260
 261% -----------------------
 262  \subsection*{Electric fields}
 263
 264    \[F=qE \tag{$E$ = strength} \]
 265    \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
 266    \[E=k{q \over r^2} \tag{field on point charge} \]
 267    \[E={V \over d} \tag{field between plates}\]
 268    \[F=BInl \tag{force on a coil} \]
 269    \[\Phi = B_{\perp}A\tag{magnetic flux} \]
 270    \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
 271    \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
 272
 273    \textbf{Lenz's law:}  $I_{\operatorname{emf}}$ opposes $\Delta \Phi$
 274
 275    \textbf{Eddy currents:} counter movement within a field
 276
 277    \textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
 278
 279    \textbf{Right hand slap:} $B \perp I \perp F$
 280
 281    \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
 282
 283    \textbf{Transformers:} core strengthens \& focuses $\Phi$
 284
 285% -----------------------
 286  \subsection*{Particle acceleration}
 287
 288    $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
 289
 290    eaccelerated with $x$ V is given $x$ eV
 291
 292    \[W={1\over2}mv^2=qV \tag{field or points}\]
 293    \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
 294
 295
 296% -----------------------
 297  \subsection*{Power transmission}
 298
 299    % \begin{align*}
 300      $$V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}}$$
 301      P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \\
 302      V_{\operatorname{loss}}=IR
 303    % \end{align*}
 304
 305    Use high-$V$ side for correct $|V_{drop}|$
 306
 307    \begin{itemize}
 308      {\item Parallel $V$ is constant}
 309      {\item Series $V$ shared within branch}
 310    \end{itemize}
 311
 312    \includegraphics[height=4cm]{graphics/ac-generator.png}
 313
 314% -----------------------
 315  \subsection*{Motors}
 316% \begin{wrapfigure}{r}{-0.1\textwidth}
 317
 318    \includegraphics[height=4cm]{graphics/dc-motor-2.png}
 319      \includegraphics[height=3cm]{graphics/ac-motor.png} \\
 320% \end{wrapfigure}
 321    \textbf{DC:} split ring (two halves)
 322
 323% \begin{wrapfigure}{r}{0.3\textwidth}
 324
 325% \end{wrapfigure}
 326    \textbf{AC:} slip ring (separate rings with constant contact)
 327
 328% +++++++++++++++++++++++
 329\section{Waves}
 330
 331  \textbf{nodes:} fixed on graph
 332
 333  \textbf{Longitudinal (motion $||$ wave)}
 334  \includegraphics[height=4cm]{graphics/longitudinal-waves.png}
 335
 336  \textbf{Transverse (motion $\perp$ wave)}
 337  \includegraphics[height=4cm]{graphics/transverse-waves.png}
 338
 339  % -----------------------
 340  \subsection*{Motors}
 341  $T={1 \over f}\quad$(period: time for one cycle)
 342  $v=f \lambda \quad$(speed: displacement per second)
 343
 344  % -----------------------
 345  \subsection*{Doppler effect}
 346  When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to travel than $w_{n-1}$. Hence, $w_n$ reaches the observer sooner than $w_{n-1}$, increasing "apparent" wavelength.
 347
 348  % -----------------------
 349  \subsection*{Interference}
 350  When a medium changes character, energy is reflected, absorbed, and transmitted
 351
 352  % -----------------------
 353  \subsection*{Polarisation}
 354  \includegraphics[height=4cm]{graphics/polarisation.png}
 355
 356  % -----------------------
 357  \subsection*{Refraction}
 358  \includegraphics[height=4cm]{graphics/refraction.png}
 359
 360  Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
 361
 362  Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$
 363
 364  Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
 365
 366% +++++++++++++++++++++++
 367\section{Light and Matter}
 368
 369  % -----------------------
 370  \subsection*{Planck's equation}
 371
 372  f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c
 373
 374  h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s}
 375
 376  1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}
 377
 378  \subsection*{Force of electrons}
 379  F={2P_{\text{in}}\over c}
 380
 381  \text{photons per second}={\text{total energy} \over \text{energy per photon}}={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
 382
 383  \subsection*{Photoelectric effect}
 384
 385  \begin{itemize}
 386    \item $V_{\operatorname{supply}}$ does not affect photocurrent
 387    \item $V_{\operatorname{sup}} > 0$: eattracted to collector anode
 388    \item $V_{\operatorname{sup}} < 0$: attracted to illuminated cathode, $I\rightarrow 0$
 389    \item $v$ of edepends on ionisation energy (shell)
 390    \item max current depends on intensity
 391  \end{itemize}
 392
 393  \textbf{Threshold frequency $f_0$}
 394
 395  Minimum $f$ for photoelectrons to be ejected. $x$-intercept of frequency vs $E_K$ graph. if $f < f_0$, no photoelectrons are detected.
 396
 397  \textbf{Work function $\phi$}
 398
 399  Minimum $E$ required to release photoelectrons. Magnitude of $y$-intercept of frequency vs $E_K$ graph. $\phi$ is determined by strength of bonding.
 400
 401   $\phi=hf_0$
 402
 403  \textbf{Kinetic energy}
 404
 405  E_{\operatorname{k-max}}=hf - \phi
 406
 407  voltage in circuit or stopping voltage = max $E_K$ in eV
 408  equal to $x$-intercept of volts vs current graph (in eV)
 409
 410  \textbf{Stopping potential $V$ for min $I$}
 411
 412  $V=h_{\text{eV}}(f-f_0)$
 413
 414  \subsection*{De Broglie's theory}
 415
 416  $\lambda = {h \over \rho} = {h \over mv}$
 417  $\rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c$
 418  \begin{itemize}
 419    \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
 420    \item confirmed by similar e- and x-ray diff patterns
 421  \end{itemize}
 422
 423  \subsection*{X-ray electron interaction}
 424
 425  \begin{itemize}
 426    \item e- is only stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
 427    \item rearranging this, $2\pi r = n{h \over mv} = n \lambda$ (circumference)
 428    \item if $2\pi r \ne n{h \over mv}$, no standing wave
 429    \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
 430    \item calculating $h$: $\lambda = {h \over \rho}$
 431  \end{itemize}
 432
 433  \subsection*{Spectral analysis}
 434
 435  \begin{itemize}
 436    $n\item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
 437    $n\item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
 438    $n\item Ionisation energy - min $E$ required to remove e-
 439    $n\item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
 440  \end{itemize}
 441
 442  \subsection{Indeterminancy principle}
 443
 444  measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
 445
 446  \subsection{Wave-particle duaity}
 447
 448  wave model:
 449
 450  \item cannot explain photoelectric effect
 451  \item $f$ is irrelevant to photocurrent
 452  \item predicts delay between incidence and ejection
 453  \item speed depends on medium
 454
 455  particle model:
 456
 457  \item explains photoelectric effect
 458  \item rate of photoelectron release $\propto$ intensity
 459  \item no time delay - one photon releases one electron
 460  \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
 461  \item light exerts force
 462  \item light bent by gravity
 463
 464
 465
 466
 467
 468
 469
 470
 471\end{multicols}
 472\end{document}