fabaceff42984213d4aeeb6c305c14811ab84f8a
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 $W=Fs=Fs \cos \theta=\Delta \Sigma E$
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 $E_K_{\text{bottom}}=E_K_{\text{top}}+mgh$
85
86% -----------------------
87 \subsection*{Projectile motion}
88 \begin{itemize}
89 \item $v_x$ is constant: $v_x = {s \over t}$
90 \item use suvat to find $t$ from $y$-component
91 \item vertical component gravity: $a_y = -g$
92 \end{itemize}
93
94 % \begin{align*}
95 $v=\sqrt{v^2_x + v^2_y}$ \hfill vectors \\
96 $h={{u^2\sin \theta ^2}\over 2g}$ \hfill max height \\
97 $x=ut\cos\theta$ \hfill $\Delta x$ at $t$ \\
98 $y=ut \sin \theta-{1 \over 2}gt^2$ \hfill height at $t$ \\
99 $t={{2u\sin\theta}\over g}$ \hfill time of flight \\
100 $d={v^2 \over g}\sin \theta$ \hfill horiz. range \\
101 % \end{align*}
102
103 \includegraphics[height=3.2cm]{graphics/projectile-motion.png}
104
105% -----------------------
106 \subsection*{Pulley-mass system}
107
108 $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
109
110 $\Sigma F = m_2g-m_1g=\Sigma ma$ (solve)
111
112% -----------------------
113 \subsection*{Graphs}
114 \begin{itemize}
115 \item{Force-time: $A=\Delta \rho$}
116 \item{Force-disp: $A=W$}
117 \item{Force-ext: $m=k,\quad A=E_{spr}$}
118 \item{Force-dist: $A=\Delta \operatorname{gpe}$}
119 \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
120 \end{itemize}
121
122% -----------------------
123 \subsection*{Hooke's law}
124
125 $F=-kx$ (intercepts origin)
126
127 $\text{elastic potential energy} = {1 \over 2}kx^2$
128
129 $x={2mg \over k}$
130
131 Vertical: $\Delta E = {1 \over 2}kx^2 + mgh
132
133% -----------------------
134 \subsection*{Motion equations}
135
136 \begin{tabular}{ l r }
137 & no \\
138 $v=u+at$ & $x$ \\
139 $x = {1 \over 2}(v+u)t$ & $a$ \\
140 $x=ut+{1 \over 2}at^2$ & $v$ \\
141 $x=vt-{1 \over 2}at^2$ & $u$ \\
142 $v^2=u^2+2ax$ & $t$ \\
143 \end{tabular}
144
145% -----------------------
146 \subsection*{Momentum}
147
148 $\rho = mv$
149
150 $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
151
152 $\Sigma (mv_0)=(\Sigma m)v_1$ (conservation)
153
154 % $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
155
156 % $\Sigma E_K = \Sigma ({1 \over 2} m v^2) = {1 \over 2} (\Sigma m)v_f$
157
158 if elastic:
159 $$\sum _{i{\mathop {=}}1}^{n}E_K (i)=\sum _{i{\mathop {=}}1}^{n}({1 \over 2}m_i v_{i0}^2)={1 \over 2}\sum _{i{\mathop {=}}1}^{n}(m_i) v_f^2$$
160
161 % $n$-body collisions: $\rho$ of each body is independent
162
163% ++++++++++++++++++++++
164\section{Relativity}
165
166 \subsection*{Postulates}
167 1. Laws of physics are constant in all intertial reference frames
168
169 2. Speed of light $c$ is the same to all observers (Michelson-Morley)
170
171 $\therefore \, t$ must dilate as speed changes
172
173 {\bf high-altitude particles:} $t$ dilation means more particles reach Earth than expected (half-life greater when obs. from Earth)
174
175 {\bf Inertial reference frame} $a=0$
176
177 {\bf Proper time $t_0$ $\vert$ length $l_0$} measured by observer in same frame as events
178
179% -----------------------
180 \subsection*{Lorentz factor}
181
182 $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}, \quad v = c\sqrt{1-{1 \over \gamma^2}}$$
183
184 $t=t_0 \gamma$ ($t$ longer in moving frame)
185
186 $l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame)
187
188 $m=m_0 \gamma$ (mass dilation)
189
190% -----------------------
191 \subsection*{Energy and work}
192
193 $E_{\text{rest}} = mc^2, \quad E_K = (\gamma-1)mc^2$
194
195 $E_{\text{total}} = E_K + E_{\text{rest}} = \gamma mc^2$
196
197 $W = \Delta E = \Delta mc^2=(\gamma-1)m_{\text{rest}} c^2$
198
199% -----------------------
200 \subsection*{Relativistic momentum}
201
202 $$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$
203
204 $\rho \rightarrow \infty$ as $v \rightarrow c$
205
206 $v=c$ is impossible (requires $E=\infty$)
207
208 $$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
209
210% -----------------------
211
212% +++++++++++++++++++++++
213\section{Fields and power}
214
215 \subsection*{Non-contact forces}
216 \begin{itemize}
217 {\item electric (dipoles \& monopoles)}
218 {\item magnetic (dipoles only)}
219 {\item gravitational (monopoles only, $F_g=0$ at mid, attractive only)}
220 \end{itemize}
221
222 \vspace{1em}
223
224 \begin{itemize}
225 \item monopoles: lines towards centre
226 \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (two magnets) or $\rightarrow$ N (single)
227 \item closer field lines means larger force
228 \item dot: out of page, cross: into page
229 \item +ve corresponds to N pole
230 \item Inv. sq. ${E_1 \over E_2} = ({r_2 \over r_1})^2$
231 \end{itemize}
232
233 \includegraphics[height=2cm]{graphics/field-lines.png}
234 % \includegraphics[height=2cm]{graphics/bar-magnet-fields-rotated.png}
235
236% -----------------------
237 \subsection*{Gravity}
238
239 \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]
240 \[g={F_g \over m_2}=G{m_{1} \over r^2}\tag{field of $m_1$}\]
241 \[E_g = mg \Delta h\tag{gpe}\]
242 \[W = \Delta E_g = Fx\tag{work}\]
243 \[w=m(g-a) \tag{app. weight}\]
244
245 % \columnbreak
246
247% -----------------------
248 \subsection*{Satellites}
249
250 \[v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
251
252 \[T={\sqrt{4 \pi^2 r^3 \over {GM}}}\tag{period}\]
253
254 \[r = \sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
255
256% -----------------------
257 \subsection*{Magnetic fields}
258 \begin{itemize}
259 \item field strength $B$ measured in tesla
260 \item magnetic flux $\Phi$ measured in weber
261 \item charge $q$ measured in coulombs
262 \item emf $\mathcal{E}$ measured in volts
263 \end{itemize}
264
265 % \[{E_1 \over E_2}={r_1 \over r_2}^2\]
266
267 \[F=qvB\tag{$F$ on moving $q$}\]
268 \[F=IlB\tag{$F$ of $B$ on $I$}\]
269 \[B={mv \over qr}\tag{field strength on e-}\]
270 \[r={mv \over qB} \tag{radius of $q$ in $B$}\]
271
272 if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
273
274% -----------------------
275 \subsection*{Electric fields}
276
277 \[F=qE(=ma) \tag{strength} \]
278 \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
279 \[E=k{q \over r^2} \tag{field on point charge} \]
280 \[E={V \over d} \tag{field between plates}\]
281 \[F=BInl \tag{force on a coil} \]
282 \[\Phi = B_{\perp}A\tag{magnetic flux} \]
283 \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} = Blv\tag{induced emf} \]
284 \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
285
286 \textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
287 (emf creates $I$ with associated field that opposes $\Delta \Phi$)
288
289 \textbf{Eddy currents:} counter movement within a field
290
291 \textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
292
293 \includegraphics[height=2cm]{graphics/slap-2.jpeg}
294 \includegraphics[height=3cm]{graphics/grip.png}
295
296 % \textbf{Right hand slap:} $B \perp I \perp F$ \\
297 % ($I$ = thumb)
298
299 \includegraphics[width=\columnwidth]{graphics/lenz.png}
300
301 \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}.$
302 If $f$ increases, ampl. \& $f$ of $\mathcal{E}$ increase
303
304 \textbf{Xfmr} core strengthens \& focuses $\Phi$
305
306% -----------------------
307 \subsection*{Particle acceleration}
308
309 $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
310
311 e- accelerated with $x$ V is given $x$ eV
312
313 \[W={1\over2}mv^2=qV \tag{field or points}\]
314 \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
315
316 Circular path: $F\perp B \perp v$
317
318% -----------------------
319 \subsection*{Power transmission}
320
321 % \begin{align*}
322 \[V_{\operatorname{rms}}={V_{\operatorname{p}}\over \sqrt{2}}={V_{\operatorname{p\rightarrow p}}\over {2 \sqrt{2}}} \]
323 \[P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \]
324 \[V_{\operatorname{loss}}=IR \]
325 % \end{align*}
326
327 Use high-$V$ side for correct $|V_{drop}|$
328
329 \begin{itemize}
330 {\item Parallel $V$ is constant}
331 {\item Series $V$ shared within branch}
332 \end{itemize}
333
334 \includegraphics[height=4cm]{graphics/ac-generator.png}
335
336% -----------------------
337 \subsection*{Motors}
338% \begin{wrapfigure}{r}{-0.1\textwidth}
339
340 \includegraphics[height=4cm]{graphics/dc-motor-2.png}
341 \includegraphics[height=3cm]{graphics/ac-motor.png} \\
342
343 Force on current-carying wire, not copper \\
344 $F=0$ for front & back of coil (parallel) \\
345 Any angle $> 0$ will produce force \\
346% \end{wrapfigure}
347 \textbf{DC:} split ring (two halves)
348
349% \begin{wrapfigure}{r}{0.3\textwidth}
350
351% \end{wrapfigure}
352 \textbf{AC:} slip ring (separate rings with constant contact)
353
354% \pagebreak
355
356% +++++++++++++++++++++++
357\section{Waves}
358
359 \textbf{nodes:} fixed on graph \\
360 \textbf{amplitude:} max disp. from $y=0$ \\
361 \textbf{rarefactions} and \textbf{compressions} \\
362 \textbf{mechanical:} transfer of energy without net transfer of matter \\
363
364
365 \textbf{Longitudinal (motion $||$ wave)}
366 \includegraphics[width=6cm]{graphics/longitudinal-waves.png}
367
368 \textbf{Transverse (motion $\perp$ wave)}
369 \includegraphics[width=6cm]{graphics/transverse-waves.png}
370
371 % -----------------------
372 $T={1 \over f}\quad$(period: time for one cycle)
373 $v=f \lambda \quad$(speed: displacement / sec)
374 $f={c \over \lambda}\quad\hspace{0.7em}$(for $v=c$)
375
376 % -----------------------
377 \subsection*{Doppler effect}
378
379 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" $\lambda$).
380
381 % -----------------------
382 \subsection*{Interference}
383
384 \includegraphics[width=4.5cm]{graphics/poissons-spot.png} \\
385 Poissons's spot supports wave theory (circular diffraction)
386
387 \textbf{Standing waves} - constructive int. at resonant freq. Rebound from ends.
388
389 \textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
390
391 \textbf{Incoherent} - e.g. incandescent/LED
392
393
394
395
396
397 % -----------------------
398 \subsection*{Harmonics}
399
400 1st harmonic = fundamental
401
402 \textbf{for nodes at both ends:} \\
403 \(\hspace{2em} \lambda = {{2l} \div n}\)
404 \(\hspace{2em} f = {nv \div 2l} \)
405
406 \textbf{for node at one end ($n$ is odd):} \\
407 \(\hspace{2em} \lambda = {{4l} \div n}\)
408 \(\hspace{2em} f = {nv \div 4l} \) \\
409 alternatively, $\lambda = {4l \over {2n-1}}$ where $n\in \mathbb{Z}$ and $n+1$ is the next possible harmonic
410
411
412 % \(a=2\) for nodes at both ends, \\ \(a=4\) for node at one end
413
414 % -----------------------
415 \subsection*{Polarisation}
416 \includegraphics[height=3.5cm]{graphics/polarisation.png} \\
417 Transverse only. Reduces total $A$.
418
419 % -----------------------
420 \subsection*{Diffraction}
421 \includegraphics[width=6cm]{graphics/diffraction.jpg}
422 \includegraphics[width=6cm]{graphics/diffraction-2.png}
423 \begin{itemize}
424 % \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
425 \item Constructive: \(pd = n\lambda, n \in \mathbb{Z}\)
426 \item Destructive: \(pd = (n-{1 \over 2})\lambda, n \in \mathbb{Z}\)
427 \item Path difference: \(\Delta x = {{\lambda l }\over d}\) where \\
428 % \(\Delta x\) = fringe spacing \\
429 \(l\) = distance from source to observer\\
430 \(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
431 \item diffraction $\propto {\lambda \over d}$
432 \item significant diffraction when ${\lambda \over \Delta x} \ge 1$
433 \item diffraction creates distortion (electron $>$ optical microscopes)
434 \end{itemize}
435
436
437 % -----------------------
438 \subsection*{Refraction}
439 \includegraphics[height=3.5cm]{graphics/refraction.png}
440
441 When a medium changes character, light is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}. $\lambda$ changes, not $f$.
442
443 angle of incidence $\theta_i =$ angle of reflection $\theta_r$
444
445 Critical angle $\theta_c = \sin^{-1}{n_2 \over n_1}$
446
447 Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
448
449 ${v_1 \div v_2} = {\sin\theta_1 \div \sin\theta_2}$
450
451 $n_1 v_1 = n_2 v_2$
452
453 $n={c \over v}$
454
455
456% +++++++++++++++++++++++
457\section{Light and Matter}
458
459 % -----------------------
460 \subsection*{Planck's equation}
461
462 \[ \quad E=hf={hc \over \lambda}=\rho c = qV\]
463 \[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
464 \[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]
465
466 \subsection*{De Broglie's theory}
467
468 \[ \lambda = {h \over \rho} = {h \over mv} = {h \over {m \sqrt{2W \over m}}}\]
469 \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
470 \[ v = \sqrt{2E_K \div m} \]
471
472 \begin{itemize}
473 \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
474 \item confirmed by e- and x-ray patterns
475 \end{itemize}
476
477 \subsection*{Force of electrons}
478 \[ F={2P_{\text{in}}\over c} \]
479 % \begin{align*}
480 \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
481 \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
482 % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
483 % \end{align*}
484
485 \subsection*{X-ray electron interaction}
486
487 \begin{itemize}
488 \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ and $r$ is radius of orbit
489 \item $\therefore 2\pi r = n{h \over mv} = n \lambda$ (circumference)
490 \item if $2\pi r \ne n{h \over mv}$, no standing wave
491 \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
492 % \item calculating $h$: $\lambda = {h \over \rho}$
493 \end{itemize}
494
495 \subsection*{Photoelectric effect}
496
497 \begin{itemize}
498 \item $V_{\operatorname{supply}}$ does not affect photocurrent
499 \item $V_{\operatorname{sup}} > 0$: attracted to +ve
500 \item $V_{\operatorname{sup}} < 0$: attracted to -ve, $I\rightarrow 0$
501 \item $v$ of e- depends on shell
502 \item max $I$ (not $V$) depends on intensity
503 \end{itemize}
504
505 \subsubsection*{Threshold frequency $f_0$}
506
507 min $f$ for photoelectron release. if $f < f_0$, no photoelectrons.
508
509 \subsubsection*{Work function $\phi=hf_0$}
510
511 min $E$ for photoelectron release. determined by strength of bonding. Units: eV or J.
512
513 \subsubsection*{Kinetic energy E_K=hf - \phi = qV_0}
514
515
516 $V_0 = E_K$ in eV \\
517 % $E_K = x$-int of $V\cdot I$ graph (in eV) \\
518 dashed line below $E_K=0$
519
520
521 \subsubsection*{Stopping potential $V_0$ for min $I$}
522
523 $$V_0=h_{\text{eV}}(f-f_0)$$
524 Opposes induced photocurrent
525
526 \subsubsection*{Graph features}
527
528 \newcolumntype{b}{>{\hsize=.75\hsize}X}
529\newcolumntype{s}{>{\hsize=.3\hsize}X}
530
531 \begin{tabularx}{\columnwidth}{bbbb}
532\hline
533&$m$&$x$-int&$y$-int \\
534\hline
535\hline
536$f \cdot E_K$ & $h$ & $f_0$ & $-\phi$ \\
537$V \cdot I$ & & $V_0$ & intensity\\
538$f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ &
539\hline
540\end{tabularx}
541
542
543
544 \subsection*{Spectral analysis}
545
546 \begin{itemize}
547 \item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
548 \item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
549 \item Ionisation energy - min $E$ required to remove e-
550 \item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
551 \item No. of lines - include all possible states
552 \end{itemize}
553
554 \subsection*{Uncertainty principle}
555
556 measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
557
558 \subsection*{Wave-particle duality}
559
560 \subsubsection*{wave model}
561 \begin{itemize}
562 \item cannot explain photoelectric effect
563 \item $f$ is irrelevant to photocurrent
564 \item predicts delay between incidence and ejection
565 \item speed depends on medium
566 \item supported by bright spot in centre
567 \item $\lambda = {hc \over E}$
568 \end{itemize}
569
570 \subsubsection*{particle model}
571
572 \begin{itemize}
573 \item explains photoelectric effect
574 \item rate of photoelectron release $\propto$ intensity
575 \item no time delay - one photon releases one electron
576 \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
577 \item light exerts force
578 \item light bent by gravity
579 \item quantised energy
580 \item $\lambda = {h \over \rho}$
581 \end{itemize}
582
583 % +++++++++++++++++++++++
584 \section{Experimental \\ design}
585
586 \textbf{Absolute uncertainty} $\Delta$ \\
587 (same units as quantity)
588 \[ \Delta(m) = {{\mathcal{E}(m)} \over 100} \cdot m \]
589 \[ (A \pm \Delta A) + (B \pm \Delta A) = (A+B) \pm (\Delta A + \Delta B) \]
590 \[ (A \pm \Delta A) - (B \pm \Delta A) = (A-B) \pm (\Delta A + \Delta B) \]
591 \[ c(A \pm \Delta A) = cA \pm c \Delta A \]
592
593 \textbf{Relative uncertainty} $\mathcal{E}$ (unitless)
594 \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100 \]
595 \[ (A \pm \mathcal{E} A) \cdot (B \pm \mathcal{E} B) = (A \cdot B) \pm (\mathcal{E} A + \mathcal{E} B) \]
596 \[ (A \pm \mathcal{E} A) \div (B \pm \mathcal{E} B) = (A \div B) \pm (\mathcal{E} A + \mathcal{E} B) \]
597 \[ (A \pm \mathcal{E} A)^n = (A^n \pm n \mathcal{E} A) \]
598 \[ c(A \pm \mathcal{E} A)=cA \pm \mathcal{E} A \]
599
600 Uncertainty of a measurement is $1 \over 2$ the smallest division
601
602 \textbf{Precision} - concordance of values \\
603 \textbf{Accuracy} - closeness to actual value\\
604 \textbf{Random errors} - unpredictable, reduced by more tests \\
605 \textbf{Systematic errors} - not reduced by more tests \\
606 \textbf{Uncertainty} - margin of potential error \\
607 \textbf{Error} - actual difference \\
608 \textbf{Hypothesis} - can be tested experimentally \\
609 \textbf{Model} - evidence-based but indirect representation
610
611\end{multicols}
612
613\begin{center}
614 \includegraphics[height=2.95cm]{graphics/spectrum.png}
615\end{center}
616
617\end{document}