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[]$\OML/cmm/m/it/10 d \OT1/cmr/m/n/10 = []\OML/cmm/m/it/10 sin^^R$ \OT1/cmr/m/n/10 (hor-i-zon-tal range)
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\documentclass[a4paper]{article}
-
-\usepackage[a4paper]{geometry}
\usepackage{multicol}
\usepackage[cm]{fullpage}
\usepackage{amsmath}
+\usepackage{amssymb}
\setlength{\parindent}{0cm}
\usepackage[nodisplayskipstretch]{setspace}
-\setstretch{1.5}
+\setstretch{1.3}
\usepackage{graphicx}
\usepackage{wrapfig}
+\usepackage{enumitem}
+\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt}
\begin{document}
$y=ut \sin \theta-{1 \over 2}gt^2$ (time of flight)
$d={v^2 \over g}sin \theta$ (horizontal range)
- \includegraphics[height=4cm]{/mnt/andrew/graphics/projectile-motion.png}
+ \includegraphics[height=3.2cm]{/mnt/andrew/graphics/projectile-motion.png}
\subsection*{Pulley-mass system}
\item{Force-time: $A=\Delta \rho$}
\item{Force-disp: $A=W$}
\item{Force-ext: $m=k,\quad A=E_{spr}$}
+ \item{Force-dist: $A=\Delta \operatorname{gpe}$}
+ \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
\end{itemize}
\subsection*{Hooke's law}
\section{Relativity}
+\subsection*{Postulates}
+1. Laws of physics are constant in all intertial reference frames
+
+2. Speed of light $c$ is the same to all observers (Michelson-Morley)
+
+$\therefore , t$ must dilate as speed changes
+
+{\bf Inertial reference frame} - $a=0$
+
+{\bf Proper time $t_0$ $\vert$ length $l_0$} - measured by observer in same frame as events
+
+\subsection*{Lorentz factor}
+
+$$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$
+
+$t=t_0 \gamma$ ($t$ longer in moving frame)
+
+$l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame)
+
+$m=m_0 \gamma$ (mass dilation)
+
+$$v = c\sqrt{1-{1 \over \gamma^2}}$$
+
+\subsection*{Energy and work}
+
+$E_0 = mc^2$ (rest)
+
+$E_{total} = E_K + E_{rest} = \gamma mc^2$
+
+$E_K = (\gamma - 1)mc^2$
+
+$W = \Delta E = \Delta mc^2$
+
+\subsection*{Relativistic momentum}
+
+$$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$
+
+$\rho \rightarrow \infty$ as $v \rightarrow c$
+
+$v=c$ is impossible (requires $E=\infty$)
+
+$$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
+
+\subsection*{Fusion and fission}
+
+$1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
+
+e- accelerated with $x$ V is given $x$ eV
+\subsection*{High-altitude muons}
+\begin{itemize}
+ {\item $t$ dilation - more muons reach Earth than expected}
+ {\item normal half-life is $2.2 \operatorname{\mu s}$ in stationary frame}
+ {\item at $v \approx c$, muons observed from Earth have halflife $> 2.2 \operatorname{\mu s}$}
+ {\item slower time - more time to travel, so muons reach surface}
+\end{itemize}
+
+\section{Fields and power}
+
+
+\subsection*{Non-contact forces}
+\begin{itemize}
+ {\item electric fields (dipoles \& monopoles)}
+ {\item magnetic fields (dipoles only)}
+ {\item gravitational fields (monopoles only)}
+\end{itemize}
+
+\begin{itemize}
+\item monopoles: field lines radiate towards central object
+\item dipoles - field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (opposite in solenoid)
+\item closer field lines means larger force
+\item dot means out of page, cross means into page
+\end{itemize}
+
+\subsection*{Gravity}
+\[
+F_g=G{{m_1m_2}\over r^2}\tag{grav. force}
+\]
+
+\[
+g={F_g \over m}=G{M_{\operatorname{planet}} \over r^2}\tag{grav. acc.}
+\]
+
+\[
+E_g = mg \Delta h\tag{gpe}
+\]
+
+\[
+W = \Delta E_g = Fx\tag{work}
+\]
+
+\subsection*{Satellites}
+\[
+v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}
+\]
+
+\[
+T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}
+\]
+
+\[
+\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}
+\]
+
+
+
+\subsection*{Magnetic fields}
+% \begin{itemize}
+% \item field strength $B$ measured in tesla
+% \item magnetic flux $\Phi$ measured in weber
+% \item charge $q$ measured in coulombs
+% \item emf $\mathcal{E}$ measured in volts
+% \end{itemize}
+
+% \[
+% {E_1 \over E_2}={r_1 \over r_2}^2
+% \]
+
+\[
+F=qvB\tag{force on moving charged particles}
+\]
+
+if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
+
+
+\includegraphics[height=2cm]{/mnt/andrew/graphics/field-lines.png}
+
+\subsection*{Electric fields}
+
+\begin{align*}
+F=qE \tag{$E$ = strength} \\
+W=q_{\operatorname{point}}\Delta V \tag{in field or points} \\
+F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \\
+E=k{Q \over r^2} \tag{$r=||EQ||$} \\
+F=BInl \tag{force on a coil} \\
+\Phi = B_{\perp}A\tag{magnetic flux} \\
+\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \\
+{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \\
+\end{align*}
+
+
+\textbf{Lenz's law:} ``$-n$'' in Faraday - emf opposes $\Delta \Phi$
+
+\textbf{Eddy currents:} counter movement within a field
+
+\textbf{Right hand grip:} thumb points to north or $I$
+
+\textbf{Right hand slap:} field, current, force are $\perp$
+
+\textbf{Flux-time graphs:} gradient $\times n = \operatorname{emf}$
+
+\textbf{Transformers:} core strengthens \& focuses $\Phi$
+
+% \columnbreak
+
+\subsection*{Power transmission}
+
+\begin{align*}
+ V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \tag
+ P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R}
+\end{align*}
+
+\begin{itemize}
+ {\item Parallel - voltage is constant}
+ {\item Series - voltage is shared within branch}
+\end{itemize}
+
+\includegraphics[height=4cm]{/mnt/andrew/graphics/ac-generator.png}
+\subsection*{Motors}
+% \begin{wrapfigure}{r}{-0.1\textwidth}
+\includegraphics[height=4cm]{/mnt/andrew/graphics/dc-motor-2.png}
+\includegraphics[height=3cm]{/mnt/andrew/graphics/ac-motor.png} \\
+% \end{wrapfigure}
+\textbf{DC:} split ring (one ring split into two halves)
+% \begin{wrapfigure}{r}{0.3\textwidth}
+% \end{wrapfigure}
+\textbf{AC:} slip ring (separate rings with constant contact)
\end{multicols}
# Waves
-## Longitudinal and Transverse Waves
+## Mechanical waves
+- need a medium to travel through
+- cannot transfer energy through vacuum
+- individual particles have little movement regardless of the distance of wave
+- **transfer of energy without net transfer of matter**
-Longitudinal - direction of motion parallel to wave
-Transverse - direction of motion perpendicular to wave
+**Nodes** - point of no motion (fixed point on graph)
+**Antinodes** - point of maximum motion (peaks)
+**Crests** (peaks) & **troughs** (azimuths)
-## Wavelength & frequency
+### Longitudinal waves
+**Direction of motion is parallel to wave**
-$$v=f \times \lambda$$
+
+
+### Transverse waves
+**Direction of motion is perpendicular to wave**
+- rarefactions (expansions)
+- compressions
+
+
+### Measuring mechanical waves
+
+**Amplitude $A$** - max displacement from rest position (0)
+**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)
+**Frequency $f$** - number of cycles (wavelengths) per second
+
+$T={1 \over f}\quad$(period: time for one cycle)
+$v=f \lambda \quad$(speed: displacement per second)
+
+### Doppler effect
+- occurs when there is relative movement between source and observer
+- inverse relationship between frequency and distance: $f \propto {1 \over d}$
+- applies to all types of wave
+- only affects apparent $f$; actual $f$ is constant
+
+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.
-Nodes - point of no motion
-Antinodes - point of maximum motion (peaks)
-Crests (peaks) & troughs (azimuths)
## Interference patterns
-Standing wave - constructive interference at resonant frequency
+When a medium changes character:
+- some energy is *reflected*
+- some energy is *absorbed* by new medium
+- some energy is *transmitted*
+
+**Superposition** - stimuli add together at a given point (vector addition)
+**Standing wave** - constructive interference at resonant frequency
+
+### Reflection
+
+**Diffuse** reflection - irregular surface reflects each ray in a different angle.
+
+#### Rays
+Two- or three-dimensional *wave fronts* can be reflected, e.g. waves at a beach.
+
+Direction of motion of wave fronts can be shown by arrows, called *rays*, which are perpendicular to the wave front:
+
+
+
+Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
+- Normal: $\perp$ to wall
+- Incident wave front: $\perp$ to incident ray
+- Incident ray: $
+
+#### Transverse
+- sign of reflected transverse wave is inverted when endpoint is fixed in y-axis (equivalent to $180^\circ=\pi={\lambda \over 2}$ phase change)
+- phase is constant if endpoint is free to move in y-axis (**reflected** is same as **incident**)
+
+## Harmonics
+
+**Harmonic** - fundamental (lowest) frequency to produce a certain number of wavelengths
+**Overtone** - a multiple of the fundamental harmonic which produces the same no. of wavelengths at a different frequency (due to constructive interference)
+
+#### Wave has antinodes at both ends:
+$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$f = {nv \div 2l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
+
+#### Wave has antinode at one end:
+$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$f = {nv \div 4l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
+
+## Light
+
+Newton - light as a particle
+- light speeds up as it travels through a solid medium
+
+Hooke - light as a wave
+- light slows down through solid medium
+
+### Huygen's principle
+**Each point on a wavefront can be considered a source of secondary wavelets**
+
+
+### Refraction
+**Change in direction caused by change in speed** e.g. prism
+$\Delta v$ depends on $\lambda$, so wavelengths become "split"
+
+
+Refractive index of a medium depends $\Delta v$ from $c$
+$n={c \over v}\quad$ (refractive index of a medium)
+$n_1v_1=n_2v_2$ (equivalence between media)
+
+### Snell's law
+$n$ can be used to determine how much a ray will refract going between two media.
+
+$$n_1 \sin \theta_1=n_2 \sin \theta_2$$
+
+### Total internal reflection
+When $n_1 < n_2$, light is refracted *towards* normal ($90^\circ$ to medium border - "vertical" line in case of air/water).
+When $n_1 > n_2$, light is reflected *away* from normal.
+**Critical angle $\theta_c$** - angle of incidence $\theta_1$ at which $\theta_2 \gt 90^\circ$ to normal
+$n_1 sin \theta_c = n_2 \sin 90^\circ$
+$\therefore \theta_c = {n_2 \over n_1}$
+
+### Dispersion