$\Sigma F, a$ towards centre, $v$ tangential
- $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
+ $\Sigma F = F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}=T \sin \theta = mg \tan \theta$
\includegraphics[height=4cm]{graphics/circ-forces.png}
% -----------------------
\subsection*{Vertical circular motion}
- $T =$ tension, e.g. circular pendulum
+ % $T =$ tension, e.g. circular pendulum
- $T+mg = {{mv^2}\over r}$ at highest point
+ $T+mg = {{mv^2}\over r}, v = \sqrt{rg}$ (top)
- $T-mg = {{mv^2} \over r}$ at lowest point
+ $T-mg = {{mv^2} \over r}, v = \sqrt{2rg}$ (bottom)
+
+ $E_K_{\text{bottom}}=E_K_{\text{top}}+mgh$
% -----------------------
\subsection*{Projectile motion}
\item vertical component gravity: $a_y = -g$
\end{itemize}
- \begin{align*}
- v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\
- h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\
- x=ut\cos\theta \tag{$\Delta x$ at $t$} \\
- y=ut \sin \theta-{1 \over 2}gt^2 \tag{height at $t$} \\
- t={{2u\sin\theta}\over g} \tag{time of flight}\\
- d={v^2 \over g}\sin \theta \tag{horiz. range} \\
- \end{align*}
+ % \begin{align*}
+ $v=\sqrt{v^2_x + v^2_y}$ \hfill vectors \\
+ $h={{u^2\sin \theta ^2}\over 2g}$ \hfill max height \\
+ $x=ut\cos\theta$ \hfill $\Delta x$ at $t$ \\
+ $y=ut \sin \theta-{1 \over 2}gt^2$ \hfill height at $t$ \\
+ $t={{2u\sin\theta}\over g}$ \hfill time of flight \\
+ $d={v^2 \over g}\sin \theta$ \hfill horiz. range \\
+ % \end{align*}
\includegraphics[height=3.2cm]{graphics/projectile-motion.png}
\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{$F_g$-dist: $A=\Delta \operatorname{gpe}$}
\item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
\end{itemize}
$x={2mg \over k}$
+ Vertical: $\Delta E = {1 \over 2}kx^2 + mgh
+
% -----------------------
\subsection*{Motion equations}
% -----------------------
\subsection*{Lorentz factor}
- $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$
+ $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}, \quad v = c\sqrt{1-{1 \over \gamma^2}}$$
$t=t_0 \gamma$ ($t$ longer in moving frame)
$m=m_0 \gamma$ (mass dilation)
- $$v = c\sqrt{1-{1 \over \gamma^2}}$$
-
% -----------------------
\subsection*{Energy and work}
+ Total energy = mass energy
+
$E_{\text{rest}} = mc^2, \quad E_K = (\gamma-1)mc^2$
- $E_{\text{total}} = E_K + E_{\text{rest}} = \gamma mc^2$
+ $E_{\text{total}} = \gamma E_{\text{rest}} = E_K + E_{\text{rest}} = \gamma mc^2$
$W = \Delta E = \Delta mc^2=(\gamma-1)m_{\text{rest}} c^2$
\begin{itemize}
\item monopoles: lines towards centre
- \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (or perpendicular to wire)
+ \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (two magnets) or $\rightarrow$ N (single)
\item closer field lines means larger force
\item dot: out of page, cross: into page
\item +ve corresponds to N pole
+ \item Inv. sq. ${E_1 \over E_2} = ({r_2 \over r_1})^2$
\end{itemize}
\includegraphics[height=2cm]{graphics/field-lines.png}
\[v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
- \[T={\sqrt{4 \pi^2 r^3 \over {GM}}}\tag{period}\]
+ \[T={\sqrt{4 \pi^2 r^3 \over {GM}}}=2 \pi \sqrt{r_{\text{alt}} \over g_{\text{alt}}}\tag{period}\]
\[r = \sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
% -----------------------
\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}
+ % \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}
% \[{E_1 \over E_2}={r_1 \over r_2}^2\]
\[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
\textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
- (emf creates $I$ with associated field that opposes $\Delta \phi$)
+ (emf creates $I$ with associated field that opposes $\Delta \Phi$)
\textbf{Eddy currents:} counter movement within a field
\textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
+ \textbf{Magnet through ring:} consider $g$
+
\includegraphics[height=2cm]{graphics/slap-2.jpeg}
\includegraphics[height=3cm]{graphics/grip.png}
\textbf{Xfmr} core strengthens \& focuses $\Phi$
+ \columnbreak
+
% -----------------------
\subsection*{Particle acceleration}
e- accelerated with $x$ V is given $x$ eV
\[W={1\over2}mv^2=qV \tag{field or points}\]
+ \[V_{\text{point}} = (V_1 - V_2) \div 2 \tag{if midpoint} \]
\[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
+ Circular path: $F\perp B \perp v$
+
% -----------------------
\subsection*{Power transmission}
% \begin{align*}
- \[V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \]
+ \[V_{\operatorname{rms}}={V_{\operatorname{p}}\over \sqrt{2}}={V_{\operatorname{p\rightarrow p}}\over {2 \sqrt{2}}} \]
\[P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \]
\[V_{\operatorname{loss}}=IR \]
% \end{align*}
{\item Series $V$ shared within branch}
\end{itemize}
- \includegraphics[height=4cm]{graphics/ac-generator.png}
-
% -----------------------
\subsection*{Motors}
% \begin{wrapfigure}{r}{-0.1\textwidth}
\includegraphics[height=4cm]{graphics/dc-motor-2.png}
- \includegraphics[height=3cm]{graphics/ac-motor.png} \\
+ % \includegraphics[height=3cm]{graphics/ac-motor.png} \\
+ \includegraphics[height=4cm]{graphics/ac-generator.png} \\
- Force on current-carying wire, not copper \\
+ Force on I-carying wire, not Cu \\
$F=0$ for front & back of coil (parallel) \\
Any angle $> 0$ will produce force \\
% \end{wrapfigure}
% -----------------------
\subsection*{Polarisation}
\includegraphics[height=3.5cm]{graphics/polarisation.png} \\
- Reduces total amplitude
+ Transverse only. Reduces total $A$.
% -----------------------
\subsection*{Diffraction}
% \(\Delta x\) = fringe spacing \\
\(l\) = distance from source to observer\\
\(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
- \item diffraction $\propto {\lambda \over d}$
+ \item diffraction $\propto {\lambda \over d} \propto$ fringe spacing
+ \item $d(|\overrightarrow{S_1W}|-|\overrightarrow{S_2W}|)=d \Delta x = \lambda l$
\item significant diffraction when ${\lambda \over \Delta x} \ge 1$
- \item diffraction creates distortion (electron $>$ optical microscopes)
+ \item diffraction creates distortion (electron $\gt$ optical microscopes)
\end{itemize}
\subsection*{Refraction}
\includegraphics[height=3.5cm]{graphics/refraction.png}
- When a medium changes character, energy is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}
+ When a medium changes character, light is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}. $\lambda$ changes, not $f$. $n$ changes slightly with $f$ (dispersion)
angle of incidence $\theta_i =$ angle of reflection $\theta_r$
$n_1 v_1 = n_2 v_2$
+ $n={c \over v}$
+
% +++++++++++++++++++++++
\section{Light and Matter}
\item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
\item Ionisation energy - min $E$ required to remove e-
\item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
- \item No. of lines - include all possible states
+ \item No. of lines - include all possible states. \Delta E \ne |\Delta E|
\end{itemize}
\subsection*{Uncertainty principle}
- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
+ $\Delta x \approx {\text{slit width} \over 2$}
+
+ measurement: $\rho$ transferred to e-\\ slit: possibility of diff. before slit
\subsection*{Wave-particle duality}
\subsubsection*{wave model}
\begin{itemize}
- \item cannot explain photoelectric effect
- \item $f$ is irrelevant to photocurrent
+ % \item cannot explain photoelectric effect
+ \item any $f$ works, given $t$
\item predicts delay between incidence and ejection
\item speed depends on medium
\item supported by bright spot in centre
\subsubsection*{particle model}
\begin{itemize}
- \item explains photoelectric effect
+ % \item explains photoelectric effect
\item rate of photoelectron release $\propto$ intensity
\item no time delay - one photon releases one electron
+ \item threshold frequency
\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
\item light exerts force
\item light bent by gravity
\end{multicols}
+\begin{center}
+ \includegraphics[height=2.95cm]{graphics/spectrum.png}
+\end{center}
+
\end{document}