$T-mg = {{mv^2} \over r}$ at lowest point
+ $E_K_{\text{bottom}}=E_K_{\text{top}}+mgh$
+
% -----------------------
\subsection*{Projectile motion}
\begin{itemize}
\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}
$x={2mg \over k}$
+ 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}
\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
\[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
\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
\[W={1\over2}mv^2=qV \tag{field or points}\]
\[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
+ Circular path: $F\perp B \perp v$
+
% -----------------------
\subsection*{Power transmission}
% -----------------------
\subsection*{Polarisation}
\includegraphics[height=3.5cm]{graphics/polarisation.png} \\
- Reduces total amplitude
+ Transverse only. Reduces total $A$.
% -----------------------
\subsection*{Diffraction}
\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}
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}
\end{multicols}
+\begin{center}
+ \includegraphics[height=2.95cm]{graphics/spectrum.png}
+\end{center}
+
\end{document}