$T-mg = {{mv^2} \over r}$ 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}
% -----------------------
\subsection*{Hooke's law}
- $F=-kx$
+ $F=-kx$ (intercepts origin)
$\text{elastic potential energy} = {1 \over 2}kx^2$
$x={2mg \over k}$
+ 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}
\subsection*{Non-contact forces}
\begin{itemize}
- {\item electric fields (dipoles \& monopoles)}
- {\item magnetic fields (dipoles only)}
- {\item gravitational fields (monopoles only)}
+ {\item electric (dipoles \& monopoles)}
+ {\item magnetic (dipoles only)}
+ {\item gravitational (monopoles only, $F_g=0$ at mid, attractive only)}
\end{itemize}
\vspace{1em}
\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
% -----------------------
\subsection*{Satellites}
- \[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
+ \[v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
- \[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\]
+ \[T={\sqrt{4 \pi^2 r^3 \over {GM}}}\tag{period}\]
\[r = \sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
% -----------------------
\subsection*{Electric fields}
- \[F=qE \tag{$E$ = strength} \]
+ \[F=qE(=ma) \tag{strength} \]
\[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
\[E=k{q \over r^2} \tag{field on point charge} \]
\[E={V \over d} \tag{field between plates}\]
\[F=BInl \tag{force on a coil} \]
\[\Phi = B_{\perp}A\tag{magnetic flux} \]
- \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
+ \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} = Blv\tag{induced emf} \]
\[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
\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{Flux-time graphs:} $m \times n = \operatorname{emf}.$
If $f$ increases, ampl. \& $f$ of $\mathcal{E}$ increase
- \textbf{Transformers:} core strengthens \& focuses $\Phi$
+ \textbf{Xfmr} core strengthens \& focuses $\Phi$
% -----------------------
\subsection*{Particle acceleration}
\[W={1\over2}mv^2=qV \tag{field or points}\]
\[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
+ Circular path: $F\perp B \perp v$
% -----------------------
\subsection*{Power transmission}
\includegraphics[width=4.5cm]{graphics/poissons-spot.png} \\
Poissons's spot supports wave theory (circular diffraction)
- \textbf{Standing waves} - constructive int. at resonant freq
+ \textbf{Standing waves} - constructive int. at resonant freq. Rebound from ends.
\textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
% -----------------------
\subsection*{Polarisation}
- \includegraphics[height=3.5cm]{graphics/polarisation.png}
+ \includegraphics[height=3.5cm]{graphics/polarisation.png} \\
+ 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}
\subsection*{De Broglie's theory}
- \[ \lambda = {h \over \rho} = {h \over mv} \]
+ \[ \lambda = {h \over \rho} = {h \over mv} = {h \over {m \sqrt{2W \over m}}}\]
\[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
\[ v = \sqrt{2E_K \div m} \]
+
\begin{itemize}
\item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
\item confirmed by e- and x-ray patterns
\subsubsection*{Stopping potential $V_0$ for min $I$}
$$V_0=h_{\text{eV}}(f-f_0)$$
+ Opposes induced photocurrent
\subsubsection*{Graph features}
\item predicts delay between incidence and ejection
\item speed depends on medium
\item supported by bright spot in centre
+ \item $\lambda = {hc \over E}$
\end{itemize}
\subsubsection*{particle model}
\item light exerts force
\item light bent by gravity
\item quantised energy
+ \item $\lambda = {h \over \rho}$
\end{itemize}
% +++++++++++++++++++++++
\end{multicols}
+\begin{center}
+ \includegraphics[height=2.95cm]{graphics/spectrum.png}
+\end{center}
+
\end{document}