\begin{multicols}{3}
% +++++++++++++++++++++++
-
+
{\huge Physics}\hfill Andrew Lorimer\hspace{2em}
% +++++++++++++++++++++++
\section{Motion}
+ $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$
+
\subsection*{Inclined planes}
$F = m g \sin\theta - F_{frict} = m a$
\includegraphics[height=4cm]{/mnt/andrew/graphics/banked-track.png}
- $\theta = \tan^{-1} {{v^2} \over rg}$ (also for objects on string)
+ $$\theta = \tan^{-1} {{v^2} \over rg}$$
$\Sigma F$ always acts towards centre, but not necessarily horizontally
- $\Sigma F = {{mv^2} \over r} = mg \tan \theta$
+ $\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
Design speed $v = \sqrt{gr\tan\theta}$
$\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
% -----------------------
- \subsection*{Horizontal motion}
-
- $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$
+ \subsection*{Horizontal circular motion}
$v = {{2 \pi r} \over T}$
$a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
- $\Sigma F$ towards centre, $v$ tangential
+ $\Sigma F, a$ towards centre, $v$ tangential
$F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
\begin{align*}
v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\
h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\
- y=ut \sin \theta-{1 \over 2}gt^2 \tag{time of flight} \\
+ 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*}
$\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
- Momentum is conserved.
+ $\Sigma mv_0=\Sigma mv_1$ (conservation)
$\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
$$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
-% -----------------------
- \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}
+ {\item normal half-life $2.2 \operatorname{\mu s}$ in stationary frame, $> 2.2 \operatorname{\mu s}$ observed from Earth}
\end{itemize}
% +++++++++++++++++++++++
\item monopoles: lines towards centre
\item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (or perpendicular to wire)
\item closer field lines means larger force
- \item dot means out of page, cross means into page
+ \item dot: out of page, cross: into page
\item +ve corresponds to N pole
\end{itemize}
+ \includegraphics[height=2cm]{/mnt/andrew/graphics/field-lines.png}
+
% -----------------------
\subsection*{Gravity}
\[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]
-
- \[g={F_g \over m}=G{M_{\operatorname{planet}} \over r^2}\tag{grav. acc.}\]
-
+ \[g={F_g \over m_2}=G{m_{1} \over r^2}\tag{field of $m_1$}\]
\[E_g = mg \Delta h\tag{gpe}\]
-
\[W = \Delta E_g = Fx\tag{work}\]
-
\[w=m(g-a) \tag{app. weight}\]
+ % \columnbreak
+
% -----------------------
\subsection*{Satellites}
- \[v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
+ \[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
\[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
\item emf $\mathcal{E}$ measured in volts
\end{itemize}
- \[{E_1 \over E_2}={r_1 \over r_2}^2\]
+ % \[{E_1 \over E_2}={r_1 \over r_2}^2\]
- \[F=qvB\tag{force on moving charged particles}\]
+ \[F=qvB\tag{$F$ on moving $q$}\]
+ \[F=IlB\tag{$F$ of $B$ on $I$}\]
+ \[r={mv \over qB} \tag{radius of $q$ in $B$}\]
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*}
+ \[F=qE \tag{$E$ = strength} \]
+ \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
+ \[E=k{q \over r^2} \tag{field on point charge} \]
+ \[E={V \over d} \tag{field between plates}\]
+ \[F=BInl \tag{force on a coil} \]
+ \[\Phi = B_{\perp}A\tag{magnetic flux} \]
+ \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
+ \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
- \textbf{Lenz's law:} ``$-n$'' in Faraday - emf opposes $\Delta \Phi$
+ \textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$
\textbf{Eddy currents:} counter movement within a field
- \textbf{Right hand grip:} thumb points to north or $I$
+ \textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
- \textbf{Right hand slap:} field, current, force are $\perp$
+ \textbf{Right hand slap:} $B \perp I \perp F$
- \textbf{Flux-time graphs:} gradient $\times n = \operatorname{emf}$
+ \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
\textbf{Transformers:} core strengthens \& focuses $\Phi$
+% -----------------------
+ \subsection*{Particle acceleration}
+
+ $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
+
+ e- accelerated with $x$ V is given $x$ eV
+
+ \[W={1\over2}mv^2=qV \tag{field or points}\]
+ \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
+
+
% -----------------------
\subsection*{Power transmission}
- \begin{align*}
- V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \\
+ % \begin{align*}
+ $$V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}}$$
P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \\
- \end{align*}
+ V_{\operatorname{loss}}=IR
+ % \end{align*}
Use high-$V$ side for correct $|V_{drop}|$
\begin{itemize}
- {\item Parallel - voltage is constant}
- {\item Series - voltage is shared within branch}
+ {\item Parallel - $V$ is constant}
+ {\item Series - $V$ shared within branch}
\end{itemize}
\includegraphics[height=4cm]{/mnt/andrew/graphics/ac-generator.png}