% +++++++++++++++++++++++
\section{Motion}
- $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$
+ $\operatorname{m/s} \, \times \, 3.6 = \operatorname{km/h}$
\subsection*{Inclined planes}
- $F = m g \sin\theta F_{frict} = m a$
+ $F = m g \sin\theta - F_{\text{frict}} = m a$
% -----------------------
\subsection*{Banked tracks}
\includegraphics[height=4cm]{graphics/banked-track.png}
- $$\theta = \tan^{-1} {{v^2} \over rg}$$
+ $\theta = \tan^{-1} {{v^2} \over rg}$
- $\Sigma F$ always acts towards centre, but not necessarily horizontally
+ $\Sigma F$ always acts towards centre (horizontally)
$\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
Design speed $v = \sqrt{gr\tan\theta}$
+ $n\sin \theta = {mv^2 \div r}, \quad n\cos \theta = mg$
+
% -----------------------
\subsection*{Work and energy}
$F=-kx$
- $E_{elastic} = {1 \over 2}kx^2$
+ $\text{elastic potential energy} = {1 \over 2}kx^2$
+
+ $x={2mg \over k}
% -----------------------
\subsection*{Motion equations}
\begin{tabular}{ l r }
+ & no \\
$v=u+at$ & $x$ \\
$x = {1 \over 2}(v+u)t$ & $a$ \\
$x=ut+{1 \over 2}at^2$ & $v$ \\
2. Speed of light $c$ is the same to all observers (Michelson-Morley)
- $\therefore , t$ must dilate as speed changes
+ $\therefore \, t$ must dilate as speed changes
{\bf Inertial reference frame} $a=0$
\end{itemize}
\includegraphics[height=2cm]{graphics/field-lines.png}
+ % \includegraphics[height=2cm]{graphics/bar-magnet-fields-rotated.png}
% -----------------------
\subsection*{Gravity}
\[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
- \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
+ \[T={\sqrt{4 \pi^2 r^3} \over {GM}}\tag{period}\]
\[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
\[F=qvB\tag{$F$ on moving $q$}\]
\[F=IlB\tag{$F$ of $B$ on $I$}\]
+ \[B={mv \over qr}\tag{field strength on e-}\]
\[r={mv \over qB} \tag{radius of $q$ in $B$}\]
if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
\[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
\[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
- \textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$
+ \textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
+ % emf is gradient of flux-time graph
\textbf{Eddy currents:} counter movement within a field
\textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
- \textbf{Right hand slap:} $B \perp I \perp F$
+ \includegraphics[height=2cm]{graphics/slap-2.jpeg}
+ \includegraphics[height=3cm]{graphics/grip.png}
+
+ % \textbf{Right hand slap:} $B \perp I \perp F$ \\
+ % ($I$ = thumb)
\textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
$1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
- eaccelerated with $x$ V is given $x$ eV
+ 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*}
- $$V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}}$$
- P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \\
- V_{\operatorname{loss}}=IR
+ \[V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \]
+ \[P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \]
+ \[V_{\operatorname{loss}}=IR \]
% \end{align*}
Use high-$V$ side for correct $|V_{drop}|$
% \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} \\
+
+ $F=0$ for front & back of coil (parallel) \\
+ Any angle $> 0$ will produce force \\
% \end{wrapfigure}
\textbf{DC:} split ring (two halves)
% \end{wrapfigure}
\textbf{AC:} slip ring (separate rings with constant contact)
+% \pagebreak
+
% +++++++++++++++++++++++
\section{Waves}
- \textbf{nodes:} fixed on graph
+ \textbf{nodes:} fixed on graph \\
+ \textbf{amplitude:} max disp. from $y=0$ \\
+ \textbf{rarefactions} and \textbf{compressions} \\
+ \textbf{mechanical:} transfer of energy without net transfer of matter \\
+
\textbf{Longitudinal (motion $||$ wave)}
- \includegraphics[height=4cm]{graphics/longitudinal-waves.png}
+ \includegraphics[width=6cm]{graphics/longitudinal-waves.png}
\textbf{Transverse (motion $\perp$ wave)}
- \includegraphics[height=4cm]{graphics/transverse-waves.png}
+ \includegraphics[width=6cm]{graphics/transverse-waves.png}
% -----------------------
- \subsection*{Motors}
$T={1 \over f}\quad$(period: time for one cycle)
- $v=f \lambda \quad$(speed: displacement per second)
+ $v=f \lambda \quad$(speed: displacement / sec)
% -----------------------
\subsection*{Doppler effect}
- 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.
+ When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to travel than $w_{n-1}$. $w_n$ reaches observer sooner than $w_{n-1}$ ("apparent" $\lambda$).
% -----------------------
\subsection*{Interference}
- When a medium changes character, energy is reflected, absorbed, and transmitted
+
+
+
+ \textbf{Standing waves} - constructive int. at resonant freq
+
+ \subsection*{Harmonics}
+
+
+ \(\lambda = {{al} \div n}\quad\) (\(\lambda\) for \(n^{th}\) harmonic)\\
+ \(f = {nv \div al}\quad\) (\(f\) for \(n_{th}\) harmonic at length
+ \(l\) and speed \(v\)) \\
+ where \(a=2\) for antinodes at both ends, \(a=4\) for antinodes at one end
% -----------------------
\subsection*{Polarisation}
- \includegraphics[height=4cm]{graphics/polarisation.png}
+ \includegraphics[height=3.5cm]{graphics/polarisation.png}
+
+ % -----------------------
+ \subsection*{Diffraction}
+ \includegraphics[width=6cm]{graphics/diffraction.jpg}
+ \includegraphics[width=6cm]{graphics/diffraction-2.png}
+ \begin{itemize}
+ % \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
+ \item Constructive: \(pd = n\lambda, n \in \mathbb{Z}\)
+ \item Destructive: \(pd = (n-{1 \over 2})\lambda, n \in \mathbb{Z}\)
+ \item Path difference: \(\Delta x = {{\lambda l }\over d}\) where \\
+ % \(\Delta x\) = fringe spacing \\
+ \(l\) = distance from source to observer\\
+ \(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
+ \item significant diffraction when ${\lambda \over \Delta x} \ge 1$
+ \end{itemize}
+
+
% -----------------------
\subsection*{Refraction}
- \includegraphics[height=4cm]{graphics/refraction.png}
+ \includegraphics[height=3.5cm]{graphics/refraction.png}
+
+ When a medium changes character, energy is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}
- Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
+ angle of incidence $\theta_i =$ angle of reflection $\theta_r$
Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$
Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
+
% +++++++++++++++++++++++
\section{Light and Matter}
% -----------------------
\subsection*{Planck's equation}
- f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c
-
- h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s}
-
- 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}
+ \[ f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c \]
+ \[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
+ \[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]
\subsection*{Force of electrons}
- F={2P_{\text{in}}\over c}
-
- \text{photons per second}={\text{total energy} \over \text{energy per photon}}={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
+ \[ F={2P_{\text{in}}\over c} \]
+ % \begin{align*}
+ \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
+ \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
+ % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
+ % \end{align*}
\subsection*{Photoelectric effect}
\begin{itemize}
\item $V_{\operatorname{supply}}$ does not affect photocurrent
- \item $V_{\operatorname{sup}} > 0$: eattracted to collector anode
+ \item $V_{\operatorname{sup}} > 0$: e- attracted to collector anode
\item $V_{\operatorname{sup}} < 0$: attracted to illuminated cathode, $I\rightarrow 0$
- \item $v$ of edepends on ionisation energy (shell)
+ \item $v$ of depends on ionisation energy (shell)
\item max current depends on intensity
\end{itemize}
$V=h_{\text{eV}}(f-f_0)$
+ % \columnbreak
+
\subsection*{De Broglie's theory}
- $\lambda = {h \over \rho} = {h \over mv}$
- $\rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c$
+ \[ \lambda = {h \over \rho} = {h \over mv} \]
+ \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
\begin{itemize}
\item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
\item confirmed by similar e- and x-ray diff patterns
\subsection*{Spectral analysis}
\begin{itemize}
- $n\item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
- $n\item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
- $n\item Ionisation energy - min $E$ required to remove e-
- $n\item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
+ \item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
+ \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
\end{itemize}
- \subsection{Indeterminancy principle}
+ \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.
- \subsection{Wave-particle duaity}
+ \subsection*{Wave-particle duaity}
- wave model:
+ \subsubsection*{wave model}
+ \begin{itemize}
+ \item cannot explain photoelectric effect
+ \item $f$ is irrelevant to photocurrent
+ \item predicts delay between incidence and ejection
+ \item speed depends on medium
+ \end{itemize}
- \item cannot explain photoelectric effect
- \item $f$ is irrelevant to photocurrent
- \item predicts delay between incidence and ejection
- \item speed depends on medium
+ \subsubsection*{particle model}
- particle model:
+ \begin{itemize}
+ \item explains photoelectric effect
+ \item rate of photoelectron release $\propto$ intensity
+ \item no time delay - one photon releases one electron
+ \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
+ \item quantised energy
+ \end{itemize}
+
+ % +++++++++++++++++++++++
+ \section{Experimental \\ design}
+
+ \textbf{Absolute uncertainty} $\Delta$ \\
+ (same units as quantity)
+ \[ \Delta(m) = {{\mathcal{E}(m)} \over 100} \cdot m \]
+ \[ (A \pm \Delta A) + (B \pm \Delta A) = (A+B) \pm (\Delta A + \Delta B) \]
+ \[ (A \pm \Delta A) - (B \pm \Delta A) = (A-B) \pm (\Delta A + \Delta B) \]
+ \[ c(A \pm \Delta A) = cA \pm c \Delta A \]
- \item explains photoelectric effect
- \item rate of photoelectron release $\propto$ intensity
- \item no time delay - one photon releases one electron
- \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
+ \textbf{Relative uncertainty} $\mathcal{E}$ (unitless)
+ \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100 \]
+ \[ (A \pm \mathcal{E} A) \cdot (B \pm \mathcal{E} B) = (A \cdot B) \pm (\mathcal{E} A + \mathcal{E} B) \]
+ \[ (A \pm \mathcal{E} A) \div (B \pm \mathcal{E} B) = (A \div B) \pm (\mathcal{E} A + \mathcal{E} B) \]
+ \[ (A \pm \mathcal{E} A)^n = (A^n \pm n \mathcal{E} A) \]
+ \[ c(A \pm \mathcal{E} A)=cA \pm \mathcal{E} A \]
+ Uncertainty of a measurement is $1 \over 2$ the smallest division
+ \textbf{Precision} - concordance of values \\
+ \textbf{Accuracy} - closeness to actual value\\
+ \textbf{Random errors} - unpredictable, reduced by more tests \\
+ \textbf{Systematic errors} - not reduced by more tests
+
+ \columnbreak
+
+ \quad
\end{multicols}
+
+% \includegraphics[height=5cm]{graphics/em-spectrum.png}
+
\end{document}