\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{enumitem}
+\usepackage{supertabular}
+\usepackage{tabularx}
\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt}
% +++++++++++++++++++++++
\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}
- $W=Fx=\Delta \Sigma E$ (work)
+ $W=Fs=Fs \cos \theta=\Delta \Sigma E$
$E_K = {1 \over 2}mv^2$ (kinetic)
% -----------------------
\subsection*{Projectile motion}
\begin{itemize}
- \item{horizontal component of velocity is constant if no air resistance}
- \item{vertical component affected by gravity: $a_y = -g$}
+ \item $v_x$ is constant: $v_x = {s \over t}$
+ \item use suvat to find $t$ from $y$-component
+ \item vertical component gravity: $a_y = -g$
\end{itemize}
\begin{align*}
$F=-kx$
- $E_{elastic} = {1 \over 2}kx^2$
+ $\text{elastic potential energy} = {1 \over 2}kx^2$
+
+ $x={2mg \over k}$
% -----------------------
\subsection*{Motion equations}
\begin{tabular}{ l r }
+ & no \\
$v=u+at$ & $x$ \\
$x = {1 \over 2}(v+u)t$ & $a$ \\
$x=ut+{1 \over 2}at^2$ & $v$ \\
$\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
- $\Sigma mv_0=\Sigma mv_1$ (conservation)
+ $\Sigma (mv_0)=(\Sigma m)v_1$ (conservation)
+
+ % $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
- $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
+ % $\Sigma E_K = \Sigma ({1 \over 2} m v^2) = {1 \over 2} (\Sigma m)v_f$
- $n$-body collisions: $\rho$ of each body is independent
+ if elastic:
+ $$\sum _{i{\mathop {=}}1}^{n}E_K (i)=\sum _{i{\mathop {=}}1}^{n}({1 \over 2}m_i v_{i0}^2)={1 \over 2}\sum _{i{\mathop {=}}1}^{n}(m_i) v_f^2$$
+
+ % $n$-body collisions: $\rho$ of each body is independent
% ++++++++++++++++++++++
\section{Relativity}
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 high-altitude particles:} $t$ dilation means more particles reach Earth than expected (half-life greater when obs. from Earth)
{\bf Inertial reference frame} $a=0$
% -----------------------
\subsection*{Energy and work}
- $E_0 = mc^2$ (rest)
+ $E_{\text{rest}} = mc^2, \quad E_K = (\gamma-1)mc^2$
- $E_{total} = E_K + E_{rest} = \gamma mc^2$
+ $E_{\text{total}} = E_K + E_{\text{rest}} = \gamma mc^2$
- $E_K = (\gamma 1)mc^2$
-
- $W = \Delta E = \Delta mc^2$
+ $W = \Delta E = \Delta mc^2=(\gamma-1)m_{\text{rest}} c^2$
% -----------------------
\subsection*{Relativistic momentum}
$$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$
% -----------------------
- \subsection*{High-altitude muons}
- \begin{itemize}
- {\item $t$ dilation more muons reach Earth than expected}
- {\item normal half-life $2.2 \operatorname{\mu s}$ in stationary frame, $> 2.2 \operatorname{\mu s}$ observed from Earth}
- \end{itemize}
% +++++++++++++++++++++++
\section{Fields and power}
\end{itemize}
\includegraphics[height=2cm]{graphics/field-lines.png}
+ % \includegraphics[height=2cm]{graphics/bar-magnet-fields-rotated.png}
% -----------------------
\subsection*{Gravity}
\[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
- \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
+ \[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\]
- \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
+ \[r = \sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
% -----------------------
\subsection*{Magnetic fields}
\[F=qvB\tag{$F$ on moving $q$}\]
\[F=IlB\tag{$F$ of $B$ on $I$}\]
+ \[B={mv \over qr}\tag{field strength on e-}\]
\[r={mv \over qB} \tag{radius of $q$ in $B$}\]
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 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{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)
+
+ \includegraphics[width=\columnwidth]{graphics/lenz.png}
- \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
+ \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}.$
+ If $f$ increases, ampl. \& $f$ of $\mathcal{E}$ increase
\textbf{Transformers:} core strengthens \& focuses $\Phi$
% \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} \\
+
+ Force on current-carying wire, not copper \\
+ $F=0$ 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{amplitude:} max displacement from $y=0$
- \textbf{rarefactions} (expansions) / \textbf{compressions}
- \textbf{mechanical:} transfer of energy without net transfer of matter
-
+ \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[width=6cm]{graphics/longitudinal-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 / sec)
+ $f={c \over \lambda}\quad\hspace{0.7em}$(for $v=c$)
% -----------------------
\subsection*{Doppler effect}
+
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
+
+ \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{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
+
+ \textbf{Incoherent} - e.g. incandescent/LED
+
+
+
+
+
+ % -----------------------
+ \subsection*{Harmonics}
+
+ 1st harmonic = fundamental
+
+ \textbf{for nodes at both ends:} \\
+ \(\hspace{2em} \lambda = {{2l} \div n}\)
+ \(\hspace{2em} f = {nv \div 2l} \)
+
+ \textbf{for node at one end ($n$ is odd):} \\
+ \(\hspace{2em} \lambda = {{4l} \div n}\)
+ \(\hspace{2em} f = {nv \div 4l} \) \\
+ alternatively, $\lambda = {4l \over {2n-1}}$ where $n\in \mathbb{Z}$ and $n+1$ is the next possible harmonic
+
+
+ % \(a=2\) for nodes at both ends, \\ \(a=4\) for node at one end
% -----------------------
\subsection*{Polarisation}
\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 diffraction $\propto {\lambda \over d}$
+ \item significant diffraction when ${\lambda \over \Delta x} \ge 1$
+ \item diffraction creates distortion (electron $>$ optical microscopes)
+ \end{itemize}
+
+
% -----------------------
\subsection*{Refraction}
\includegraphics[height=3.5cm]{graphics/refraction.png}
- Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
+ When a medium changes character, energy is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}
+
+ angle of incidence $\theta_i =$ angle of reflection $\theta_r$
- Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$
+ Critical angle $\theta_c = \sin^{-1}{n_2 \over n_1}$
Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
+ ${v_1 \div v_2} = {\sin\theta_1 \div \sin\theta_2}$
+
+ $n_1 v_1 = n_2 v_2$
+
+
% +++++++++++++++++++++++
\section{Light and Matter}
% -----------------------
\subsection*{Planck's equation}
- \[ f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c \]
+ \[ \quad E=hf={hc \over \lambda}=\rho c = qV\]
\[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
\[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]
+ \subsection*{De Broglie's theory}
+
+ \[ \lambda = {h \over \rho} = {h \over mv} \]
+ \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
+ \[ v = \sqrt{2E_K \div m} \]
+ \begin{itemize}
+ \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
+ \item confirmed by e- and x-ray patterns
+ \end{itemize}
+
\subsection*{Force of electrons}
\[ F={2P_{\text{in}}\over c} \]
% \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}
+ \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
+ \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
+ % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
% \end{align*}
+ \subsection*{X-ray electron interaction}
+
+ \begin{itemize}
+ \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ and $r$ is radius of orbit
+ \item $\therefore 2\pi r = n{h \over mv} = n \lambda$ (circumference)
+ \item if $2\pi r \ne n{h \over mv}$, no standing wave
+ \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
+ % \item calculating $h$: $\lambda = {h \over \rho}$
+ \end{itemize}
+
\subsection*{Photoelectric effect}
\begin{itemize}
\item $V_{\operatorname{supply}}$ does not affect photocurrent
- \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 depends on ionisation energy (shell)
- \item max current depends on intensity
+ \item $V_{\operatorname{sup}} > 0$: attracted to +ve
+ \item $V_{\operatorname{sup}} < 0$: attracted to -ve, $I\rightarrow 0$
+ \item $v$ of e- depends on shell
+ \item max $I$ (not $V$) depends on intensity
\end{itemize}
- \textbf{Threshold frequency $f_0$}
+ \subsubsection*{Threshold frequency $f_0$}
- Minimum $f$ for photoelectrons to be ejected. $x$-intercept of frequency vs $E_K$ graph. if $f < f_0$, no photoelectrons are detected.
+ min $f$ for photoelectron release. if $f < f_0$, no photoelectrons.
- \textbf{Work function $\phi$}
+ \subsubsection*{Work function $\phi=hf_0$}
- Minimum $E$ required to release photoelectrons. Magnitude of $y$-intercept of frequency vs $E_K$ graph. $\phi$ is determined by strength of bonding.
+ min $E$ for photoelectron release. determined by strength of bonding. Units: eV or J.
- $\phi=hf_0$
+ \subsubsection*{Kinetic energy E_K=hf - \phi = qV_0}
- \textbf{Kinetic energy}
- E_{\operatorname{k-max}}=hf - \phi
+ $V_0 = E_K$ in eV \\
+ % $E_K = x$-int of $V\cdot I$ graph (in eV) \\
+ dashed line below $E_K=0$
- voltage in circuit or stopping voltage = max $E_K$ in eV
- equal to $x$-intercept of volts vs current graph (in eV)
- \textbf{Stopping potential $V$ for min $I$}
+ \subsubsection*{Stopping potential $V_0$ for min $I$}
- $V=h_{\text{eV}}(f-f_0)$
+ $$V_0=h_{\text{eV}}(f-f_0)$$
- \columnbreak
+ \subsubsection*{Graph features}
- \subsection*{De Broglie's theory}
+ \newcolumntype{b}{>{\hsize=.75\hsize}X}
+\newcolumntype{s}{>{\hsize=.3\hsize}X}
- \[ \lambda = {h \over \rho} = {h \over mv} \]
- \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
- \begin{itemize}
- \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
- \item confirmed by similar e- and x-ray diff patterns
- \end{itemize}
+ \begin{tabularx}{\columnwidth}{bbbb}
+\hline
+&$m$&$x$-int&$y$-int \\
+\hline
+\hline
+$f \cdot E_K$ & $h$ & $f_0$ & $-\phi$ \\
+$V \cdot I$ & & $V_0$ & intensity\\
+$f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ &
+\hline
+\end{tabularx}
- \subsection*{X-ray electron interaction}
- \begin{itemize}
- \item e- is only stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
- \item rearranging this, $2\pi r = n{h \over mv} = n \lambda$ (circumference)
- \item if $2\pi r \ne n{h \over mv}$, no standing wave
- \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
- \item calculating $h$: $\lambda = {h \over \rho}$
- \end{itemize}
\subsection*{Spectral analysis}
\item No. of lines - include all possible states
\end{itemize}
- \subsection{Uncertainty 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 duality}
- 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
+ \item supported by bright spot in centre
\end{itemize}
- particle model:
+ \subsubsection*{particle model}
\begin{itemize}
\item explains photoelectric effect
\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{Uncertainty}
+ \section{Experimental \\ design}
- \textbf{Absolute uncertainty} - $\Delta$ - same units as quantity.
+ \textbf{Absolute uncertainty} $\Delta$ \\
+ (same units as quantity)
\[ \Delta(m) = {{\mathcal{E}(m)} \over 100} \cdot m \]
-
\[ (A \pm \Delta A) + (B \pm \Delta A) = (A+B) \pm (\Delta A + \Delta B) \]
\[ (A \pm \Delta A) - (B \pm \Delta A) = (A-B) \pm (\Delta A + \Delta B) \]
\[ c(A \pm \Delta A) = cA \pm c \Delta A \]
- \textbf{Relative uncertainty} - $\mathcal{E}$ - unitless.
- \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100} \]
+ \textbf{Relative uncertainty} $\mathcal{E}$ (unitless)
+ \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100 \]
\[ (A \pm \mathcal{E} A) \cdot (B \pm \mathcal{E} B) = (A \cdot B) \pm (\mathcal{E} A + \mathcal{E} B) \]
\[ (A \pm \mathcal{E} A) \div (B \pm \mathcal{E} B) = (A \div B) \pm (\mathcal{E} A + \mathcal{E} B) \]
\[ (A \pm \mathcal{E} A)^n = (A^n \pm n \mathcal{E} A) \]
Uncertainty of a measurement is $1 \over 2$ the smallest division
\textbf{Precision} - concordance of values \\
- \textbf{Accuracy} - closeness to actual value
-
-
-
-
-
-
-
-
+ \textbf{Accuracy} - closeness to actual value\\
+ \textbf{Random errors} - unpredictable, reduced by more tests \\
+ \textbf{Systematic errors} - not reduced by more tests \\
+ \textbf{Uncertainty} - margin of potential error \\
+ \textbf{Error} - actual difference \\
+ \textbf{Hypothesis} - can be tested experimentally \\
+ \textbf{Model} - evidence-based but indirect representation
\end{multicols}
+
\end{document}