$\theta = \tan^{-1} {{v^2} \over rg}$
- $\Sigma F$ always acts towards centre, but not necessarily horizontally
+ $\Sigma F$ always acts towards centre (horizontally)
$\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
$\text{elastic potential energy} = {1 \over 2}kx^2$
+ $x={2mg \over k}$
+
% -----------------------
\subsection*{Motion equations}
\[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
- \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
+ \[T={\sqrt{4 \pi^2 r^3} \over {GM_\text{planet}}}\tag{period}\]
\[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
\[F=qvB\tag{$F$ on moving $q$}\]
\[F=IlB\tag{$F$ of $B$ on $I$}\]
+ \[B={mv \over qr}\tag{field strength on e-}\]
\[r={mv \over qB} \tag{radius of $q$ in $B$}\]
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
% \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)
% -----------------------
\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}
+ \textbf{Standing waves} - constructive int. at resonant freq
+ \textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser
- \textbf{Standing waves} - constructive int. at resonant freq
+ \textbf{Incoherent} - e.g. incandescent bulb
- \subsection*{Harmonics}
+ % -----------------------
+ \subsection*{Harmonics}
\(\lambda = {{al} \div n}\quad\) (\(\lambda\) for \(n^{th}\) harmonic)\\
\(f = {nv \div al}\quad\) (\(f\) for \(n_{th}\) harmonic at length
\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 \(pd = |S_1P-S_2P|\) 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 Fringe separation: \(\Delta x = {{\lambda l }\over d}\) where \\
- \(\Delta x\) = fringe spacing \\
- \(l\) = distance from slits to screen\\
- \(d\) = slit separation (\(=S_1-S_2\))
+ \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 ${\lambda \over \Delta x} \ge 1$
+ \item diffraction creates distortion (electron $>$ optical microscopes)
\end{itemize}
\textbf{Work function $\phi$}
- Minimum $E$ required to release photoelectrons. Magnitude of $y$-intercept of frequency vs $E_K$ graph. $\phi$ is determined by strength of bonding.
+ Minimum $E$ required to release photoelectrons. Magnitude of $y$-intercept of $f$ vs $E_K$ graph. $\phi$ is determined by strength of bonding. Units: eV or J.
- $\phi=hf_0$
+ $\phi=hf_0$
\textbf{Kinetic energy}
E_{\operatorname{k-max}}=hf - \phi
- voltage in circuit or stopping voltage = max $E_K$ in eV
- equal to $x$-intercept of volts vs current graph (in eV)
+ voltage in circuit or stopping voltage = max $E_K$ in eV \\
+ $E_K = x$-int of $V\cdot I$ graph (in eV) \\
+ dashed line below $E_K=0$ ($E_K < 0$ is impossible)
+
\textbf{Stopping potential $V$ for min $I$}
$V=h_{\text{eV}}(f-f_0)$
- % \columnbreak
-
\subsection*{De Broglie's theory}
\[ \lambda = {h \over \rho} = {h \over mv} \]
Uncertainty of a measurement is $1 \over 2$ the smallest division
\textbf{Precision} - concordance of values \\
- \textbf{Accuracy} - closeness to actual value
-
- \columnbreak
-
- \quad
-
-
-
-
-
+ \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}
-% \includegraphics[height=5cm]{graphics/em-spectrum.png}
-
\end{document}