$\theta = \tan^{-1} {{v^2} \over rg}$
- $\Sigma F$ 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}
\[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
- \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
+ \[T={\sqrt{4 \pi^2 r^3} \over {GM}}\tag{period}\]
\[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
\[F=qvB\tag{$F$ on moving $q$}\]
\[F=IlB\tag{$F$ of $B$ on $I$}\]
+ \[B={mv \over qr}\tag{field strength on e-}\]
\[r={mv \over qB} \tag{radius of $q$ in $B$}\]
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
% \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)
\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 ${\lambda \over \Delta x} \ge 1$
\end{itemize}
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
\columnbreak