lenz's law clarifications

diff --git a/physics/final.pdf b/physics/final.pdf
index 904bb0126cbaade84fa9bd6f2271d8003e71b759..c1e97902bafe7af2b3ecbc2f1ba968a5d6890829 100644 (file)
Binary files a/physics/final.pdf and b/physics/final.pdf differ

diff --git a/physics/final.tex b/physics/final.tex
index b37f67c8200e0d80eea913ff55dc54b24b10471e..4dbb0bdb06faa1810395d9a9431a09ec7b2bbdff 100644 (file)
--- a/physics/final.tex
+++ b/physics/final.tex
@@ -91,14 +91,14 @@
       \item vertical component gravity: $a_y = -g$
     \end{itemize}
 
-    \begin{align*}
-      v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\
-      h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\
-      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*}
+    % \begin{align*}
+      $v=\sqrt{v^2_x + v^2_y}$ \hfill vectors \\
+      $h={{u^2\sin \theta ^2}\over 2g}$ \hfill max height \\
+      $x=ut\cos\theta$ \hfill $\Delta x$ at $t$ \\
+      $y=ut \sin \theta-{1 \over 2}gt^2$ \hfill height at $t$ \\
+      $t={{2u\sin\theta}\over g}$ \hfill time of flight \\
+      $d={v^2 \over g}\sin \theta$ \hfill horiz. range \\
+    % \end{align*}
 
     \includegraphics[height=3.2cm]{graphics/projectile-motion.png}
 
@@ -128,6 +128,8 @@
 
   $x={2mg \over k}$
 
+  Vertical: $\Delta E = {1 \over 2}kx^2 + mgh
+
 % -----------------------
   \subsection*{Motion equations}
 
@@ -177,7 +179,7 @@
 % -----------------------
   \subsection*{Lorentz factor}
 
-    $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$
+    $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}, \quad v = c\sqrt{1-{1 \over \gamma^2}}$$
 
     $t=t_0 \gamma$ ($t$ longer in moving frame)
 
@@ -185,8 +187,6 @@
 
     $m=m_0 \gamma$ (mass dilation)
 
-    $$v = c\sqrt{1-{1 \over \gamma^2}}$$
-
 % -----------------------
   \subsection*{Energy and work}
 
@@ -223,7 +223,7 @@
 
     \begin{itemize}
       \item monopoles: lines towards centre
-      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (or perpendicular to wire)
+      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (two magnets) or $\rightarrow$ N (single)
       \item closer field lines means larger force
       \item dot: out of page, cross: into page
       \item +ve corresponds to N pole
@@ -413,7 +413,7 @@
   % -----------------------
   \subsection*{Polarisation}
   \includegraphics[height=3.5cm]{graphics/polarisation.png} \\
-  Reduces total amplitude (transverse only)
+  Transverse only. Reduces total $A$.
 
   % -----------------------
   \subsection*{Diffraction}
@@ -609,4 +609,8 @@ $f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ &
 
 \end{multicols}
 
+\begin{center}
+  \includegraphics[height=2.95cm]{graphics/spectrum.png}
+\end{center}
+
 \end{document}