\section{Appendices}
\subsection{Precision Measurement Devices}

\noindent
{\bf Vernier Calipers:} \label{vernier_info} \latex{\medskip} \\

\begin{figure}[h]  
\ \ \includegraphics[width=6.2in]{figs/l103/m01-1.eps}
\caption{The vernier caliper}
\end{figure}
     
\pind{
A Vernier consists of a fixed scale and a moving {\em vernier} scale.  In a
metric vernier the fixed scale is marked in centimeters and millimeters, the
vernier scale is nine millimeters long, and is divided into ten parts each 0.9
millimeters long.  The distances of each line from the first are therefore 0.9,
1.8, 2.7, $\dots,${\it mm} or generally: $d_i=0.9 \times i,$ where $d_i$ is the
distance between the zero line and the $i^{th}$ line of the vernier scale.  If
the vernier caliper is closed, so that the two jaws touch each other, the zero
of the fixed scale should coincide with the zero of the vernier scale. Opening
the jaws 0.03 {\it cm} = 0.3 {\it mm} will cause the fourth line (the {\em
three} line which is a distance of 2.7 {\it mm} from the zero line of the of the
vernier scale) to coincide with the 3 {\it mm} line of the fixed scale as shown
below.  }

\begin{figure}[h] \centering 
\includegraphics[width=2.60in]{figs/l103/m01-9.eps}
\caption {The vernier reads 0.03 {\it cm}}
\end{figure}
\pind{Below is another example of vernier reading; the arrow shows which
mark on the vernier scale is being used.
 }
\begin{figure}[h] \centering
\ \ \includegraphics[width=2.6in]{figs/l103/m01-2.eps}
\caption{The vernier reads  9.13 {\it cm} }
\end{figure}

\noindent EXERCISES: 
\begin{enumerate}
\epind{ 
\item[A.] Close the vernier
and observe that the first vernier mark coincides with the zero of the 
centimeter scale.  
\item[B.] Open the jaws of the vernier very slowly and observe how the 
different vernier marks coincide successively with the millimeter 
marks on the fixed scale: the first mark coincides with the 1 {\it mm} mark on the 
fixed scale; then the second mark coincides with the 2 {\it mm} mark on the 
fixed scale; then the third mark coincides with the 3 {\it mm} mark on the 
fixed scale and so on.
\item[C.] Estimate the dimension of an object using a meter stick and then
Use the vernier caliper to measure the dimension precisely. 
\item[D.] In the four examples of Fig.~4 determine the actual reading.
}
\end{enumerate}
\begin{figure}[h] \centering
\includegraphics[width=3.6in]{figs/l103/m01-3.eps} \\
\includegraphics[width=3.6in]{figs/l103/m01-4.eps}
\caption{Test cases} 
\end{figure}

\noindent
{\bf Micrometer:} \label{caliper_info} \latex{\medskip} \\ 


\begin{figure}[h]  \centering
\includegraphics[width=5.3in]{figs/l103/m01-5.eps}
\caption{The micrometer calipers}
\end{figure}     
\pind{

A micrometer can measure distances with more precision than a vernier caliper.
The micrometer has a 0.5 {\it mm} pitch screw, this means that you read millimeters
and half millimeters along the barrel. The sleeve is divided into 50 divisions
corresponding to one hundredth of a millimeter (0.01 {\it mm}) or 10 $\mu$
each. The vernier scale on the micrometer barrel has ten divisions, marked from
2 to 10 in steps of two.  The ``zero'' line is not marked `0', but is longer
than the others.  The vernier allows you to read to the nearest thousandth of a
millimeter, {\it i.e.}, to the nearest micron (0.001 {\it mm} = 1 $\mu$).} \\ \\

\begin{rawhtml}  
<IMG SRC="http://romano.physics.wisc.edu/lab/gif/caution.gif" ALIGN=center> <br><br> 
\end{rawhtml} 
\hspace*{.48in} {\bf Precaution:} %\vspace*{.1in}  
\parbox{4.5in}{ Great care must be taken in using the
micrometer caliper; A ratchet knob is provided for closing the caliper on the
object being measured without exerting too much force.  Treat the micrometer
with care, {\bf ALWAYS} close the calipers using the ratchet knob, this prevents
tightening the screw too strongly.  Closing the calipers too hard damages the
precision screw.} 
\begin{rawhtml}  
<IMG SRC="http://romano.physics.wisc.edu/lab/gif/caution.gif" ALIGN=center> <br><br> 
\end{rawhtml}

\newpage

\pind{Below are two examples of micrometer reading; the arrow shows which mark
 on the vernier scale is being used. }

\begin{figure}[h] \centering 
\includegraphics[width=3.65in]{figs/l103/m01-6.eps}
\caption{The micrometer reads 20.912 {\it mm}}
\end{figure}   
  \begin{figure}[h!]  \centering 
\includegraphics[width=3.25in]{figs/l103/m01-10.eps}
\caption{The micrometer reads 3 $\mu$}
\end{figure}     

\noindent
\pind{In Fig.~7 the zero line on the barrel is barely visible, and the vernier reads
0.003 {\it mm} = 3 $\mu$; the zero error is $\epsilon_0\ =\ 3 \mu$.}

\newpage
\noindent
\pind{A negative zero error, as shown below requires a moment of thought.}
\begin{figure}[h!]   \centering 
\includegraphics[width=3.65in]{figs/l103/m01-8.eps}
\caption{The micrometer reads -4 $\mu$}
\end{figure}     

\pind{In Fig.~8 the zero line on the barrel of the micrometer is obscured by 
the sleeve, (the ``zero" line on the sleeve is above the  ``zero" line
on the barrel) 
this corresponds to a reading
of -0.5 {\it mm}; the vernier 
reads  0.496 {\it mm} the zero error is then $\epsilon_0 = -0.5 + 0.496 = -0.004 \mbox{\it mm}\ = -4\ \mu$.}
