\subsection{MC-1b~~Errors and the Density of a Solid}

%\begin{htmlonly}
\begin{rawhtml}
<hr><table border="0" cellpadding="2">
<tr><td>Quick links on this page:</td> 
<td><A HREF="#introduction">  <strong> Introduction  </strong> </A></td> 
<td><A HREF="#micrometer">  <strong> Micrometer  </strong> </A></td> 
<td><A HREF="#vernier">  <strong> Vernier  </strong> </A> </td>
<td><A HREF="#toolbar">  <strong> Toolbar  </strong> </A></td> 
<td><A HREF="#density">  <strong> Density  </strong> </A></td>
</tr> </table>
<hr>
\end{rawhtml}
%<td><A HREF="../../lab/info/Information.html">
%<IMG SRC="../../PHYS207/gif/info2.gif" ALT="Information"></A>
%</td>
%\end{htmlonly}

\noindent
OBJECTIVES:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}
To learn about systematic and random errors; to understand 
significant figures; to estimate the reliability of one's measurements; 
and to calculate the reliability of the final result. 

NOTE:  This experiment illustrates the earlier sections on Errors 
and Significant Figures.  The actual density of the metal is incidental.  
However, the accuracy of your estimate of reliability will 
show whether you have mastered the material in the earlier sections.
\end{enumerate}

\noindent APPARATUS:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}
Metal cylinders of varying sizes, micrometer and vernier 
calipers, precision gauge blocks, precision balance. \\
\latex{\noindent {\bf Precautions:}}  
\end{enumerate}

\begin{rawhtml}
<HR> <P>
    <a name="Demo"><h3><u> <font color="ff0000">Pre-lab Quiz</font> </u></h3></a>
     <blockquote>You should be able to complete this brief quiz before proceeding.
     </blockquote>

<ul><ul>
    <li><a href="../js/q207_m01.html" target="_blank" 
    >Expt. M01 Quiz</a>

</ul></ul>
</P>
<HR>
\end{rawhtml}

\begin{rawhtml}  
<IMG SRC="../gif/caution.gif" ALIGN=center> <br>
PRECAUTIONS:  
\end{rawhtml}

\begin{enumerate}
\item[] \latex{\vspace*{-.05in}\fbox{\parbox{5.2in}{\sloppy Avoid dropping or deforming in any other way the metal  cylinders.  Avoid damage to the precision screw of the micrometer by turning only the friction head to open or close the caliper jaws. Be sure  to disengage the caliper lock before using. (The caliper lock lets you  preserve a reading.)  
\underline{Improper weighing procedures may damage the precision balance.}
Consult your instructor if in doubt.  In handling the gauge 
blocks avoid touching the polished surfaces since body acids are corrosive.}}}
\html{Avoid dropping or deforming in any other way the metal  cylinders.  Avoid damage to the precision screw of the micrometer by turning only the friction head to open or close the caliper jaws. Be sure  to disengage the caliper lock before using. (The caliper lock lets you  preserve a reading.)  
\underline{Improper weighing procedures may damage the precision balance.}
Consult your instructor if in doubt.  In handling the gauge 
blocks avoid touching the polished surfaces since body acids are corrosive.}

\end{enumerate}
\begin{rawhtml}  
<IMG SRC="../gif/caution.gif" ALIGN=center>  <br> 
\end{rawhtml}

\begin{rawhtml}  
<a name="introduction"><br></a>
\end{rawhtml}

\noindent INTRODUCTION:
\begin{enumerate}
\item[]\latex{\vspace*{-.05in}}First read the material on \hyperref{Errors and 
Significant Figures in the Manual Introduction}{Errors and 
Significant Figures in the Manual Introduction}
{)}{error_intro}. Since density is the mass per unit 
volume, you must measure the mass (on a balance) and  compute the volume ($h \pi r^2  = h \pi d^2 / 4$) from measurements of the cylinder's dimensions where $d$ is the diameter and $h$ is the height.
Any one of three length measuring devices may be used. These include a
micrometer, a vernier caliper and/or a simple metric ruler.  The micrometer will permit the highest precision measurements but using one can be cumbersome, especially when reading the vernier scale.  All methods will demonstrate the aforementioned objectives. Your instructor will give you guidance in choosing an appropriate measurement device.
\end{enumerate}

\begin{rawhtml}  
<a name="micrometer"> </a>
\end{rawhtml}


\noindent THE MICROMETER:
\begin{enumerate}
\item[]\latex{\vspace*{-.05in}}Record the serial number of your micrometer.  Then  familiarize yourself with the operation of the caliper and the reading of  the scales: \hyperref{work through 
Appendix}{{\bf work through Appendix} }{}{caliper_info} on the Micrometer.   Note that use of the ``friction'' head in closing the jaws insures  the same pressure on the measured object each time.  Always estimate tenths of the smallest division.  Some micrometers have verniers to assist the estimation.
\end{enumerate} \latex{\vspace*{-.05in}}

\begin{rawhtml}  
<a name="vernier"> </a>
\end{rawhtml}

\noindent THE VERNIER CALIPER:
\begin{enumerate}
\item[]\latex{\vspace*{-.05in}}\hyperref{Work through
Appendix}{{\bf Work through Appendix} }{}{vernier_info} on the Vernier Caliper.
You may also wish to try this \htmladdnormallink{java applet vernier}{http://webphysics.davidson.edu/Applets/TaiwanUniv/ruler/vernier.html}
\latex{at  http://webphysics.davidson.edu/}.
Experiment with one of the large verniers in the lab until you  are sure you understand it. Note that verniers need not be decimal:  for  many inch scales the vernier estimates 1/8's of the 1/16 inch  division, i.e. 1/128's of an inch.  However  vernier calipers divide  the inch into 50 divisions and the vernier estimates 1/25 of the 1/50 inch  divisions, i.e. 1/1000 inch or 1 mil.  The vernier was invented 1631 by
Pierre Vernier.
\end{enumerate} \latex{\vspace*{-.05in}}

\htmlrule
\begin{rawhtml}
<IMG SRC="../gif/red.gif" ALIGN=MIDDLE> <br>
\end{rawhtml}

\vspace*{.05in}
{\bf Precautions on use of the calipers:}  

\begin{enumerate}
\item Unclamp {\bf both} top thumbscrews to permit moving caliper jaws.

\item Open caliper to within a few mm of the dimension being measured.
 
\item Close right thumbscrew to lock position of lower horizontal knurled
cylinder which executes fine motions of caliper jaw.  
{\bf Never over tighten!}
\end{enumerate} 

\vspace*{.1in} 
\begin{rawhtml}
<IMG SRC="../gif/red.gif" ALIGN=MIDDLE>
\end{rawhtml}

\noindent CALIBRATION OF THE MICROMETER (or VERNIER, ETC.): 
\begin{enumerate}

\item OPTIONAL:  Wipe the micrometer caliper jaws with 
cleaning paper.  Then determine the zero error by closing the jaws.  Make and record five readings.  The variation of these repeated readings gives you an estimate of the reproducibility of the measurements.  (For those using the micrometers they have been given a small zero error.
Thus a zero error  correction is necessary.)  In general any measurement device can have a zero error.

\item Measure all four calibration gauge blocks  (6, 12, 18 and 24 mm): Set the gauge blocks on end, well-in from the edge of the table, and thus freeing both hands to handle the caliper.   Record the actual (uncorrected)  reading.  A single measurement of each block will suffice.  
 
\item Plot a correction curve for your micrometer, i.e. plot errors as 
ordinates and nominal blocks sizes (0, 6, 12, 18, and 24 mm) as abscissa.   Normally the correction will not vary from block to block by more than 0.003 mm (for the micrometer).  If it is larger, consult your instructor.

\end{enumerate} 

 

\begin{rawhtml}  
<a name="toolbar"> </a>
\end{rawhtml}

\input{toolbar/m01_tb1}

\begin{rawhtml}  
<a name="density"><br></a>
\end{rawhtml}


\noindent DENSITY DETERMINATION:  
\begin{enumerate} 

\item Make five measurements (should be in millimeters) of the height and five of the diameter.  Since our object is to determine the volume 
of the cylinder, distribute your measurements so as to 
get an appropriate average length and average diameter.  Avoid 
any small projections which would result in a misleading measurement.  If  not possible to avoid, estimate their importance to the result.  Record actual readings and indicate, in your lab book, how you distributed them.

\item Calculate the average length, average diameters and the
respective standard deviation.  

\item Use your correction curve to correct these average readings.  If you were to use the uncorrected values, how much relative error would this introduce?

\item Weigh the cylinder twice  
      on the electronic balance; estimate to 0.1 mg.

\item From the average dimensions and the mass, calculate the 
volume and density.  Make a \hyperref{quantitative}{quantitative (refer
to the Error and Uncertainties section }{on page~\pageref{error_top})}{error_top}
estimate of the uncertainty in 
the density. The sample worksheet asks for both the maximum and minimum values.   You should use, as your starting point:  
$$  \mbox{Density}(\rho\pm \Delta \rho) =\rho(h\pm\Delta h,d \pm \Delta d, m \pm \Delta
m)=\frac{m}{\pi (d/2)^2~h}  $$


\item Compare the density with the tabulated value.  Tabulated values are  averages over samples whose densities vary slightly depending upon how the  material was cast and worked; also on impurity concentrations.


\begin{rawhtml} <br>
<color=red>Link to a web-based</color> <A HREF="http://www.webelements.com/webelements/scholar/index.html" 
target="_blank"> periodic table</A>.
\end{rawhtml} 

\item In your notebook or lab form summarize the data and results.  
Also record  
your result on the blackboard.  Is the distribution of blackboard values 
reasonable, i.e.~\hyperref{``normal'' distribution}{``normal'' distribution
(refer to the section }{on  page~\pageref{normaldist})}{normaldist}? 


\end{enumerate}

\begin{enumerate}
\item[]\latex{\vspace*{-.05in}}
To test how accurately you can estimate a fraction of a 
division, estimate the fractions on the vernier caliper before reading the 
vernier.  Record both your estimate and the vernier reading.
\end{enumerate}
  
\noindent Related facts and URL links:
\begin{enumerate}
\item[]\latex{\vspace*{-.05in}}{\bf Question:} Why are there are {\em exactly} 25.4 mm in 1 inch?

{\bf Answer}(not verified): The Treaty of the Meter (Convention du Metre) in the 
late 19th century
established the first centralized international system of metrology.  This
defined the meter.\\

In 1959, the countries of the world that were using Imperial units defined them
uniformly based on the metric units.  The inch was simply defined that way and
agreed to by all. Before 1959, different countries related inches to meters in
other ways.  Among them was the United States.  The Metric Act of 1866 defined
the meter in terms of inches (i.e., before the Treaty of the Meter), and that
relationship had continued to been used even after the Treaty fixed the length
of the meter.\\

Changing the definition in the U.S. in 1959 caused very little problem,
except for the U.S. Geological Survey.  When you deal with things 10$^5$
meters big (like the sizes of the states), even 1 part in 10$^5$ changes
affect the specifications of boundary lines in significant ways.  So, to
this day the U.S. has two different systems of inches, feet, and yards (in
the ratio of 36:3:1, for both). There's the usual inch, foot, and yard;
and there's the survey foot (and inch and yard), which is based on the
pre-1959 definition.  (The ambiguity goes away, of course, when metric
specification is used.  Newer USGS maps are metric.) \\
%jhardis@tcs.wap.org (Jonathan E. Hardis)

Links to metrology  site(s): \\ 
\htmladdnormallink{National Institute of Science \&
Technology}{http://ts.nist.gov/}
\begin{latexonly} at http://www.nist.gov/
\end{latexonly} 

\end{enumerate}

\begin{latexonly}
\newpage
\begin{center}
{\Large Experiment 1b Worksheet}
\end{center}


\begin{enumerate}
\item{} 
\begin{tabular}[t]{|rcc|}
\hline 
CALIBRATION TABLE &&\\
Micrometer Seriel \# = & \underline{\hspace*{1in}} &\\
MEASUREMENTS  ~~~~ &&\\
Zero gap 1 = &\underline{\hspace*{1in}}& $\pm$ \underline{\hspace*{1in}}\\
         2 = &\underline{\hspace*{1in}}&\\
         3 = &\underline{\hspace*{1in}}&\\
         4 = &\underline{\hspace*{1in}}&\\
         5 = &\underline{\hspace*{1in}}&\\
Mean$\pm  {\sigma}$= &\underline{\hspace*{1in}}&$\pm$ \underline{\hspace*{1in}}\\ 
Gauge block 6 mm  &\underline{\hspace*{1in}}&\\
12 mm &\underline{\hspace*{1in}}&\\
18 mm &\underline{\hspace*{1in}}&\\
24 mm &\underline{\hspace*{1in}}&\\
\hline  
\end{tabular}  

\item
Make a plot of Gauge block thickness (x-axis) versus Micrometer reading.\\ 
Next fit the data to a line; this generates a correction curve 
where \\Corrected value = actual $\times$ slope + intercept\\
slope = \underline{\hspace*{1in}}\\
intercept = \underline{\hspace*{1in}} 

\item Now for the unknown value \\ \\
\begin{tabular}{|r|c|c|c|}
\hline  CYLINDER  & Height ({\underline{$\pm$\hspace*{0.5in}}}) & Width (\underline{$\pm$\hspace*{0.5in}}) & Mass (\underline{$\pm$\hspace*{0.5in}})\\
1= &\underline{\hspace*{1in}}  &\underline{\hspace*{1in}} & \underline{\hspace*{1in}} \\
2= & \underline{\hspace*{1in}}  &\underline{\hspace*{1in}} & \underline{\hspace*{1in}} \\
3= & \underline{\hspace*{1in}}  &\underline{\hspace*{1in}} &\underline{\hspace*{1in}} \\
4= & \underline{\hspace*{1in}}  &\underline{\hspace*{1in}} &\underline{\hspace*{1in}} \\
5= & \underline{\hspace*{1in}}  & \underline{\hspace*{1in}} & \underline{\hspace*{1in}} \\
\hline           Mean$\pm  {\sigma}$ =  &\underline{\hspace*{0.6in}$\pm$\hspace*{0.4in}}
&\underline{\hspace*{0.6in}$\pm$\hspace*{0.4in}}
&\underline{\hspace*{0.6in}$\pm$\hspace*{0.4in}}
\\ 
Corrected  Mean$\pm  {\sigma}$=   
&\underline{\hspace*{0.6in}$\pm$\hspace*{0.4in}}
&\underline{\hspace*{0.6in}$\pm$\hspace*{0.4in}}
&\underline{\hspace*{0.6in}$\pm$\hspace*{0.4in}}
\\ 
Density = &\underline{\hspace*{1in}} &&\\
Max. Density = &\underline{\hspace*{1in}} && \\
Min. Density = &\underline{\hspace*{1in}} && \\
\hline   
\end{tabular} 

\item Final result, density = \underline{\hspace*{0.6in}$\pm$\hspace*{0.4in}}

\item Cut and tape this into you lab notebook.

\item Answer the following questions in your notebook. \\
A. Identify  two sources of systematic error and give their magnitude. \\
B. Identify two examples of random error and give their magnitude.
\end{enumerate}
\end{latexonly}
\newpage 

 
%\vspace{.2in}
%\hrulefill 
%\vspace{.2in}
