\input{labpreamble}
\begin{document}
\subsubsection*{PHSX 101L\hfill Experiment 4\hfill Name:\underline{\hspace*{2in}}}

\underline{Title:} Acceleration of a Freely-Falling Body

\underline{Object:} To study the motion of a freely-falling 
body including to measure its acceleration.

\underline{Introduction:} The law of falling bodies says that all 
objects in free fall (motion under the influence of gravity only) have 
the same acceleration; near the surface of the earth, it has a value 
of $a_{g}$ (often simply called $g$).

Recently developed techniques in 
measurement have resulted in very precise experimental results on the 
value of $a_{g}$.  At the National Bureau of Standards Labs in 
Maryland, free fall acceleration has been determined to 
be $a_{g}=9.810424$~m/s$^{2}$, while at CSIRO National Laboratory in 
Sydney, Australia, $a_{g}$ has been measured as $a_{g}= 
9.7967134$~m/s$^{2}$.  The high precision of these measurements is 
indicated by the large number of significant figures reported.  Such 
accuracy is not always possible, mostly because of difficulties in 
measuring short fall times.  For many years the most reliable 
measurements were made on slowly accelerating systems, from which 
$a_{g}$ could be calculated; an example is the pulley system on 
display at the Smithsonian Institute.  The difference in the above 
values of $a_{g}$ is due to the fact that the value actually varies 
from point to point on the earth's surface.  One formula, which takes 
into account latitude and altitude, predicts that $a_{g}$ in Ephraim 
should be $a_{g}=9.7959$~m/s$^{2}$ [CRC, 56th~ed.  p.~F87].

Your experiment today uses a fast timing mechanism to observe a metal 
cylinder as it falls freely under the pull of gravity.  A high voltage 
between two vertical wires causes a spark to jump between the wires 
every 1/30 second.  The spark passes through the falling cylinder and 
then through the special paper tape, making a small dot on the tape, 
thus marking the position of the falling cylinder at each 1/30 
second.  The rate at which the spark is produced is controlled by the 
60 hertz (or 60 cycles per second) power line, which is a very stable 
timing source.

\underline{Apparatus:}  Draw a diagram of the apparatus.
\vspace{3.0in}

\underline{Procedure:} 
\vspace{-.25in}
\begin{enumerate}
\item Level the apparatus.  Verify levelness by dropping the cylinder 
in a dry run with no sparks.
\item Start the sparker, release the cylinder, and then release the 
sparker button after the cylinder has landed, all in quick succession.
\end{enumerate}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline
assumed&  displacement  $\boldsymbol{d}$ &$\Delta\boldsymbol{d}$          &$ \boldsymbol{v} $& $\Delta\boldsymbol{v}$&$\boldsymbol{a}$ \\
 time  & (from 1st  dot)                 &$ \ \ \ \boldsymbol{d}_{2}-\boldsymbol{d}_{1}$\ \ \ &\ \ \ $\Delta \boldsymbol{d}/\Delta t \ \ \ $  
 &$ \ \ \  \boldsymbol{v}_{2}-\boldsymbol{v}_{1}$  \ \ \ & \ \ \  $\Delta \boldsymbol{v}/\Delta t $ \ \ \   \\
  (s)  &(cm,$\downarrow$)&(cm,$\downarrow$)&(cm/s,$\downarrow$)
  &(cm/s,$\downarrow$)&(cm/s$^{2}$,$\downarrow$)  \\ \hline
       &        & xxxxx & xxxxx  & xxxxx  & xxxxx \\ \cline{3-4}
  0    &   0.00 &       &        & xxxxx  & xxxxx \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 1/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 2/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 3/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 4/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 5/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 6/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 7/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 8/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
 9/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
10/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
11/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
12/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
13/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &        &       \\ \cline{3-4}
14/30  &        &       &        &        &       \\ \cline{1-2} \cline{5-6}
       &        &       &        &  xxxxx & xxxxx \\ \cline{3-4}
15/30  &        & xxxxx &  xxxxx &  xxxxx & xxxxx \\ \hline 
\end{tabular}
\vspace{.1in}
\caption{\hfill \hspace*{3.6in}\label{results} ave. $\boldsymbol{a}$ = {\underline{\hspace{.8in}}}}
\end{center}
\end{table}

\underline{Analysis:} 
\vspace{-.25in}
\begin{enumerate}
\item Call the first \emph{good} dot time zero and location zero (you 
may not want to use the very first dot).  Then 
record in table~\ref{results} the displacements for the rest of the 
dots relative to \emph{that} dot.

\item Subtract successive displacements in pairs to obtain the 
information for the $\Delta\boldsymbol{d}$ column in table~\ref{results}.  
Verify the $\Delta\boldsymbol{d}$ by measuring between successive dots with a 
ruler.  Then divide each $\Delta\boldsymbol{d}$ by $\Delta t$ (which is 1/30 
s) to obtain the average velocity for each time interval ({\it i.e.}, 
between successive dots).

\item Subtract successive velocities in pairs to obtain the 
information for the $\Delta\boldsymbol{v}$ column in table~\ref{results}.  
Then divide each $\Delta\boldsymbol{v}$ by $\Delta t$ to obtain the 
acceleration for each interval ({\it i.e.}, between successive 
velocities).  Compute the average of all of the numbers in the acceleration 
column and record it just below table~\ref{results}.  (Don't forget 
units.)

\item Make a graph of velocity versus time.  The value on the vertical 
axis for each data point is the average velocity (acquired from table 
\ref{results}) during that time interval.  The value on the horizontal 
axis is halfway between the tic marks representing the times 1/30~s 
apart.  See example below.

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\vspace{-.1in}
% \item Make a graph of velocity versus time.  Make horizontal bars 
% between the known times at a height to indicate the average velocity 
% between successive dots.  (In reality the velocity is changing all 
% the time, and what we have is the \emph{average} \boldsymbol{v} during each 1/30~s 
% time interval.)

% \begin{picture}(500,150)
% \thicklines
% \put (15,15){\line(0,1){130}}
% \put (15,15){\line(1,0){180}}
% %\put (15,35){\line(2,3){90}}
% \put (80,0){\makebox(0,0)[b]{time (s)}}
% \put (0,50){\shortstack{v\\e\\l\\o\\c\\i\\t\\y}}
% \thinlines
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% \put(115,100){\line(1,0){25}}
% \put(140,121){\line(1,0){25}}
% \thinlines
% \end{picture}
%\vspace{.3in}

% \item Draw a point at each 1/30~s time on the graph at a height half-way 
% between adjacent horizontal velocity bars; these points represent the 
% velocity \emph{at} the times corresponding to the dots on the data tape.
 
\item Draw a best-fit straight line between the points on the graph and 
compute its slope and read the intercept off the $v$ axis.  Think of 
$v=v_{0}+at$ in terms of $y=mx+b$; the slope of your line will be 
$a$ and the intercept will be $v_{0}$.\label{straightline}

\item What do you expect a graph of displacement versus time to look 
like?  \underline{\hspace{1.2in}} Make a graph of displacement vs.\ 
time using the first two columns of table \ref{results} as your data.  
Draw a best-fit smooth curve through the data points.  Is the shape of 
the curve what you expected?


\end{enumerate}

\goodbreak
\underline{Analysis:}
\vspace{-.25in}
\begin{enumerate}

\item 
\begin{enumerate}

% \item Did the graph of your velocity vs.~time data conform to a 
% straight line?
% \vspace{.2in}

\item Did your straight line in step \ref{straightline} above go 
through the origin?  Why or why not?  
\vspace{.4in}

\item What value of acceleration did you compute from the slope of 
this line?
\vspace{.4in}
\end{enumerate}

\item Compute the percent difference between each of your two measured 
values of $a_{g}$ (the slope of the line and the average from table 
\ref{results}) and the value of $a_{g}=979.59$~cm/s$^{2}$ predicted by 
the formula for Ephraim.

\vspace{.6in}

\item On the graph of displacement vs.\ time also plot the 
theoretical curve of $x=v_{0}t + \frac{1}{2}at^{2}$ using values of 
$v_{0}$ and $a$ obtained from step \ref{straightline} of the analysis.  How does 
this theoretical curve compare with the experimental one?

\vspace{.6in}

\end{enumerate}

\underline{Questions:}
\vspace{-.25in}
\begin{enumerate}
\item
If an object were thrown straight up
\begin{enumerate}
	\item would it be in free fall (neglecting air resistance)?\vspace{.4in}
	\item what would its acceleration be on the way up (both magnitude 
	and direction)?\vspace{.4in}
\end{enumerate}

\item In another experiment a body falling straight down at one instant 
has a speed of 3.00~m/s.
\begin{enumerate}
\item Compute its speed 0.50~s later.
\vspace{.4in}

\item Compute the distance it would fall during this 0.50~s.
\vspace{.4in}
\end{enumerate}

\end{enumerate}

\vspace{-.25in}
\underline{Conclusions:}
\vspace{-.25in}


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\end{document}
