\section{MC-7~~Simple Pendulum}

\noindent OBJECTIVES:
\begin{enumerate}
\item \latex{\vspace*{-.05in}} To measure how the period of a simple pendulum depends on amplitude.
\item To measure how the pendulum period depends on length if the amplitude 
is small enough that the variation with amplitude is negligible. 
\item To measure the acceleration of gravity. 
\end{enumerate}


\noindent VIRTUAL PRE-LAB EXPERIMENT:
\begin{enumerate} 
\item\latex{\vspace*{-.05in}} For students wishing to try this experiment
on-line,  there is a simulation of a pendulum included in the web version of the lab manual...click on the Launch Virtual Pendulum
button. 
\input{toolbar/m07_tb2}

\item\latex{\vspace*{-.03in}} Start the pendulum swinging and then let it swing for about 10 periods.  Estimate the mean and standard deviation of a single measurement. 

\item\latex{\vspace*{-.03in}} Perform the required investigations  as below except use the virtual pendulum.
% If you wish to, you may enter your result into the interactive write-up form.
% When done, you may submit the form for inspection by your lab instructor.


\end{enumerate} 

\noindent  APPARATUS:
 
\latex{
\noindent\parbox{0.55\linewidth}{
\begin{itemize}
\item[] {\it Basic equipment:} Pendulum ball with bifilar support 
so ball swings in a plane parallel to wall, protractor, infrared photogate and mount, single pan balance. 
\item[] {\it Computer equipment:}   Personal computer set
to the MC-7 lab manual web-page; PASCO interface module;  photogate sensor (plugged into DIGITAL input \#1).   
\item[] NOTE:The period is the time for a complete swing of the pendulum.  For the most sensitivity the start and stop of the photogate timer should occur at the bottom of the swing where the velocity is maximum. \\ \end{itemize}}
\parbox{0.40\linewidth}{\hspace*{.1in}
\includegraphics[width=2.in]{figs/m07-01rev.eps}
\begin{center} Fig.~1: Side view of pendulum ball and bifilar support.
NOTE: $L$ and $l$ are equivalent.
\end{center}}}

\html{
\begin{rawhtml} <table width="750"> <tr><td width=50%> 
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\begin{itemize}
\item[] {\it Basic equipment:} Pendulum ball with bifilar support 
so ball swings in a plane parallel to wall, protractor, infrared photogate and mount, single pan balance. 
\item[] {\it Computer equipment:}   Personal computer set
to the MC-7 lab manual web-page; PASCO interface module;  photogate sensor and
extension jack.  
\item[] NOTE:The period is the time for a complete swing of the 
pendulum.  For the most sensitivity the start and stop of the photogate timer should occur at the bottom of the swing where the velocity is maximum.  \end{itemize}
\begin{rawhtml}
</td><td>
\end{rawhtml}

\begin{center} 
\includegraphics[width=2.2in]{figs/m07-01rev.eps} \\
Fig.~1: Side view of pendulum ball and bifilar support. NOTE: $L$ and $l$ are equivalent.
\end{center}
\begin{rawhtml}
</td></tr></table>
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}

\noindent SUGGESTED EXPERIMENTAL TECHNIQUE:
\begin{enumerate} 
\item \latex{\vspace*{-.05in}} Adjust the infrared photogate height so that the bob interrupts the beam at the bottom of the swing. (Make sure the PASCO interface is on and that the phone jack is plugged into the first slot.)  There is a red LED on the photogate that will light up when the bob interrupts the beam.  Rotating the photogate may help you to intercept the bob at the bottom of the swing (but perfect alignment is
not essential).

\item To initiate the PASCO interface software click the computer mouse on the telescope icon in the ``toolkit'' area. There will be just a single table for recording the measured period.

\input{toolbar/m07_tb1}

\item Start the pendulum swinging and then start the data acquisition by clicking the \includegraphics[height=0.18in]{figs/start_button.eps} icon.  Let the bob swing for about 17 periods.  Calculate the mean and standard deviation by simply clicking on the table statistics icon (i.e.~\fbox{$\Sigma$}).  For a single measurement of the period the standard deviation is a reasonable measure of the uncertainty.  With 17 measurements the {\em uncertainty} of the mean is  $\sigma/\sqrt{N-1}$ with $N=17$.   (This is the ``standard deviation of the mean." Refer to the \hyperref{section in this manual on errors}{error section in this manual}{on  page~\pageref{normaldist})}{normaldist}.)

\end{enumerate} 

\noindent{\bf REQUIRED INVESTIGATIONS:} (Error analysis required only for item
3) 
\begin{enumerate} 
\item\latex{\vspace*{-.05in}} Period {\it vs} Amplitude: For a pendulum of convenient length $L$ (about 0.5 m) determine the dependence of period on angular amplitude. (Do {\bf not} cut string to decrease $L$; there are clips on the strings to adjust the length.) Use  at least amplitudes between 5 and 50 degrees.  Measuring the angle is a bit hard; to avoid \hyperref{parallax}{parallax (see Appendix~}{ on page~\pageref{parallax})}{parallax} effects, position your eye so the two strings are aligned with each other and read the protractor.  [Of course, the amplitude of the swing will decrease slowly because of friction. Keep the number of swings that you time small 
enough that the amplitude changes (because of friction) by well under 5$^\circ$ during the timing. This effect is especially apparent at large amplitudes.] For each group of swings timed, record the average angular amplitude and the standard deviation in
a table in your lab book.  (Check with your instructor to see if an Excel template
document is available.  If then the columns will already have a layout including the
period versus amplitude formula.)

{\em Plot the measured period as a function of angular amplitude including
a few error bars.}

The accurate formula for period as a function of amplitude $\theta$ is:
\vspace*{-.05in}$$T = T_0\left(1 + \frac{1}{2^2} \sin^2 \frac{\theta}{2} + 
     \frac{1 (3^2)}{2^2(4^2)} \sin^4 \frac{\theta}{2} + \cdots \right) $$
\vspace*{-.05in}

where $T_0= 2\pi\sqrt{(L/g)}$ and $\theta$ is the angular amplitude. 
The results follow:

\parbox{0.6\linewidth}{
 
\begin{center}
\begin{tabular}{ccc}
$\theta$(deg) & $T/T_0$  & \% Change \\ 
~~5  &  ~~~~~~1.0005~~~~~  &  0.05 \\
 10  &  1.0019  &  0.19 \\
 15  &  1.0043  &  0.43 \\
 20  &  1.0077  &  0.77 \\
 25  &  1.0120  &  1.20 \\
 30  &  1.0174  &  1.74 \\
 35  &  1.0238  &  2.38 \\
 40  &  1.0313  &  3.13 \\
 45  &  1.0400  &  4.00 \\
 50  &  1.0498  &  4.98 \\
\end{tabular} \end{center}
}
\parbox{0.40\linewidth}{\hspace*{.05in}
\includegraphics[width=1.8in]{figs/m07.eps}
}

Compare your plot (above) with values predicted from this table.

\item \latex{\vspace*{-.05in}}{\em Period~vs.~Length:}  For an amplitude small enough that the period is  almost independent of amplitude, determine the variation of period with  length. (The length of the pendulum is the vertical distance from the  support to the center of the bob.)  Try at least five (or more) lengths 
from 0.10 m to 0.60 m.  Remember to correct $T$ to $T_0$ to account for the
offset.\\
Note: Change lengths by loosening the two spring-loaded clamps above 
protractor.

I.~Plot period ($T_0$) versus length ($L$) and extend the curve to $L=0$.  
Can you tell at a glance how $T_{0}$ depends on $L$?

II.~Plot $T_{0}^{2}$ vs L. What is the shape of this curve? What can you tell 
from this curve about the dependence of $T_{0}$ on $L$?
\item\latex{\vspace*{-.05in}} {\em Measurement of} {\bf g}:  With a pendulum length at its maximum make a  measurement of {\bf g}, the acceleration of gravity. (See your text for proof  that a simple pendulum swinging through a {\bf small} angle has $T=2\pi\sqrt{(L/g)}$ where $T$ is the period, $L$ the length and $g$ is the acceleration of gravity.)  Take enough measurements to estimate the reliability of your period 
determination. 
%\end{enumerate} 
%\begin{enumerate} 
\item \vspace*{-.05in}{\em Calculate the uncertainty in your determination of $g$.} \\ Note that $\Delta g = g~\sqrt{\left( {\Delta L}/{L} \right)^2
+ \left( {2\Delta T}/{T} \right)^2 }$. (Take a look at the ``Errors and Uncertainties'' section of this manual.) Is the accepted  value within the limits you have set? If not,  can you explain it?  (Consider optional items below. The string lineal density is 0.5 g/m, and  $ M_{\mbox{\small bob}}=62$ g.)  

\end{enumerate} 

\latex{
\noindent\parbox{0.75\linewidth}{
\noindent EXERCISE:  At right is a sketch of a compound pendulum.  There is a bar that you can swing into place which will give half the swing a length $L_1 + L_2$ and the half of the swing a length $L_2$.  The bar should be positioned so the right face just touches the string when the pendulum is at rest and hangs freely.   In your lab book first estimate the period of the motion (explain your logic) and then conduct the experiment (stating the steps in your experiment).  Is your measured value close to what you expected?  \\


}~

\parbox{0.23\linewidth}
{ \includegraphics[width=1.8in]{figs/m07-02.eps} \begin{center}  Fig.~2: A compound pendulum. \end{center}}
}

\html{
\begin{rawhtml} 
<table width="750"> <tr><td width=50%> 
\end{rawhtml}
\noindent EXERCISE:  At right is a sketch of a compound pendulum.  There is a bar
that you can swing into place which will give half the swing a length
$L_1 + L_2$ and the other half of the swing a length $L_2$.  The bar should
be positioned so the right face just touches the string when the pendulum
is at rest and hangs freely.   In your lab book first estimate the period of the
motion (explain your logic) and then conduct the experiment (stating the
steps in your experiment).  Is your measured value close to what you
expected?

\noident OPTIONAL CALCULATIONS (these pertain item 4 about): 

\begin{enumerate}
\item \vspace*{-.08in}Show that the buoyant force of air increases the period to $$T=T_{0} \left(1+ \frac{\rho_{\mbox{\small air}}}{2\rho_{\mbox{\small bob}}} \right) $$ where $T_{0}$ is the period in vacuum and $\rho$ is the density.  Test by swinging simultaneously two pendula of equal length but with bobs of quite different densities:  aluminum, lead, and wooden pendulum balls are available.  (Air resistance will also increase $T$ a comparable amount. See Birkhoff ``Hydrodynamics,''  p.~155.)
\item \vspace*{-.08in}The finite mass of the string, $m$, decreases the period to 
$$T=T_{0}\left(1-\frac{m}{12M}\right)$$ where $M$ is the mass of the bob (S.T.~Epstein  and M.G.~Olsson, {\it American Journal of Physics} 45, 671, 1977).  Correct your value of $g$ for the mass of the string. 
\end{enumerate}
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</td><td>
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\begin{center} 
\includegraphics[width=1.5in]{figs/m07-02.eps} \\
Fig.~2: A compound pendulum
\end{center}
\begin{rawhtml}
</td></tr></table>
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\begin{enumerate}
\item[3.] The finite size of a spherical bob with radius $r$ increases the period  slightly. When $L$ is the length from support to center of the sphere, then  the period becomes (see Tipler ``Physics'' 2nd ed.~p.~346, problem 26): $$T=T_{0}\sqrt{1+\frac{2r^2}{5L^2}}~\simeq~ T_0 
\left(1+\frac{r^2}{5L^2}\right)~.$$
What is the resulting percent error in your determination of ``$g$''?
\end{enumerate}

\noindent OPTIONAL EXPERIMENT:  \\
In addition to changing the angle or the length of the string there is also a box
of differing mass bobs (of approximately the same radius) that can be used to
verify the mass portion of the equation and the impact of air resistance/buoyancy.

\begin{description}
\item NOTE:
For a comprehensive discussion of pendulum corrections needed for 
an accurate measurement of $g$ to four significant figures, see
R. A. Nelson and M. G. Olsson, {\it American Journal of Physics} 54, 112, 
(1986).
\end{description}


