\input{labpreamble}
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\subsubsection*{PHSX 101L\hfill Experiment 11\hfill Name:\underline{\hspace*{2in}}}

\underline{Title:} Simple Harmonic Motion
\vspace{-.1in}

\underline{Object:} To measure the spring constant, $k$, for a spring; and 
to measure the period of oscillation for a mass on a spring and for a 
simple pendulum, and compare the results with theory.
\vspace{-.1in}

\underline{Theory:} Simple harmonic motion is a subject of great interest 
and importance in all fields of physical science.  This phenomenon occurs 
in countless mechanical, electrical, and atomic systems of interest.  We 
will consider two simple examples of systems where simple harmonic motion 
can be easily studied: that of an object with mass $m$ oscillating at the end of a 
spring, and of a bob swinging back and forth at the end of a cord to 
form a simple pendulum.  In both of these examples the motion is a 
steady oscillation with the period $T$ given by standard formulas:
\begin{equation}
{\rm for\ the\ mass\ and\ spring:\ \ } T=2\pi\sqrt{m/k}\label{spring}
\end{equation}
\begin{equation}
{\rm for\ the\ simple\  pendulum:\ \ } T=2\pi\sqrt{L/g}\label{pendulum}
\end{equation}
\vspace{-.1in}

\underline{Procedure and Results:} Show all work in each step below.  Measure periods 
by letting the system oscillate for several periods and dividing the 
total time by the number of oscillations.

\vspace{-.1in}

\begin{enumerate}
\vspace{-.2in}

\item Hooke's Law for springs is $F=-kx$.  (The negative sign merely 
indicates that it is a restoring force.)  The spring constant $k$ is 
the applied force per distance stretched (units of N/m).  Determine 
the spring constant of your spring by hanging different amounts of 
weight on the spring and note the resting stretch in each case.  Make 
a graph of force vs.\ stretch on the next page.  The slope of the 
best-fit straight line through $(0,0)$ is $k$.  You will use this 
value of $k$ in procedure \ref{springgraph}.

\vspace{.2in}
\hspace{1in}
\begin{tabular}{|c|c|}                   \hline
 \ \ Stretch\ \   & \ \ Force \ \    \\  
       (m)        &      (N)         \\  \hline
        0         &       0          \\  \hline
                  &                  \\  \hline
                  &                  \\  \hline
                  &                  \\  \hline
                  &                  \\  \hline
\end{tabular}       
\hspace{1in}  $k=$ \underline{\hspace{1.2in}}

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\vspace{.2in}
\item With your spring hanging vertically, hang different masses on it 
and measure the period for small oscillations.  Do this by beginning with 
50~g (the hanger alone) and increasing by 50~g each trial for about six 
trials.  Use your calculator to square the period for the third column.

\hspace{.7in}
\begin{tabular}{|c|c|c|}                                       \hline
\ \ \ Mass\ \ \       & \ \ Period \ \   &  Period$^2$     \\  
      (kg)            &      (s)         &    (s$^2$)      \\  \hline
                      &                  &                 \\  \hline
                      &                  &                 \\  \hline
                      &                  &                 \\  \hline
                      &                  &                 \\  \hline
                      &                  &                 \\  \hline
                      &                  &                 \\  \hline
\end{tabular} 


\marginparwidth=3in \marginparsep=-3in \marginpar{You will get better 
results on your graph if you record the hanging mass plus 1/3 the mass 
of the spring on each line of the table.}

\vspace{.2in}
\item\label{springgraph} Using graph paper (next page), plot period squared vs.\ mass 
and compute the slope of the best-fit straight line.  Compare with the 
theoretical value obtained by squaring both sides of equation~\ref{spring}, 
and give the percent difference.

\vspace{.2in}
Slope = \underline{\hspace{1in}}\hfill Theoretical 
Value = \underline{\hspace{1in}}\hfill \% diff = 
\underline{\hspace{1in}}

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\vspace{.2in}
\item Using a bob and string, form a simple pendulum.  Measure the 
period (for small oscillations, $\theta < 20^\circ$) for different lengths 
$L$ (measured to the middle of the bob) of the pendulum.  Start with a 
length of 15~cm and increase by 15~cm each trial until maximum length is 
reached.  Use your calculator to square the period for the third column.

\begin{center}
\begin{tabular}{|c|c|c|}                                   \hline
 \ \ Length\ \    & \ \ Period \ \   &  Period$^2$     \\  
     (m)          &      (s)         &    (s$^2$)      \\  \hline
                  &                  &                 \\  \hline
                  &                  &                 \\  \hline
                  &                  &                 \\  \hline
                  &                  &                 \\  \hline
                  &                  &                 \\  \hline
                  &                  &                 \\  \hline
\end{tabular}       
\end{center}

\item Using graph paper (next page), plot period squared vs.\ length
and compute the slope of the best-fit straight line.  Compare with the
theoretical value obtained by squaring both sides of
equation~\ref{pendulum}, and give the percent difference.

\vspace{.2in}
Slope = \underline{\hspace{1in}}\hfill Theoretical 
Value = \underline{\hspace{1in}}\hfill \% diff = 
\underline{\hspace{1in}}

\item Experimentally determine if the mass of the bob affects the 
period of a simple pendulum.  Experimentally determine if the angle of 
swing (amplitude of oscillation) affects the period of of a simple 
pendlumu, first just for small angles, but then for angles clear up to 
90\degree.  Observations:
\vspace{.9in} 

\end{enumerate}

\underline{Questions:}
\vspace{-.1in}
\begin{enumerate}
\item When a 5.0~kg mass is hung from a certain spring it stretches down 
35~cm to reach equilibrium.  If the mass is then set into oscillatory 
motion, what will be the period of these oscillations?
%Answer: T = 1.1874 s
\vspace{.8in}
\item Suppose you have a clock with a pendulum that has a period of 
oscillation of 2~s.  How much should you add to its length in order to 
increase its period to 2.5~s?
%Answer: \Delta L = 0.5586 m
\vspace{.8in}
\end{enumerate}

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\newpage
\underline{Conclusions:}



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