\input{labpreamble}
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\subsubsection*{PHSX 221L\hfill Simple Harmonic Motion\hfill Name:\underline{\hspace*{2in}}}

% \noindent
% \underline{Title:} Simple Harmonic Motion

\noindent
\underline{Object:} 
\vspace{-.25in}
\begin{enumerate}

\item To measure the spring constant of a spring, and then to check 
the dependence of the period of oscillation on spring constant and on 
mass.

\item To study the motion of a simple small-angle pendulum and check the 
dependence of the period of motion on the length of the pendulum and 
gravity.

\item To measure the torsional constant of a rod, and then to check 
the dependence of the period of oscillation of a torsional pendulum on 
the torsional constant and on the rotational inertia of the disc.

\end{enumerate}

\noindent \underline{Theory:} Many mechanical systems, when analyzed 
using Newton's laws of motion, are found to result in an equation of 
motion where the acceleration and the displacement functions of time 
differ only by a negative constant: $\ddot{x} + \omega^{2}x = 0$.  The 
solution of this differential equation predicts a periodic motion 
described by a sine (or a cosine) function with a well-defined period.  
For each of the following cases derive the differential equation, and 
determine the period of the oscillation.

\vspace{-.25in}
\begin{itemize}
\item A mass on a spring
\item A simple pendulum
\item A torsional pendulum
\end{itemize}

\noindent
\underline{Apparatus:}  Include a labled diagram of the apparatus used in 
each case.

\noindent
\underline{Procedure:} 

\underline{For the mass and spring:}
\vspace{-.25in}
\begin{enumerate}

\item Measure the spring constant $k$ by using Hooke's law and finding 
the slope of a graph of suspended weight versus stretch distance.

\item Measure the period of oscillation for each of several different 
masses (include 1/3 the mass of the spring each time).

\item Plot a graph of the period squared versus mass.  Compute the 
slope of this graph and compare with theory.

\end{enumerate}

\underline{For the simple pendulum:}
\vspace{-.25in}
\begin{enumerate}

\item Measure the period of oscillation for several different string 
lengths (with small amplitude of oscillation).

\item Plot a graph of the period squared versus string length.  Compute the 
slope of this graph and compare with theory.

\item Vary the angle of swing of a ``seconds pendulum'' (0.993 m long)
up to 90\degree\ in 10\degree\ increments to experimentally determine
the dependence of the period on it.

\item Vary the mass of the bob to experimentally determine 
the dependence of the period on it.
\end{enumerate}

\underline{For the torsional pendulum:}
\vspace{-.25in}
\begin{enumerate}

\item Measure the torsional constant $\kappa$ of the rod by using the 
rotational version of Hooke's Law: $\tau = -\kappa\theta$ and finding 
the slope of a plot of $\tau$ versus\ $\theta$.

\item Measure the period of oscillation for each of several different 
rotational inertias of the disc (create different $I$'s by adding 
hoops to the disc).

\item Plot a graph of the period squared versus rotational inertia.  Compute the 
slope of this graph and compare with theory.

\end{enumerate}


\noindent 
\underline{Conclusions:} 
\vspace{-.25in}
\begin{enumerate}

\item Does the period of a simple small-angle pendulum depend on the
angle of swing?  Should it?  What if the angle of swing is not kept
small?  In the large-angle case is it still simple harmonic motion?

\item Does the period of a simple pendulum depend on the mass of the 
bob?  Should it?

\item Note that instead of plotting period squared, you could do a 
least squares fit of your data expecting a parabolic fit.  You should 
be able to match the constants obtained by the least squares parabolic 
fit with the appropriate information obtained on your graphs above.

\item Summarize and evaluate your experiment.  Include mention of 
likely sources of error and how you dealt with them.
\end{enumerate}


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