\input{labpreamble}
\begin{document}


\subsubsection*{PHSX 221L\hfill Rotational Inertia\hfill Name:\underline{\hspace*{2in}}}

% \noindent
% \underline{Title:} Rotational Inertia

\noindent
\underline{Object:} To measure the rotational inertia of a wheel and 
axle combination.

\noindent
\underline{Theory:} The rotational inertia (or moment of 
inertia) of an object is 
defined by the integral:
\begin{equation}
I=\int_{}^{}r^{2}dm.
\end{equation}
The result for a disc is $I= \frac{1}{2}MR^{2}$ (around the axis of 
symmetry), with different 
formul\ae\ for other geometrical shapes.

\noindent
Newton's second law of motion for rotating objects can be written as
\begin{equation}
\tau_{\rm net}=I\alpha.
\end{equation}

\noindent
\underline{Apparatus:} The apparatus for this experiment consists of a 
heavy metal wheel with a cylindrical axle attached.  The wheel is 
mounted on a frame which allows it to rotate freely about a fixed 
axis.  A string is wrapped around the outside of either the wheel or 
the axle with a known mass attached and allowed to hang vertically.  
When the hanging mass is released the resulting gravitational force 
exerts a torque on the wheel, causing it to rotate with an angular 
acceleration as the mass falls.

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

\item As usual, draw a picture of the apparatus.  Indicate on the 
diagram the important dimensions of the system.

\item Wrap a string around the outside of the wheel and attach a 
small mass that will cause the wheel to rotate without acceleration.  
From this determine the frictional torque in the bearings, and then 
treat it as you did the frictional force in the Atwood Machine lab.

\item\label{large} Attach a larger mass to the string and measure the
time required to fall a predetermined distance while accelerating (use
something soft to cushion the fall at the bottom).  From the time and
distance compute the linear acceleration (assumed constant) using
$y=\frac{1}{2}at^{2}$.  The angular acceleration $\alpha$ is then
simply $a/r$.  Also compute the tension $T$ in the string by making a
free body diagram for the hanging mass; this tension is the force in
the applied torque $\boldsymbol{\tau} =
\boldsymbol{r}\times\boldsymbol{F}$.  See example 10.12 in Serway.

\item\label{repeat} Collect more data points by repeating step
\ref{large} for different hanging weights and also for the string
wrapped around the axle (smaller radius).


\end{enumerate}

\noindent
\underline{Analysis:} 
\vspace{-.23in}
\begin{enumerate}

\item The mass of the wheel-axle combination is printed on the wheel;
however you cannot use the simple formula $I= \frac{1}{2}MR^{2}$ for
the compound system at once.  Instead, compute the theoretical value
of $I$ for the system by adding the rotational inertias of the various
parts (wheel and axle) of the system--each of which is a disk or
cylinder to which $I= \frac{1}{2}MR^{2}$ can be applied individually. 
Assume a constant density $\rho$ so that the ratio of the masses of
the wheel and axle is equal to the ratio of the volumes.  Don't count
the chunk of metal in the very middle of the system twice.

\item From the results of procedures \ref{large} and \ref{repeat},
make a graph of $\tau_{\rm net}$ vs.~$\alpha$ (this should remind you
of a graph of $F$~vs.~$a$ that you made in a previous lab).  The
least-squares slope is your experimental value of the rotational
inertia of the wheel-axle combination; report it to the proper number
of significant digits.

\item How do your computed and measured values of $I$ for
the system compare?

\end{enumerate}


\noindent
\underline{Conclusions:} 
\vspace{-.23in}
\begin{enumerate}

\item What problems did you encounter in this lab and how did you 
overcome them?

\item How has this lab helped you understand rotational inertia?

\item Could the rotational inertia for oddly shaped objects be 
determined using the methods in this lab?  How?

\end{enumerate}
\end{document}
