\section{M-6~~Uniform Circular Motion}

OBJECTIVE:  To measure the centripetal force, $F_{c}$, and compare to
$$F_{c}=\frac{mv^{2}}{r}=m\omega^{2}r$$

\noindent APPARATUS:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}Fig.~1 is a schematic of the equipment.  The bobs and 
springs are removable for weighing.  Not shown are table clamp and pulley, 
slotted masses and weight hanger.
\end{enumerate}\latex{\vspace*{-.05in}}
\begin{figure}[h] \centering \includegraphics[height=4in]{figs/l103/m06-1.eps} 
\caption{The UCM apparatus.} \end{figure}
\latex{\vspace*{-.05in}}
\noindent INTRODUCTION:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}} A variable speed motor drives the rotating system which 
has two slotted bobs which slide on a low-friction bar.  One 
adjusts the speed until one bob just covers the optical light pipe and thus 
reduces the signal seen at the center of the rotating system to zero.
 A revolutions counter is on the shaft.  
The counter operates by sensing the rotating magnetic poles and 
electronically reads out 
directly the frequency of revolution in rpm. 
A spring (plus any friction) supplies the centripetal 
force required to keep the bob 
traveling in a circle.   \\
 If you measure first the frequency of rotation required 
to make the bob just cover the optical light pipe,
and if you then measure the force required to pull the bob out the same 
distance when the system is not rotating, you can determine $F_c$ and compare
your result to
$$F=m\omega^{2}r\,,$$
where $r$ is the distance from the axis of rotation to the center of mass 
of the bob.
\end{enumerate}\latex{\vspace*{-.05in}}



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    <a name="Demo"><h3><u> Pre-lab Quiz </u></h3></a>
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    <li><a href="../../js/q207_m06.html">Expt. M06 Quiz</a>

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\noindent SUGGESTIONS:\vspace*{.1in} 
\begin{enumerate} 
\item Find the mass of the nickel plated brass bob; also the 
aluminum bob.
\item {\bf Dynamic measurement of the force}:  Attach the brass bob to 
the spring.  Replace the lucite cover, and adjust the motor speed until the
light from the light pipe at the center of the rotating system goes to zero.
Record the rotation frequency. \\ \\
{\bf To correct for frictional effects of the bob on the bar}, record the
frequencies both as the speed is slowly {\em increased} to the correct value and and
as the speed is {\em decreased} from too high a value.  Since the direction of
the frictional force reverses for the two cases, the average should eliminate
the frictional effect. \\ \\
Repeat several times so you can estimate the 
average and the standard deviation of your values.
\item {\bf Static measurement of the force}:  Use the string, pulley and weight
holder plus slotted weights to measure the force required to stretch the spring
so that the optical light pipe is again just covered.  Devise a way to avoid
error caused by the friction at the pulley and of the sliding bob on the bar.
\item While the spring is stretched to its proper length (item 3 above), measure the 
distance r from the axis of rotation to the center of mass of the bob. The 
center of mass is marked on the bob. 
\begin{figure}[h] \centering \includegraphics[height=3.5in]{figs/l103/m06-2.eps} 
\caption{Static measurement of the force using hanging weights} \end{figure} \\
 \item Compare the measured centripetal force to the computation $F_c = m\omega^{2}r$.
In computing the centripetal force, also take into consideration the mass 
of the spring. One 
can show (Weinstock, {\it American Journal of Physics}, 32,p.\ 370, 1964) 
that $\approx (1/3)$ of the spring mass should be 
added to the mass of the bob to obtain the total effective mass.  
\item Repeat the above item 1 through item 5 but for the aluminum bob.
\end{enumerate}  


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\noindent QUESTIONS:  
\begin{enumerate}
\item Estimate the reliability of your measurements.  How well do the 
measured and computed forces agree?  Try to account for any discrepancy.

 
\end{enumerate}


