\section{M-13~~Gyroscope}
\noindent { OBJECTIVE:}  
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}To study gyroscopes and to check the 
relationship between angular momentum, applied torque and rate of 
precession.
\end{enumerate}

\noindent {REFERENCES:}  
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}Don't do this experiment before reading about
angular momentum and gyroscopes; see also the Mitac 
Instruction Manual, especially pages 1-7, experiments 1-4, pages 
11-14, and Sec.~6 on ``Operation''.
\end{enumerate}

\noindent {APPARATUS:} Mitac Gyroscope, 150 g and 300 g snap on 
slotted masses, and stroboscope. \\

\noindent {SUGGESTIONS:} 
\begin{enumerate}
\item \latex{\vspace*{-.05in}}Read the sections of the Mitac Manual referred to 
above, start the gyroscope, and experiment with it.  For example, predict the 
direction of precession when a certain torque is applied; then check it 
experimentally.  
\item When you have become familiar with the operation of the 
gyroscope perform
the following quantitative experiment: 
\begin{enumerate} 
\item[i.] Calculate the rotor's angular momentum from the following data: \\

\latex{\hspace{0.5in}}
 \begin{tabular}{|lll|} \hline
Outer diameter of rotor &  ~~20.0 & cm \\
Inner diameter of rotor &  ~~15.1 & cm \\
Thickness of rotor      &  ~~~~2.54 & cm \\
Rotor webb thickness    & ~~~~0.435 & cm \\
Hub diameter            & ~~~~3.01 & cm \\
Mass of rotor           & ~~~~2.73 & kg \\
Rotor rpm               & 150 & ~~  \\ 
\multicolumn{3}{|l|}{(Check rotor rpm with the  stroboscope)}  \\ \hline
\end{tabular} \smallskip

\item[ii.] Place one of the supplied weights into the groove on the rotor axis 
so as to create a torque on the axis and cause the gyroscope to precess.  
From this weight and its position, calculate the torque. 
Predict the rate of 
precession by using the calculated torque and angular momentum.  
Check by experiment.
\item[iii.] Ask your instructor what other quantitative measurement you should make.  
The experiments in the Mitac Manual are good possibilities. 
\end{enumerate}
\end{enumerate}
\newpage
{\Large{\noindent \bf Alternative M-13 Gyroscope:}} \\

\noindent{OBJECTIVE:}  To study gyroscopic effects especially precession and nutation. \\

\noindent {APPARATUS:}  Air bearing gyroscope, torque weights, timer, stroboscope. \\

\noindent {INTRODUCTION:}  Review gyroscope treatment in text (or in
handout). \\

\html{\begin{tabular}{p{4.8in}c}}
\latex{\parbox{0.40\linewidth}}
{\noindent
{\bf CAUTION:}  
{\em The air bearing surfaces and torque 
shaft can easily suffer 
damage so handle carefully:}  e.g.~~when the air pressure is on (and ball 
friction negligible) the {\em glued-on torque shaft may break off if 
the shaft bangs against the stationary ball holder}.
}
\html{ & }
 \latex{\parbox{0.55\linewidth}}{\hspace*{.1in}
\latex{\includegraphics[width=2.9in]{figs/m13-01.eps}}
\html{\includegraphics[height=3.3in]{figs/m13-01.eps}}
}
\html{ \\ \end{tabular}}

\input{toolbar/m13_tb1}


\noindent { SUGGESTED STUDIES:}
\begin{enumerate}
\item {\em Qualitative explorations:}  With air flow on to minimize friction,
spin ball by twisting the torque shaft with thumb and forefinger.  Observe 
{\em precession} about the vertical $Z$ axis, and observe {\em nutation} 
(torque shaft 
bobbing 
up and down) for different torque weights on the shaft and for different 
spin angular velocities, $\omega_s$.  Upon what factors does the precession 
depend? 

\noindent Observe the three basic {\em nutation} patterns when the total torque 
weight, $m$, has: \vspace*{.2in}

\begin{tabular}{ll}
a) no initial vertical velocity: \vspace*{-.4in}\\ &
\includegraphics[height=0.6in]{figs/m13-02.eps}
\vspace*{.2in}\\
b) a small initial vertical velocity:\vspace*{-.4in} \\ & 
\includegraphics[height=0.6in]{figs/m13-03.eps}
\vspace*{.2in}\\
c) an initial small horizontal velocity:\vspace*{-.4in}\\ &
\includegraphics[height=0.6in]{figs/m13-04.eps}
\vspace*{-.10in}\\
\end{tabular} 


\item {\em Measure the angular acceleration, $\alpha$, from the air friction:}  For 
low $\omega_s$ use a stopwatch and time the revolutions of a mark on the 
rotating shaft.  Plot spin angular velocity, $\omega_s$, versus time (remember to 
plot {\em average} $\omega_s$ at the middle of the time interval).  
Calculate $\alpha$ from 
the slope.  For higher $\omega_s$ use the stroboscope.  To avoid errors from 
subharmonics synchronizing with the mark on the shaft, start with the strobe 
at too high a frequency.  If the synchronization is at the {\em n$^{th}$} 
{\em higher 
harmonic}, you will see the shaft marks appear at $n$ different places.

\item {\em Calculate mass of the unweighted torque shaft from observed precessional 
angular velocity, $\omega_p$.}
Spin the shaft at about $\omega_s  \sim 2\pi$ rad/s and minimize nutation by starting 
the shaft with a small horizontal (or slightly upward) velocity to 
approximate the $\omega_p$ expected.  Since the torque 
$\tau = (mgh)\sin\theta$ and 
$\omega_p = \tau/(L_s sin\theta)$, we predict $\omega_p$ is independent of 
$\theta$.  
Check this experimentally.  
With spin angular momentum $L_s = I_s \omega_s$ the result is
$$\omega_p = \frac{mgh}{I_s \omega_s}  \mbox{~~~~~~or~~~~~~~} 
m = \frac{\omega_p \omega_s I_s}{gh} . $$ 

For the shaft mass $m \ll M$ of the ball, $I_s \sim (2/5)MR^2$ where $R$ is 
radius of the ball.  Measure the appropriate quantities and calculate $m$. 
Compare with $m$ and $h$ measured directly for an unattached torque shaft.
\item {\em Sleeping Top:}  If 
$$\omega_s^2 \;\geq \; 4mgh \left[I_s \; + \; mh^2 \left(\frac{M}{m+M}\right)
\right] / I_s^2  ,$$ 
then one can show (see handout) that the gyroscope will spin 
without nutation if released carefully 
at  $\theta$ = 0.  You can observe the qualitative effect by starting with a 
higher $\omega_s$ and then gradually slowing down $\omega_s$
by touching the rotating 
ball symmetrically in the horizontal plane until $\omega_s$ becomes critical 
when the shaft will suddenly fall out of the vertical line.  Calculate and 
check this critical $\omega_s$.
\end{enumerate}

\noindent {OPTIONAL:}
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}} 
\noindent {\em Nutation frequency}, $\omega_N$:  One can show (see handout) that:
$$\omega_N = \frac{\omega_s I_s}{ I_s + mh^2 M/(m+M) } .$$
Check this expression when $\omega_s \sim 2\pi$ rad/s.  If one has time, change 
m and h  by adding torque weights and check the new ratio of 
$\omega_N / \omega_s$. \\ \\

\noindent {\em Amplitude of nutation:}  The handout shows that the 
amplitude of nutation varies as 1/L$_s^2$.  Since $L_s = I_s \omega_s$, 
spinning the 
gyroscope faster should reduce nutation amplitude at the same time the 
nutation frequency increases.  Check this at least qualitatively. 
Use strobe to measure high $\omega_s$. 
\end{enumerate}

