\section{MC-15--Simple Harmonic Motion and Resonance}

\latex{\vspace*{-.02in}}
\subsection{MC-15a~~Simple Harmonic Motion and Resonance (Rolling Carts)}

\noindent {NOTE TO INSTRUCTORS:} This lab uses PASCO rolling carts instead of the more problematic 
gliders on the air track.  The older airtrack version of this lab is MC-15b.

\input{toolbar/m15_tb0}

\noindent {OBJECTIVES:} \latex{\vspace*{-.02in}}   
\begin{itemize}
\item \latex{\vspace*{-.05in}}To study the period of Simple Harmonic Motion (SHM) 
as a function of oscillation amplitude.
\item{\vspace*{-.08in}To demonstrate Hooke's Law, $F=-kx$, where $k$ is the
spring constant, and $x$ is the elongation.}
\item{\vspace*{-.08in}To study the period $T$ of SHM as a function of the mass
$m$ which is oscillating, investigating $\sqrt{m/k} \propto T (=
2\pi/\omega=1/f$).}
\item \latex{\vspace*{-.08in}}To observe the relationships between the
potential and kinetic energy.
\end{itemize} 

\noindent
THEORY: 
\begin{enumerate}
\item[] \vspace*{-.05in}The restoring force ($F$) on an object 
attached to a 
``simple'' one-dimensional spring is proportional to the 
displacement from equilibrium and has the form, $F= -k(x-x_0)$, where
$k$ is the spring constant (or stiffness in N/m), $x_0$ is the equilibrium
position (i.e., no net force) and $x$ is the position of the object.  This is Hooke's Law.  
Remember that the simple harmonic oscillator is a good
approximation to physical systems in the real
world, so we want to understand it well.  That's the purpose of this lab!

The expression $F= ma = -k(x-x_0) $ is a 2nd order differential
equation with $$a =\frac{d^2 x'}{dt^2}= -\frac{k}{m} (x') $$ where $x'\equiv x-x_0$.
The most general solution for this expression is often given as
$$ x'(t)= A \cos(\omega_0 t) + B \sin(\omega_0 t)$$ where $\omega_0\equiv \sqrt{k/m}$.
$\omega_0$ is defined as the {\it natural frequency} for the {\it undamped}
harmonic oscillator.  $A$ and $B$ are arbitrary  initial displacement parameters.
Alternatively the solution is often specified as 
$$ x'(t)=  C \sin(\omega_0 t + \phi)  \mbox{~~~~~or~~~~~}  
   x'(t)=  C \cos(\omega_0 t + \phi) 
$$ where $\phi$ is the starting phase and $C$ is the displacement.
It is also possible to write down these solutions using complex numbers as in
$$ x'(t)= A' e^{i\omega_0 t} + B' e^{-i\omega_0 t}~~.$$
The relative merit of these equivalent expressions will become clearer
in EXPTS.~V and VI.
\end{enumerate} 

\noindent FUNDAMENTAL CONCEPTS: 
\begin{enumerate} 
\item \vspace*{-.05in}The solution to the undamped (i.e., no frictional or 
drag forces) harmonic
oscillator is time dependent and periodic.  When the $\omega_0 t$ term varies
by $2\pi$ (or one period $T = 1/f = 2\pi/\omega_0$) 
both the position, $x'(t+T)=x'(t)$, and velocity, $v'(t+T)=v'(t)$ return to
their previous values.

\item Total energy (TE) is conserved in SHM motion.  As time evolves kinetic 
energy (KE) 
is transferred to (and from) potential energy (PE).  
Thus at any time $t$:
$$ TE = \mbox{constant} = KE(t) + PE(t) = \frac{1}{2}m [v'(t)]^2 + \frac{1}{2}k
[x'(t)]^2 ~~~~.$$

\end{enumerate}

\noindent APPARATUS:
\begin{enumerate}
\item[] \vspace*{-.05in}
{\it Basic equipment:} PASCO dynamic track, PASCO cart with ``picket fence'', 
PASCO cart with aluminum plate, adjustable stop, assorted masses,  springs:  
{\em this experiment works best with a pair of short 
(~10 mm unstretched length) springs},
timer, photogate \& support stand, knife edge assembly.

\item[] {\it Computer equipment:}   Personal computer set
to the MC15a lab manual web-page; PASCO interface module;  photogate sensor and
extension jack, PASCO sonic position sensor, speaker with driver stem, power
amplifier module. 
\end{enumerate}

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\vspace*{.1in} 
\noindent {\em PRECAUTIONS:}  \vspace*{.1in} 
\begin{enumerate}
\item See MC-14a
\item {\bf When setting up the springs, use the adjustable end stop on the track so the springs are stretched properly:  don't stretch them beyond their elastic limit and don't let them sag and drag on the track}.  Stretching the springs to reach the ends of the track will damage them.
\item {\em Keep the amplitudes small enough that a
slack spring doesn't touch the track nor a stretched spring exceed its 
elastic limit}. 
\end{enumerate}

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\noindent {SUGGESTIONS:}  To measure the period of the oscillating cart:
\begin{enumerate}

\item Install a ``picket fence'' on the cart.
\item Locate the photogate so the black stripe on the fence just cuts off the LED beam (see MC-14a) when the cart is in the
equilibrium position ($ x-x_0 = 0$).  The photogate phone jack should
be in the first PASCO interface position.  
Choose two springs having a similar length and refer to the above precaution.  
  

\item  To initiate the PASCO interface software click the
mouse on the telescope icon in the ``toolkit'' area
below. There will be just a single table for recording the measured period.

\input{toolbar/m15_tb1}

\item Start the cart by displacing it from equilibrium and then releasing
it. Then start the data acquisition by clicking the START icon.  Let the
cart oscillate for about 10 periods.  Calculate the mean and standard deviation by simply clicking on the statistics icon (i.e.~\fbox{$\Sigma$}) on the data table.  


\end{enumerate}

\begin{figure}[h!] 
\centering 
\includegraphics[width=5.2in]{figs/m15a-1rev.eps}\\
\vspace*{-.1in}\caption{Sketch of the dynamic track configuration for SHM.}
\end{figure}


\noindent {UNDAMPED SIMPLE HARMONIC MOTION:}
\begin{enumerate}
\item[1.] \latex{\vspace*{-.05in}}Using sketches in your lab book, both at equilibrium and
after a displacement from equilibrium, show that the effective force 
constant for 2 identical springs of force constant {$k$} on either side of an oscillating 
mass is {$2k$}.
\item[2.] {\bf EXPERIMENT I:} Show experimentally that the 
period is independent of the amplitude.  Try amplitudes 
of {\em approximately} 10 cm, 20 cm, and 30 cm.  (Friction may be a 
problem at very small amplitudes).


\item[3.] {\bf EXPERIMENT II:} By adding mass to the cart, study the period versus total oscillating 
mass.  The latter must include a correction for the oscillating springs 
whose effective oscillating mass (see the note below) is approximately $m_s$/3 
where m$_s$ is the mass of the {\em two} springs.  Explain, in words, how this
correction may be qualitatively justified (Why not $m_s$ or $m_s/2$?)
By inspection of your 
$T$ vs $[M + (m_s/3)]$ curve, what function of $T$ might 
yield a straight line when plotted versus $ [M + (m_s/3)]$?  Prepare this 
plot.  Calculate the effective force constant, $ k'$, (equal to $k_1+k_2$, the
two springs will actually differ slightly)
of the system from the slope 
of this straight line graph.  (Remember 
$ T = 2\pi \sqrt{M_{\mbox{\tiny eff}}/k'}$ where $ M_{\mbox{\tiny eff}} \; \simeq$ M + m$_s$/3.)
Estimate the uncertainty in {$k'$} by solving for $k'$ at each cart mass and
calculating the standard deviation of the mean.  \smallskip \\
NOTE: Theoretically {$ m_{\mbox{\tiny eff}}=m/3$} only for
{$M/m_s=\infty$}  For {$ M = 0$}, the effective mass is 
{$4m/\pi^2= 0.405\,$m}.  
However for {$M/m_s$ = 5}, the effective mass is already {\bf $\simeq 
0.336 \; m_s$}.  See 
Fig.~1 of J.G. Fox and J. Makanty, {\em American Journal of Physics}, {\bf 38}, 
98 (1970).


\item[4.] {\bf EXPERIMENT III :} Also measure $k$ directly by hanging the two
springs vertically on a
``knife edge'' assembly in the laboratory.  Record the stretch produced by a series 
of weights, {\em but do not exceed the elastic limit of either spring!}  Graph 
$F$ vs $y$ for each spring and obtain a best-fit straight line.  
The slope should be the spring constant.  
% or  the method of differences  (as used in the alternative method in M4), 
%results in the most effective use of the data.  
Compare this value of $k'$ with that obtained in part \#3. 

%Determine in the same way $k$ for a spring of different stiffness and 
%predict the period of the glider when it oscillates between two of these 
%springs.

\item[5.] QUESTION:  Assuming the spring constant doubles, how would T vary? 
\end{enumerate}

%\noindent {OPTIONAL (6-8) only:} 
%\begin{enumerate} \epind{
%\item[6.] Using experimental values of {$T$, $k$} and various 
%masses of the glider, 
%determine the effective oscillating mass of the springs.  How does this 
%compare with {$ m_s$/3?} 

%\item[7.] Predict theoretically and then check the period of a glider held 
%between springs of {\em different} force constant.
%\item[8.] Measure the coefficient of friction, $\mu$, for a 
%paper clip on top of glider by finding the 
%maximum amplitude of SHM for which there is no relative 
%motion between clip and glider.}
%\end{enumerate}


\noindent {FURTHER INVESTIGATIONS OF ``UNDAMPED'' SIMPLE HARMONIC MOTION:}  
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}Up to this point you have characterized the SHM of a spring-mass assembly in terms of only a single parameter
$T$ (the period).  One of the possible time-dependent expressions for 
describing the motion was 
 $$ x'(t)= A \cos(\omega_0 t +\phi_0) ~~\mbox{and}~~
    v'(t)= -A\omega_0 \sin(\omega_0 t +\phi_0) $$
    or
 $$ v'(t)= ~A\omega_0 \cos(\omega_0 t +\phi_0-\frac{\pi}{2}) \hspace*{2in}$$
 
 
 
where $A$ is the amplitude (displacement from equilibrium),
$\omega_0$ is the {\em natural} angular frequency and $\phi_0$ is 
the starting phase.  Notice that the velocity has a {\em phase} shift of
$\frac{\pi}{2}$ relative to the displacement. \\

Equally characteristic of SHM is the process of energy transference: 
kinetic energy of motion is transferred into potential energy (stored in
the spring) and back again.  Friction is an
ever present energy loss process so that the total energy always diminishes
with time. \\

To capture this rather rapid cyclic process we will again use the 
PASCO interface while replacing the photogate sensor with the sonic 
position sensor.
\end{enumerate}

\noindent {\bf EXPERIMENT IV: Measuring the $x$ vs $t$ behavior:}

\begin{enumerate}
\item Replace the cart with the picket fence by a cart with an aluminum vane attached to it.
You will want to measure the mass of this new cart and predict the new natural frequency.

\begin{figure}[h!] 
\centering 
\includegraphics[width=5.2in]{figs/m15a-2rev.eps}\\
%\includegraphics[width=5.2in]{figs/m15a-02.eps}\\
\vspace*{-.1in}\caption{Sketch of the dynamic track with the sonic position sensor.}
\end{figure}


\item Place the position sensor approximately 60 cm from the vane in the
direction of oscillatory motion.  Make sure the yellow phone jack is in
the third slot and the black phone jack is in the fourth slot. 
Alignment is very important so that the
sensor senses only the vane and not the cart.  A slight upward tilt may
help (or raising the vane up slightly as well).

\item CLICK on the telescope icon below to initiate the PASCO$^{\mbox{\copyright}}$
interface software.  

\input{toolbar/m15_tb2}

\item Displace the cart approximately 20 cm from equilibrium to initiate
the oscillatory motion.
CLICK on the the START icon to start your data acquisition.  The graph
will simultaneously display both absolute position and velocity versus time.

\item Practice a few times to make sure you can obtain smoothly varying
sinusoidal curves.   Then run the data acquisition for just over ten cycles
and use the cross-hair feature to read out the time increment for ten
full cycles.  How does your prediction check out?  Determine the initial
phase (i.e., at $t=0$).  Record the equilibrium position ($x_0$) as well.

\item Use the "Zoom select" feature of the PASCO graph toolbar to better view a single full cycle by clicking on the  magnifying glass icon (4th from
left) and then select two points in the graph using a CLICK and DRAG motion of the mouse.

\item Print out (click in graph region and then type ALT, CTRL-P)  
or, alternatively, sketch the position and velocity curves in your lab book 
identifying key features in the time dependent curves.  In particular identify the characteristic(s) which demonstrate the $\pi/2$ phase difference between the velocity and displacement curves.
 
\item Using the PASCO cross-hair option to read out the relevant time, position and velocity, make a table as below and fill in the missing entries (identify units). \\
\hspace*{0.5in}
\begin{tabular}{ccccccc}
\hline
time & phase( $\omega_0 t+\phi_0$ ) & $~~x'(t)~~$  & ~~$v'(t)$~~  & ~~$KE$~~  & ~~$PE$~~  & ~~$TE$~~ \\ 
 & 0~~~~~~~~~~~~                & & & & &  \\
 & ~$\pi/4$ ~(~~45 deg)  & & & & &  \\
 & ~$\pi/2$ ~(~~90 deg)  & & & & &  \\
 & $3\pi/4$ (135 deg)  & & & & &  \\
 & ~~$\pi $ ~~(180 deg)    & & & & &  \\
\hline 
\end{tabular}

\item  Is the total energy a constant of the motion?

\item How much displacement amplitude and energy are lost after five full cycles? What is the approximate friction coefficient?

\end{enumerate} 

\noindent {OPTIONAL INVESTIGATIONS OF ``UNDER-DAMPED'' SIMPLE HARMONIC MOTION:} 
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}} In the real world friction is an ever present process.  In the case of SHM friction can have a profound effect.  Damping of unwanted vibrations is important in a myriad of situations.  (Imagine what driving a car would be like if there were no shock absorbers!) 

Introducing friction can be done by simply adding one more term in the
force expression, $F_{\mbox{\small drag}} \equiv -R v$,  a drag force which is proportional to the velocity where $R$ is the drag coefficient [units of kg m/s or N/(m/s)] . This is appropriate for motion thru a viscous fluid but it is really  only a rough approximation for the frictional forces in the track.  The modified force expression now becomes
$$ F= ma = -k x' - R v'  ~~\mbox{or}~~  
0=\frac{d^2x'}{dt^2}+\frac{R}{m}~\frac{dx'}{dt}+\omega^2 x' $$
Since energy is continuously lost, solutions of this expression will be {\em time dependent} but NOT periodic.  Adding this ``simple'' term dramatically complicates the process of finding appropriate solutions.  The most general form of the solution is
$$ x' = e^{-(R/2m)t}~~\left[ A e^{+\sqrt{(\frac{R^2}{4m^2}-\omega_0^2)}~~t} +
 B e^{-\sqrt{(\frac{R^2}{4m^2}-\omega_0^2)}~~t}\right] $$
which is quite formidable. Since the track drag is low (i.e., R is relatively small) the solution is said to be {\em underdamped} and oscillatory when $\omega_0^2 > R^2/4m^2$. The solution in this case becomes:
$$ x' = C e^{-(R/2m)t}~ \cos(\omega_1t+D) ~~\mbox{where}~~ 
\omega_1^2\equiv \omega_0^2-\frac{R^2}{4m^2}$$
where $\omega_1$ is the {\em natural frequency} of the {\em underdamped} system and C and D are the initial displacement and phase.
Because the frequency is lowered, the period lengthens.  This is consistent with one's intuition;  drag works against oscillatory motion.   
\end{enumerate}

\begin{figure}[h!] 
\centering 
\includegraphics[height=2.6in]{figs/m15fig_3.eps}\\
\vspace*{-.1in}\caption{Sketch of underdamped harmonic motion with $\omega_0^2 > R^2/4m^2$.}
\end{figure}

\noindent{\bf EXPERIMENT V:}
\begin{enumerate}
\item \latex{\vspace*{-.05in}}CLICK on the telescope icon below (web-version) to initiate the next PASCO$^{\mbox{\copyright}}$ interface application.  In addition to the graph there will be a ``two'' column table displaying the time and position then the time and velocity. 

\input{toolbar/m15_tb3}

\item \latex{\vspace*{-.05in}}Displace the cart approximately 20 cm from equilibrium to initiate the oscillatory motion. CLICK on the the START icon to start your data acquisition and record enough cycles to see the amplitude diminish by two thirds.

\item \latex{\vspace*{-.05in}}As the amplitude of oscillation decreases, does the period $T$ change? Base the answer on measurements of $T$ using the cursor in the PASCO display.

\item \latex{\vspace*{-.05in}}Select six representative times using the table 
(identifying where the velocity changes sign) and make a table of $t$ vs maximum displacement. Use these points in the graphical analysis package and fit these points to the expression $C \exp{-(R/2m)t}$ or, in terms of the 
explicit analysis formula, $y= C *\exp[-B*(x-x_0)]$. 

\item \latex{\vspace*{-.05in}}How good or poor is the assumption that the drag force is proportional to the velocity?
      
\item \latex{\vspace*{-.05in}}Sketch out an approximate curve of $x$ vs $t$ if $R$ were significantly larger. Which $R$ would be more appropriate for absorbing and dissipating a physical ``shock'' (and why)?      
      
\end{enumerate}

\noindent {OPTIONAL INVESTIGATIONS OF RESONANCE:} 
\vspace*{.1in} \\
One of the most important situations of the harmonic oscillator is
that of FORCED, damped harmonic motion.  In one-dimension the applied force
is typically sinusoid  and when $\omega$ approaches the natural frequency of the system  (nominally $\omega_1$ and, if $R^2/4m^2 \ll 1$, also
$\omega_0$) the energy of the driver  is additively coupled to the moving
mass (e.g., a glider) and resonance occurs.  Resonance is a {\em very} important aspect of the world around us and many mechanical and electronic devices employ resonant behavior as a fundamental aspect of their operation (e.g., musical instruments, radios, televisions). \\

Along with the frictional drag ($R v'$ where $R$ is the drag coefficient) one
more force term must be added, that of the mechanical driver, with
$F_{\mbox{\small driver}}= F_d~\cos(\omega t)$.  The new force expression is
conventionally written as: 
$$ F= ma = - k x' - R v' + F_d \cos \omega t 
~~~~\mbox{or}~~ ~~  \frac{F_d}{m} \cos \omega t
=\frac{d^2x'}{dt^2}+\frac{R}{m}~\frac{dx'}{dt}+\omega^2 x' ~~.$$ 

Since the system energy is lost through friction and may be gained through the driver action, solutions of this expression will be {\em time dependent} but with both {\em transient} and {\em steady-state} attributes.   In many instances resonant systems respond so quickly that one only views the steady-state behavior.  In this lab you will be able to observe BOTH the {\it transient} and {\em steady-state} processes.
 

As you may expect the most complete solution of this new differential equation
has a rather complicated form and so is not reproduced here.  Since we are interested only in resonance we can simplify the expression by assuming solutions that apply to the {\em underdamped case} (those with oscillatory behavior). Thus the analytic solution reduces to:  $$ x' = Ce^{-(R/2m)t}~
\cos(\omega_1t+D) + \frac{F_d}{[m^2(\omega_0^2-\omega^2)^2+\omega^2 R^2]^{1/2}}
\cos(\omega t - \phi) $$ where the first term is the {\em transient} behavior,
identical to that of the simple damped harmonic oscillator (described in the
last section) and the second term is the  {\em steady-state} solution.  At large times $t$ the first terms dies out exponentially so that $x'$ is approximated by only  $$ x'((R/2m)t \gg 1) = \frac{F_d}{[m^2(\omega_0^2-\omega^2)^2+\omega^2 R^2]^{1/2}}
\cos(\omega t - \phi) $$ where $\omega_0$ is the natural frequency of the undamped harmonic oscillator, $\omega$ is the mechanical driver frequency, $R$ is the drag coefficient and $\phi$ is a measure of the phase difference between the driver motion and the cart motion.  

In this lab we will only investigate  the nature of the cart displacement with driver frequency ($\omega$) in the vicinity of the resonant frequency.  Thus the only relationship of interest becomes amplitude$\,\propto  [m^2(\omega_0^2-\omega^2)^2+\omega^2]^{-1/2} \equiv Z^{-1/2}$.  $Z$ is a minimum when the driver frequency is set to $ \omega_0^2- R^2/(2m^2)$ or $ \omega_1^2 - R^2/(4m^2) $ which is defined to be $\omega_2$.  Since $R^2/2m^2$ is small (the PASCO track is a low friction experiment) $\omega_0$ is nearly the {\em same} and, in addition, $x'$ will be sharply peaked about $\omega_0$.\\

\begin{figure}[h!] 
\centering 
\includegraphics[height=2.6in]{figs/m15-04.eps}\\
\vspace*{-.1in}\caption{Sketch of the amplitude squared vs driver frequency.}
\end{figure}
\latex{\vspace*{-.03in}}
\noindent{\bf EXPERIMENT VI:} {(SUGGESTED PROCEDURE:)}  \latex{\vspace*{-.03in}}

\begin{enumerate}

\item \latex{\vspace*{-.07in}}
Use the rolling cart with the aluminum plate.  Weigh the cart and predict the natural frequency for this new arrangement.

\begin{figure}[h!] 
\centering 
\includegraphics[width=5.2in]{figs/m15a-5rev.eps}\\
%\includegraphics[width=5.2in]{figs/m15a-05.eps}\\
\vspace*{-.1in}\caption{Sketch of the dynamic track with the position sensor and speaker.}\vspace*{-.2in}
\end{figure}

\item \latex{\vspace*{-.07in}} Place the position sensor approximately 60 cm from the vane in the direction of oscillatory motion.  Make sure the yellow phone jack is in the third slot and the black phone jack is in the fourth slot. Alignment is very important so that the sensor senses only the vane and not the cart.  A slight upward tilt may help.

\item \latex{\vspace*{-.08in}}Detach the fixed spring end stop and place the speaker as shown in the figure above with the spring looped through the small slot in the speaker driver stem using the same considerations for the spring extension as in the previous experiments.

\item \latex{\vspace*{-.05in}}Make sure the speaker power leads are plugged into the amplifier module and that its power is turned on.  Also verify that the DIN-9 pin connector (from the amplifier module) is plugged into the A position in the PASCO interface module.

\item \latex{\vspace*{-.05in}}CLICK on the telescope icon (if web version) below to initiate the PASCO$^{\mbox{\copyright}}$ interface software.  

\input{toolbar/m15_tb4}

\item \latex{\vspace*{-.05in}}Make sure the speaker driver window has been switched to the {\bf Off}  position. Displace the cart approximately 20 cm from equilibrium to initiate the oscillatory motion. CLICK on the the START icon to start your data acquisition.  The graph will simultaneously display glider position and velocity versus time and the amplifier current (which should be zero).   Practice a few times to make sure you can obtain smoothly varying sinusoidal curves.    

\item \latex{\vspace*{-.05in}}Displace the cart approximately 20 cm from equilibrium to initiate the oscillatory motion. CLICK on the the START icon to start your data acquisition and record enough cycles to see the amplitude diminish by 90\%.  Compare your data to Fig.~3 and verify that your cart has a similar transient behavior.  Determine the {\em natural frequency} of the system and compare to your prediction. (This frequency is actually $\omega_1$ but it is, for this low friction set-up, nearly the same as  either $\omega_2$ or $\omega_0$.)  To observe the steady-state properties of resonance you will have to wait for times {\em longer} than those required for this step.

\item[8.] \latex{\vspace*{-.05in}}Moving to the PASCO Signal Generator window (shown in Fig.~6), set the amplifier frequency (which is in Hertz or cycles per second) to the closest to resonance  (0.01 Hz steps) and engage the speaker but CLICKing the {\bf Auto} icon.  The driver output should use the sinusoid AC waveform and, if the overload light on the Power Amplifier flashes on, reduce the voltage setting slightly. Start the data acquisition (and speaker motion) by CLICKing on the START icon and record data until you achieve steady-state behavior.  You have to wait a couple of minutes before you reach steady state.\\

\begin{figure}[h!] 
\centering 
\includegraphics[height=2.4in]{figs/s01-08rev.eps}\\
\vspace*{-.1in}\caption{Image of the PASCO software Signal Generator window.}
\vspace*{-.1in}\end{figure}

NOTE: The nominal step sizes for adjusting the amplifier frequency and voltage
are {\em very} large.  To alter the step size use the \fbox{$\blacktriangleleft$} or \fbox{$\blacktriangleright$} buttons. To alter the current or voltage (which of these depends on configuration) use the \fbox{\bf{\scriptsize +}} or \fbox{\bf{\LARGE -}} buttons.  


\item[9.] \latex{\vspace*{-.07in}}Set the amplifier frequency 0.01 Hz steps above and below resonance and record data until you achieve steady-state behavior.  Repeat for 0.10 Hz and (if time permits) 0.40 Hz steps.  Plot out a few representative sets of data.


\item[10.] \latex{\vspace*{-.07in}}Determine the maximum steady-state displacement of the cart for each of the measured frequencies and plot the relative amplitude squared  [($x'(\omega)/x'(\omega_0))^2$] vs frequency offset ($\omega-\omega_0$). Estimate the full width at half maximum for this curve.   This value should be equal to $R/m$.

\item[11.] \latex{\vspace*{-.07in}}Discuss the nature of this resonance curve. If you adjust the $R/m$ ratio to further sharpen the resonance curve can you identify a compensating complication if you are interested in achieving steady-state behavior? If the speaker were attached to an amplifier playing audible music (nominally 20-20,000 Hz), what do you think this mass will do?


\end{enumerate}




