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\begin{document}

\title{Torsion Pendulum}
\author{Ross Terhaar}
\email{rterhaar@csbsju.edu}
\affiliation{Department of Physics, Saint John's University, Collegeville, MN 56321}
\date{\today}

\begin{abstract}
This experiment used a torsion pendulum
apparatus to calculate the damping constant, $\beta$, at points
where the system is under dampened and critically
dampened. This experiment also used a forcing motor to
drive the pendulum at frequencies near resonance for three
different brake currents and used the calculated resonance
frequencies to determine $\beta$ for each case. In this
experiment, we determined that $\beta$ is not constant for free
rotational oscillations but is for forced rotational
oscillations.
\end{abstract}

\maketitle

\section{Introduction}
This experiment\cite{crumley} examines the behavior of a torsion
pendulum, a special case among mechanical oscillators. 

Note that angular frequency ($\omega$ in rad/s) and frequency ($f$ in
Hz.) are not the same.
In the damped case, the torque balance for the torsion 
pendulum yields the differential equation\cite{LDa}:
\begin{equation}
J\;\frac{d^2\theta}{dt^2}+b\;\frac{d\theta}{dt}+c\;\theta=0
\label{eq:diffeq0}
\end{equation}
where $J$ is the moment of inertia of the pendulum, $b$ is the
damping coefficient, $c$ is the restoring torque constant, and $\theta$
is the angle of rotation. This
equation can be rewritten in the standard form\cite{marion}:
\begin{equation}
\ddot{\theta}+2\beta\;\dot{\theta}+\omega_0^2\;\theta=0
\label{eq:diffeq}
\end{equation}
where the damping constant is $\beta=b/2J$ and the natural
frequency is $\omega_0=\sqrt{c/J}$. The general solution to this
differential equation is:
\begin{equation}
\theta(t)=e^{-\beta t}\left[A_1 e^{\sqrt{\beta^2-\omega_0^2}\; t}+A_2 e^{-\sqrt{\beta^2-\omega_0^2}\; t}\right]
\label{eq:gen_solution}
\end{equation}
with three different types of solutions possible depending on
the relationships between $\omega_0$ and $\beta$. In the underdamped
case ($\beta < \omega_0$):
\begin{equation}
\theta(t)=\theta_0 e^{-\beta t} \cos(\omega_1 t -\gamma)
\label{eq:underdamped_solution}
\end{equation}
with the oscillation frequency $\omega_1=\sqrt{\omega_0^2-\beta^2}$, initial
amplitude $\theta_0$, and phase $\gamma$. In the critically damped case
($\beta= \omega_0$):
\begin{equation}
\theta(t)=(A+B t) e^{-\beta t} 
\label{eq:critdamped_solution}
\end{equation}
In the overdamped case ($\beta > \omega_0$):
 \begin{equation}
\theta(t)=e^{-\beta t}\left[A_1 e^{\omega_2 t}+A_2 e^{-\omega_2 t}\right]
\label{eq:overdamped_solution}
\end{equation}
where $\omega_2=\sqrt{\beta^2-\omega_0^2}$ 

For the forced oscillation case, an external torque is added 
to Eq.~(\ref{eq:diffeq0}): 
\begin{equation}
J\;\frac{d^2\theta}{dt^2}+b\;\frac{d\theta}{dt}+c\;\theta=\tau_0\sin(\omega t)
\label{eq:diffeq0+drive}
\end{equation}
where $\omega$ is the driving frequency and $\tau_0$ is the driving
torque\cite{LDb}. The general solution
to the differential equation is the sum of the homogeneous
solutions (which are the solutions to the damped case
above) plus a particular solution. The particular solution
has the form:
\begin{equation}
\theta(t)=\theta_m(\omega) \sin(\omega t -\phi)
\label{eq:driven_solution}
\end{equation}
with
\begin{equation}
\theta_m(\omega)=\frac{\tau_0}{J\sqrt{(\omega_0^2-\omega^2)^2+(b\omega/J)^2}} 
\label{eq:amplitude_driven_solution}
\end{equation}
In this case the resonance frequency is
\begin{equation}
\omega_r=\sqrt{\omega_0^2-2\beta^2}
\label{eq:wr}
\end{equation}
and the phase shift between the
pendulum and the external oscillator is:
\begin{equation}
\tan\phi=\frac{2\beta\omega}{\omega_0^2-\omega^2} 
\label{eq:phase}
\end{equation}

\section{The Experiment}
The experiment was conducted with the torsion pendulum
apparatus set up as shown in Fig.~\ref{fig1}, using a low voltage
power supply and two digital multimeters.
\begin{figure}[t]
\begin{center}
\includegraphics[width=3in]{RossTfig1.eps}
\caption{\label{fig1} A schematic of the torsion pendulum apparatus\cite{LDb}.}
\end{center}
\end{figure}

For the first half of the experiment, the driving motor was
not active.

We first determined was the period of the torsion
pendulum to find the natural frequency by taking 3
measurements of 10 oscillations of the pendulum with the
damping magnet turned off.

Next, we determined the damping constant, $\beta$, at two
values of the damping current, $0.1\;\rm{A} < I_1 < 0.3\;\rm{A}$ and 
$0.3\;\rm{A} < I_2 < 0.6\;\rm{A}$. First, we calculated the period of
oscillation at each current and then started the pendulum
at its furthest point of rotation and measured the amplitude
after each period. We then fit the amplitude vs.\ time data to
an exponential curve and used Eq.~(\ref{eq:underdamped_solution}) to determine the
damping constant.

The damping current was increased until the pendulum was
near critically dampened, or when it only completed one
oscillation after release from its furthest rotation point. For
this case, we took the average oscillation time. We then
further increased the current to the point where the
pendulum was critically dampened, or did not cross zero
after release from its furthest point of rotation. We used the
critically dampened case to estimate the damping constant
from Eq.~(\ref{eq:critdamped_solution})
by measuring the time it took the
pendulum to reach 1/10 of its initial value after being
released from its furthest rotational point. In the critically 
dampened case, $B = 0$, $A = $furthest rotational point value at
time $t = 0$. We used this $\beta$ to get estimates of the damping
coefficient, $b$, and restoring torque, $c$, where the moment of
inertia of the pendulum.

The latter part of this experiment was conducted with the
forcing motor turned on.

To determine a relationship between voltage and frequency
of the forcing motor we determined the frequency of the
motor at various voltages by finding the average period of
the motor at several voltages and making a fit of frequency
to obtain the relation between them.
The resonance frequency curve for currents of, $I = 0$~A, $I \sim
0$~A, $I\sim 0.4$~A, $I \sim 0.8$~A were determined by taking
measurements of amplitude of oscillations in intervals of
0.005~Hz from $f = 0.465$~Hz to $f = 0.560$~Hz for each
damping current by varying the voltage of the motor
accordingly.

\section{Data Presentation and Intrepretation Of Results}

Determination of the natural angular frequency, $\omega_0$, with
the damping current at $I = 0$~A:

\begin{table}[h!]
\caption{\label{tab1}Results: the period, natural frequency and natural angular frequency
of the torsion pendulum for braking $I = 0$~A.}
\begin{center}
\begin{tabular}{|l|c|}
\hline
Mean period for 10 oscillations &1.927 $\pm$ 0.005 s \\ \hline
Natural frequency & 0.519 $\pm$ 0.0012 Hz\\ \hline
Natural angular frequency, $\omega_0$ & 3.260 $\pm$ 0.008 rad/s \\
\hline
\end{tabular}
\end{center}
\label{tab:results}
\end{table}

Mean period at damping current $I = 149 \pm 3$~mA: $1.927 \pm
0.005$ seconds.

The data for amplitude vs.\ time for a damping current of $I =
149 \pm 3$~mA were taken at each consecutive period starting
after the pendulum had gone through one oscillation. The
uncertainty in measurement of amplitude was assumed to be
0.2 at each measurement with an uncertainty in time given
by: $\delta t = |n|\delta T$. When these data were fit to an
exponential curve, the fit equation was: $y = Ae^{Bx}$ , where
$A = 21.2 \pm 0.5$ and $B = -0.040 \pm 0.001$~rad/s. $\beta$ was found
to be $-B$ as per Eq.~(\ref{eq:underdamped_solution}): $\beta = 0.040 \pm 0.001$~rad/s.

For a damping current of $490 \pm 7$~mA, the mean period
was found to be $1.927 \pm 0.007$~seconds

The data of amplitude vs.\ per consecutive period with a
damping current of $490 \pm 7$~mA were taken in a similar
fashion. Again, the uncertainty in amplitude was taken to
be 0.2 and uncertainty in time was : $\delta t = |n|\delta T$. These
data were also fit to an exponential curve as seen in Fig.~\ref{fig2}
with a reduced chi-squared of 4.7. The equation of the
fit line was of the form $y = Ae^{Bx}$ where $A = 26.6 \pm 0.4$
and $B = -0.17 \pm 0.03$~rad/s which, as per Eq.~(\ref{eq:underdamped_solution}),
gives the value of the damping constant to be $\beta = 0.17 \pm
0.03$~rad/s, a reasonable answer as one would expect the
damping constant to increase as the damping current
increases.

\begin{figure}[t]
\begin{center}
\rotatebox{-90}{\includegraphics[width=2.5in]{RossTfig2.ps}}
\end{center}
\caption{\label{fig2} Plot of Amplitude vs.\ time fit to an exponential, where
each data point is spaced apart by one period ($1.927\pm0.007$~seconds)}
\end{figure}

The system was near critical dampening at a damping
current of $1.69 \pm 0.02$~A with a mean stopping time of $2.3
\pm 0.1$~seconds. We found that the system was critically
dampened at a damping current of $1.90 \pm 0.02$~A with a
mean stopping time of $1.6 \pm 0.3$~seconds. The results for
the time to stop agree with the idea that the pendulum
comes to rest at sooner when the damping is increased

For the critically dampened case, the mean time for $\theta$ to
decrease from its furthest point of rotation $\theta_0=19$ to
$\theta=1.9$ was found to be $1.00 \pm 0.08$~seconds. We used this in
conjunction with Eq.~(\ref{eq:critdamped_solution}) to get estimates of $\beta$, $b$,
and $c$. The damping constant was found using
following equation:
\begin{equation}
\beta=\frac{-\ln(\theta/\theta_0)}{\Delta t}
\label{eq:beta_overdamped}
\end{equation}
and uncertainty in $\beta$ was
\begin{eqnarray}
\delta\beta&=&\sqrt{\left(\frac{\partial\beta}{\partial\theta}\delta\theta\right)^2+
\left(\frac{\partial\beta}{\partial\theta_0}\delta\theta_0\right)^2+
\left(\frac{\partial\beta}{\partial\Delta t}\delta\Delta t\right)^2}\\
&=&\frac{1}{\Delta t}\;\sqrt{\left(\frac{\delta\theta}{\theta}\right)^2+
\left(\frac{\delta\theta_0}{\theta_0}\right)^2+
\left(\beta\;\delta\Delta t\right)^2}\label{eq:dbeta_overdamped}
\end{eqnarray}
where $\Delta t=$the mean time for $\theta$ to decrease from $\theta_0$. The
estimate the damping coefficient, $b$, was found by from
$b = 2J\beta$ and
\begin{equation}
\delta b = \sqrt{\left(\frac{\partial b}{\partial J}\;\delta J\right)^2+ 
\left(\frac{\partial b}{\partial \beta}\;\delta \beta\right)^2} 
\label{eq:db}
\end{equation}
where $J = 3.0 \pm 0.1\;{\rm kg \cdot m^2}$.
An estimation of the restoring torque was found from $c = J\beta^2$ and
\begin{equation}
\delta c = \sqrt{\left(\frac{\partial c}{\partial J}\;\delta J\right)^2+ 
\left(\frac{\partial c}{\partial \beta}\;\delta \beta\right)^2} 
\label{eq:dc}
\end{equation}
See Table~\ref{tab2} for the obtained
values. In the critically dampened case, it is expected that
$\beta=\omega_0$, which is not exactly the case as seen in comparing 
$\beta$ from Table~\ref{tab2} and $\omega_0$ from Table~\ref{tab1}.
\begin{table}[ht]
\caption{\label{tab2}Results: estimations of $\beta$, $b$, and $c$ for a near critically dampened
torsion pendulum (braking $I = 1.90\pm0.2$~A).}
\begin{center}
\begin{tabular}{|c|}
\hline
$\beta = 2.3 \pm 0.2\; {\rm rad/s}$\\ \hline
$b = 14 \pm 1 \;{\rm kg \cdot m^2 \cdot s^{-1}}$\\ \hline
$c = 16 \pm3\; {\rm N\cdot m}$\\
\hline
\end{tabular}
\end{center}
\end{table}

The latter part of this experiment dealt with forced
oscillations of the pendulum.

A relationship between frequency and voltage was
established by determining the mean period and thus
frequency at a range of dial settings along with the voltage
across the motor measured by the multimeter and fitting that
data to a linear function.

The equation relating frequency and voltage is of linear of
the form $y = A + Bx$ , where $A = -0.01 \pm 0.006$~Hz, $B =
(0.071 \pm 0.11)\times10^{-2}$~Hz/V, $y$ is the frequency and $x$ is the
voltage across the motor.

We then chose the range of frequencies to look at resonance
to be $0.465\;{\rm Hz} \leq f \leq 0.560\;{\rm Hz}$ for each damping current.
Measurements of amplitude vs.\ frequency were taken by
setting the forcing motor to the particular voltage to give the
desired frequency, stopping the pendulum before each
measurement and letting the pendulum settle for a few
minutes for each measurement.

% Figure 3: Plot of the resonance curves for damping currents of
%$I = 0$~mA, $I = 72 \pm 2$~mA, $I = 424 \pm 6$~mA

% Figure 4: Plot of the resonance curve for a damping current of
% $I=424\pm6$~mA

\begin{figure*}[t]
\centering
\subfloat[\label{fig3a}Plots of the resonance curves for damping currents of
$I = 0$~mA ($\blacksquare$) and  $I = 424 \pm 6$~mA ($\square$)]{
\rotatebox{-90}{\includegraphics[width=2.3in]{RossTfig3.ps}}}\qquad
\subfloat[Plot of the resonance curve for a damping current of
$I=0.830\pm0.01$~A]{
\rotatebox{-90}{\includegraphics[width=2.3in]{RossTfig5.ps}}}
\caption{\label{fig3-5} Plots of amplitude vs.\ frequency for
various braking currents.  The curves are fits to Eq.~(\ref{eq:amplitude_driven_solution})}
\end{figure*}

%Figure 5: Plot of the resonance curve for a damping current of
%$I=0.830\pm0.01$~A

Figure \ref{fig3-5} displays the resonance curves for braking
currents $I = 0$~A, $I = 424 \pm 6$~mA, and $I =
0.83 \pm 0.01$~A. (The resonance curve for braking $I = 72 \pm 2$~mA 
was indistinguishable from the resonance curve for $I=0$~A.) 
The plots were used to determine the
resonance frequency. The uncertainties in resonance
frequency were estimated based on accuracy to which we
could determine the frequency of resonance for each
braking current. Using Eq.~(\ref{eq:wr}), the damping constant was estimated by
\begin{equation}
\beta = \sqrt{\frac{\omega_0^2-\omega_r^2}{2}}
\end{equation}
and the uncertainty of $\beta$ was given by
\begin{equation}
\delta \beta = \sqrt{\left(\frac{\partial \beta}{\partial \omega_r}\delta \omega_r\right)^2+ 
\left(\frac{\partial \beta}{\partial \omega_0}\delta \omega_0\right)^2} \\
\end{equation}
with $\omega_0 = 3.26 \pm 0.02$~rad/s . The values of $f_r$ , $\omega_r$, and $\beta$ obtained 
are shown in Table~\ref{tab3} below.

\begin{table}
\caption{\label{tab3}Resonance frequencies and the associated angular resonance
frequencies, in addition to the calculated damping constant at each brake
current for the driven oscillator.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
braking $I$, mA &0 & $71\pm2$ &$424\pm6$ &$830\pm10$ \\ \hline
$f_r$, Hz &$0.528 $&$ 0.527$&$0.523$&$0.510$\\ 
          &$\pm0.003$&$\pm0.003$&$\pm0.004$&$\pm0.007$\\ \hline
$\omega_r$, rad/s &$3.34$&$ 3.31$&$3.29 $&$3.20 $\\ 
                  &$\pm0.02$&$\pm0.02$&$\pm 0.03$&$ \pm 0.04$\\ \hline
$\beta$, rad/s &$0.51$&$0.41 $&$0.3$&$0.4 $\\ 
               &$\pm0.07$&$\pm 0.09$&$\pm 0.1$&$\pm0.1$\\ \hline
\end{tabular}
\end{center}
\label{tab:results3}
\end{table}
%braking current, mA &0 & $71\pm2$ &$424\pm6$ &$830\pm10$ \\ \hline
%$f_r$, Hz &$0.528\pm0.003$&$ 0.527\pm0.003$&$0.523\pm0.004$&$0.510\pm0.007$\\ \hline
%$\omega_r$, rad/s &$3.34\pm0.02$&$ 3.31\pm0.02$&$3.29 \pm 0.03$&$3.20 \pm 0.04$\\ \hline
%$\beta$, rad/s &$0.51\pm0.07$&$0.41 \pm 0.09$&$0.3 \pm 0.1$&$0.4 \pm0.1$\\ \hline

As expected, the resonance frequency differs less from the
natural frequency with lower damping.

%Figure 6: A plot of phase vs.\ frequency around the natural frequency of
%$0.519 \pm 0.001$~Hz

\begin{figure}
\begin{center}
\rotatebox{-90}{\includegraphics[width=2.3in]{RossTfig6.ps}}
\end{center}
\caption{\label{fig6} A plot of phase vs.\ frequency 
for $I = 0$~mA (solid) and  $I = 424 \pm 6$~mA (dashed)
using Eq.~(\ref{eq:phase}) and
the fit paramters used in the curves of
Fig.~\ref{fig3a} }
\end{figure}

It can be seen in Fig.~\ref{fig6} that as the difference between the
angular driving frequency and the natural angular frequency
approaches zero, the phase difference between the
pendulum and the forcing motor approaches $\pm \pi/2$ , and at
high and low frequencies, the phase approaches zero. The
phase plotted in Fig.~\ref{fig6} was given by Eq.~(\ref{eq:phase})
with range properly shifted to 0 to $pi$.
%\begin{equation}
%\phi = \tan^{-1}\left(\frac{2\beta\omega}{\omega_0^2-\omega^2}\right)
%\end{equation}
%where $\omega$ is the angular frequency of the
%forcing motor, and $\omega_0$ is the natural angular frequency.

\begin{table}
\caption{A comparison of $\beta$ calculated for dampened oscillations at
different damping currents.}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
braking $I$, mA &$149\pm3$& $490\pm7$& $1900\pm20$ \\ \hline
$\beta$, rad/s &$0.040\pm0.001$&$ 0.17\pm0.03$&$2.3\pm0.2$\\ \hline
\end{tabular}
\end{center}
\label{tab:results4}
\end{table}

\begin{table}
\caption{A comparison of $\beta$ calculated using the resonance curve of forced
oscillations at different braking currents.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
braking $I$, mA &0 & $72\pm2$ &$424\pm6$ &$830\pm10$ \\ \hline
$\beta$, rad/s &$0.51\pm0.07$&$0.41 \pm 0.09$ & $0.3 \pm 0.1$& $0.4 \pm0.1$\\ \hline
\end{tabular}
\end{center}
\label{tab:results5}
\end{table}

Comparing the damping constants found for dampened
and forced oscillation at similar values of damping
current, as shown in Tables \ref{tab:results4} and \ref{tab:results5}, it is evident that $\beta$ is
not consistent. This could be a result of the poor fit for
the damping current of $I = 149\pm3$~mA. The damping
constant for damping currents of $\sim0.45$~A are marginally
close but do not agree within the bounds of uncertainty,
which may again be a result of inaccuracies of the fit of
the data.

\begin{thebibliography}{99}

\bibitem{crumley}J.P.\ Crumley, {\it Torsion Pendulum}, \url{http://www.physics.csbsju.edu/~jcrumley/370/tp/torsion_pendulum.pdf}

\bibitem{LDa}Leybold Scientific, {\it Free Rotational Oscillations}, \url{http://www.leybold-didactic.de/literatur/hb/e/p1/p1531_e.pdf}, 
(LD Didactic, Huerth, Germany, 2006)

\bibitem{marion} S.T.\ Thornton and J.B.\ Marion,
{\it Classical Dynamics of Particles and Systems}, 5th Ed. (Brook/Cole-Thomson, Belmont, CA, 2003)

\bibitem{LDb}Leybold Scientific, {\it Forced Rotational Oscillations}, \url{http://www.leybold-didactic.de/literatur/hb/e/p1/p1532_e.pdf}, 
(LD Didactic, Huerth, Germany, 2006)

\end{thebibliography}

\end{document}
\bye
