\experiment{The Cavendish Experiment}

\section{Background }

Isaac Newton's (1642--1727) theory of gravitation explained the
motion of terrestrial objects and celestial bodies by positing a
mutual attraction between all pairs of massive objects proportional
to the product of the two masses and inversely proportional to the
square of the distance between them. In modern notation, the law of
universal gravitation is expressed
\begin{equation}
	\label{eq:Cav1}
	F = \frac{G M m}{r^{2}} 
\end{equation}
where $M$ and $m$ are the masses of the two objects, $r$ the distance
separating them, and $G$ is the universal constant of gravitation.
Newton was not particularly concerned to evaluate the constant of
proportionality, $G$, for two reasons. First, a consistent unit of
mass was not in widespread use at the time. Second, he judged that
since the gravitational attraction was so weak between any pair of
objects whose mass he could sensibly measure, being so overwhelmed by
the attraction each feels toward the center of the Earth, any
measurement of $G$ was impractical.

Notwithstanding Newton's pessimism, towards the latter half of the
18th century several scientists attempted to weigh the Earth by
observing the gravitational force on a test mass from a nearby large
mountain. These efforts were hampered, however, by very imperfect
knowledge of the composition and average density of the rock
composing the mountain. Spurred by his interest in the structure and
composition of the interior of the Earth, Henry Cavendish in a 1783
letter to his friend Rev.\ John Michell discussed the possibility of
devising an experiment to ``weigh the Earth.'' Borrowing an idea from
the French scientist Coulomb who had investigated the electrical
force between small charged metal spheres, Michell suggested using a
torsion balance to detect the tiny gravitational attraction between
metal spheres and set about constructing an appropriate apparatus. He
died in 1793, however, before conducting experiments with the
apparatus.

The apparatus eventually made its way to Cavendish's home/laboratory,
where he rebuilt most of it. His balance was constructed from a
6-foot wooden rod suspended by a metal fiber, with 2-inch-diameter
lead spheres mounted on each end of the rod. These were attracted to
350-pound lead spheres brought close to the enclosure housing the
rod, roughly as illustrated in the figure below. He began his
experiments to ``weigh the world'' in 1797 at the age of 67, and
published his result in 1798 that the average density of the Earth is
5.48 times that of water. His work was done with such care that this
value was not improved upon for over a century. The modern value
forthe mean density of the Earth is 5.52 times the density of water.
Cavendish's extraordinary attention to detail and to the
quantification of the errors in this experiment has lead C.W.F.\ 
Everitt to describe this experiment as the first modern physics
experiment. In this experiment you will use a torsional balance
similar to Cavendish's to ``weigh the Earth'' by determining a value
for $G$.\footnote{In 1998 the accepted value for $G$ was known to only
  0.15\% precision, a surprisingly crude number, reflecting the
  miniscule forces involved. In May 2000 in an experimental \emph{tour de force}, Jens
Gundlach and Stephen Merkowitz of University of Washington improved
on that precision by a factor of 100, finding a value of 
$\val{(6.67390 \pm 0.00009) \times 10^{-11}}{N m^{2}kg^{-2}}$.} 

\section{Theory}

% \begin{wrapfigure}[nlines]{<rl>}[overhang into margin]{width or 0pt}
\begin{wrapfigure}[14]{r}[.2in]{0pt}
	\igw[1.8in]{Cavendish1}
	\caption{The Cavendish torsional balance.}
	\label{fig:Cavendish1}
\end{wrapfigure}

The Cavendish torsional balance is illustrated in
\fref{Cavendish1}. Two small metal balls of mass $m$ are attached to
opposite ends of a light, rigid, horizontal rod which is suspended
from a torsion fiber. When the ``dumbbell'' formed by the rod and
masses is twisted away from its equilibrium position (angle), the
fiber generates a restoring torque proportional to the angle of twist,
$\tau = -\kappa \theta$. In the absence of damping, the dumbbell
executes an oscillatory motion whose period is given by $T = 2\pi
\sqrt{\frac{I}{\kappa}}$, where $I$ is the rotational inertia of the
dumbbell, $I = 2m \left(d^{2} + \frac{2}{5} r^{2}
\right)$.  In this expression, $r$ is the radius of the small masses
$m$, and $d$ is the distance from the center of the rod to the center
of one of the masses, and we have neglected the mass of the thin
rod. Knowledge of $m$, $d$, and $r$, and a careful measurement of the
period of oscillation $T$ allows one to calibrate the torsion fiber,
obtaining its spring constant $\kappa$. From $\kappa$ and a
measurement of the twist caused by the large masses $M$ you can deduce
the gravitational force between the masses, and hence $G$.



% \begin{wrapfigure}[nlines]{<rl>}[overhang into margin]{width or 0pt}
\begin{wrapfigure}[11]{r}[.2in]{0pt}
  \vspace*{-.5in}
	\igw[1.8in]{Cavendish2}
	\caption{Top view.}
	\label{fig:Cavendish2}
\end{wrapfigure}

\subsection{Gravitational Torque} 

When the large metal spheres are positioned as shown in the figure,
the gravitational attraction between the large and small spheres
produces a torque that rotates the dumbbell clockwise. Only the
component of the force on each mass that is perpendicular to the
horizontal bar produces a torque about the center of the rod. The
magnitude of the torque between the two adjacent masses is given by
$\tau_{g} = 2 F_{\perp}d$, where the factor of $2$
comes from the fact that the torque is equal on the two masses $m$.
This torque displaces the equilibrium angle of the dumbbell by an
amount given by $\tau = -\kappa \theta_{0}$.
%
Hence, if one can measure the equilibrium angle $\theta_{0}$ very carefully,
one can deduce the gravitational force that produces the torque and
finally $G$. 

\subsection{Light Lever}

Cavendish mounted a finely ruled scale near the end of the dumbbell,
which he could read with a telescope to one-hundredth of an inch.  The
telescope allowed him to remain outside the experimental chamber, thus
eliminating air currents and his gravitational influence on the
oscillator.

We will take advantage of a light lever to magnify the dumbbell's
tiny rotation into an easily observed displacement on a far screen.
The light lever is produced by bouncing a laser beam off a mirror
mounted to the dumbbell. When light bounces off a mirror, the angle
the incoming beam makes with the normal (perpendicular) of the mirror
is equal to the angle the outgoing beam makes with the normal. If the
mirror rotates through a small angle $\alpha$, the outgoing beam rotates
through the angle $2\alpha$, since both the incoming and outgoing angles
change by the same amount. By measuring the motion of the laser spot
on a far screen, and knowing the distance between the mirror and the
screen, you can determine the angle $\alpha$, from which you can infer the
rotation of the dumbbell, $\theta$. 

\subsection{Damping}

In the absence of damping, the motion of the dumbbell is a sinusoidal
oscillation with the period given by 
\begin{equation}
	\label{eq:Cavendish2}
	T = 2 \pi \sqrt{\frac{I}{\kappa}} 
\end{equation}
Viscous damping of the pendulum's motion caused by air resistance
produces a drag torque proportional to the angular speed of the
dumbbell. This causes the sinusoidal oscillation to decay
exponentially, with a time constant $\tau$ that is long compared to the
oscillation period. The motion is described by the function 
\begin{equation}
	\label{eq:Cavendish3}
	\theta = \theta_{0} + A e^{-t/\tau} \sin(2 \pi t/T + \phi_{0})
\end{equation}
where $\phi_{0}$ is the initial phase of the motion, $\theta_{0}$ is the equilibrium
position, $A$ is the amplitude of the motion, and $\tau$ is the time for
the oscillation to fall to $1/e$ of its original value. (Note that this
time has nothing to do with the torques, which, unfortunately use the
same symbol!) By measuring $\theta(t)$ and fitting your data to this
equation, you can determine both $T$ and $\theta_{0}$, from which you can
determine $G$. 

\subsection{Parameters}

According to the Pasco manual, the parameters of the
apparatus include the following: 
\begin{center}
\begin{tabular}{cc}
  \toprule
  Variable & Value \\
  \midrule
  $m$ & \val{(38.3 \pm 0.2)}{g} \\
  $r$ & \val{9.53}{mm} \\
  $d$ & \val{50.0}{mm} \\
  $b$ & \val{46.5}{mm} \\
  \bottomrule
\end{tabular}
\end{center}

No uncertainties are given for the last three values. Since $m \propto
r^{3}$, one could infer an uncertainty in $r$ of 0.02~mm resulting
from the uncertainty in $m$ given above. Based on the precision of the
value given for $d$ one might reasonably assume an uncertainty of 0.1
mm in that value. The uncertainty in $b$ depends on the accuracy of
the horizontal alignment of the pendulum (See the Pasco manual for the
proper alignment procedure; your check of the alignment should give
you an estimate of the uncertainty in $b$.)

\subsection{Safety}

\caution{The laser pointer that forms the light lever for this
experiment is a class III laser capable of damaging
retinas. \scream{Do not look directly into the beam.} Please ensure
that nobody looks into the beam. Note that it is safe to look at the
diffuse spot the beam produces as it reflects from an object, such as
a meter stick.}

\section{Procedure}

\begin{enumerate}
\item See the instructions next to the apparatus for proper alignment
of the oscillator. You should be sure that when the freely suspended
dumbbell remains at rest, it is equidistant from the front and back
faces of the enclosure. This alignment should be done with the large
lead balls removed. 

\item Carefully weigh the large lead balls. Place a styrofoam tray on
the electronic scale and tare it. Then gently place the lead ball
into the tray. Ifthe lead balls are dropped, they will become
misshapen, which will severely compromise the accuracy ofthe
experiment. 

\item 	Lock the dumbbell using the screws on the bottom of the
enclosure, then gently place the large lead balls in the armature.
Rotate the armature until the balls just touch the sides of the
enclosure. Gently lower the support screws until the dumbbell rotates
freely. Ifit rotates so much that the dumbbell bounces off the sides
of the enclosure, use the support screws to settle the motion. 

\item Once the dumbbell oscillates freely, begin recording the
  position of the reflected laser spot as a function of time. Record
  at least two full periods. 

\item Gently rotate the armature until the large balls once again
  touch the sides of the enclosure in the opposite orientation and
  record the position of the reflected laser spot as a function of
  time for at least another two periods.

\item If you have time you should finally return the armature to its
original position and again record the position of the laser spot as
a function of time for at least another two periods. 

\item From the distance between the mirror and the screen on which you
measured the laser spot position, you can deduce the angle of
rotation of the dumbbell. As described above, fit a damped sinusoid
to each set of data and extract the centers of rotation. From their
difference and the data given in the parameters table above, you can
deduce the value of $G$.

\end{enumerate}
\vfill

\noindent Created 10/22/99 by Peter N.\ Saeta

\noindent updated 1/17/02 by Daniel C.\ Petersen. 

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