| Sideways
Gravity in the Basement: Norman Scheinberg's Cavendish
Experiment
John W. Dooley,
Physics Department, Millersville University
Norman Scheinberg is a professor of
electrical engineering at The City College of the City
University of New York. He built a Foucault pendulum
in his basement just to see if he could get it to work.
He also built a "Cavendish experiment" for his son as
a demonstration of the gravitational force between two
laboratory-sized objects. His Cavendish implementation
should be achievable by at least some amateur scientists.
It also turns out that his video of the experiment (see
below) can be analyzed to give fair agreement with more
sophisticated measurements.
Dr. Scheinberg discovered that his
was not the only Foucault pendulum in New Jersey when
he learned of Mark
Streitman's pendulum. In conversation with Mark,
he mentioned his Cavendish experiment. Mark was interested,
so he sent a video of the experiment, and Mark forwarded
the video to me. Figure 1 is a composite from two frames
of the video that shows the overall construction of
the apparatus.
The tripod legs are made from electrical
conduit, bent to shape. The upper segment of the tripod
supports a tungsten wire which is fastened to the rod
seen lying across the top plate. This upper short rod
can be rotated to adjust the equilibrium location of
the small masses at the bottom.
Figure 2 shows a closer view of the
small masses. These 0.07 kg lead masses are hung on
opposite ends of a horizontal rod. The rod is suspended
at its center by a 0.025 mm (0.001 inch) diameter tungsten
wire that is about 1.5 meters (5 feet) long. The entire
rod-mass assembly is surrounded by a metal chamber.
The small masses will be attracted to two large lead
masses by the force of gravity. In the first two figures,
the large masses are so far away from the small masses
that the gravitational attraction is negligible.
The two large (32 kg) lead spheres
are mounted on a heavy duty "lazy susan." When rotated
to a position near the metal trough, these two masses
will exert a force of about 10 micronewtons on the small
spheres. (That's about the weight of one crumb of hamburger.)
When they feel this force, the small
balls move towards the large balls, twisting the tungsten
wire. The force is so small that the metal chamber must
be covered by Plexiglas to eliminate the much larger
force that can be exerted by air currents.
Ordinarily, such a sudden pull would
set the rod into oscillation, swinging to and fro, twisting
the tungsten wire clockwise and anti-clockwise, in the
same way that an "anniversary clock" oscillates. For
this reason, the apparatus is called a "torsion pendulum."
Norman prevents endless torsional oscillation with a
"damper." The damper is a small paddle wheel hanging
below the middle of the rod, dipping into a (pink) mixture
of dish soap and water.
When the large balls are up against
the metal chamber, the small balls move almost 2.5 cm
(1 inch), hit the wall of the chamber, and come to rest
there (see Fig. 3). This is the fundamental fact of
the demonstration. The balls are pulled together by
the (sideways) force of gravity between them.
Figure 4 shows the essential parts
for those who want to build the apparatus. The heavy
weights were cast into a hemispherical cooking pot,
and the two hemispheres joined together.
The video is available here as Scheinberg_divix.avi.
I had to download the player at
www.divx.com to view it. Although intended just
as a demonstration of the force of gravity between two
ordinary objects, the video plus the size of the masses
and the distance traveled by the small mass is enough
to analyze the motion in some detail.
The first step is a check to see if
it's worthwhile to do a careful analysis. When the small
ball in the foreground is pulled to the left side of
the chamber and released, it swings to the right side
in about 10 minutes (as seen in the video), provided
that the large mass is at the wall of the chamber as
in Fig. 3. (If that experiment is done with the large
mass missing, the time is longer, about 16 minutes.)
We start with the simplest possible
force model: We assume that the force by the twisted
tungsten wire is constant as the small mass swings from
one side of its chamber to the other in 16 minutes.
In this motion it travels about 0.05 meters (2 inches).
Using the formula
from introductory physics:
x = 1/2 a t 2
to estimate the acceleration, a, caused
by the wire.
With the heavy mass pulling it, the
small ball has larger average acceleration and takes
less time (10 minutes). Using the same formula with
the new time, we find an increased acceleration, aG.
The difference between these two is
the change caused by the gravitational force of the
large mass. We find
a - aG ~ 1.6 x 10
-7 m/sec2.
Using Newton's
law for gravitation
F = GmM/(R)2
to estimate the increase in acceleration,
we find (using the average value for R, the separation
of the masses m and M)
a - aG ~ 2.3 x 10
-7 m/sec2 .
This is close enough agreement to justify
more careful analysis. First, we recognize that the
torsion pendulum executes harmonic
motion, with an acceleration that varies greatly
with time.
The gravitational force of the large
mass changes two things about the motion of the torsion
pendulum. First, the center of oscillation is pulled
towards the large mass. Second, because the gravitational
force gets weaker as the separation increases, it appears
that the spring force (strictly, a torque) by the tungsten
wire is weakened. The result is an increase in period
of oscillation.
This time we use Newton's law for gravitation
F = GmM/(R)2
with R allowed to vary more realistically.
With this model we calculate that it takes the small
ball about 11 minutes to move from the left side to
the right side of its chamber, when forced by a large
ball on the right. This compares well with the approximately
10 minutes observed in the video.
The agreement means that Newton's law
for gravitation works for this particular case. To seriously
test it, the measurement and prediction must be repeated
for a variety of values for M, and also for a variety
of values for the distance, D, between the relaxed position
of the small ball and the center of the large ball.
The Basis
for the Prediction:
Dr. Scheinberg reported that x can
shift .025 meters (1 inch) to the left and to the right
of its relaxed position, before hitting the wall of
its chamber. The video shows that the period T0
of the small ball in the absence of the large ball is
about 32 minutes. The decay constant, gamma, is found
from the video to be 0.0007. (The units of gamma are
1/second.) The video also allows us to estimate the
distance, D, to be 0.10 meter (4 inches).
The addition of the gravity force from
the large ball makes two changes in the solution. It
pulls the center point of the oscillation closer to
the large mass, to a point called x0. It
also changes the period of oscillation from T0
= 32 minutes to T = 41 minutes. The extra gravity force
does not change the decay constant, gamma.
Because the small mass moves only 0.025
m (1 inch) on a radius arm of about 0.180 m (7 inches),
its path is nearly a straight line. Assuming that the
tungsten wire provides a torque proportional to the
twist angle, we can argue that the small mass feels
a (tangential) force proportional to its displacement.
That is, for a displacement, x, the
small mass feels a force F = -kx, where k is the effective
spring constant. The natural period of oscillation is
related to this spring constant by
T2 = (4 pi2)(m/k)
where m is the mass, m=0.07 kg.
The pendulum is heavily damped, and
this changes the observed period. But the effect is
only about 3% and will be ignored.
Newton's equation of motion for such
a system, without the gravitational force of the large
lead balls is
m a = -c
v -k x
where m is the mass of the small ball,
x is its position,
v is its velocity, and a
is its acceleration. c is a constant
representing the soap solution, which produces a force
proportional to velocity, like a liquid friction.
The solution to the equation is discussed
at the hyperphysics
site.
If we include the gravitational force
of the large ball, Newton's equation is
ma = -cv
-kx + GmM/(D-x)2
where D is the distance from the center
of the heavy lead ball to the relaxed position (x=0)
of the small ball. M is the mass of the large ball (32
kg) and G is Newton's gravitation constant, 6.67 x 10-11
newton m 2 /kg2.
A graph of the two terms that depend
on x shows that for x within the chamber, the sum of
the two terms is approximately proportional to x. This
allows calculation of a new effective spring constant
and also a new equilibrium point. When it is all put
into the standard damped oscillator solution, the time
to travel from one side of the chamber to the other,
starting from rest, is about 10 minutes. I will provide
details of the calculation upon request. 
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