The Cavendish Experiment
by U. E. Kruse and T. N. Tran,
preliminary version, March 1995.
from
http://complex.ccsr.uiuc.edu/~alfred/Ph101/Notes.htmd/cavendish/
Introduction
The basic idea of the experiment: Sound.au
or Sound.snd
(900kbyte)
Sir
Isaac Newton's universal law of gravitation says the gravitational force of
attraction F between two masses m and M separated by a
distance r is:
.
As a simple way to check this law and determine G, we could try to
attach a small mass (m = 0.15 kg) to a spring of known stiffness and
measure the stretching of the spring by the gravitational force on the small
mass when a second mass (M = 1.5 kg) is brought close (r = 0.05
m). In this example, the force F is so weak that a typical spring
(k = 6 N/m) is only stretched 0.1 nm, about the size of an atom. (The
values of m, M, and r chosen above are the values actually
used in our experiment described below.) Despite the weakness of the attraction,
Henry
Cavendish was able to devise an experiment to measure the force, and hence
G. In 1798, one hundred and eleven years after Newton proposed his law of
gravitation, Cavendish constructed a "torsion balance", in which he measured the
small twist of a dumbbell hung from a fine fiber when two spheres are placed
nearby. Below, we present a modern version of the Cavendish experiment, with
which we measured G.
The Torsion Balance
A dumbbell consisting of two small lead spheres
fastened to a thin rod is hung from a fine fiber. The dumbbell will twist until
the torque from the fiber is equal to any external torque applied to the
dumbbell. A small mirror is attached to the dumbbell to help determine the angle
of the twist.
We shine a laser beam at the mirror, reflecting onto a screen. The location
of the laser spot on the screen allows us to determine the twist of the
dumbbell. By placing the screen at a reasonable distance away, we can detect
even a small twist.
Two big lead spheres are placed near the small spheres on the dumbbell. The
gravitational force between the large spheres and the nearby small spheres
twists the dumbbell. The laser spot consequently shifts to one side. When the
dumbbell reaches equilibrium, the torque from the fiber balances the torque from
the gravitational forces on the dumbbell.
A Simulation of the Experiment
First, we allow the balance to come to
equilibrium with a clockwise torque as seen from a topview. Next, we move the
big spheres to the opposite side to give an equal torque in the counterclockwise
direction. The dumbbell then moves and after oscillating settles onto a new
equilibrium.
75K MPEG.
Using the Torsion Balance to find G
At equilibrium, the gravitational
torque on the dumbbell is balanced by the torque 
due to the fiber resisting the twist:
.
The larger the twist 
, the stronger the torque (with D being the
torsion constant of the fiber). By measuring the separation S between
equilibrium positions of the spot, and knowing the distance L to the
screen, we can determine the twist . The torsion constant D can
be found from the period of oscillation T of the balance and the moment
of inertia I of the dumbbell using:
.
Thus we find the gravitational attraction.
The Actual Balance
The actual balance is enclosed in a metal case with
glass windows to protect the torsion balance from air currents while allowing
the laser beam to bounce from the mirror.
755K MPEG.
The Various Parts of the Experiment
The various components of the
experiment are: the torsion balance, the laser, and a ruler, which acts as the
screen. In the last part of the movie, one can see the laser spot at about 34
cm. There is a counter below the ruler, which will measure the elapsed time in
seconds in our actual run. 
746K MPEG.
Start of Measurements
To start our measurement, we allow the dumbbell to
settle at equilibrium 1, with the big spheres twisting the dumbbell clockwise.
We then move the big spheres to the opposite positions, so that they will twist
the dumbbell with an equal torque counterclockwise.
567K MPEG.
Time-lapse Measurement of Motion
After the big spheres are moved, the
dumbbell leaves equilibrium 1, oscillates, and settles onto the new equilibrium
2. The movie shows the position of the laser spot on the ruler. In the lower
part of the picture, we plot the spot's position against the elapsed time in
seconds, which can be read from the counter.
226K MPEG.
The Observed Value of G
We determined G from the measurements of:
- the masses of the spheres: m and M,
- the separation of the spheres: r,
- the length of the moment arm of the torsion balance: d,
- the stiffness of the suspension fiber: D,
- the distance from the mirror to the screen: L,
- the separation of the two equilibrium positions: S,
- and a correction factor to account for the attractions between each small
sphere and its more distant big sphere: beta.
We found
,
about 89% of the standard value of
.