```MEASURING THE GRAVITATIONAL CONSTANT, G

5015 S. Blackstone Ave.
Chicago, IL 60615
1-312-536-8907

BEHAVIORAL OBJECTIVE:
Students will see that the gravitational force is indeed universal; The
gravitational force acts not only on, but also between objects of normal size and
mass.  Advanced students will measure the proportionality constant G which appears
in Newton's expression for the gravitational force acting between two spherical
objects.

MATERIALS:
A large Cavendish or torsion balance was constructed; Bags of sand (nominal
mass 27 kg) acting on 0.3 kg end masses provide the torque.  The torsion fiber is
a stainless steel wire with a diameter of 0.003 inch (0.0012 cm) and a length of
approximately 1.75 m. The lever arm of the balance is approximately 0.9 m. A large
frame was constructed of wood and enclosed with plastic windows. Aluminum window
screen was used to control electrostatic forces acting on the end masses. The
torsion bar was a piece of aluminum edging (of the sort used to encircle kitchen
sinks) obtained from a local hardware store.

STRATEGY:
When students study the gravitational force in the usual manner (Kepler's
Laws of Planetary Motion are presented as empirical deductions from data; The
nature of the required force is deduced from the centripetal force law and
Newton's Third Law), the instructor is normally forced to make two concessions
which are directly contrary to the phenomenological approach to teaching.  First,
he must make the unsupported assertion that the law derived from planetary data is
"universal," and can be applied to any two objects, including those of ordinary
size and mass. Secondly, the instructor must provide the value of the constant G
which appears in the Newtonian expression.  Since the treatment of gravitation
usually occurs in the middle of the first semester, shortly after the students
have finally begun to appreciate the phenomenological approach to physics
instruction, the traditional treatment runs contrary to the phenomenological
teaching philosophy.
An alternative strategy which is consistent with the phenomenological
approach would be to provide at least a demonstration of the universality of
Newtonian gravitation by showing that objects of ordinary mass do in fact attract
one another with a force which cannot be attributed to other causes. It should be
possible to extend the demonstration to include actual measurement of the
gravitational constant, though this could (and perhaps should) be reserved for
advanced students or special projects. Apparatus to accomplish this objective is
commercially available.  Unfortunately, the cost is prohibitive in the light of
most public school budgets. Thus, an attempt was made to build a large-scale
version of the traditional torsion balance apparatus which would be economically
feasible and provide a clearly visible demonstration of the gravitational
attraction between objects of ordinary mass. Drawings and a complete list of
materials used are available from the author.
In practice, the torsion balance is allowed to come into equilibrium with the
two bags of sand "kitty-corner" from each other, as shown in the diagram.  Once
equilibrium has been achieved with the sand bags in this position, the bags are
carried to the other end of the apparatus and a new equilibrium position is
determined. The change in equilibrium position shows the existence of the force.
2
The dimensions of the apparatus were chosen to make the actual displacement of the
end masses as large as possible. Permanent records of the original and final
positions can be made by placing a sheet of graph paper over the top window and
recording the point directly over the lever arm at each of the two equilibrium
positions.  This approach will not result in data precise enough to allow a
measurement of G.
To make the demonstration quantitative, a more precise method of measuring
the angular displacement between the two equilibrium positions must be used. At
present, the author has not attempted to improve the apparatus to include such
measurements. Usually, an optical lever is used, though Cavendish himself used
ordinary ruled markings.

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