"However, my question is this: Can this be solved using ONLY the
condition of
translational equilibrium - such as finding contradictory statements,
or any
other logical procedure?? Perhaps I'm just getting too old to think of
a
simple explanation in this case (except for what is implied above)."
My answer is yes. This is not an electrostatics problem. It's a
center of mass problem. The two masses are going to be held apart at
some separation by the electrostatic repulsion. It does not matter
what that separation is. As far as we are concerned, the balls could
be held apart by a spring. If we initially treat both balls together
as one mass, it is clear we have a pendulum, and the equilibrium
position of this pendulum is with the center of mass directly
underneath the pivot point.
Since the equilibrium position has the center of mass under the pivot,
the angles are determined solely by the masses of the balls. For
equal-mass balls the center of mass is in the middle of the two balls.
That means the angle of each string from vertical must be the same.
This is totally independent of the ratio of the charges or whatever
else that holds the balls apart (assuming whatever holds them apart
doesn't alter the CM).
We can also solve the theta-1 theta-2 ratio for non-equal mass balls.
The only difference is that the center of mass is now closer to the
more massive ball. For example, if one ball has three times the mass
of the other, the center of mass is located one-fourth the total
separation (center to center) from the center of the more massive ball.
That point must be vertically in-line with the pivot. That makes two
triangles sharing a common central vertical line. The two outer lines
are length L (string length) and the bottom lengths are in the ratio
1:3. We have sin(T1) = b/L and sin(T2) = 3b/L. sin(T1)/sin(T2) = 1/3.
T is theta and b is some length. This is interesting, and I am going
to go ahead and post this message without thinking any more about what
this ratio means.
Notice that I didn't mention charge. I just assumed the balls would
come into equilibrium at some non-zero separation.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817