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Re: Electrostatics problem



Hi all-
In response to John Mallinckrodt-
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The "proofs" I have seen here so far have been flawed, incomplete, or
overly difficult. Allow me to propose an exceedingly simple geometric
argument:

Adopting Bob's notation, i.e.,

T1,T2 = angles of the strings from the vertical
F = the coulomb force magnitude.

1. Note that the strings and the line joining the masses form an
isosceles triangle. Call the two equal angles T3.

2. Draw vector force triangles for each mass recognizing the
equality of the two electrostatic forces and the two
gravitational forces and recognizing the angle T3 at the
vertices formed by the electrostatic and tension forces.

3. From the triangle of forces for body 1, the law of sines says:

sinT1/F = sinT3/mg

From the triangle of forces for body 2, the law of sines says:

sinT2/F = sinT3/mg

With only the most obvious added assumption that T1 and T2 are both acute,
it can be ssen by inspection that T1 = T2.

QED?
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There is, in fact a second solution to the static problem for sufficiently
large force. You have excluded it with your assumption that T1 and T2 are both
acute. The second solution is unstable, but the equations of static equilibrium
do not address stability questions.
To set up the general equations let, again, A be the angle between the
line of centers and the the horizontal. Note that
A = (T2-T1)/2.
This latter equation lets you solve all the trig equations in closed form, and
taking ratios to eliminate all forces leads to two alternative trigonometric
conditions on the angles. One is equality, as already pointed out. What is
the other?
Edmiston's "elegant solution" let's you see it immediately. The center
of mass of the system lies along the line of centers, but the angles are not
acute and, strictly speaking, they are unequal.
Recall the original question.

Why the restriction to small angles? Here is the answer.
Regards,
Jack

"I scored the next great triumph for science myself,
to wit, how the milk gets into the cow. Both of us
had marveled over that mystery a long time. We had
followed the cows around for years - that is, in the
daytime - but had never caught them drinking fluid of
that color."
Mark Twain, Extract from Eve's
Autobiography