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Electrostatics problem



In Serway's 4th ed. of Physics for Scientists and Engineers, problem 57 of
Chapter23 (Electric Fields) states:
Two small spheres of mass m are suspended from strings of length l that are
connected at a common point. One sphere has charge Q; the other has charge
2Q. Assume the angles theta1 and theta2 that the strings make with the
vertical are small. (a) How are theta1 and theta2 related? (b).......

The solutions manual then shows these angles to be equal - but only by
making an assumption that FORCES this situation; i.e. by assuming the
electrical force on either to be horizontal. This should have nothing to
do with the original assumption that the angles are small, and I feel sorry
for anyone who tries to learn physics from such presentations. If you simply
allow yourself to think beyond translational equilibrium, the solution becomes
trivial.

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).

Also, it doesn't matter if the charges are equal or not. In fact, for equal
charges, this is a standard problem in many texts (even algebra based) which
begins by stating that the angles are equal and then typically asking a
question relating the charges to the separation distance. In other words,
MANY students are led to solve a certain class of problem in a trivial way
without being allowed to more fully understand how things work. Even four
co-authors of a solutions manual may have been led astray.

Tom Walkiewicz
Edinboro University of PA