(1) I have not seen the AJP that Peter Schoch mentions, but I'm not
sure I need to. Isn't this fairly obvious? Our analysis has to
provide the same result regardless of whether we solve the problem in
Cartesian coordinates or polar coordinates. Some problems are easier
to solve in one coordinate system while other problems become simpler
in the other coordinate system. It seems that Tom Walkiewicz was
apparently asking for someone to present a solution in Cartesian
coordinates; to which I reply, why would you want to? The balls are
constrained to move around a fixed point at fixed radius from that
point. If that doesn't shout "please view me as a rotational problem"
I don't know what would. As we teach our students various insights to
solve problems, I think it is important that we (a) tell them the
choice of coordinate system doesn't change the outcome of the physics,
but (b) various things about the problem might tip us off as to which
coordinate system might yield a simpler analysis; therefore, watch for
those clues.
(2) The reason I mentioned my assumption of small angles in my
post-script message was because in my original posting I clearly solved
the problem using right triangles by stating that the sine of the angle
was the opposite side over the hypotenuse. This is true when the
masses are the same. If the masses are not equal, the triangles to
which I applied this procedure are not quite right triangles, although
they are close to right triangles at small mass separations. To remove
the small angle restriction you notice that the overall triangle
(string, string, line-between-masses) is an isosceles triangle, and
then you use the law of sines for the two triangles formed inside the
isosceles triangle by the vertical line from the pivot to the center of
mass. As I mentioned in my PS, that's a bit more trig than assuming
right triangles, but it's not difficult.
(3) Forcing the center of mass below the pivot (forcing the line from
the top of the strings to the center of mass to be a vertical line) is
clearly the way to go when the masses are not equal, but I maintain it
is the way to go regardless. Of course, by my statement (1) above, I
acknowledge there must exist a valid analysis in Cartesian coordinates.
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