Re: Help with bob!

[Hugh Haskell <hhaskell@mindspring.com>]

> There is a problem in Halliday, Resnick, and Walker that asks for the
> deviation from vertical that a plumb bob experiences when it is hung at a
> latitude of 40 degrees.  I consider the pendulum to be composed of two
> component pendula, one that is perpendicular to the rotational surface,
> which requires taking the sine of the weight and the tension.  The other
> component is parallel to the surface and has no effect.  If the Earth did
> not rotate, this component pendulum would have a weight pulling down of w
> sin(theta) and a tension pulling up of T sin(theta), where theta is the
> latitude.  In the presence of rotation, the weight term remains the same
> and the centripetal term comes from the additional tension force that is
> created as the pendulum moves outward by the deviation angle.  I calculate
> an angle of 0.23 degrees, while the back of the book lists 0.09 degrees.
> This is Problem # 71 in Chapter 5 of the previous edition.  The one with
> the streaming lights on the front.
>
>
> Tom McCarthy
> Saint Edward's School
> 1895 St. Edward's Drive
> Vero Beach, FL  32963
> 561-231-4136
> Physics and Astronomy
>
> ----------

My students do this problem every year, and have trouble with it every year
(don't they ever learn??). I think you are making it unnecessarily
complicated. You'll get the book's answer if you deal with the force
diagram carefully. Gravity is pulling the bob toward the center of the
earth, tension in the cord is pulling it toward the support, and these two
forces have to give rise to a resultant which is the mass of the bob times
the centripetal acceleration, which must be directed perpendicular to the
earth's axis of rotation (not toward the center of the earth, but tilted
away from the vertical by an angle equal to the latitude). The resulting
triangle can be solved with the law of sines. Solving the problem this way
gives an answer of .099 degrees at a latitude of 45 degrees (It also gives
0 at both the pole and the equator). The algebra is a bit messy and I had
to redo the problem about four times just now in order to get a sensible
answer, but it works.

If it still doesn't work for you, and you have the capability to read
BinHex files (i.e., a Mac with an appropriate e-mail program like Eudora or
Claris e-mailer), I can send you a complete solution with equations and
diagrams.

Hugh

To get random signatures put text files into a folder called "Random
Signatures" into your Preferences folder.The box said "Requires Windows 95
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