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RE: Help with bob!




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I too usually assign this problem and the problem and my students rarely =
complete it. I agree, the free body diagram is the key to the problem. =
I couldn't find the problem in the 5 th edition of Halliday, Resnick and =
Walker but it is number 74 in chapter 6 of the 3rd edition. The answer =
is not listed because it is an even problem. I worked it out again and =
came up with an answer of .0971 degrees. At 45 degrees I get a slightly =
larger angle (it should be max at 45) of .0985.

I simply applied the second law to the fbd. This resulted in the =
equations

Tsin(40+theta)-mgsin(40)=3D0 along a direction parallel to the =
earth's axis

And mgcos(40)-Tcos(40+theta)=3DmRcos(40)w^2 along a direction =
perpindicular to the earth's axis.

Where T =3D tension, m =3D mass of the bob, R =3D radius of the earth, w =
=3D the angular rate of the earth in rad/sec, and theta =3D the =
deflection angle.

I get a result of theta =3D atan((gtan(40))/(g-Rw^2)) - 40

To calculate for a differnent latitude simple change the 40 to 45=20
If interested I could send a diagram as an attached file also. I did not =
use the law of cosines but did assume that the length of the bob is =
small compared to the radius of the earth. Using this method, which I =
think is very straight forward does not require a great deal of algebra. =
Just a solution of two simultaneous equations and two unknowns.

Brian

-----Original Message-----
From: Hugh Haskell [SMTP:hhaskell@mindspring.com]
Sent: Sunday, November 02, 1997 12:23 AM
To: phys-l@atlantis.uwf.edu
Subject: Re: Help with bob!

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
or better." So I bought a Macintosh.
************************************************************


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