Re: Spreadsheet Analysis of Rotating Stick

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Sun, 28 Nov 1999, Robert Carlson wrote:

> Here is a question for anyone on the list.  What is the period of a point
> mass attached to a string of length one meter, released from rest from the
> horizontal?
>
> The small angle approximation gives 2.01 s which is of course not correct
> here.  In Tipler, an approximation for T is given as:
>
> T = To(1+ 0.25sin(0.5B)^2+0.1406sin(0.5B)^4+ . . .)
>
> Where To = 2.01 s, and B is the release angle measured from the vertical of
> the bob at its lowest point.
>
> For B = 90 degrees this gives T = 2.33 s.  However, this only includes terms
> through sin(0.5B) to the fourth power.
>
> The spreadsheet gives T = 2.37 s.

For an initial angle of 90 degrees, the series becomes

           1^2    1    1^2 * 3^2    1
T = To(1 + --- * --- + --------- * --- + ... )
           2^2   2^1   2^2 * 4^2   2^2

  = To(1.1803406...)

So, for L = 1 m and g = 9.8 m/s^2 T = 2.36904972... s in agreement with
your spreadsheet solution.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm