Saturday, March 17, 2007

g whiz

I am somewhat curious as to why my method of determining local g seems not to come up with good values. I suppose it has to do with irregularities in distribution of the earth's mass and possibly with problems of measurement of big G and the earth's shape.

For example, the standard value of g is given as 9.80665 m/s2, taken at sea level at latitude 45.5o.

Here's what I get:

The expression for an ellipse (the earth's shape) is

(x/a)2 + (y/b)2 = 1,

where a and b are the semimajor and semiminor axes. Hence, to determine the radius by angle, we have

r = [((cos K)/a)2 + ((sinK)/b)2 ]-0.5.

Letting the polar radius = 6364630 meters and the equatorial radius = 6378000, and setting earth mass at 5.98(10)24 kg and G at 6.725(10-11), I get g = 9.907 at sea level.

Of course, this seemingly obvious formula is not the one used. For details on the actual method of calculation, see Wikipedia article on "standard gravity."
Using somewhat more precise values of G and M, here are some other values of g to three decimal places (including altitude of the U.S. cities listed):

Nashville TN: 9.891
Knoxville TN 9.890
Albany NY: 9.896
New York NY: 9.894
London: 9.902
Jerusalem: 9.873
North or South Pole: 9.918
Equator: 9.877

I suppose the real problem is that the earth isn't a perfect ellipsoid and that its mass is not quite unformly distributed. Dunno.