# Some notes on gravitational acceleration

The "constant" acceleration due to gravity varies due to several effects.
One of these effects is the latitude of the observer. Another is the altitude.

### Latitude

At a latitude of L, the acceleration due to gravity at sea level is approximately

g= 9.780327 [ 1 + .0053024*sin^{2}(L) - .0000058*sin^{2}(2L) ]
meters per second per second.

G310 is at approximately 39 degrees 29.01 minutes N latitude, giving
an angle L of approximately .689117 radians,
and an acceleration due to gravity (at sea level)
of approximately 9.81012397 meters per sec^{2}

### Altitude

Acceleration due to gravity decreases as the inverse of the square of the distance from the center of mass of the body imparting the gravitational acceleration; g=k/x^{2}

Because the earth is not perfectly spherical, the distance from the center of the earth for any person standing on the surface depends upon the latitude L.

The equatorial radius of the earth is approximately 6,378,140 meters.

The polar radius is approximately 6,356,755 meters.

If we model the earth as an ellipsoid, this means that the radius of the earth at latitude L is given by
R= 6,356,755* sqrt( 1 + .0067396*cos^{2}(L) ).

At the latitude of Terre Haute, this is approximately
6,369,502 meters.

Thus, a person at altitude of H meters above sea level experiences and acceleration due to gravity of

a= g*R^{2}/(R+H)^{2}.

G310 is at approximately 590 feet above sea level. This is roughly 168 meters above sea level.

This gives an acceleration due to gravity of approximately
**9.800723** meters per second^{2}.

Moving to the classroom ceiling, this constant becomes
9.800713 meters per second^{2}.

Moving to the circle in front of Hadley Hall (L-> .689103, H->180m),
we get an acceleration due to gravity of approximately
9.800722 meters per second^{2}.

### Rates of change

The perceived acceleration changes in response to changes in latitude and altitude.

The rate of change with respect to altitude is given by
da/dH= -2g*R^{2}/(R+H)^{3} = -2ga/(R+H).

Thus the acceleration due to gravity decreases by approximately one part in 3,000,000 for each meter risen. This is approximately 1/300,000 meters per second^{2} per meter.

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