|THERMO Spoken Here! ~ J. Pohl © ~ 2017||www.ThermoSpokenHere.com (A4500-047)|
The attention of physics and engineering texts to the purported "significance" of the variation of Earth's gravity with altitude is exaggerated. Those text problems ignore the inconsequentiality the effect has for the overwhelming majority of events of engineering systems.
Before investing time in regard to the "variation of gravity with altitude" consider that but for space activities, virtually every engineering system operates at altitudes of less than 30 kilometers ( ~ 16 miles). Up to 30 kilometers approximates the homo sphere or zone of our uniformly mixed atmosphere. The peak of Mount Everest has an elevation of 8.85 kilometers
( ~ 5.5 miles). There's not much engineering going on even that high. So what error would one incur by assuming a constant, surface value, go,Earth for all altitudes up to 30 kilometers?
Gravity expressed mathematically in its "inverse-square" form and its approximated, Earth-surface, value are written below.
Leftmost is the actual value of gravity at an elevation. Immediately left is our approximation.
The percent error of using the constant, surface-value of gravity up to some elevation r, is written below then algebraically reduced. approximation that a constant (surface value) for the acceleration of gravity for all elevations (of altitudes) reduces algebraically to:
Some algebra makes the error|
of approximation clearer.
This result makes sense, it says the error increases with altitude. Using some numbers, the Earth radius is about 6367 km, so the error for use of the constant at an altitude of 30 kilometers is approximately:
This is a very small error. We should use the surface value of gravity for every engineering event that occurs at an elevation less than the peak of Mt. Everest.
The error is less than 0.9% for all altitudes less that 30 kilometers. The "positiveness" of the error indicates that the surface approximation is too large a number. But in error by only 0.9%.
A further conclusion is: to include the variation of gravity in any initial engineering analysis is unwise.