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So g is not a constant, but varies with elevation, latitude rotation of the earth, etc. So what? I see that as even a more reason to disconnect g's association with weight. It is also highly confusing to beginning students to call g an acceleration. It may have the units of acceleration but it is clearly *not* an acceleration, since it is still operational in situations where the object is at rest (in the relevant reference frame). It is the (adjusted, or corrected, or whatever adjective is convenient) strength of the local gravitational field. It is there whether you are standing still, free falling, exiting the mouth of a cannon at high speed, in a rocket leaving the launch pad, in an airplane doing aerobatics, in orbit, riding a carnival ride, or floating atop a stream of water from a fire hose. It's value may vary from place to place but it never goes away. But weight does go away, It also increases or decreases depending on what your reference frame of choice is.
True weight is mg ...
The problem with this definition is that mg is the force measured on a ROTATING earth.
g ‚ GM/R^2
"Although the symbol g is sometimes incorrectly used for standard gravity, g (without a suffix) strictly means the local acceleration due to local gravity and centripetal acceleration, which varies depending on one's position on Earth (see Earth's gravity)."