g is also referred to as the gravitational field strength at a given position, as it is the force of gravity per unit mass on a small object at that position.
So the gravitational field strength at the Earth's surface is 9.81N/kg.
Note that the weight of a fixed mass depends on its location. For example, the weight of a 1kg object is 9.81N on the Earth's surface and 1.62N on the Moon's surface.
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