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Gravity itself is an acceleration. The standard equation for force is F=ma, where F is force, m is mass and a is acceleration. w=mg is the same equation applied to a particular type of force: weight. In w=mg, w is weight, m is mass and g is gravity, or the acceleration due to gravity if you prefer. The value of g for ALL objects near Earth's surface is 9.8 meters per square second.
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