Universal Gravitation and Apparent Weight

Newton's law of gravitation

Every pair of masses attracts along the line joining them, with equal and opposite forces (a third-law pair, so the Earth pulls on you exactly as hard as you pull on the Earth). The distance $r$ is measured between centres, not from the surface. Because of the $r^2$, doubling the distance divides the force by four, and tripling it divides by nine.

The familiar $g$ is just this law with the Earth as one of the masses:

At one Earth radius above the surface, $r=2R_E$, so $g$ falls to $\frac{9{,}8}{4}\approx2{,}45$ m/s². Astronauts in low orbit at $400$ km altitude still feel about $90\%$ of surface gravity — they float because they are in free fall, not because gravity has vanished.

Mass is not weight

Mass is the amount of matter and does not change from place to place; it appears in $F=ma$ as inertia and in $F_g=Gm_1m_2/r^2$ as the source of gravity. That these two masses are the same number is why all objects fall with the same $g$. Weight is the gravitational force $mg$ and changes with location.

Apparent weight: what a scale actually reads

A bathroom scale does not measure gravity — it measures the normal force it exerts on you. Standing in a lift, the free-body diagram has only two forces, $N$ up and $mg$ down, and Newton's second law with up positive gives

Accelerating upward (or slowing while moving down) makes $N \gt mg$: you feel heavier. Accelerating downward makes $N \lt mg$. In free fall $a=-g$ and $N=0$: "weightless", though the weight $mg$ is unchanged. A $60$ kg person in a lift accelerating upward at $2$ m/s² presses on the scale with $60(10+2)=720$ N instead of $600$ N.