system
Given the following post, respond as if you're commenting on Reddit:
human
If Earth were the same mass but half as dense, would we weigh the same?

Curious if the increased volume would put us further from the center of mass making gravity less strong at the surface, or if that wouldn't be a factor.
gpt
The "acceleration due to gravity", g, which on the Earth's surface is 9.8 m/s^2 .Is given by GM/R^2  where M is the mass of the object (for the Earth this is about 6x10^24 kg), and R is the distance to the center of mass (about 6,400 km on Earth). G is the gravitational constant 6.67x10^-11 .  If the Earth had half the density but the same volume (i.e. R is the same, M is half), then g is just halfed to 4.9 m/s^2 . So you'd weigh have as much.

If you changed the volume, rather than the mass, then the volume of a sphere is V= (4/3)pi R^3 , which is (3/4pi)V^(2/3) = R^2  so we have g = GM/(3/4pi V^(2/3))   = (4pi/3)GMV^(-2/3)  so it changes proportional to the power of -2/3 with volume (assuming it remains a sphere).