Question

By what factor would your weight change if the earth's diameter were doubled and its mass were also doubled?

Answer 1

Aha! This is a fun question, likely AP Physics 1 if I'm not mistaken. This is how you solve it, you need the following equation.

$F=G\frac{m_{1} *m_{2}}{r^2}$

Now on this question you don't need to actually use the mass and diameter of the earth as we are only looking at the factor of how much your weight would change, so you can just plug in some random numbers for the mass and radius. In my case I will use the original radius as being 2 meters and the original mass of the earth as being 20kg. You can actually just omit the mass of your body and the gravitational constant isn't necessary to be multiplied by because they will remain the same in both scenarios.

Original weight:

$F= \frac{20kg}{2^2m}$
$F=5N$

So the original weight in this circumstance is 5N. Now then if we double the mass to 40kg and the radius to 4m you will have the following.

$F= \frac{40kg}{(4)^2m}$
$F=2.5N$

So the original weight was 5N and after doubling the mass and radius the weight reduced to 2.5N. This means the factor which your weight would change by is .5 or decrease by 1/2.

I hope this helps you :) You can ask me if you have any additional physics questions.