Answer
$F = [mr~(6.67\times 10^{-7})]~N$
Work Step by Step
If the distance from the center is $r$, such that $r \leq 1$, then the volume of the sphere within a distance of $r$ from the center is a fraction of $r^3$ of the total volume of the sphere. The mass of the solid sphere of radius $r$ is $(1.0\times 10^4~kg)~r^3$
We can find $a_g$ at $r$:
$a_g = \frac{GM}{r^2}$
$a_g = \frac{(6.67\times 10^{-11})~(1.0\times 10^4)~(r^3)}{r^2}$
$a_g = (6.67\times 10^{-11})~(1.0\times 10^4)~(r)$
$a_g = (6.67\times 10^{-7})~r$
We can find the magnitude of the gravitational force on the particle:
$F = m~a_g$
$F = [mr~(6.67\times 10^{-7})]~N$
