Answer
$F = m~(3.34\times 10^{-7})~N$
Work Step by Step
A distance of $0.50~m$ from the center is half of the radius. Then the volume within $0.50~m$ of the center is $\frac{1}{8}$ of the total volume of the sphere. The mass of the solid sphere of radius $0.50~m$ is $\frac{1.0\times 10^4~kg}{8} = 1250~kg$
We can find $a_g$ at $r = 0.5~m$:
$a_g = \frac{GM}{r^2}$
$a_g = \frac{(6.67\times 10^{-11}~N~m^2/kg^2)(1250~kg)}{(0.5~m)^2}$
$a_g = 3.34\times 10^{-7}~m/s^2$
We can find the magnitude of the gravitational force on the particle:
$F = m~a_g$
$F = m~(3.34\times 10^{-7})~N$
