Question

The density of mass inside a solid sphere of radius $a$ is given by $\rho ={\rho }_{0}a/r$, where ${\rho }_0$ is the density at the surface and $r$ denotes the distance from the centre. Find the gravitational field due to this sphere at a distance $2a$ from its centre

Answer 1

Let us calculate the mass of the sphere.

$dm=\rho \left(4\pi r^{2}dr\right)=4\pi {\rho }_{0}ardr$

$m={\int }_0^{a}dm={\int }_0^{a}4\pi {\rho }_{0}ardr=2\pi {\rho }_0{a}^3$

Thus, gravitational field at a distance of $2a$ from the center is

$g=\frac{Gm}{{\left(2a\right)}^2}=\frac{\pi G{\rho }_{0}a}{2}$

Answer is $\frac{\pi G{\rho }_{0}a}{2}$