Question

The density inside a solid sphere of radius 'a' is given by $\rho ={\rho }_0\frac{a}{r}$, where ${\rho }_0$ is the density at the surface and r denotes the distance from the centre. The gravitational field due to this sphere at a distance 3a from its centre is $\frac{2\pi }{x}{\rho }_{0}Ga$. Find the value of x.

Answer 1

Mass of the sphere will be

$M={\int }_0^a\left(4\pi r^2\right)\left({\rho }_0\frac{a}{r}\right)dr=2\pi {\rho }_0{a}^3$

Since the sphere behaves as a point mass at its center when the field outside is calculated. Thus,

$g=\frac{GM}{{\left(3a\right)}^2}=\frac{2\pi {\rho }_{0}Ga}{9}$

Answer is 9.