Question

The density inside a solid sphere of radius a is given by $\rho ={\rho }_0(1-\frac{r}{a})$, 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 $2a$ from its centre is $\frac{\pi G{\rho }_{0}a}{x.}$ Find $x$.

Answer 1

Find the gravitational field due to this sphere. at a distance $2a$
Formula used: $E=\frac{GM}{R^2}$

Volume of spherical element,
$dV=4\pi r^2\text{dr}$
Mass of element,
$dM=\rho dV=p_0(1-\frac{r}{a}).4\pi r^{2}dr$
Mass of sphere,
$M=4\pi {\rho }_0{\int }_0^a(r^2-\frac{r^3}{a})dr$
$=4\pi {\rho }_0{|\frac{r^3}{3}-\frac{r^4}{4a}|}_0^a=\frac{\pi {\rho }_0{a}^3}{3}$
Gravitational field at $P$
,
$E=\frac{GM}{(2a{)}^2}=\frac{G.\pi {\rho }_0{a}^3}{12a^2}$
$=\frac{1}{12}\pi G{\rho }_{0}a$
Final Answer: 12