Question

Consider earth to be a homogeneous sphere of mass $M$ and radius $R$. If we consider another spherical planet of mass $M$ and radius $R$ whose density $\rho$ changes linearly from ${\rho }_0$ at centre to $5{\rho }_0$ at surface. Choose the correct statements

• A. the ratio of acceleration due to gravity on the surface of earth and that on the surface of that planet is $1$.
• B. the ratio of acceleration due to gravity on the surface of earth and that on the surface of that planet is $4$
• C. the value of ${\rho }_0$ is $\frac{3M}{4\pi R^3}$
• D. the value of ${\rho }_0$ is $\frac{3M}{16\pi R^3}$

Answer 1

Answer: (A), (D)

The correct options are
A the ratio of acceleration due to gravity on the surface of earth and that on the surface of that planet is $1$.
D the value of ${\rho }_0$ is $\frac{3M}{16\pi R^3}$

Acceleration due to gravity on the surface in both cases is given by,
$g_1=\frac{GM}{R^2}$ and $g_2=\frac{GM}{R^2}$
$\frac{g_1}{g_2}=1$ (Since mass is same)

As density varies linearly from ${\rho }_o$ to $5{\rho }_0$,



Variation of density of planet can be written as
$\rho ={\rho }_0+\left[\frac{5{\rho }_0-{\rho }_0}{R}\right]r={\rho }_0+\frac{4{\rho }_0}{R}r$
$\Rightarrow \rho ={\rho }_0[1+\frac{4r}{R}]$

The mass of the planet is $M$
$\Rightarrow M={\int }_0^R\rho 4\pi r^{2}dr$
$M=4\pi {\rho }_0{\int }_0^R[r^2+\frac{4r^3}{R}]dr=4\pi {\rho }_0{[\frac{r^3}{3}+\frac{4r^4}{4R}]}_0^R$
$\Rightarrow M=4\pi {\rho }_0[\frac{R^3}{3}+R^3]=\frac{16\pi {\rho }_0{R}^3}{3}$
$\Rightarrow {\rho }_0=\frac{3M}{16\pi R^3}$