QJEEM20Ph76e0: JEE Main 2020 Physics - Class 11 JEEM20Ph76e0 Solution: The mass density of a planet of radius R varies with the…

Question

Verified Solution • 2025–26 Session moderate

The mass density of a planet of radius R varies with the distance r from its centre as

\rho

(r) =

{\rho _0}\left( {1 - {{{r^2}} \over {{R^2}}}} \right)

.
Then the gravitational field is maximum at :

Accepted Answer

dm =

\rho

dv

\int {dm}

= \int {{\rho _0}} 4\pi {r^2}dr

\Rightarrow

M =

\int {{\rho _0}} 4\pi {r^2}dr

Also E =

{{GM} \over {{r^2}}}

\Rightarrow E{r^2} = 4\pi G\,\int\limits_0^r {{\rho _0}} \left( {1 - {{{r^2}} \over {{R^2}}}} \right){r^2}dr

\Rightarrow E = 4\pi G{\rho _0}\left( {{{{r}} \over 3} - {{{r^3}} \over {5{R^2}}}} \right)

For maximum E,

{{dE} \over {dr}} = 0

{1 \over 3} - {{3{r^2}} \over {5{R^2}}}

= 0

\therefore

r = \sqrt {{5 \over 9}} R

Step-by-Step Verified Solution