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  4. The mass density of a planet of radius R varies with the distance r from its centre as ρ( r )=ρ0(1-( r 2/ R 2)) Then the gravitational field is maximum at :

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Q. The mass density of a planet of radius $R$ varies with the distance $r$ from its centre as $\rho( r )=\rho_{0}\left(1-\frac{ r ^{2}}{ R ^{2}}\right) $ Then the gravitational field is maximum at :

JEE MainJEE Main 2020Gravitation

Solution:

$ E \,4 \pi r ^{2}=\int \rho_{0} \,4 \pi r ^{2} \,dr$
$\Rightarrow Er ^{2}=4 \pi G \int\limits_{0}^{ r } \rho_{0}\left(1-\frac{ r ^{2}}{ R ^{2}}\right) r ^{2} dr$
$\Rightarrow E =4 \pi G \rho_{0}\left(\frac{ r ^{3}}{3}-\frac{ r ^{5}}{5 R ^{2}}\right)$
$\frac{ dE }{ dr }=0$ $ \therefore r =\sqrt{\frac{5}{9}} \,R$