The mass density of a planet of radius R varies with the distance r from its centre as ( r )= _ 0 (1- r ^ 2 R ^ 2 ) . Then the gravitational field is maximum at

The mass density of a planet of radius R varies with the distance…

MCQ

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

A
$r =\frac{1}{\sqrt{3}} R$
✓
$r=\sqrt{\frac{5}{9}} R$
C
$r=\sqrt{\frac{3}{4}} R$
D
$r=R$

✓

Answer

Correct option: B.

$r=\sqrt{\frac{5}{9}} R$

b
$E 4 \pi r ^{2}=\int \rho_{0} 4 \pi r ^{2} dr$

$\Rightarrow Er ^{2}=4 \pi G \int \limits_{0}^{ r } \rho_{0}\left(1-\frac{ r ^{2}}{ R ^{2}}\right) r ^{2} dr$

$\Rightarrow E =4 \pi G \rho_{0}\left(\frac{ r ^{3}}{3}-\frac{ r ^{5}}{5 R ^{2}}\right)$

$\frac{ d E }{ dr }=0 \therefore r =\sqrt{\frac{5}{9}} R$

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