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 :

Question

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 : (A) r =(1/√3) R (B) r =√(5/9) R (C) r =√(3/4) R (D) r = R

Tardigrade Admin · Accepted Answer

E 4 π r 2=∫ ρ0 4 π r 2 dr ⇒ Er 2=4 π G ∫ limits0 r ρ0(1-( r 2/ R 2)) r 2 dr ⇒ E =4 π G ρ0(( r 3/3)-( r 5/5 R 2)) ( dE / dr )=0 ∴ r =√(5/9) R

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

$r = \frac{1}{3} R$

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

$r = \frac{3}{4} R$

$r = R$

$E 4 π r^{2} = \int ρ_{0} 4 π r^{2} d r$
$\Rightarrow E r^{2} = 4 π G 0 \int r ρ_{0} (1 - \frac{r ^{2}}{R ^{2}}) r^{2} d r$
$\Rightarrow E = 4 π G ρ_{0} (\frac{r ^{3}}{3} - \frac{r ^{5}}{5 R ^{2}})$
$\frac{d E}{d r} = 0$ $∴ r = \frac{5}{9} R$

Q. The mass density of a planet of radius R varies with the distance r from its centre as ρ(r)=ρ0​(1−R2r2​) Then the gravitational field is maximum at :

r=3​1​R

r=95​​R

r=43​​R

r=R