The density of core of a planet is ρ1 and that of the outer shell is ρ2. The radii of core and that of the planet are R and 2R respectively. Gravitational acceleration at the surface of planet is same as at a depth. The ratio between (ρ 1/ρ 2) is

Question

The density of core of a planet is ρ1 and that of the outer shell is ρ2. The radii of core and that of the planet are R and 2R respectively. Gravitational acceleration at the surface of planet is same as at a depth. The ratio between (ρ 1/ρ 2) is (A) 2.3 (B) 4.5 (C) 3.2 (D) 5.4

Tardigrade Admin · Accepted Answer

(Gm1/R2) = (G(m1+m2)/(2R)2) 4m1 = m1 + m2 3m1 = m2 3((4/3)π R3ρ1) = (4/3)π[8R3-R3]ρ2 3ρ1 = 7ρ2 (ρ 1/ρ 2) = (7/3) = 2.3

Q. The density of core of a planet is $ρ_{1}$ and that of the outer shell is $ρ_{2}$ . The radii of core and that of the planet are $R$ and $2 R$ respectively. Gravitational acceleration at the surface of planet is same as at a depth. The ratio between $\frac{ρ _{1}}{ρ _{2}}$ is

$2.3$

$4.5$

$3.2$

$5.4$

$\frac{G m _{1}}{R ^{2}} = \frac{G ( m _{1} + m _{2} )}{( 2 R ) ^{2}}$
$4 m_{1} = m_{1} + m_{2}$
$3 m_{1} = m_{2}$
$3 (\frac{4}{3} π R^{3} ρ_{1})$
$= \frac{4}{3} π [8 R^{3} - R^{3}] ρ_{2}$
$3 ρ_{1} = 7 ρ_{2}$
$\frac{ρ _{1}}{ρ _{2}} = \frac{7}{3} = 2.3$

Q. The density of core of a planet is ρ1​ and that of the outer shell is ρ2​. The radii of core and that of the planet are R and 2R respectively. Gravitational acceleration at the surface of planet is same as at a depth. The ratio between ρ2​ρ1​​ is

2.3

4.5

3.2

5.4