Question

The density of core of a planet is $\rho_1$ and that of the outer shell is $\rho_2$. 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 $\frac{\rho _{1}}{\rho _{2}}$ is

• A. $2.3$
• B. $4.5$
• C. $3.2$
• D. $5.4$

Answer 1

Answer: (A)

$2.3$

Answer 2

$\frac{Gm_{1}}{R^{2}} = \frac{G\left(m_{1}+m_{2}\right)}{\left(2R\right)^{2}}$ $4m_{1} = m_{1} + m_{2}$ $3m_{1} = m_{2}$ $3\left(\frac{4}{3}\pi R^{3}\rho_{1}\right)$ $= \frac{4}{3}\pi\left[8R^{3}-R^{3}\right]\rho_{2}$ $3\rho_{1} = 7\rho_{2}$ $\frac{\rho _{1}}{\rho _{2}} = \frac{7}{3} = 2.3$