Question

cinsider a planet in some solar system which has mass double the mass of earth and density equal to average density of earth . if an object weights W on earth ,its weight on the planet will be?

Answer 1

Relation between masses:

The mass of the planet is twice to that of earth.

$$\begin{gathered} m_p=2m_e \\ \rho \left[\frac{4}{3}\pi {\left(r_p\right)}^3\right]=2\rho \left[\frac{4}{3}\pi {\left(r_e\right)}^3\right] \\ {\left(r_p\right)}^3=2{\left(r_e\right)}^3 \\ {r}_p=2^{1/3}{r}_e \end{gathered}$$

Acceleration due to gravity:

$$\begin{gathered} g=\frac{Gm}{r^2} \\ \frac{g_p}{g_e}=\frac{m_p}{m_e}{\left(\frac{r_e}{r_p}\right)}^2 \\ =\frac{2m_e}{m_e}{\left(\frac{r_e}{2^{1/3}{r}_e}\right)}^2 \\ =\left(2\right)\frac{1}{2^{2/3}} \\ =2^{1/3} \end{gathered}$$

Relation between weights:

$$\begin{gathered} \frac{W_p}{W_e}=\frac{m{g}_p}{m{g}_e} \\ =\frac{g_p}{g_e} \\ =2^{1/3} \\ {W}_p=2^{1/3}{W}_e \\ =2^{1/3}W \end{gathered}$$