Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is \({W}\), then weight of the same object on that planet will be

\(\rho_{ p }=\rho_{ e }\)

\(\Rightarrow \frac{ M _{ p }}{\frac{4}{3}}\pi R _{ p }^{3}=\frac{ M _{ e }}{\frac{4}{3}}\)

\(\pi R _{ e }^{3}\)

\(\Rightarrow \frac{ R _{ p }}{ R _{ e }}=\left(\frac{ M _{ p }}{ M _{ e }}\right)^{1 / 3}\)

The value of gravitational acceleration \(=g=\frac{G M}{R^{2}}\)

\(\Rightarrow \frac{W_{p}}{W_{e}}=\frac{m g_{p}}{m g_{e}}\)

\(\Rightarrow \frac{g_{p}}{g_{e}}=\frac{M_{p}}{M_{e}} \frac{R_{e}^{2}}{R_{p}^{2}}\)

\(=\frac{M_{p}}{M_{e}}\left(\frac{M_{e}}{M_{p}}\right)^{2 / 3}=\left(\frac{M_{p}}{M_{e}}\right)^{1 / 3}=2^{1 / 3}\)

\(\Rightarrow W_{p}=2^{1 / 3}\; W\)