The ratio of the weights of a body on the Earth's surface to that on the surface of a planet is \(9 : 4\). The mass of the planet is \(\frac{1}{9}^{th}\) of that of the Earth. If \('R'\) is the radius of the Earth, what is the radius of the planet ? (Take the planets to have the same mass density)

\(\therefore \) Weight of object will be proporational to \('g'\) \((acceleration\, due\, to\,gravity)\) Given ; 

\(\frac{{{W_{earth}}}}{{{W_{planet}}}} = \frac{9}{4} = \frac{{{g_{earth}}}}{{{g_{planet}}}}\)

Also,

\({g_{surface}} = \frac{{GM}}{{{R^2}}}\) \((M is mass  planet,

G is unversal gravitational constant, R is radius of planet)\)

\(\therefore \frac{9}{4} = \frac{{G{M_{earth}}R_{planet}^2}}{{G{M_{planet}}R_{earth}^2}} = \frac{{{M_{earth}}}}{{{M_{planet}}}} \times \frac{{R_{planet}^2}}{{R_{earth}^2}}\)

\( = 9\frac{{R_{planet}^2}}{{R_{earth}^2}}\)

\(\therefore {R_{planet}} = \frac{{{R_{earth}}}}{2} = \frac{R}{2}\)