Question

Two planets have radii $ {{r}_{1}} $ and $ {{r}_{2}} $ and densities $ {{d}_{1}} $ and $ {{d}_{2}} $ respectively. Then the ratio of acceleration due to gravity on them will be:

• A. $ {{r}_{1}}{{d}_{1}}:{{r}_{2}}{{d}_{2}} $
• B. $ {{r}_{1}}{{d}_{2}}:{{r}_{2}}{{d}_{1}} $
• C. $ r_{1}^{2}{{d}_{1}}:r_{2}^{2}{{d}_{2}} $
• D. $ {{r}_{1}}:{{r}_{2}} $
• E. $ {{r}_{1}}/\sqrt{{{d}_{1}}}:{{r}_{2}}\sqrt{{{d}_{2}}} $

Answer 1

Answer: (A)

The relation between density (D) and acceleration due to gravity (g) is
$ d=\frac{3g}{4\pi {{R}_{e}}G} $
$ \therefore $ $ \frac{{{d}_{1}}}{{{d}_{2}}}=\frac{{{g}_{1}}}{{{r}_{1}}}\times \frac{{{r}_{2}}}{{{g}_{2}}} $
$ \Rightarrow $ $ \frac{{{g}_{1}}}{{{g}_{2}}}=\frac{{{d}_{1}}{{r}_{1}}}{{{d}_{2}}{{r}_{2}}} $