Two planets A and B have densities ′ ϱ 1 ′ , ′ ϱ 2 ′ and have radii ′ r 1 ′ , ′ r 2 ′ respectively. The ratio of acceleration due to gravity on A to that of B is

Question

Two planets A and B have densities ′ ϱ 1 ′ ​ , ′ ϱ 2 ′ ​ and have radii ′ r 1 ′ ​ , ′ r 2 ′ ​ respectively. The ratio of acceleration due to gravity on A to that of B is

TestHub · Accepted Answer

The acceleration due to gravity on the surface of a planet is given by $g = \frac{GM}{r ^{2}}$ .

The mass of the planet can be expressed in terms of its density $ϱ$ and radius $r$ as $M = \frac{4}{3} π r^{3} ϱ$ .

Substituting the expression for mass into the formula for $g$ , we get:

$g = \frac{G}{r ^{2}} (\frac{4}{3} π r^{3} ϱ) = \frac{4}{3} π G r ϱ$

This shows that $g \propto r ϱ$ .

For planets A and B, the ratio of acceleration due to gravity is:

$\frac{g _{A}}{g _{B}} = \frac{r _{1} ϱ _{1}}{r _{2} ϱ _{2}}$

Answer: $r_{1} ϱ_{1} : r_{2} ϱ_{2}$

Physics - Gravitation Question with Solution | TestHub

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