Two planets of radii $r_{1}$ and $r_{2}$ are made from the same material. The ratio of the acceleration due to gravity $g_{1} / g_{2}$ at the surfaces of the two planets is:

r_{1} / r_{2}

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r_{2} / r_{1}

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(r_{1} / r_{2})^{2}

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(r_{2} / r_{1})^{2}

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Solution

The correct option is C $r_{1} / r_{2}$
According to Gravitational Law
$F = \frac{G M_{1} M_{2}}{r^{2}}$ $. . . . . . (i)$
Where $M_{1}$ is mass of planet & $M_{2}$ is the mass of any body. Now according to Newton's second law, a body of mass $M_{2}$ feels gravitational acceleration $g$ which is
$F = M_{2} g$ $. . . . . . (i i)$
So from $(i)$ & $(i i)$ , we get $g = \frac{G M_{1}}{r^{2}}$
So the ratio of gravitational acceleration due to two planets is
$\frac{g_{1}}{g_{2}} = \frac{M_{1}}{r_{1}^{2}} \times \frac{r_{2}^{2}}{M_{2}} = \frac{(4 / 3) π r_{1}^{3} \times ρ}{r_{1}^{2}} \times \frac{r_{2}^{2}}{(4 / 3) π r_{2}^{3} \times ρ}$
$\frac{g_{1}}{g_{2}} = [\frac{r_{1}}{r_{2}}]$ (both planets have same material, so density is same)

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r_{1}

and

r_{2}

are made from the same material.The ratio of the acceleration of gravity

\frac{g_{1}}{g_{2}}

at the surface of the two planets is-

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