State Newtons law of gravitation. Write an expression for acceleration due to gravity on the surface of the earth. If the ratio of acceleration due to gravity of two heavenly bodies is $1 : 4$ and the ratio of their radii is $1 : 3$ , what will be the ratio of their masses?

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Solution

Newton's law of gravitation states that every particle in the universe with a force which is directly proportional to the product of their masses and universally proportional to the square of the distance between their centers.
$g = \frac{G M}{R^{2}}$
$M \to$ Mass of earth.
$R \to$ Radius of earth.
$g_{1} : g_{2} = 1 : 4$
Let $g_{1} - x$ and $g_{2} = 4 x$
$∴ \frac{G M_{1}}{R_{1}^{2}} = x ⟶ (i)$
$\frac{G M_{2}}{R_{2}^{2}} = 4 x ⟶ (i i)$
$\frac{e q u a t i o n (i)}{e q u a t i o n (i i)}$
$\frac{M_{1}}{M_{2}} = \frac{x R_{1}^{2}}{G} \frac{G}{4 x R_{2}^{2}}$
$= \frac{R_{1}^{2}}{4 R_{2}^{2}}$
$R_{1} : R_{2} = 1 : 9$
$∴ \frac{M_{1}}{M_{2}} = \frac{1}{4 (9)} = \frac{1}{36}$

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