If the distance between the centres of earth and moon is D and mass of earth is 81 times that of moon. At what distance from the centre of earth gravitational field will be zero:

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If the distance between the centres of earth and moon is $D$ and mass of earth is $81$ $t i m e s$ that of moon. At what distance from the centre of earth gravitational field will be zero:

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$\frac{D}{2}$

$\frac{2 D}{3}$

$\frac{4 D}{5}$

$\frac{9 D}{10}$

answer is D.

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Detailed Solution

At point $' p'$ for gravitational field to be zero $field due to earth = field due to moon$
$\Rightarrow \frac{G M_{e}}{x^{2}} = \frac{G M_{m}}{{(D - x)}^{2}} \Rightarrow \frac{81 M_{m}}{x^{2}} = \frac{M_{m}}{{(D - x)}^{2}} \Rightarrow \frac{x}{D - x} = 9 \Rightarrow 9 (D - x) = x \Rightarrow x = \frac{9 D}{10}$