Question

At what distance (in metre) from the centre of the moon, the intensity of gravitational field will be zero?
(Take, mass of earth and moon as 5.98 x 10²⁴ kg and 7.35 x 10²² kg respectively and the distance between
moon and earth is 3.855x 108 m.) [JIPMER 2018]

Moderate

Solution

Given, $M_{e} = 5.98 \times 10^{24} kg, M_{m} = 7.35 \times 10^{22} kg$ and
$r = 3.855 \times 10^{8} m$
Let x be the distance of the point from the centre of earth, where gravitational intensity is zero. Therefore,
$\frac{G M_{e}}{x^{2}} = \frac{G M_{m}}{{(3855 \times 10^{8} - x)}^{2}}$
or    $\frac{x}{3.855 \times 10^{8} - x} = \sqrt{\frac{M_{e}}{M_{m}}} = \sqrt{\frac{598 \times 10^{24}}{7.35 \times 10^{22}}} ≅ 9$
or    $\frac{x}{9} + x = 3855 \times 10^{8}$
or $x = 9 \times 3.855 \times 10^{8} / 10 = 3.469 \times 10^{8} m$
Distance from moon $= 3.855 \times 10^{8} - 3.469 \times 10^{8}$
$= 3.86 \times 10^{7} m$
   $≅ 3.85 \times 10^{7} m$

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