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(D) Let $M_e$ be the mass of the Earth and $M_m$ be the mass of the Moon. Given $M_e = 81 M_m$.Let the point where the net gravitational field is zero

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$\frac{9D}{10}$

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(D) Let $M_e$ be the mass of the Earth and $M_m$ be the mass of the Moon. Given $M_e = 81 M_m$.Let the point where the net gravitational field is zero be at a distance $r$ from the centre of the Earth.The gravitational field due to the Earth at this point is $E_e = \frac{G M_e}{r^2}$.The gravitational field due to the Moon at this point is $E_m = \frac{G M_m}{(D-r)^2}$.For the net gravitational field to be zero,$E_e = E_m$.$\frac{G M_e}{r^2} = \frac{G M_m}{(D-r)^2}$Substituting $M_e = 81 M_m$:$\frac{81 M_m}{r^2} = \frac{M_m}{(D-r)^2}$$\frac{81}{r^2} = \frac{1}{(D-r)^2}$Taking the square root on both sides:$\frac{9}{r} = \frac{1}{D-r}$$9(D-r) = r$$9D - 9r = r$$10r = 9D$$r = \frac{9D}{10}$

If the distance between the centres of the Earth and the Moon is $D$ and the mass of the Earth is $81$ times the mass of the Moon,then at what distance from the centre of the Earth will the net gravitational field be zero?

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