The distance between the centres of the moon | vedclass

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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 distance from the centre of the earth where the

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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 distance from the centre of the earth where the gravitational force is zero be $x$.The gravitational force exerted by the earth on a test mass $m$ at distance $x$ is $F_e = \frac{G M_e m}{x^2}$.The gravitational force exerted by the moon on the same test mass at distance $(D - x)$ is $F_m = \frac{G M_m m}{(D - x)^2}$.For the net gravitational force to be zero,$F_e = F_m$.$\frac{G M_e m}{x^2} = \frac{G M_m m}{(D - x)^2}$$\frac{81 M_m}{x^2} = \frac{M_m}{(D - x)^2}$Taking the square root on both sides:$\frac{9}{x} = \frac{1}{D - x}$$9(D - x) = x$$9D - 9x = x$$10x = 9D$$x = \frac{9D}{10}$.

The distance between the centres of the moon and the earth is $D$. The mass of the earth is $81$ times the mass of the moon. At what distance from the centre of the earth will the net gravitational force on a test mass be zero?

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