The distance between the centers 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 $M_E = 81 M_M$ and the distance between them is $D$.Let the point wher

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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 $M_E = 81 M_M$ and the distance between them is $D$.Let the point where the gravitational field is zero be at a distance $x$ from the center of the earth.At this point,the gravitational field due to the earth must be equal in magnitude to the gravitational field due to the moon.$\frac{G M_E}{x^2} = \frac{G M_M}{(D - x)^2}$Substituting $M_E = 81 M_M$:$\frac{81 M_M}{x^2} = \frac{M_M}{(D - x)^2}$$\frac{81}{x^2} = \frac{1}{(D - x)^2}$Taking the square root on both sides:$\frac{9}{x} = \frac{1}{D - x}$$9(D - x) = x$$9D - 9x = x$$10x = 9D$$x = \frac{9D}{10}$

The distance between the centers of the moon and the earth is $D$. If the mass of the earth is $81$ times that of the moon,at what distance from the center of the earth will the gravitational field be zero?

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