The mass of the Earth is 81 times the mass of | vedclass

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(A) 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 $x$ be the distance from the centre of the Earth wh

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$\frac{9 R}{10}$

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(A) 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 $x$ be the distance from the centre of the Earth where the net gravitational force on a test mass $m$ is zero.At this point,the gravitational pull from the Earth must be equal in magnitude to the gravitational pull from the Moon.$\frac{G M_E}{x^2} = \frac{G M_M}{(R - x)^2}$Substituting $M_E = 81 M_M$:$\frac{G (81 M_M)}{x^2} = \frac{G M_M}{(R - x)^2}$$\frac{81}{x^2} = \frac{1}{(R - x)^2}$Taking the square root on both sides:$\frac{9}{x} = \frac{1}{R - x}$$9(R - x) = x$$9R - 9x = x$$9R = 10x$$x = \frac{9}{10} R$

The mass of the Earth is $81$ times the mass of the Moon and the distance between their centres is $R$. The distance from the centre of the Earth where the gravitational force will be zero is

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