Knowing that mass of Moon is (M/81) where M is the mass of Earth, find the distance of the point where gravitational field due to Earth and Moon cancel each other, from the Moon. Given that distance between Earth and Moon is 60 R, where R is the radius of Earth.

Question

Knowing that mass of Moon is (M/81) where M is the mass of Earth, find the distance of the point where gravitational field due to Earth and Moon cancel each other, from the Moon. Given that distance between Earth and Moon is 60 R, where R is the radius of Earth. (A) 2 R (B) 4 R (C) 6 R (D) 8 R

Tardigrade Admin · Accepted Answer

Let C be the point where the gravitational field due to earth and the moon be equal. C is at a distance x from the moon and (60R - x) from the earth. We know that the gravitational field at a distance r from mass M is given by

E = G \frac{M}{r ^{2}}

In our case
E₁ = E₂

G \frac{M}{( 60 R - x ) ^{2}} = \frac{M}{G \frac{81}{x ^{2}}}

\Rightarrow 81 x^{2} = (60 R - x)^{2}

\Rightarrow (9 x)^{2} = (60 R - x)^{2}

∴ 9 x = \pm (60 R - x) < b r / > ∴ 9 x = 60 R - x or 9 x = x - 60 R

\Rightarrow 10 x = 60 R or 8 x = - 60 R

\Rightarrow x = 6 R

or

x = - \frac{60}{8} R

which is not possible as distance cannot be negative.

Q. Knowing that mass of Moon is 81M​ where M is the mass of Earth, find the distance of the point where gravitational field due to Earth and Moon cancel each other, from the Moon. Given that distance between Earth and Moon is 60R, where R is the radius of Earth.

2 R

4 R

6 R

8 R

Q. Knowing that mass of Moon is $\frac{M}{81}$ where $M$ is the mass of Earth, find the distance of the point where gravitational field due to Earth and Moon cancel each other, from the Moon. Given that distance between Earth and Moon is $60 R$ , where $R$ is the radius of Earth.