Knowing that mass of Moon is M/81 M 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

To find the distance from the Moon where the gravitational fields due to the Earth and the Moon cancel each other, we can set up an equation where the magnitudes of the two gravitational fields are equal.

Let:
- $M_E=M$ be the mass of the Earth.
- $M_M=\frac{M}{81}$ be the mass of the Moon.
- $d=60R$ be the total distance between the centers of the Earth and the Moon.
- $x$ be the distance from the Moon's center to the point where the gravitational fields cancel each other.
Consequently, the distance from the Earth's center to this point is $d-x$ (which is $60R-x$).

At this neutral point, the gravitational field strength due to the Earth ($E_E$) must equal the gravitational field strength due to the Moon ($E_M$):

$E_E=E_M$

Using the formula for the gravitational field of a point mass, $E=\frac{Gm}{r^2}$, we have:

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

We can simplify this equation by canceling the universal gravitational constant $G$ and the mass of the Earth $M$ from both sides:

$\frac{1}{{\left(60R-x\right)}^2}=\frac{1}{81x^2}$

Taking the square root on both sides:

$\frac{1}{60R-x}=\frac{1}{9x}$

Now, cross-multiply to solve for $x$:

$9x=60R-x$

Add $x$ to both sides:

$10x=60R$

Divide by 10:

$x=6R$

Therefore, the distance of the point where the gravitational fields cancel each other, measured from the Moon, is 6 R.