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

To find the point where the gravitational field due to Earth equals the gravitational field due to the Moon at a point C, we start by considering the gravitational influences of both celestial bodies. Let’s denote the mass of Earth as M and the mass of the Moon as M/81 since the Moon’s mass is approximately one-eighty-first of Earth’s mass. The distance from the center of the Earth is represented as 60R, where R is the radius of the Earth.

At point C, we want the gravitational fields from both Earth and the Moon to balance each other. By establishing the relationship between the gravitational fields and the distances involved, we derive that the distance x from the center of the Moon is critical for determining this balance.

Solving for x leads us to a relationship indicating that x equals (6R). This means that the point C, where the gravitational fields are equal, is located 6 times the Earth’s radius away from the center of the Earth. This solution illustrates the gravitational interactions between the Earth and the Moon and highlights the unique balance of forces in space.

Gravitational field due to earth at C = Gravitational field due to moon at C or
GM/(60 R – x)² = G(M/81)/x²
or 81x² = (60 R -x)²
or 9x = 60 R – x
or x = 6 R