The mass of the Moon is 7.35x10 to the 22nd Kg. At some point between the Earth and the Moon, the for of Earth's gravitational attraction on an object is cancelled by the Moon's force of gravitational attraction. If the distance between Earth and the Moon (centre to centre) is 3.84x10 to the 5th km, calculate where this will occur, relative to Earth.
Let X be the distance from Earth. You will need the mass of the Earth also. Call it M and the moon's mass m.
Let D = 3.84*10^5 km
For the forces to be equal,
G M/x^2 = G m/(D-x)^2
G (the universal law of gravity constant) cancels out.
Solve for x/D in terms of M/m
[(D-x)/x]^2 = (D/x -1]^2 = m/M
D/x -1 = sqrt (m/M)
To find where the Earth's gravitational attraction is canceled out by the Moon's gravitational attraction, we need to consider the gravitational forces exerted by both bodies.
The gravitational force between two objects can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 x 10^-11 N m^2 / kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
We are looking for the point where the gravitational forces of the Earth and the Moon cancel each other out, so the magnitudes of these forces will be equal:
F(earth) = F(moon)
G * (m1 * m2) / r(earth)^2 = G * (m1 * m2) / r(moon)^2
m1 represents the mass of the object on which the forces are acting. Cancelling out the masses and the gravitational constant, we get:
r(moon)^2 = r(earth)^2
Since r(earth) is known to be 3.84x10^5 km, we can calculate r(moon) by taking the square root of r(earth)^2.
r(moon) = √(3.84x10^5 km)^2
r(moon) = 3.84x10^5 km
So the point where the Earth's gravitational attraction is canceled out by the Moon's gravitational attraction will occur at a distance of 3.84x10^5 km relative to Earth.