Question

Is there a point between earth and the moon where the gravitational force acting on a spaceship is zero? If so, where is that point?

Earth Moon
Mass 5.98 x 10 raise to 24 kg 7.348 x 10 raise to 22 kg
Radius 6.389 x 10 raise to 6 m 1.738 x 10 raise to 6 m

Answer 1

It is called the Lagrange point and may be calculated by equating the gravitational law of the Moon and Earth. The Lagrange point is where the gravitational pull of the Earth equals the gravitational pull of the Moon. It is safe to ignore the gravitational pull of the Sun because the centre of mass of Earth and Moon is traveling at approximately 67,000 miles per hour, thus canceling the pull of the Sun with centripetal force. Any object, which leaves the Earth or Moon, retains this velocity.

Assume the Lagrange point is a distance R from centre of the Moon and that the distance between the Earth’s centre and the Moon’s centre is D, therefore, the gravitational laws for the two masses can be equated; G is the gravitational constant and m is the mass stationary in the Lagrange point; Mm is the mass of the Moon and Me is mass of the Earth, hence:

G x Mm x m/(R^2) = G x Me x m/[(D - R)^2] … (1)

The G, and m cancel and it is known that the Earth is approximately 81 times the mass of the Moon, therefore,

Me/Mm = 81 (a dimensionless number)

The equation at (1) reduces to:

(D - R)^2/(R^2) = 81

Square root both sides and with an approximate value for D of 245,000 miles, then:

(D - R)/R = 9 or D - R = 9R or D = 10R,

Therefore, the distance (R) of the Lagrange point from centre of the Moon is 24,500 miles. The radius of the Moon is about 1000 miles and so the distance of the Lagrange point from surface of the Moon is about 23,500 miles. Note also that the Lagrange point will be along the straight line connecting the centres of the Earth and Moon and, as the Moon orbits the Earth, the Lagrange point will also move correspondingly.

As stated above the system of the Earth , Moon, their barycentre or centre of gravity, and the Lagrange point are moving at about 67,000 mph in orbiting the Sun.