Earth-Moon Center Line of Gravity

[uno]

1. Locate the position of a spaceship on the Earth-Moon center line such that, at that point, the tug of each celestial body exerted on it would cancel and the craft would literally be weightless. Answer in meters from the moon



2. Moon mass = 7.36 x 10^22 and Earth Mass = 5.97 x 10^24



3. My professor gave my class the following advice for this problem. You have probably written an equation in which the force exerted by the Earth is set equal to the force exerted by the Moon on the spaceship. That simplies, but leaves two unknown distances. Since there are two unknowns, a second equation is needed. What is the sum of the two distances equal to? Now you have a second equation. Substitute it into the first, eliminating one of the distances. Notice the unknown variable is quadratic. Expand the equation into the form ax^2 +bx + c =0. The apply the quadratic formula to solve for the unknown distance (from spaceship to Moon).

I am still very confused, please help.

 

[Rake-MC]

Hmm interesting, this is how I would have done it:

$$a = \frac{GM}{R^2}$$

Therefore for the force to be equal (because the spaceship is of constant mass), $$\frac{GM}{R^2} = \frac{Gm}{r^2}$$

However here we have two variables for radius (R and r). One is measured from the moon, and the other is measured from the earth. What we must do is get the radius from the earth, in terms of radius from the moon.

 

[baba89]

I understand your confusion and am happy to help clarify this problem for you. The Earth-Moon center line of gravity is the imaginary line that connects the center of mass of the Earth and the center of mass of the Moon. This line is important because it represents the point where the gravitational forces of the Earth and Moon are balanced, resulting in a weightless state for any object located at that point.

To find the position of the spaceship on the Earth-Moon center line, we can use the concept of gravitational force and Newton's Law of Universal Gravitation. This law states that the force of gravity between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them.

In this case, we know the masses of the Earth and Moon (given in scientific notation) and we are trying to find the distance between the spaceship and the Moon. Let's call this distance x.

Using the advice from your professor, we can set up the following equations:

Force exerted by Earth = Force exerted by Moon

(G * m1 * m2) / (r^2) = (G * m3 * m4) / (x^2)

Where G is the gravitational constant (6.67 x 10^-11 N*m^2/kg^2), m1 is the mass of the Earth, m2 is the mass of the spaceship, m3 is the mass of the Moon, m4 is the mass of the spaceship (since it is the same object in both forces), and r is the distance between the Earth and the spaceship.

The second equation we need is the sum of the distances between the spaceship and the Earth and between the spaceship and the Moon. This is equal to the distance between the Earth and the Moon, which is approximately 384,400,000 meters.

So now we have two equations:

(G * m1 * m2) / (r^2) = (G * m3 * m4) / (x^2)

r + x = 384,400,000 meters

We can rearrange the first equation to solve for x:

x = sqrt[(G * m3 * m4 * r^2) / (G * m1 * m2)]

Substituting this value for x into the second equation, we get:

r + sqrt[(G * m3 * m4 * r^2) / (G * m1