magine a straight line connecting the centers of the earth and the moon. At some point along this line the gravitational forces pulling a spacecraft towards the moon and towards the earth exactly balance each other, and the craft could just sit there in (an unstable) equilibrium. How far is this point from the center of the moon? (Neglect the gravitational field of the sun and the relative motion of earth and moon for this problem.)
The Law of Universal Gravitation states that each particle of matter attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Expressed mathematically,
F = GM(m)/r^2
where F is the force with which either of the particles attracts the other, M and m are the masses of two particles separated by a distance r, and G is the Universal Gravitational Constant. The product of G and, lets say, the mass of the earth, is sometimes referred to as GM or mu (the greek letter pronounced meuw as opposed to meow), the earth's gravitational constant. Thus the force of attraction exerted by the earth on any particle within, on the surface of, or above, is F = 1.40766x10^16 ft^3/sec^2(m)/r^2 where m is the mass of the object being attracted and r is the distance from the center of the earth to the mass.
The gravitational constant for the earth, GM(E), is 1.40766x10^16ft^3/sec^2. The gravitational constant for the moon, GM(M), is 1.7313x10^14ft^3/sec^2. Using the average distance between the earth and moon of 239,000 miles, let the distance from the moon, to the point between the earth and moon, where the gravitational pull on a 32,200 lb. satellite is the same, be X, and the distance from the earth to this point be (239,000 - X). Therefore, the gravitational force is F = GMm/r^2 where r = X for the moon distance and r = (239000 - X) for the earth distance, and m is the mass of the satellite. At the point where the forces are equal, 1.40766x10^16(m)/(239000-X)^2 = 1.7313x10^14(m)/X^2. The m's cancel out and you are left with 81.30653X^2 = (239000 - X)^2 which results in 80.30653X^2 + 478000X - 5.7121x10^10 = 0. From the quadratic equation, you get X = 23,859 miles, roughly one tenth the distance between the two bodies from the moon. So the distance from the earth is ~215,140 miles.
Checking the gravitational pull on the 32,200 lb. satellite, whose mass m = 1000 lb.sec.^2/ft.^4. The pull of the earth is F = 1.40766x10^16(1000)/(215,140x5280)^2 = 10.91 lb. The pull of the moon is F = 1.7313x10^14(1000)/(23858x5280)^2 = 10.91 lb.
This point is sometimes referred to as L2. There is an L5 Society which supports building a space station at this point between the earth and moon. There are five such points in space, L1 through L5, at which a small body can remain in a stable orbit with two very massive bodies. The points are called Lagrangian Points and are the rare cases where the relative motions of three bodies can be computed exactly. In the case of a body orbiting a much larger body, such as the moon about the earth, the first stable point is L1 and lies on the moon's orbit, diametrically opposite the earth. The L2 and L3 points are both on the moon-earth line, one closer to the earth than the moon and the other farther away. The remaining L4 and L5 points are located on the moon's orbit such that each forms an equilateral triangle with the earth and moon.
To determine the distance from the center of the moon to the point where the gravitational forces of the Earth and the Moon are balanced, we need to consider the gravitational forces acting on the spacecraft.
Let's break down the problem step by step:
1. Start by calculating the gravitational force acting on the spacecraft due to Earth. The force of gravity between two bodies can be calculated using Newton's law of universal gravitation:
F1 = G * (m1 * m_spacecraft) / r1^2
where F1 is the gravitational force due to Earth, G is the gravitational constant, m1 is the mass of the Earth, m_spacecraft is the mass of the spacecraft, and r1 is the distance between the center of the Earth and the spacecraft.
2. Similarly, calculate the gravitational force acting on the spacecraft due to the Moon:
F2 = G * (m2 * m_spacecraft) / r2^2
Here, F2 is the gravitational force due to the Moon, m2 is the mass of the Moon, and r2 is the distance between the center of the Moon and the spacecraft.
3. Since we are looking for the point where the gravitational forces balance each other, we can set F1 equal to F2:
F1 = F2
4. Substitute the expressions for F1 and F2 from steps 1 and 2:
G * (m1 * m_spacecraft) / r1^2 = G * (m2 * m_spacecraft) / r2^2
5. Cancel out m_spacecraft from both sides of the equation:
m1 / r1^2 = m2 / r2^2
6. Rearrange the equation to solve for r2, the distance from the center of the Moon to the spacecraft:
r2^2 = (m2 * r1^2) / m1
Taking the square root of both sides, we get:
r2 = sqrt((m2 * r1^2) / m1)
Now, to find the values needed to substitute into this equation, we can use the following approximate values:
- Mass of the Earth (m1) = 5.97 × 10^24 kg
- Mass of the Moon (m2) = 7.35 × 10^22 kg
- Distance between the Earth and the spacecraft (r1) = Distance between the Earth and Moon (3.84 × 10^8 m) minus the radius of the Earth (6,371 km) or 6,371,000 m.
Once you substitute these values into the equation, you can calculate the distance from the center of the Moon to the point where the gravitational forces balance.