A spacecraft is on a journey to the moon. At what point, as measured from the center of the earth, does the gravitational force exerted on the spacecraft by the earth balance that exerted by the moon? This point lies on a line between the centers of the earth and the moon. The distance between the earth and the moon is 3.85 × 108 m, and the mass of the earth is 81.4 times as great as that of the moon.
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 point where the gravitational forces exerted by the Earth and the Moon balance each other on the spacecraft, we can use Newton's Law of Universal Gravitation.
The gravitational force between two objects can be calculated using the formula:
F = G * (m1 * m2) / r^2,
where F is the gravitational force between the objects, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.
Let's denote the distance between the spacecraft and the center of the Earth as x. The distance between the Moon and the center of the Earth is given as 3.85 × 10^8 m.
Since the mass of the Earth is 81.4 times that of the Moon, we can denote the mass of the Moon as M and the mass of the Earth as 81.4M.
To find the point where the forces balance, we need to set up an equation of gravitational forces.
For the gravitational force exerted by the Earth on the spacecraft:
F1 = G * (81.4M * m) / (x^2),
where m represents the mass of the spacecraft.
For the gravitational force exerted by the Moon on the spacecraft:
F2 = G * (M * m) / ((3.85 × 10^8 - x)^2).
Since the forces balance each other, we have:
F1 = F2.
Setting the two equations equal to each other:
G * (81.4M * m) / (x^2) = G * (M * m) / ((3.85 × 10^8 - x)^2).
Cancelling out G and m from both sides of the equation, we get:
(81.4M / x^2) = (M / (3.85 × 10^8 - x)^2).
Cross multiplying and simplifying, we have:
81.4M * (3.85 × 10^8 - x)^2 = M * x^2.
Expanding the square term, we get:
81.4M * (3.85 × 10^8 - 2 * (3.85 × 10^8) * x + x^2) = M * x^2.
Distributing and rearranging the equation, we have:
310.6 × 10^8 * M - 624 * 10^8 * x = M * x^2.
Finally, rearranging again, we get:
M * x^2 + 624 * 10^8 * x - 310.6 × 10^8 * M = 0.
Now, we have a quadratic equation. By solving it, we can find the value of x at which the gravitational forces balance each other.