Somewhere between the Earth and the Moon, the gravitational pull exerted by these two bodies on a space pod would cancel. Is this location nearer the Earth or nearer the Moon?

A) Nearer the Earth.
B) Nearer the Moon.
C) At the half-way point.

Stated another way, "At what distance from the earth toward the moon will the gravitational force of attraction towards the moon be equal and opposite to the gravitational force of attraction towards the earth?

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.

The location where the gravitational pull exerted by the Earth and the Moon on a space pod cancel each other out is known as the Lagrange point or the L1 point. This point is closer to the larger gravitational body, which in this case is the Earth. Therefore, the answer is A) Nearer the Earth.

To determine whether the location where the gravitational pulls of the Earth and the Moon cancel out is nearer the Earth or nearer the Moon, we can consider the concept of the gravitational force exerted by a body.

The gravitational force between two objects depends on two factors: the masses of the objects and the distance between them. Mathematically, the force of gravity can be calculated using Newton's law of universal gravitation:

F = G * (m1 * m2) / r^2

Where:
F is the gravitational force between the two objects,
G is the gravitational constant,
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.

In this case, we know that the gravitational pulls of the Earth and the Moon cancel each other out. This means that the gravitational force exerted by the Earth on the space pod is equal in magnitude and opposite in direction to the gravitational force exerted by the Moon on the space pod. Therefore, we can set up the following equation:

F_earth = -F_moon

Since the magnitude of the gravitational force is proportional to the product of the masses of the objects, we can rewrite the equation as:

G * (m_earth * m_pod) / r_earth^2 = -G * (m_moon * m_pod) / r_moon^2

Where:
m_earth and m_moon are the masses of the Earth and the Moon, respectively,
m_pod is the mass of the space pod, and
r_earth and r_moon are the distances from the center of the Earth and the center of the Moon to the location of the space pod, respectively.

We can cancel out the gravitational constant (G) and simplify the equation:

m_earth / r_earth^2 = -m_moon / r_moon^2

Now, let's analyze the equation. Since the masses of the Earth and the Moon are constants, the only variables are the distances from the space pod to the Earth and the Moon. If we assume that the mass of the space pod (m_pod) is negligible compared to the masses of the Earth and the Moon, we can simplify the equation further:

1 / r_earth^2 = -1 / r_moon^2

Now, we need to consider the signs in the equation. The negative sign indicates that the gravitational force exerted by the Moon is in the opposite direction compared to the force exerted by the Earth. This implies that the distance to the Moon (r_moon) will be negative, indicating that it is in the opposite direction to the Earth.

Therefore, to cancel out the gravitational pulls of the Earth and the Moon, the location must be closer to the body with a greater gravitational pull, which is the Earth. Hence, the correct answer is:

A) Nearer the Earth.

nearer the earth