Lesson 8 The Apple and the Moon

Isaac Newton's Law of Univeral Gravitation explains why all bodies near the surface of the earth fall with constant acceleration and (in context) why an apple but not the moon falls from the sky.

Early Greek astronomers attempted to explain the motion of the bodies in space through uniform circular motion. Copernicus built upon the Greeks studies and placed the sun, rather than the earth, at the center of the "universe" or solar system. However, Copernicus still followed the idea of a uniform universe. Galileo then studied how bodies act under the influence of earth's gravity on earth. Utilizing all of these discoveries, Kepler developed three laws explaining the planetary motions "outside" of the earth in space:

First Law:

Second Law:

The closer a body is to the sun, the faster it moves and the farther a body is from the sun, the slower it moves.

Third Law:

The larger a body's orbit, the longer period of time it takes for the body to complete a revolution along its orbit.

Here is a video desbribing Kepler's three laws:


From these laws Newton deduced that the reason the planets orbit the sun at a constant rate and path, that the reason all bodies fall at the same rate, and that the reason that the moon orbits the earth were all related. Newton found that every two bits of matter in the universe attract each other and that the force of attraction is proportional to each of their masses. This force of attraction has a concentrated net force in the downward position, corresponding to the masses of the two objects. The force weakens inversely as the square of the distance between the two bodies.

F = -G m1m2 r^
               r^2
Let's consider and apple falling from a tree on earth.

F = -G m(apple)m(earth) r^      
              r^2
r = square of distance from the apple to the center of the earth.

F = m(a)a

m(a)a = -G m(apple)m(earth) r^
                               r^2
a = -Gm(E)
       R(E)^2

Gravity does not depend on the mass of the apple, thus gravity has the same effect on all bodies near earth. The constant force of gravity is evident in this video below when two objects are dropped in a vacuum:


g = -Gm(E)    which is 32ft/sec/sec
        R(E)^2

On the moon, the acceleration may be different but it still relates to the universal gravitational constant. The acceleration of a falling body on the moon is 1/6 the acceleration of a falling body on earth. However, why doesn't the moon fall from the sky like the apple?

If a projectile is thrown fast enough it will orbit the earth. Therefore, just as astronauts are held in orbit of the moon, gravity holds the moon in orbit of the earth. 0g or weightlessness is not the absence of gravity; objects are held in orbit as they "fall for eternity." The rate at which the moon falls is much smaller than the rate at which  the apple falls. In fact, the moon falls slower than the apple by a factor of 3,600. Therefore the moon falls 1/20 of an inch every second:

Moon falls towards earth

a(m) = G M(E) mass of earth
                r^2(m) square distance to earth
g = G M(E) mass of earth
          R^2(E) radius of earth
am/g = GM(E)/r^2(m)
            GM(E)/R^2(E)
am/g = R^2(E)
             r^2(m)
am/g = (R(E))2
              r(m)

Every month the moon travels 2(pi)r(m)

2(pi)r(m) = speed
1month

The Law of Inertia states that the moon wants to fly out of the orbit of the earth in a straight line, however, it is held in by the force of gravity.

The speed of the moon is 40,281 in./sec

Using the Pythagorean theorem with the orbit of the moon:

r(m)^2 + d^2 = (s(m) + r(m))^2

d^2 = [s(m)^2] + 2r(m)s(m)       [s(m)^2] is so small it is ignored.

d^2 = 2r(m)s(m)

d^2 = s(m)
2r(m)

s(m) = 1/20 in. (The rate at which the moon falls towards the earth as it "falls for eternity").