Galileo and Newton were both mystified by the forces and motions that occurred in pendulums. But only Newton spent a significant amount of time attempting to analyze what was happening with a pendulum.
Newton assumed that the force acting on the bob which pulled it down toward Earth was the gravitational force exerted by the Earth in attracting the bob, and that the Earth’s centripetal force exerted upon the bob was proportional to the inertial mass of the bob. Newton also assumed that the upward motion or momentum of the bob was proportional to the bob’s inertia (i.e. the inertial resistance offered by the bob to the gravitational force trying to alter its state of motion).
Evidently, Newton then decided that the period of the bob’s swing to and fro must depend upon the ratio between these two different phenomena exhibited by the bob’s mass. The fact that these two different phenomena were measured by the same number (the mass) was to Newton “a most remarkable accidental coincidence, something like a miracle.” Why? Because gravitational force and inertial resistance were assumed to be very different phenomena of nature. So how could they be measured by the same number: the same mass of the same body?
Thereafter, a bewildered Newton conducted pendulum experiments to determine the ratio between these two qualities or phenomena of the same mass, and to try to confirm the apparently equal results of Galileo’s Leaning Tower experiments. Newton decided to swing two identical pendulums together, each with a hallow bob that could be filled with different materials. Newton filled one bob with one substance (i.e. gold, silver, lead or sand), and filled the other bob with an equal weight of a different substance (i.e. salt, wood, water or wheat). Each bob was suspended from an 11-foot cord. He then caused the two bobs to swing back and forth together, and they swung in unison “for a very long time.”
Based on these pendulum experiments, Newton concluded that there was the same amount of matter in each substance, and that this was the reason why all heavy bodies fall toward the Earth in equal times. The fact that the period of the swing always has the same time interval regardless of the material substance in the bob for some reason convinced Newton that the mass which caused the bob’s weight and the mass that caused the bob’s inertial resistance were equal.
Newton then postulated in Proposition 6 of Book 3 of his Principia, that:
“[T]he weight of anybody whatever toward [a] planet is proportional to the quantity of matter which the body contains.”
As an example of this postulate, Newton went on to state, that:
“Others have long since observed that the falling of heavy bodies toward the earth…takes place in equal times, and it is possible to discern that equality of the times, to a very high degree of accuracy, by using pendulums.”
By “the falling of heavy bodies,” Newton meant the “gravitational acceleration” of the mass of such bodies toward the relatively enormous mass of the Earth.
What was wrong with Newton’s analysis of pendulum experiments, and his Proposition 6 of Book 3? The answer is: just about everything. One should begin by asking: What did identically swinging pendulums of equal weight, or with the same amount of matter, have to do with Galileo’s experiment where he dropped unequally weighted cannonballs with different amounts of matter from the Leaning Tower in apparently the same time? The answer, of course, is nothing.
What was the validity of anyone characterizing the one and only mass of the bob as having two or three different types of mass (i.e. inertial mass, gravitational accelerating mass and energy mass), and then deciding that the ratio between these theoretical swinging masses was equal because different substances of the same weight swing back and forth in equal times? The answer, again, is: nothing. The only way to rationalize Newton’s pendulum experiments, and his deductions from them, is to conclude that: Newton was very confused by the equal result of Galileo’s Leaning Tower experiment.
Newton’s Proposition 6 of Book 3 should have stated, as follows: The gravitational force emitted by any body is proportional to its inertial mass, and the gravitational accelerations of any body receiving the appropriate quantity of such force are inversely proportional to the inertial mass of such receiving body. These concepts would have been consistent with Newton’s three laws of motion, but they also would have contradicted Galileo’s Leaning Tower of Pisa result: that unequal masses accelerate equally toward the Earth.
If the correct wording of Proposition 6 of Book 3 was applied to the motions of a pendulum, the correct explanation of its forces, resistances and motions would be as follows. The gravitational acceleration of the bob toward the Earth (caused by the equivalent centripetal force of the Earth applied to the bob) is restrained by the resistance of the rope and causes the bob to angularly accelerate in an arc from position A to position C (Figure 18). These forces, resistances, actions, reactions, accelerations and motions are the result of Newton’s law of mutual gravitational attraction in conjunction with his three laws of motion.
At position C the restraining rope caused the angular acceleration of the bob to terminate and, in conjunction with the Earth’s centripetal force, the bob’s angular inertial momentum begins to decelerate in an arc from position C to position B where it terminates. At position B the aforementioned process repeats itself in the opposite direction. During all of these back and forth motions the velocity of the bob gradually decreases because of the friction and resistance of the restraining rope and the air through which the bob and rope of the pendulum moves. The dubious and artificial concept of the equality of inertial and gravitational mass has nothing to do with this process.
What were the real explanations for the paradoxes which Galileo and Newton attributed to pendulums? The primary reason that a light bob and a heavy bob appear to swing back and forth during the same time interval, is the same reason that Galileo’s two unequal masses appeared to free-fall from the Leaning Tower of Pisa to Earth during the same time interval. The ratio of the slightly different masses between each tiny bob (or each relatively tiny cannonball) was almost identical when compared to the enormous mass of the Earth which was pulling both of them toward it with almost the same force.
The reason why Newton’s and Galileo’s pendulums always appeared to swing to and fro during the same equal time regardless of the distance of the swing, was primarily due to the simple fact that as the distance of the swing gets progressively shorter the fall of the bob becomes less steep, and the speed of the bob becomes progressively slower over a progressively shorter distance.
The reason why such equal times had a longer or shorter duration (or period) depending upon the length of the rope was because each bob had a longer or shorter distance to swing depending upon the length of the rope. In other words, if the rope was shortened the bob would have a shorter distance to travel in less time. If the rope was lengthened the bob would have a longer distance to travel, and this would take more time. It is just that simple.
The reason why pendulums slowed down slightly at the Earth’s equator or at higher altitudes was because the Earth’s centripetal force decreased slightly (due to the greater distance from the pendulum to the center of the Earth) at such more distant locations, so that the downward force and action on the bob was not as great at such more distant locations. Thus the distance and period of such back and forth motion was never quite as long at a location closer to the center of the Earth.
The real reason why the two different qualities or phenomena of the same mass were equal was because in reality there was only one mass of any body which performed several different functions. Any theory of two or three different masses to explain the paradoxical motions and forces of a pendulum was false and irrelevant to the reality of the situation.
Gradually, during the 18th and 19th centuries, physicists became so accustomed to Newton’s meaningless pendulum experiments that they forgot what such experiments were based upon and what they were supposed to mean. However, we now know that such equality of theoretical masses was a meaningless concept for yet another reason: because Galileo’s conclusions concerning falling unequal masses were not correct.
Nevertheless, in Einstein’s 1915 General Theory of Relativity, the equality of gravitational mass and inertial mass was not just a miracle. It became a necessity, because Einstein premised his entire General Theory upon the following invalid concepts:
“The fundamental property of the gravitational field of giving all bodies the same acceleration, or, what comes to the same thing, on the law of the equality of inertial and gravitational mass.”
In Einstein’s General Theory of Gravity, gravitation and inertial were not just equal; they became identical. For all of these reasons, Einstein’s General Theory was also meaningless.