8.) Galileo Dropped the Ball

In fact he rolled balls of different sizes, weights and densities down a ramp, because he wanted to understand exactly how things fell to earth. Here's what he figured out:

• The widely held belief of Aristotle from the fourth century B.C. that heavier objects fall faster than lighter ones was incorrect.
• In a vacuum, falling objects would accelerate downward uniformly, regardless of their size, weight or density. He understood that air and water would create resistance, (friction), affecting the acceleration of objects of varying sizes, shapes and densities.
• Friction: a force that resists motion. Though Galileo did not understand it fully at the time, friction is the result of electromagnetic force between charged particles, including electrons, protons, atoms, and molecules. It is caused by chemical bonding between surfaces.
• That any ball requires twice as long to roll four times as far down a ramp, deriving the "Time-Squared Law".
• Time-Squared Law: the distance a uniformly accelerating object travels is proportional to the square of the elapsed time.

Calculating using Time-Squared Law
(D) = Distance Traveled by a Uniformly Accelerating Object
(T) = Elapsed Time - time interval over which the object travels
(D) ∝ (T)²

Galileo's discoveries were not "entirely original. The time-squared law for uniformly accelerated change was already known to Nicole Oresme in the 14th century." What's important is that "Galileo expressed the time-squared law using geometrical constructions and mathematically precise words, adhering to the standards of the day."

"Ideas relating to inertia had been proposed by Ibn al-Haytham centuries earlier." "According to Joseph Needham, Mo Tzu had proposed it centuries before." In Galileo's case, it "was the first time" these ideas "had been mathematically expressed" and "verified experimentally."

He "introduced the idea of frictional force, the key breakthrough in validating inertia. Galileo's Principle of Inertia stated: "A body moving on a level surface will continue in the same direction at constant speed unless disturbed." This principle was incorporated into Newton's laws of motion (first law)." (from "Galileo" at Wikipedia: http://en.wikipedia.org/wiki/Galileo_Galilei)

• Forty or fifty years after Galileo's experiments, Robert Boyle proved Galileo's theory by dropping objects in a tall cylinder that he'd pumped the air out of to create a vacuum. The table below is intended as a visual representation of what occurred:

• On Earth and in a vacuum, falling objects accelerate downward uniformly at the rate of 32 ft. / second², (sometimes stated as 32 feet per second, per second), regardless of their size, weight or density. Let's apply the three formulas we've learned so far to see how they work and to make sure we understand them correctly.

Calculating using Time-Squared Law
(D) = Distance Traveled by a Uniformly Accelerating Object
(T) = Elapsed Time - time interval over which the object travels
(D) ∝ (T)²

Calculating Average Acceleration
(A) = Average Acceleration, (V) = Change in Velocity, (T) = Time interval over which velocity changes: (A) = (V) / (T)

Calculating Average Velocity
(V) = Average Velocity, (D) = Displacement, (T) = Time:
(V) = (D) / (T)

Oooops! What exactly does ∝ (proportional) mean? How would we express velocity? 32 ft. /second, downward? Looks like Donkey's got more work to do.

Next blog entry: "A Degree in Gravity"