There are many stories that educated people are “supposed” to know. These are the stories that tell us the axioms or postulates of our culture. Two examples are Archimedes running naked through the street of Syracuse yelling, “Eureka,” and Galileo dropping two cannon balls of different weights from the Leaning Tower of Pisa. Although historians of science may question the accuracy of the stories, there is no doubt that the stories help us remember the physical principles involved. The first is buoyancy, why boats float, and the second tells us that objects fall at a constant acceleration that is not dependent on their weight.
What have I learned that I would like to share with my students? There are a number of stories that illustrate physical principles that everyone is supposed to know. Let us tell them.
There are fundamental principles that need to be known as well. Let us mention them even if we do not teach the topics completely so the students will be prepared to listen more carefully when they meet the principles the next time an instructor presents them.
One topic that I would like to share with my students is called dimensional analysis. Physics is what we can hold, see, and measure. Just what can we measure? We can use a ruler to measure the length of something, we can use a balance to find the mass of something, we can use a watch to time something, and we can use a thermometer to measure the temperature of something. All other measurements are combinations of length, mass, and time. In fact, a thermometer is using length to measure temperature; the length of the column of mercury tells us the temperature. The words “dimensions” and “units” have distinct meanings. When I say a board is six feet long, I have given the units feet to the dimension length.
A. Archimedes: The Lever, Density, and Buoyancy
Archimedes was a Greek mathematician. He was killed by a Roman soldier during the invasion of Syracuse in 212 B.C. You may read about this in James R. Newman’s collection,
The World of Mathematics
. Two concepts that Archimedes worked on are of interest to naval architects: The lever and buoyancy.
The lever is the first machine. It is the principle of the beam balance, which allows us to weigh things. Scientists look for formulas to describe phenomena. The formula for a lever can be best explained by looking at a seesaw. Two children take a board and support it near the center. Each one sits on an end with the support between them. They now can go up and down. The support is called the fulcrum. What if one child weighs much more than the other? The heavy end will go down and only go up with much effort on the part of the heavy child. What can be done? Move the board on the support so the heavy child is closer to the fulcrum and the lighter child is farther from the fulcrum. Let w1 be the weight of the first child, let d1 be the length of the seesaw from the fulcrum to the first child, let w2 be the weight of the second child, and let d2 be the length of the seesaw from the fulcrum to the second child. The formula for this law of mechanics is
(figure available in print form)
When physicists multiply a weight or force times a distance they call the product a moment or a torque. So if you move a weight with a lever and you have to move another weight that is twice as heavy as the first, just use a lever that is twice as long and it will be just as easy to move as the first. This led Archimedes to say, “Give me a place to stand and I will move the earth,” the principle being that to apply more force, use a longer lever.
The second principle is illustrated by the story of how Archimedes tested the crown of king Hiero for the purity of its gold content. Rumor said the goldsmith had replaced some of the gold with an equal weight of silver. So the crown weighed as much as it should. How could it be tested for purity? Archimedes was consulted.
Archimedes’ moment of insight occurred while he was taking a bath. He jumped out of the tub and ran throughout the streets yelling, “Eureka!” which means “I have found it!” in Greek. That’s the story, anyway. So what had he found?
Let us ask ourselves some questions. What happened when his body went into the water? Water ran out of the tub. Water was displaced. How much water was displaced? Two things cannot take up the same space at the same time, so the amount of water displaced has the same volume as the submerged body. Do equal weights of gold and silver take up the same amount of space? No, gold is denser than silver. Equal volumes of gold and silver would have different weights; the gold would weigh more than the silver. So equal weights of gold and silver would have different volumes, with the gold, being denser, taking up a smaller volume.
Archimedes found that the goldsmith, indeed, had cheated. Archimedes measured the displacements of the crown and of two weights, one of pure gold, the other of pure silver, both equal in weight to the crown. Archimedes found the crown’s displacement to be between the displacements for the pure gold and the pure silver.
There is more to the story. Archimedes also weighed the crown in air and suspended under water. He found that it weighed less when submerged. How much less? The weight loss was equal to the weight of the displaced water. What if a body displaced a volume of water that weighed more than the body itself? The body would weigh less than nothing? No, the body would float. That is the principle of buoyancy. A body floats when the weight of the water displaced is equal to the weight of the body.
If we divide the mass of the body by its volume we get its density. Knowing the density of an object and the density of water we can tell if the object will float. If its density is less than the density of water then the object floats, greater than water it sinks.
With these principles it is possible to determine the water line of a boat. Here is a sample problem. I have a block of wood 8 cm x 10 cm x 15 cm, that has a mass of 780 gm. I would like to float it with the 8 cm dimension being the height. How deep would it sink into the water? What is the density of this block?
One convenient feature of the metric system is that one cubic centimeter of water has a mass of one gram, so by definition the density of pure water is one gram per cubic centimeter. The density of the block is
(figure available in print form)
So the block is a typical piece of wood; its density is less than one. In other words, its density is less than the density of water, so it will float.
Since the block weighs 780 grams it will have to displace 780 cubic centimeters of water in order to float. If the block is to float with the 8 cm dimension as the height, then the submerged portion will have dimensions of 10 cm by 15 cm by X cm and will have a volume of 780 cubic centimeters. So find X.
(figure available in print form)
Consequently the block will float with 5.2 cm of the 8 cm vertical dimension submerged.
This is a demonstration of the principle that allows a naval architect to determine the water line of a boat before the boat is built. Of course, the submerged portion will be more complex than a block. Calculus will then be needed to calculate the volume of the submerged portion. The architect will also need to know the densities of the materials out of which the boat is to be so that the weight of the boat can be calculated. Again by calculus the volume of the whole boat can be determined.
Archimedes was one of the Greeks who contributed to classical geometry. In fact, they called themselves philosophers, no matter what they were studying—mathematics, science, or what we call philosophy today. The axiomatic method of thinking was their legacy to us. Consider the statement, “Two things cannot take up the same space at the same time.” That is a self-evident truth, that is an axiom! So the laws of science are axioms that we accept because our experience says they seem to be self-evident. If the axioms are true then the consequences of the axioms are true as well. So investigate by doing experiments to see if the consequences are true. If the consequences do not work out, then our axioms do not apply to the real world. The investigations are called experiments.
What Archimedes was doing in our discussion is known as statics, the study of bodies at rest, in equilibrium. Statics is a branch of the larger field of physics called mechanics which includes the study of motion.
B. Other Contributors
Among the people who contributed to the field of mechanics we find Copernicus, Brahe, Kepler, Galileo, and Newton. Yes, those are the names associated with the concept of the solar system. Mechanics is the study of motion, any kind of motion, even the motion of celestial bodies.
Here is a short version of the story. Copernicus said the planets revolved about the sun. Brahe made astronomical observations. Kepler analyzed Brahe’s data and came up with three laws.
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1. Planets follow elliptical paths with the sun as a focus.
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2. A line from the sun to a planet will sweep out equal areas in equal times.
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3. The square of the period of revolution about the sun is proportional to the cube of the major axis of the orbit.
Some friends went to Isaac Newton (1642-1727) with the idea that gravitational attraction between two bodies was inversely proportional to the square of the distance between the bodies. The story goes that he had the solution already written out for them. The law of gravity with Newton’s laws of motion result in proving Kepler’s laws as theorems. Newton had used the calculus that he had invented to do the problem. His friends induced him to publish his results in a book called Newton’s Principia.
C. Newton’s Three Laws:
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1. A body at rest will remain at rest and a body in motion will remain in straight line motion, unless it is acted upon by a force to change that motion.
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2. The force causing the motion of a body is equal to the product of the mass of the body and the acceleration of the body. (F = ma) where F is the force, m is the mass of the body, and a is the acceleration of the body. Acceleration is defined as the rate of change of the velocity, that is, how fast the speed changes.
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3. For every action there is an equal and opposite reaction.
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Newton also proposed the Universal Law of Gravity. He stated that the force of attraction, F, between two bodies varies jointly as the product of their masses, m1 and m2, and inversely as the square of the distance, r, between them:
(figure available in print form)
where G is the universal gravitational constant.
There is much more to the story. Who was Leibniz, and what did Galileo have to do with the story?
So Newton’s scheme worked; it was applied to celestial bodies and earthly bodies. His work in this area is now called Newtonian Mechanics or Classical Mechanics. When we apply the rules of mechanics to fluids we have fluid mechanics.
