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Terrestrial and Celestial Motion

From Greek Science to the Age of Enlightenment
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When did science begin? Egyptians used wheeled chariots, measured mass using balances and built the pyramids. they also moved from a lunar calendar to one with 12 thirty day months. This also gave them a good excuse for a 5 day holiday at the end of the year. Was this science? It depends on what you call science. One thing is clear: A missing ingredient in all the advances of those early times was any formulation of physical laws. There was no structure generated that would allow new knowledge to be built from basic principles. This was to be the legacy of the Greeks. It began some time between 800 and 600 BCE.

Among the most well known of the Greek thinkers is Aristotle. He taught that there were two types of motion, natural and violent. Every object had a natural motion, depending on its composition. If the object was made of celestial material (as were the stars and planets), its natural motion was circular, and so it moved around Earth under the influence of the Prime Mover. If made of earth or water it had the motion of heavy objects, which fell towards the center of the universe and so accumulated on Earth. If the object consisted of air or fire its motion was lightness and it moved upward. All things were made of earth, water, air, fire or celestial matter, and so moved accordingly. Violent (or forced) motion occurred when an external agent overcame an object's natural motion. Once the external force was removed the violent motion ceased.  Although this view has obvious and fatal flaws (How, for example, can an arrow continue upward after the force of the bowstring no longer pushes the arrow along?) it remained the prevailing theory of motion for almost two thousand years.

There was little progress on such matters during the middle ages, with the notable exception of the work of St. Thomas Aquinas in the 13th century, who reconciled the teachings of Aristotle with the doctrine of the Church. This alignment set the stage for the collision between the authority of the Church and the inevitability of scientific discovery. In his work "On the Revolutions of the Celestial Spheres", published in 1543, Nicolaus Copernicus moved Earth from its ordained position at the center of the universe into an orbit about the sun, not unlike that of the other planets. In so doing he ushered in an era dominated by the likes of Galileo, Brahe and Kepler who were to build a new way of doing science

The Copernican Revolution

Of all the motions there were, motions of the heavenly bodies most occupied the thoughts of ancient thinkers. The Pythagoreans put Earth at the center of the universe and the planets (including our Sun and Moon) on spheres turning around it. Variations of this model were proposed (most notably by Aristarchus in about 200 BC), but the matter was essentially settled by Plato when he posed the problem of finding the best combinations of circular orbits to explain the motions of the planets. The most successful attempt was by Claudius Ptolemy in about 100 AD. Using the idea of epicycles proposed by Hipparchus over 200 years earlier, he created a complex model (the Ptolemaic system) that could predict the motions fairly accurately - Plato's problem had been solved. The Ptolemaic system was accurate enough (although modified - and made more complex - over time) to stand for fifteen centuries until, looking for a simpler way, Nicolaus Copernicus moved the center of the World to the Sun.

From Impetus to Inertia

He was a major contributor to the Copernican Revolution, but a greater contribution of Galileo Galilei is his work on the motion of falling bodies.  He was born in Pisa, Italy in the same year as Shakespeare, on the day that Michelangelo died: 1564. He died in 1642, the year Newton was born. His work "Discourses on Two New Sciences" was published in 1638 while he was under house arrest for his support of the Copernican view of planetary motion. This work was much more that an explanation of the mechanics of how objects moved, it was the beginnings of modern science. For the first time a theory was valid only when verified by experiment, and the scientific method was established. Here is a brief description of his "new science" - the presentation is streamlined, but the ideas are Galileo's.

Uniform Motion: Wherein equal distances are covered in equal times.

Uniformly Accelerated Motion: Wherein the motion gains equal increments of speed in equal time intervals.

These relations he acquired through experimentation and reduction of data - what we call the experimental method began here.

The connections between distance, velocity and acceleration can be seen directly when presented in graphical form:

Here, a graph of acceleration with a constant value of 2 meters/sec2 is shown - it's just a straight line of value 2, which never changes. This is a particularly important case: First, it's simple; second it is the same situation we find all the time when something falls under the influence of gravity. (If it were the case of Earth's gravity, the acceleration would be equal to 9.8 meters/sec2.)

OK: If you accelerate (from rest) at 2 meters/sec2 for say, 1 second, how fast are you going? You should be able to figure it out - the answer is 2 meters/sec. After 2 seconds your speed will be 4 meters/sec., after 3 seconds it will be 6 meters/sec., etc. A graph of this is shown below. For completeness I have added a constant starting velocity to the motion. That is, rather than starting from rest I start at an initial speed of 1 m/s. Of course, this starting speed just pushes the curve upward by 1 m/s everywhere along the curve.

One more step: From the velocity we should be able to find the distance covered. This is shown in the third graph. In the first second we move at 1 m/s due to our initial velocity; 1 m/s for 1 second gets us to the 1 meter point. We must add to this the distance traveled due to the acceleration. The velocity changes during the first second, but our average velocity (due to the acceleration only - we already have counted our starting velocity) is 1 m/s (We start at 0 m/s and end at 2 m/s. The average is (2 + 0)/2 = 1). The distance covered after 1 second is then 2 meters (1 meter due to our initial velocity and 1 due to the acceleration). Keep doing this and you get the curve shown. A chart of values is shown below.

Distance covered due to a constant acceleration of 2 m/s2 and an initial velocity of 1 m/s.
t (seconds) 1 2 3 4
x (due to v0) 1 2 3 4
x (due to a) 1 4 9 16
x (total) 2 6 12 20

 Galileo found this out through experimentation - rolling balls down an inclined plane. Newton derived the same from basic principles - his laws of motion.

 

Before leaving this matter, there's one important observation to make. If you followed the calculations above you arrived at the results shown, but it took some careful reasoning and an understanding of how things move. Put that aside now and look at a purely geometrical problem related to the curves: Start with the acceleration curve and calculate the area under the curve from time zero until time 1 second. It's easy to see that the area is (height times width) 2 x 1 = 2. After 2 seconds the total area is 2 x 2 = 4. After 3 seconds the area is 6, then 8 then 10. These numbers are exactly the velocity achieved because of the acceleration! of course, when you multiply length times height you must also multiply the units, which are m/second2 times seconds = m/second, the units for velocity. Conclusion: the area under the acceleration curve is a measure of the velocity. As an exercise you should calculate the area under the velocity curve. Guess what - it provides the values for x, the displacement. Geometry solves problems in physics! Finally, if you remember what the slope of a line is, you can see that the slope of the velocity curve is exactly 2, the value of acceleration. The language of physics is mathematics.

Basic Ideas about Motion

We need to be able to describe the motion of things (for obvious reasons). To be coherent in a discussion of motion, we first must agree on a common vocabulary, so I show here a glossary of terms we will use:

Speed:              How fast our object is moving.
Velocity:           How fast (or more formally, the rate at which) our object is moving, and the direction in which it is moving.
    (Already, we have come upon an important distinction - speed can be described by a number, say 20 m/s, but for velocity we require more. We take the velocity of our object to be its speed combined with the direction of its motion. This distinction is generalized into the mathematical ideas of scalars and vectors, which we define next).
Scalars:            A scalar is something whose measure is defined by its magnitude. Examples are speed (how fast), temperature (how hot), length (how long), and many more. (Think of a few).
Vectors:            Vectors require direction as well as magnitude to describe their measure. Examples are velocity (speed, and the direction of the motion; 20 m/s to the northeast), Force (how hard you push, and in what direction - to convince yourself that force is a vector rather than a scalar, try pushing on something without pushing in a particular direction). There are many more. (Again, try to think of some).

Now we can get back to motion:

Acceleration:    The rate at which velocity changes. Now we may be getting into new territory. Not that you haven't heard of acceleration, but have you really thought about it. You drive along at 40 miles per hour. Your speed is constant and you are on a straight section of road , so your direction and speed, and so your velocity does not change. Your acceleration is zero, right? Sure it is. Now you step on the gas, and you accelerate - you go faster. How much faster? If for every second you keep accelerating your velocity increases by, say, 2 miles per hour, your acceleration is 2 miles per hour per second.  Using more consistent (if less familiar) units, lets say you accelerate at a rate of 2 meters per second every second. Your acceleration is 2 m/s per sec, or 2 m/s2. (Recall our discussion of dimensional consistency.)
Position:             Where our object is, relative to some reference frame. The reference frame defines the origin of our coordinate system, from which all measurements are made.

With these definitions we can describe the motion of an object. By an object I mean a simple particle, consisting of a single point in space. It can also be a point representing an object whose size makes its extent negligible compared to the rest of the system under consideration. For example, a baseball is essentially a point compared to a baseball stadium, as is the earth compared to the solar system.

First we establish our coordinate system. We choose a rectilinear, orthogonal system (i.e. a Cartesian system) with origin at (0,0,0). Using this coordinate system we can quantitatively discuss the motion of our object. Motions, of course, can be very complex - so complex in fact, that a head-on approach to analysis can be daunting. The trick that scientists continually play is to break complex things down into simple things, and then work on those. Once the simple components of our complex problem are understood, we join the solutions to finally understand the initial problem. Here's a case in point: The picture shows the motion of an ant as it walks on a table. (A fly buzzing around the room would be the three-dimensional analogy). The motion is complex, but think about it: things move in straight lines or they curve. But any curve, at least over a small enough part if it, can be thought of as part of a circle. Click on the curve to the right and see what I mean. If we address small parts at a time, the complicated path of the ant becomes simple, and we can build up an understandable (i.e. quantitatively describable) picture of its motion.

Once we simplify the motion into straight lines and circles, we further break down the straight line motion into its x-, y- and z-components, and the circular motion into rotation about the center of the circle. We will look at the mechanics of doing this later.

 

 Newton's theory of motion

In the "Principia" (Mathematical Principles of Natural Philosophy), Newton's monumental work published in 1687, he presented the motions of planets and things terrestrial in a form both compact and comprehensive. He generalized the work of Galileo into two laws, added a third law, and explained the law by which the gravitational force between massive objects work. Because of this the workings of the world became thought of as a giant clockwork which, once started, could be predicted from that point forward by a simple application of his laws of motion. Few others in the history of mankind contributed more to its direction. In formulating his laws he first defined some basic quantities:

Mass:    "The quantity of matter is the measure of the same, arising from its density and bulk conjointly."
     In modern terms, the mass of an object is its density times its volume (m =
D V).

Momentum:    "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly."
    In modern terms, the quantity of motion is called momentum, and we write momentum as the product of mass and velocity (p = m v).

Force:    "An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right (we would say straight) line."
    Thus the tendency of objects to maintain their state of motion (inertia) is overcome by the application of a force.

He then postulates his three laws of motion:

The law of inertia (Newton's first law):
    "Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it."

The means by which an object's motion is changed (Newton's second law):
    "The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed."

The law of action and reaction (Newton's third law):
    "To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."

In terms of momentum (the quantity of motion) we can say that a force F applied to an object (of some mass m) over some time interval )t, will change the object's momentum. As an equation, F)t = )(m v). If this change in momentum involves a change in the velocity of an object of fixed mass (the usual case of, for example, someone throwing a baseball), we can divide our equation by )t and have F = m)v/)t.  Since )v/)t, the change of an object's velocity over a time interval, is just that object's acceleration, we have the common expression of Newton's second law, F = m a..

These three laws form the basis for the entire subject we call mechanics today. Except for corrections pointed out by Einstein, they are all we need.

To see how his law of gravitation is formulated, we need to look at what is called "uniform circular motion": Think of a ball going around on the end of a string or, as Newton did, the Moon going around the Earth. If the motion is uniform (i.e. at constant speed as it goes around) it nonetheless must be accelerating because its direction of motion is changing, even if its magnitude is not. It is accelerating without changing speed - how can that be? It must be accelerating at right angles to its direction of motion (think about it)! Conclusion: If an object moves with uniform motion it must accelerate towards the center of its circular orbit. What is the magnitude of this central (centripetal) acceleration? Well, that takes a bit of mathematics (click here if you want to see how it is done), resulting in the expression ac = v2/r. In order for this acceleration to occur (according to Newton's 2nd law), there must be a force. thus, if an object of some mass (m) is to move in a circular path, a center-seeking force of magnitude Fc = m v2/r must be acting on it. This is the famous centripetal force you may have heard of. In the case of the ball and string it is the tension in the string that provides the force; for the Earth and Moon, gravity does the job. Just how gravity provides the necessary force Newton deduced from Kepler's laws combined with his second law, along with the assumption that Nature didn't care if the forces were between Sun and planet (Kepler), ball and string (on Earth), or Earth and Moon (Newton's quest). The result is Newton's universal law of gravitation, written as

.

Here the left side represents the force due to gravity - the mutual attraction of any two objects of masses m and M. The right side states that this force is proportional to the product of the two masses (m and M) and inversely proportional to the square of the distances between them.  With this single equation the motions of the heavenly bodies become known - Plato's problem is solved (without the circles). How accurate is this universal law? Well, we know that on 24 July in the year 3991 there will be a total eclipse of the sun beginning at 22:20 O’clock (Universal Time), and it will last 7 minutes, 18 seconds (we also know just where on Earth one should be to see it). The power of Newton’s laws should be obvious.

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