Topic 8

Physics of Motion

From Aristotle to Newton (2000 years of thought)

Aristotle

Aristotelian physics contained a group of theories which had to accord with observable motion.From astronomy came the view that the Earth is fixed in the Universe and all planets, moons, and stars revolve around it. From material properties came the notion that all matter is made up of four elements: fire, water, earth, and air. Natural motion of all matter on Earth is up or down depending on the reletive quantities of the four elements. All other motion is then unnatural and requires a motive force (e.g., push or pull).

Key questions that were asked by the Greeks and later scientists were:

What happens when an object is dropped?

What happens when one drops two objects of similar shape but different weight?

What happens when an arrow (later cannon) is fired straight up?

What happens when one drops an object on a moving ship?

Aristotle thought that all motion is subject two two factors: motive force (F) and resistance (R). He thought that 'natural' vertical motion was due to a force proportional to weight (F~Weight). Heavy objects had proportionally more earth or water and fell fast. The lightest objects had relatively more air or fire and rose fast. For any motion to occur, F >R (Force is greater than Resistance).

Most scientific experiments or tests up until the 16th Century AD were either based on the natural observation of actions happening around the observer or 'Thought Experiments'. For example, Aristotle argued that is one drops an object in air and then in water it will fall faster in air. This must mean that there is more resistance in water. One could also drop the same object in oil or syrup and find that it falls even more slowly. Thus the oil or syrup must have even more resistance. Then Rair<Rwater<Roil. It is not clear that he ever did this experiment in any systematic way, yet common experience would have anyone agree that the 'though experiment' should be true.

Another thought experiment would be to drop two objects of similar shape but one twice the weight of the other. Aristotelian physics argued that the objects will drop a distance D, with velocities (V1, V2) proportional to weight and Times (T1, T2) inversly proportional to weight. Velocity is defined as distance travelled per unit time (V=D/T).(Velocity is normally written today as V=dx/dt; where x is a measure of position and t is a measure of time. Velocity is then the rate of change of position, x, per unit time.)

Aristotle and other Greeks never record actually doing this experiment. The first actual recorded test of this thought experiment which got the right answer (!) was carried out by Joannes Philoponus (Byzantium, ~600 AD). This result was largely forgotten or ignored and the same experiment and result was carried out by Galileo almost 1000 years later, who got credit for it.

Copernicus

The Copernican universe required the Sun to revolve around the Sun each year and rotate about its axis each day. The Earth had to be in motion! (Today we know that objects on the Earth's surface travel about 100,000 feet/se relate to the Sun due to revolution, and up to 1500 feet/se relative to the Earth's center due to rotation.)

Copernicus knew motion of the Earth had to occur, but he found no easy explanation for why all objects near the Earth's surface travel with the Earth. He assumed that all objects near the Earth have a natural tendency to move with the Earth. This, however, could not explain why the Moon revolves around the Earth (too far away).

Copernicus also argued that it was 'natural' (ala the Aristotelians) that spherical objects like the Earth rotate about an axis. He had no other way to explain it.

Overall, Copernicus had a hard time integrating the notion of Earth movement into his world view even though it was a natural consequence of his astronomical observations.

Copernicus also had difficulty with arguing the Earth was just another planet, while having an Aristotelian view of the Earth being made up of four elements (water ,fire, air, earth) while the other planets are made up of aether.

Galileo

Once Galileo accepted the Copernican model for movement of the planets around the Sun, he also had to reconsider ideas about motion. He wrote a book entitled 'Discources and Demonstrations Concerning Two New Sciences' describing his views on motion while in prison (1632-1642). The book was published in the year of his death (1642).

Galileo started with the simplest motion possible (other than being at rest, ala the Aristotelians). He defined any uniform linear motion as movement along a straight line where the distance (D) is travelled in the time (T). We would say this today as follows:

In words, the velocity is the distance travelled per unit of time; the change of velocity as a function of time is zero.

Both Galileo and Copernicus knew that if one drops a ball while standing on a moving ship, the same thing happens as if one were standing on land. The ball falls straight down. Thus, being in uniform linear motion does not change what happens to a ball being dropped. They also both knew that a person on a ship watching another ship pass by cannot be sure which ship is moving without seeing some other fixed object like the coastline to give a better 'frame of reference'.

The idea of 'frames of reference' is a key nugget of Einstein's theory of relativity, but it was clearly understood 300 years earlier and first clearly formulated by Galileo. From his definition of uniform linear motion and the simple observations noted above, Galileo knew that one could not prove whether the Earth was at rest with all other planets and the Sun revolving around it (Aristotelian view) or whether the Sun was a rest with the planets revolving around it (Cpoernican view). Thus, the argument that 'we see nothing on Earth to lead us to suspect we are in motion' is not sufficient to disprove the Copernican model.

Galileo also knew that some types of motion did not fit the idea of uniform linear motion.Careful observation of a ball being dropped makes it clear that the ball increases velocity (speed) as it falls. Galileo defined this as continuous uniform acceleration. Acceleration (A) is defined as the rate of change of velocity. Mathematically, we now write that as

Galileo knew that there was a pattern to the continuous acceleration of an object dropped. He tried several different mathematical relationships for the incremental velocity (that is, the velocity at any specific time) as a function of time or distance. He considered the following formulas:

In the end, general observation and insight caused him to conclude that the proper relationships were:

In words, the incremental velocity (velocity at any one time) is proportional to the total time elapsed (since dropping the ball); the total distance travelled is proportional to the square of the total time elapsed. He guessed this based on the obsevation that the incremental velocities tended to follow the sequential odd numbers as follows:

 Time 1 second 2 seconds 3 seconds 4 seconds D (incremental) 1 3 5 7 D (total) 1 4 9 16

This says that if one drops a ball, it travels 1 unit of distance in the first second, 3 units of distance in the second second, 5 units in the third second, etc. The total distance travelled after two seconds is four units and the total distance travelled after four seconds is 16 units.

Only after developing a hypothesis (his formulas), did he test the idea rigorously through a series of experiments. The experiments used an inclined plane and brass ball. He systematically recorded the distance travelled by the ball down the inclined plane. He changed the angle of the plane and redid the experiments. In all cases, the formulas closely matched the actual results. (He knew that the experimental results might not always be exactly the same because of measurement errors and resistance (friction) of the ball on the inclined plane.)

Galileo also tried to explain why the path of a cannon or arrow fired horizontally seems to follow the path of a parabola. He suggested that the motion of the ball or arrow in flight is the result of two independent motions, uniform linear motion horizontally and uniform continuous acceleration vertically. Let the horizontal direvtion of motion be along an x-axis and the vertical motion along ay-axis. Then the position at time T should be

for the horizontal axis (c is a constant; this is simply uniform linear motion) and

for the vertical axis (k is another constant; this is simply uniform continuous acceleration). Then

where K is another constant. The last formula (y~x2) is the formula for a parabola. Thus a cannon or arrow fired horizontally is subject to a horizontal linear motion and a vertical continuous acceleration creating a parabolic path.

Galileo tested this idea by experimentally rolling a ball on a modified incline plane called a wedge.In all of this however, Galileo paid less attention to WHY the motions occurred. That is left to Newton.

Interlude

While Galileo was working, Kepler was developing his three laws (see Lecture 5). The third law explicitely describes the relationship between orbital period (T, which is intimately related to velocity) and distance (D) from the Sun for all planets. Moreover, it clearly states that

for all planets. Thus knowing the distance and orbital period for one planet, lets one calculate the orbital period for any other planet if one knows its distance from the Sun (or visa versa). Kepler did not specifically deal with why this should be the case.

Robert Hooke, Edmond Halley, and Christopher Wren realized that Kepler's third law must mean that there is a force (F) acting on all the planets and centered at the Sun such that

to account for it. But, they did not take the idea any further. (In fact, Newton beat them to the reason.)

Newton

Newton was an eminant mathematician as well as general natural philosopher. He devised intergral and differential calculus, mathematics which are necessary to properly deal with the physics of today. Newton developed three laws which further codify the physics of motion started by Galileo and also explain more fully how the Copernican model of planetary motion works.

Newton's first law: All objects remain at rest or uniform linear motion unlessexternal forces act on them. Such a condition(rest or uniform linear motion) is now termed inertia.

Newton's second law: Any change in motion is proportional to the force and takes place in the linear direction along which the force is impressed. (If a body in uniform linear motion along a straight line, a force acting at right angles to the direction of motion will NOT affect the forward motion. This expains the flight of a cannonball or arrow shot horizontally. One force casues horizontal motion, but some other force (see the third law) acting at rightr angles causes the ball or arrow to fall to Earth.)

Newton's third law: The law of universal gravitation. There is a force acting between two bodies with masses m and m' which is proportional to the product of their masses and inversely proportional to the square of the diatance between them. The force acts on a straight line between the two objects.This law can be stated mathematically as

where G is a constant (the universal gravitational constant). This law holds for all bodies in the universe which have mass.

One can apply this law to the orbits of the planets and show that all planets with move about the Sun with elliptical orbits due to the force of gravity of the Sun acting on all of the planets.