E.1 Models of the universeEdit
Aristotle's model of the universe was that the universe was finite, spherical and above it was heaven or whatever. Everything was divided into 4 elements: earth, water, air and fire (the fire being the sun). Everything below the orbit of the moon was the earth, everything above the ether. The ether moved independently of the earth, which is a nice way to explain everything without thinking about it. Aristotle believed everything moved in circular paths, and that the earth was at the center of the universe.
Ptolomy's model was that 'heavenly bodies', such as the sun, moon planets and stars, orbited around the earth in circles, but there were little circles within these, to account for the fact that they didn't appear to match circles (called epicycles). There were eight levels in his model, with the outermost being like a fixed roof of stars.
Aristachis'/Copernicus' model had the earth orbiting the sun, along with the other planets, but the moon orbiting the earth. The stars were still like a fixed roof, out beyond the planets. Everything orbited in circles.
Objections to Copernicus' model were based primarily on religious issues, and enforced due to the close ties between the church and government. Gallileo defended the model, with observations of moons orbiting Jupiter, however the church eventually forced him to publicly withdraw his support for the model.
Tycho Brahe's model had the planets orbiting the sun, but the sun orbiting the earth, and was intended to be like a compromise between the church and Copernicus' model.
Kepler's model was an extension of Copernicus', but had simple mathematical laws to describe the motions of the planets.
- The planets orbited on elliptical paths, with the sun at the focus.
- The planets orbited a segment of equal area in equal amounts of time, which meant when they were closer to the sun, they moved faster.
- T2 is proportional to R3. (T is period, R is radius). This means T12/T22 = R13/R23.
Newton's law of universal gravitation gave a mathematical, and theoretical basis for Kepler's model.
F = Gm1m2/r2, and F = mv2/r and since v=rw, F = mrw2. Equating the two, we get the following.
Gm1m2/r2 = mrw2, and w = 2π / T, so Gm1m2/r3 = m2π / T2, so T2 is proportional to R3.
Gravity provides the centripetal force necessary for Kepler's third law.
Simple models are considered as better than complex ones. Therefore the Copernican model was good, because it was simple, and allowed predictability (although it wasn't really all that accurate).
One other thing, since it was in the spec paper. Retrograde motion was the name for the observation that the planets did not move in predictable paths in relation to the stars, and even reversed directions, due to the relative motion of the earth.
E.2 Mechanical determinismEdit
Aristotle : To move an object, a force was required. To keep an object moving also required a force. Thus, the natural state of objects is at rest (there was no concept of friction).
Galileo examined the motion of objects on ramps. If an object rolled down one, then up another, then it always ended at the original height, irrespective of the slopes (because friction was minimal). This lead to the conclusion that if an object rolled down a slope and along a flat surface, it should go on to infinity, as it would never reach it's original height. This was also important for the fact that he was using experiment, rather than theoretical reasoning based on untestable assumptions.
Galileo also applied mathematics to accelerating objects, based on the distance covered in constant time intervals, and proved that falling objects have constant acceleration (and that acceleration under gravity was independent of mass).
At the time of Newton's birth, the heavens were considered separate from the earth, obeying different, unrelated laws. There was assumed to be no connection between the laws governing planes and objects on earth, and 'up there' no force was required for continual circular motion (clearly the planets kept circling with no force, as no one knew about gravity).
(This is just getting silly)
Newton's Pricipia Mathematica Philosopha Naturalis was written in Latin, very formal, with geometric derivations of laws, and corollaries. Contained definitions used in his laws, which connected and quantified the motions of all objects in the universe.
Newton defined the following:
Mass : Quantity of matter
Momentum : Quantity of motion
Impressed force : The force that induced a change in the quantity of motion (momentum).
Newton's laws were mainly formulated in terms of changes in momentum.
Newton's arguments for universal gravitation were based on the idea that planets were pulled along, and around their orbits by a centripetal force (this idea was partly taken from the work of Robert Hooke). Newton showed that an inverse square force could provide the force necessary for such motion, and coined the term centripetal. A central force is necessary for a object to move in a circle because a tangential force would just make the plant fly off in a straight line, rather than orbiting in a circle (or actually, an ellipse).
Under the law of universal gravitation, objects on the surface of the earth accelerate towards the center, as does the moon, though with the moon, the acceleration changes its direction rather than its speed though both accelerate towards the centre of the earth.
Since F = moa and F = Gmemo/r2, we can derive a = Gme/r2, thus by finding r and a for an object on the surface of earth, and r for the moon, a for the moon can be found.
Galileo's work was confined to describing the motion of objects near the earth's surface, and specifically that objects didn't require a continual force to keep moving (in a perfect, frictionless system). Newton developed a more general, and more mathematical basis for this idea.
The difference between having a law describing something and explaining the source of something: Newton could explain the behaviour of and predict gravity, but he could not explain its source (and we still can't really).
Newtonian mechanics implied (and Newton believed) that by knowing the starting position of every object in the universe, future states could be extrapolated. Newtonian mechanics is a deterministic, mechanical view of the universe.
Maths is important in physics in that it provides a modeling system which can describe natural phenomena, and make predictions form it.
E.3 The energy conceptEdit
Phlogistin/Caloric : This was a theory of heat from the mid 1800s. Phlogistin was a substance contained within things, which, when burnt, was removed as heat, thus turning them into ashes. Caloric was the original name for heat, which was thought to be like a fluid, flowing from hot objects to cold ones. This theory could not, however, explain the production of heat through friction.
The Newcombe engine was the first steam powered engine, and although being hopelessly inefficient, it effectively sparked the industrial age. Water was heated, turned into steam, which increased the pressure in the pipes above. This pressure was then used to push a piston down. On reaching the end, some steam was allowed to escape, and the piston was pushed back up by other pistons connected to the crank shaft. The steam which had escaped was run through and back to the water tank to be reused. The piston made use of several valves, which regulated the steam flow. Steam was only allowed to enter as the piston was moving down, and only allowed to leave as it was moving up.
The most major of Watt's modifications to this system was the use of a separate condenser. After escaping from the piston, the steam was kept under pressure and carried to a separate condenser. This meant that the boiler and piston could both be kept at a high pressure and similar temperature, and reduced the energy loss due to the latent heat of vaporisation. In addition to this, he developed systems to convert the back and forth motion of the pistons into rotary motion, through a system of gears. Another major development was the ability to regulate steam production, allowing the speed of the system to be regulated without reducing pressure (i.e. leaking steam).
Sadi Carnot developed a theoretical model of the heat exchange occurring to explain how heat moved through the system. He also developed the theoretical Carnot engine, which was 100% efficient, and so could be used to compare the efficiency of real engines.
Julius Mayer was the first to propose the idea that heat was actually a form of energy, which opposed the caloric theory. This idea was developed by Rumford and Joule, who worked to find the mechanical equivalent of heat.
Joule eventually produced a successful experiment which used a falling mass to turn paddles and produced a measurable change in temperature. The change in energy of the falling mass could be found, and this energy was in part being given to the water by the turning paddles. Based on this, Joule unified the two forms of energy, and is remembered in the unit of energy, joules.
Energy is an abstract concept, a way of viewing the system rather than an observable property which can be seen and analyzed, like mass or velocity. This, however, is useful in that is works at a more fundamental level and allows different forms of energy to be unified.
E.4 The quantum conceptEdit
The wave theory fails to explain the photoelectric effect, because it is a continuous form of energy, and the photoelectric effect only makes sense if the energy from light is in quanta, and the amount of energy is determined by the frequency of the light.
Classical physics said that the energy in light was continuous, but this did not fit experimental observations. Plank attempted to modify the classical model by allowing for continuous incoming energy, but suggested that the output energy, the photo electrons, could only take certain amounts of energy. Einstein, however, created a completely new model, where both in incoming light radiation and the emission of photoelectrons were quantised (i.e., could only take certain values).
The de Broglie hypothesis was an extension of Einstein's linking of waves and quantization, by the suggestion that all particles have wave properties by the equation λ = h/p, or wavelength = plank's constant/momentum (thus it also equals h/mv).
de Broglie's ideas were eventually experimentally proved in relation to the diffraction of electrons, being particles with rest mass (they were diffracted through crystals with microscopic lattice spacing, and produced a diffraction pattern consistent with the above wavelength).
Schrodinger's wave theory analyzed the wave/particle duality, and attempted to combine it into one coherent model. One of the functions of this theory was to produce probability regions (in space-time) around atoms for where electrons would be (probably). This could then be extended to explain the diffraction pattern formed even by single electrons passing through a slit.
Schrodinger's wave theory was fundamentally an abstract model, which describes occurrences in a way that has no obvious link to the real world, and yet can effectively describe it. This de-coupling allows for modelling of events in such a way as to simply predict outcomes, rather than to represent the real world. Thus, it is easier to model the real world through abstract representations.