User:Unitfreak

http://commons.wikimedia.org/wiki/User:Unitfreak

The following table will be used in my new Bully Row Timestamp project.

Another table for my Bully Row Timestamp project.

Bully Row Time System
Here is a table from my latest project. It doesn't render well in cell phones and needs work.

https://bullyrow.eeyabo.net/index.php/Main_Page

= Work in progress =

The following images and accompanying text were initially designed for the Wikipedia mass article. However, this piece of writing has exceeded the scope of the article for which it was originally intended; therefore, I have decided to continue development offline in my own user page. These images and accompanying text are hereby made available to be placed in any appropriate Wikipedia article at the discretion of any Wikipedia editor. Unitfreak (talk) 01:08, 26 September 2009 (UTC)

Aristotelian gravity


One issue that had puzzled scientists and philosophers was the relationship between an object’s weight and its free fall velocity. The Greek philosopher, Aristotle, had reasoned that every substance and every object has a natural place in the universe, and every object will try to achieve that position. Aristotle had reasoned that there were four terrestrial elements: earth, water, air, and fire (or heat). In Aristotle’s worldview, when a rock is released in a lake it sinks to the bottom because earth belongs at the bottom, beneath water, air, and fire. Rain falls from the sky because water belongs beneath air. Air bubbles rise to the surface of water because air belongs above water. And heat from a fire rises heavenward because heat belongs above earth, water, and air. Aristotle supposed that the heavens were composed of a special "fifth element" called  aether which was unlike the other elements.

According to Aristotle, forces result when an object is taken out of its natural position. When an earthly object (one composed of earth) is lifted, one feels a force of weight from the object trying to return to the earth, and upon being released the object returns to its natural position. In Aristotle’s worldview, objects are not compelled to move by an external force such as gravity, but rather, they move according to their own independent need to be in their natural positions. Aristotle reasoned that a freely moving object would move at a constant speed, and the speed would be proportional to its weight but inversely proportional to the density of the medium through which the object was passing. For example, Aristotle believed that a rock would fall quickly through air but slowly through water, because water is denser than air, and a heavy rock would fall quicker than a light one. Aristotle’s beliefs about heavenly motion were different from his concepts of terrestrial motion. Aristotle believed that terrestrial objects would eventually come to their natural position and then stop moving, whereas heavenly objects, such as stars, planets, and the moon, continue to move forever and never come to rest.

For falling objects: $$ \frac{Distance}{Time} \propto \frac{Weight_{object}}{Density_{medium}}$$

The observational limitations which prevented Aristotle from accurately describing free fall motion were related to those which had prevented Aristotle and other early astronomers from accurately describing the solar system. Early astronomers were limited by their inability to measure large distances. They could accurately measure the periods of time required for a planet to complete a cycle, but they couldn’t measure the distances and paths followed to complete these orbits. Ironically, early scientist and philosophers were similarly restrained. They could accurately measure the distance traveled when an object was dropped a short length, but they couldn’t accurately measure the short time periods involved in free fall motion. So early astronomers had failed to accurately characterize the solar system and early scientist and philosophers had failed to accurately characterize earth's gravity. To overcome this limitation, Aristotle appears to have relied upon experiments with solid objects falling in water. However, water being dense and viscous had complicated the experimental results, thus confusing Aristotle’s comprehension of physical law.

History
Johannes Kepler was the first to give an accurate description of planetary motion, and doing so allowed the standard gravitational parameter of the Sun to be calculated. Kepler determined that the planets follow elliptical orbits under the Sun’s influence, and in 1609, he published three rules known as Kepler's laws of planetary motion. The standard gravitational parameter is an immediate consequence of Kepler's third law of planetary motion. To understand how Kepler arrived at the final form of his third law, it may be helpful to review the history of the underlying values used in this calculation, and the role that the first and second laws played in the development of the third law.

Orbital periods of the planets

 * $$\mu = \frac{4\pi^2 a^3}{\color{Red}T^2} \ $$

The periods of the planets were discovered at a very early stage of human prehistory, and played a significant role in many early religions and cultures. The word planet comes from the Greek verb πλανώμαι planōmai which means to wander around. The planets appear to move through the night sky and are thus distinguished from the stars, which appear to maintain a fixed position with respect to each other. This ability to move freely may have given the planets the appearance of self-determination. Many early cultures either directly worshipped the planets as deities or at least associated them with divinity. The modern English names for the planets were all derived from their names as Roman gods.



As the Earth orbits the Sun, to an observer on Earth the Sun appears to move with respect to the background stars. A sidereal year is the time required for the Earth to complete one orbit around the Sun, or equivalently, the time required for the Sun to appear to complete one orbit and to return to the same relative position with respect to the stars. In the western zodiac, as depicted in the image to the right, the ecliptic is divided into twelve equal zones of celestial longitude. Ancient Europeans used the zodiac to track the Earth’s orbit, and were thus aware of the duration of the sidereal year. Recording the periods of other planets in sidereal years allows a direct comparison to the Earth’s orbital period.

A stone carving of the Chinese zodiac is depicted in the image to the left. Chinese astronomers built this system (know as the Earthly Branches) from observations of the orbit of Jupiter (歳星 Suìxīng, the Year Star), which has an 11.86 yr period. Chinese astronomers divided the celestial circle into 12 sections to follow the orbit of Jupiter, and assigned an animal to each year. These Earthly Branches were cyclically paired with celestial stems, a base ten numeral system, to produce a 60 year Sexagenary cycle, and each year was assigned a Tai Sui deity to be worshipped, or at least respected during that year.



A stone carving of the Aztec calendar is depicted in the image to the right. This astronomical system, used by some early Americans, has surprising similarities with the Asian system. The Asians obtained their 60 year Sexagenary cycle by cyclically pairing the base ten celestial stems with the base twelve Earthly Branches, the least common multiple of 10 and 12 being 60. The Americans obtained a 260 day tonalpohualli (Mayan Tzolkin) cycle by pairing their base twenty numeral system with a base thirteen trecena cycle, the least common multiple of 20 and 13 being 260. The exact origin of the Mayan calendar is uncertain, but some scholars speculate that it may have been derived form the orbit of Venus, which held special significance within Mayan culture.

Early astronomers were successful in accurately measuring the duration of time required for each planet to complete its cycle, but they didn’t have the means to accurately measure the distances and path traveled by each planet during an orbit. Therefore, these early astronomers all failed to accurately describe planetary motion. Many envisioned the planets as following circular paths around the Earth. Although drawn from geographically distinct cultures, the above images all use a similar circular motif to represent the passage of time.

Orbital distances



 * $$\mu = \frac{4\pi^2 {\color{Red}a^3}}{T^2} \ $$

Kepler’s first attempt to describe planetary orbits was similar to the Wu Xing philosophy employed previously by the Chinese. The Chinese had reasoned that there are five directions on a compass, five stages in a process (each with an associated element), and five planets visible to the naked eye. The Chinese therefore associated each planet with a particular direction and element. In a similar fashion, Greek philosophers approximately two thousands years prior to Kepler had proven that there were exactly five Platonic solids. The Greek philosophers had reasoned that each solid must be associated with a specific element, and Kepler further reasoned that perhaps the distances between the planetary orbits could be determined by placing Platonic solids inside of concentric spheres. This theory was somewhat successful, but didn’t agree with available astronomical data to the level of precision that Kepler desired.

After abandoning his Platonic solids model, Kepler began working with an array of traditional astronomical methods. In 1600 AD, Kepler sought employment with Tycho Brahe and consequently gained access to astronomical data of a higher precision than any previously available. Using Brahe’s precise observations of the planet Mars, Kepler proved that the traditional astronomical methods were inaccurate in their predictions, and he spent the next five years developing his own method for characterizing planetary motion.

Orbital paths
The factor of four pi squared in kepler’s third law of planetary motion is a consequence of the orbital paths that the planets follow as defined in Kepler’s first and second laws of planetary motion.


 * $$\mu = \frac{{\color{Red}4\pi^2} a^3}{T^2} \ $$

Kepler's first law

 * The orbit of every planet is an ellipse with the Sun at one of the two foci.



In Kepler’s planetary model, he described planetary orbits as following elliptical paths with the sun at a focal point of the ellipse. The image to the right illustrates one method for characterizing ellipses. Four significant points, A, B, C, and F are identified on the illustration. Point C is located at the center of the ellipse. The line running from point C to point A is called a semi-major axis of the ellipse (meaning that it is half of the longest diameter). The line running from point C to point B is called a semi-minor axis (meaning that it is half of the shortest diameter). With a proper choice of coordinate system, an ellipse can be described as the set of all points (x,y) of the Cartesian plane that satisfy the implicit equation:


 * $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

where a and b are respectively the length of the semi-major and semi-minor axes.

Point F in the illustration is a focal point of the ellipse. A point is a focal point if and only if it is located on a semi-major axis and its distance from point B is equal to the length of the semi-major axis. The angular eccentricity of an ellipse is the angle between the line running from point B to point C, and the line running from point B to point F. Given that points B, C, and F form a right triangle with hypotenuse of length a, it is an immediate trigonometric consequence that the length of the semi-minor axis is equal to the length of the semi-major axis multiplied by the cosine of the angular eccentricity. Hence, the above implicit equation can be rewritten in terms of angular eccentricity as follows:


 * $$\frac{x^2}{a^2}+\frac{y^2}{a^2 \cos^2 \alpha}=1$$.

This equation can be expressed parametrically as the path of a point (x(E),y(E)), where
 * $$x(E)=a\,\cos E$$
 * $$y(E)=a\,\sin E \cos \alpha$$

The parameter E in this representation is called the eccentric anomaly.

Kepler's second law

 * A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.



Kepler lived in an era when there was no clear distinction between astronomy and astrology, and when these fields of study, together with geometry, were viewed as intrinsically divine. Kepler was motivated by religious convictions and incorporated religious arguments and reasoning in his work. Kepler reasoned that the sun was representative of the monotheistic God of Christianity, that the sun sat in the center of the solar system and controlled the motions of all other objects in the solar system. Kepler further reasoned that since the sun was the source of motion, then an object’s motion should be inversely proportional to its distance from the sun. In other words, the closer an object gets to the sun the faster it moves. Kepler later refined this to state that an orbit sweeps out equal areas in equal times.

Kepler’s second law was especially difficult mathematically, since no one had previously developed equations to describe the area swept out by a line joining an ellipse to one of its focal points, and Kepler lived prior to the invention of calculus, so he would have to solve this problem using geometry. The image to the left illustrates one possible solution to Kepler’s problem. In this solution, the sun is located at focal point F while a planet orbits from point A to point P along an elliptical path. The total area swept out by the planet can be obtained as the sum of two distinct areas. The points P, C, and F define a triangle (marked in green on the illustration). The points P, C, and A, together with the elliptical path, define an elliptical arc (marked in brown on the illustration).



The image to the right illustrates a second solution to Kepler’s problem. In this solution, the sun is located at focal point F while a planet orbits from point A to point P along an elliptical path. The difference between this solution and the previous solution is that in this solution the focal point F is located between the center point C and point A, whereas in the previous solution F was located opposite A. In this solution, the area swept out by the planet (marked in yellow on the illustration) is obtained by taking the difference of two distinct areas. The points P, C, and F define a triangle (marked in green on the illustration). The points P, C, and A, together with the elliptical path, define an elliptical arc.

The area of a triangle is always equal to half of its base multiplied by its height.
 * $$A_{triangle}=\frac{1}{2}\,base \times height$$

The two green triangles, one in the illustration above and the other in the illustration to the right, are dissimilar. However, these triangles have the same base lengths and the same heights, so the magnitudes of their areas are identical. The base of each green triangle is the length from the center point C to the focal point F, which for an ellipse is equal to the length of the semi-major axis multiplied by the sine of the angular eccentricity. The height of each green triangle is equal to y(E) as given in the above parametric equations:
 * $$base = a\,\sin \alpha$$
 * $$height = a\,\sin E\,\cos\alpha$$

Hence, the area of each green triangle is:
 * $$A_{triangle}=\frac{1}{2}\,a^2\,\sin E\,\cos\alpha\,\sin\alpha $$

The area swept out by an elliptical arc is equal to half of the angle multiplied by the lengths of the semi-minor and semi-major axes:
 * $$A_{arc}=\frac{\theta}{2}ab $$

In terms of angular eccentricity:
 * $$ \theta = E $$
 * $$ b = a \,\cos\alpha$$

The area of the arc thus becomes:
 * $$A_{arc}=\frac{1}{2}\,a^2 E\,\cos\alpha$$

Two solutions to kepler’s equation can now be written out, one by adding the area of the green triangle to the area of the elliptical arc, and a second by subtracting the area of the green triangle from the area of the elliptical arc:
 * Solution 1: $$\frac{1}{2}\,a^2 E\,\cos\alpha + \frac{1}{2}\,a^2\,\sin E\,\cos\alpha\,\sin\alpha $$
 * Solution 2: $$\frac{1}{2}\,a^2 E\,\cos\alpha - \frac{1}{2}\,a^2\,\sin E\,\cos\alpha\,\sin\alpha $$

A quick review of trigonometric identities reveals that these two solutions are in fact the same solution (one representing positive angular eccentricities and the other representing negative angular eccentricities). These solutions to Kepler’s equation give the area swept out by a planet as it orbits around a central mass. Kepler’s second law states that the elapsed time will be proportional to this area, hence:
 * $$t(E)\propto \frac{1}{2}\,a^2 E\,\cos\alpha - \frac{1}{2}\,a^2\,\sin E\,\cos\alpha\,\sin\alpha = \frac{1}{2}\,a^2\,\cos\alpha \left( E - \sin E\,\sin\alpha \right)$$

To get an equality it is necessary to divide both sides by some exact value, we will use T, the duration of one full orbit:


 * $$\frac{t(E)}{T} = \frac{t(E)}{t({\color{Red} 2 \pi})} = \frac{\frac{1}{2}\,a^2\,\cos\alpha \left( E - \sin E\,\sin\alpha \right)}{\frac{1}{2}\,a^2\,\cos\alpha \left({\color{Red} 2 \pi} - \sin {\color{Red} 2 \pi}\,\sin\alpha \right)} = \frac{E - \sin E\,\sin\alpha}{{\color{Red} 2\pi} - \sin {\color{Red}2\pi}\,\sin\alpha} = \frac{E - \sin E\,\sin\alpha}{\color{Red} 2\pi}$$

Multiplying both sides by the duration of one full orbit, we obtain:
 * $$t(E) = \frac{T}{\color{Red} 2\pi} \left(E - \sin E\,\sin\alpha\right)$$

From Kepler’s first and second laws, Kepler now had a complete set of parametric equations for describing orbital paths:
 * $$x(E)=a\,\cos E$$
 * $$y(E)=a\,\sin E \cos \alpha$$
 * $$t(E)=\frac{T}{\color{Red}2\pi} \left(E - \sin E\,\sin\alpha\right)$$

The only remaining task was to determine the relationship between the semi-major axis, a, and the orbital period T.

Kepler's third law

 * The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.


 * $$ \left( \frac{T}{\color{red} 2\pi} \right)^2 \propto {a}^3 \ $$

Or equivalently, the ratio of these values is constant for all planets in the solar system, a value known as the standard gravitational parameter.


 * $$\mu = \frac{{\color{red}4\pi^2} a^3}{T^2} \ $$

constants table
My original table was moved to the conventional electrical units article and a significantly altered version now resides in the natural units article. I really like this table, so I will make a copy here as backup in case someone deletes the other.

Relativistic Mass
The energy and momentum of an object with invariant mass m (also called rest mass in the case of a single particle), moving with velocity v with respect to a given frame of reference, are given by
 * $$\begin{array}{r l}

E         &= \gamma m c^2 \\ \mathbf{p} &= \gamma m \mathbf{v} \end{array}$$ respectively, where γ (the Lorentz factor) is given by
 * $$\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}.$$

Quantum Mass
The Compton wavelength, λ, of a particle is given by
 * $$ \lambda = \frac{h}{m c} \ $$

where h is the Planck constant, m is the particle's rest mass, and c is the speed of light. The significance of this formula is shown in the derivation of the Compton shift formula.

The de Broglie equations relate the wavelength $$~\lambda~$$ and frequency $$~f~$$ to the momentum $$~p~$$ and energy $$~E~$$, respectively, as


 * $$\lambda = \frac{h}{p}$$ and $$f = \frac{E}{h}$$

where $$~h~$$ is Planck's constant.

Quantum Variables