This is a posting about fundamental scientific literacy. Just about everything here has been seen in other places on this blog system, but I would like to have all of the facts in one place that anyone would need to know to be scientifically literate in today's world. Learning everything here will give you a solid background in science and able to discuss such issues. It would be ideal to read this posting along with "Scientific Perspectives And Facts" on the physics and astronomy blog, www.markmeekphysics.blogspot.com and the review of everyday technology on this blog, "The Way Things Work". Learn all three of these postings and you will really know a lot about the world and universe in which you live. This posting is basic scientific information and I do not get into any of my theories here.
This is not just a jumble of facts. I have been carefully selecting what I consider as the essentials of scientific literacy. I have tried to write it in such a way that not only provides knowledge, but also conveys a scientific and logical way of thinking. The posting is not intended as a reference, but as core material about the physical sciences which would be good for everyone to thoroughly learn. I hope for this posting to be one place where readers to come to get the essential knowledge which would benefit everyone, even if their work or course of study does not involve science. Across the world, education in science and mathematics is considered as crucial.
1) Tides are caused not by gravity, but by a difference in gravity. When the moon is overhead, the oceans are deep enough that there is a significant difference in the pull of the moon's gravity between the surface of the ocean and the bottom of the ocean. The surface of the ocean is closer to the moon so that it undergoes a stronger gravitational pull. The result is the tidal bulge, which means a high and a low tide every day as the earth rotates. The sun also exerts a tidal pull on the earth's oceans but even though the sun's gravity is far greater than that of the moon, it is 400 times as distant so there is less proportional difference between it's gravitational pull on the surface of the ocean, in comparison with the bottom, with the result that the sun's tidal influence on the earth's oceans is only about 40% that of the moon. As you might expect, the tides are strongest when the earth, moon and sun are in a line, at new moon or full moon. The tides are weakest when we see a half moon because then the sun and moon are then perpendicular relative to the earth, and their gravitational pulls are not synchronized. This is not all that there is to tides, things like the shape of the ocean basin also has an influence. Coastal land forms also play tricks with the tides, places known for tidal special effects include eastern Canada's Bay of Fundy and England's Isle of Wight. The center of our galaxy is a powerful center of gravity, but it's distance is so great that the proportion difference in gravity that it creates on earth is essentially zero, and it results in no tides.
2) There are lunar eclipses and solar eclipses. A solar eclipse is when the moon comes between the earth and the sun, and casts it's shadow on the earth. Since the angular diameters of the sun and the moon in the sky are just about exactly equal, a total solar eclipse is seen over only a limited area on earth. There is a cone-shaped zone of total eclipse extending backward from the moon, known as the umbra, and a larger outer zone of partial eclipse known as the penumbra. A lunar eclipse is when the earth is between the sun and the moon so that it casts it's shadow on the moon. I have seen many lunar eclipses, but never a solar eclipse. The moon goes around the earth every 29 days and the reason that there is not both a lunar and a solar eclipse on every lunar orbit is that the plane of the earth's orbit around the sun and the plane of the moon's orbit around the earth are not exactly the same, there is a difference of about 5 degrees between the two planes.
3) Orbits, such as that of the moon around the earth or that of the earth around the sun, are not exactly circular. They tend to form an ellipse, with the central body at one of the two foci of the ellipse. An ellipse is a kind of flattened circle, with two foci instead of one. The point at which the orbiting body is closest to the central body is called perigee and the point when it is furthest away is apogee. The orbiting body moves faster around the central body when it is closer to it so that a line between the two bodies moves over equal areas of space in equal periods of time. The energy in an orbit is proportional to the space enclosed within it, this means that a higher orbit has a higher orbital energy. But it is also true that an object in a lower orbit must orbit faster. What happens is that, if an object is given three times the orbital energy of another object, it will orbit at nine times the distance from the planet, but will move at only one-third the speed.
4) Ballistic flight means that of momentum alone, and no longer driven by any kind of continuous propulsion. A planet like the earth has both an orbital velocity and an escape velocity. If we launched a projectile into the sky at a velocity of 8 km per second (about 5 miles per second), it would go into orbit around the earth with no need of any further rocket propulsion. If we could launch the projectile at 11.25 km per second (about 7 miles per second), it would escape earth's gravity altogether and continue into space. This is a better expression of a planet's gravity than the surface weight of an object because that varies with the density of the planet, while the orbital and escape velocities are functions of the planet's overall gravity.
5) There are plenty of planets in orbit around distant stars. We cannot possibly observe them directly, not even with the most powerful telescopes. But there are indirect ways to detect such exoplanets, as they are referred to. When a planet orbits a star, the two are actually in a mutual orbit, but it appears that the planet is moving around the star simply because the star is so much more massive. However, careful measurement of the star can reveal a "wobble" in it's motion caused by the gravitational influence of the unseen planet. If the light from distant stars is received by charge-coupled devices (CCDs), the light of the star, which would usually drown out the reflected light from faint planets, can be cancelled out electronically so that the images of the planets may be visible. The sun is a single star but most stars appear to be part of star systems, with two or three stars in a mutual orbit. If this were the case with our solar system, it would make the earth's climate far more extreme and complicated and may well make our planet unlivable altogether. Rock, which is a compound of silicon and oxygen, along with iron, are very common in our solar system, but that may not be the case elsewhere.
6) We can tell a lot about the earth from how different it is from the moon. There is no atmosphere on the moon, which means that there is no sound. No atmosphere means that heat cannot move by convection from one side of the moon to the other, which means that day on the moon is extremely hot while night is extremely cold. With no atmosphere to disperse the light of the sun, the sun and stars can be seen at the same time. With no water or atmosphere, there is no erosion on the moon. The only thing that changes is from the occasional impact of meteorites. It is quite possible that the footprints of the astronauts who landed on the moon may still be visible hundreds of thousands of years from now. The moon does not have water or an atmosphere simply because it's gravity is not strong enough to retain it. You would weight only 1/6 on the moon what you weigh on earth.
7) The moon orbits the earth in the same direction in which the earth rotates, which is eastward. The moon orbits the earth every 29 days. The 24 hours in a day, divided by 29, gives us 50 minutes. This is why the moon rises 50 minutes later each day. The moon appears to us to go through phases. Actually the sun is always shining on half of the moon at any given time. If the moon is exactly opposite the earth from the sun, we see a full moon. If the moon is exactly between the earth and the sun, we cannot see the moon and refer to this as the new moon. The orbit of the moon does not coordinate with the orbit of the earth around the sun. The months were originally based on cycles of the moon, but that is only 29 days and months, except February, have 30 or 31 days. Many festivals, such as Ramadan, are based on cycles of the moon and so fall at a different time every year. The orbit of the earth around the sun does not divide evenly by the period of the rotation of the earth, a year is actually 365.25 days. This is why an extra day is added every four years as February 29.
8) There are fourteen possible calendars linking days of the week to days of the month. There are seven days in the week so that means seven possible calendars for leap years, and seven for non-leap years. Leap years retreat two days of the week every four years, if a day of the year fell on a Wednesday in 2012, it will fall on a Monday in 2016. Non-leap years advance one day of the week every year, but two days on leap years, so that the calendar repeats every six years for non-leap years as long as there was only one leap year in between. It is one day of the week later if there were two leap years in between during the six years.
9) Basic celestial navigation revolves around the fact that the two stars of the bowl of the Big Dipper, on the opposite side from the handle, point toward the North Star which is essentially directly above the north pole. The North Star is several times the angular distance between the two bowl stars away. The actual name of the North Star is Polaris, it is not the brightest star in the sky but is bright enough to use as a convenient reference point. Polaris is actually a very bright star which is a long distance away, about 600 light years. When you are facing north, south is at your back, east is to your right, and west is to your left. Telling direction by the stars is more accurate than using a compass since a magnetic compass points to the magnetic, rather than the geographic, north or south pole. The magnetic poles are fairly close to the geographic poles, but are not actually the same thing. Of course, the Big Dipper and North Star configuration is of no use in the southern hemisphere, but there is a corresponding constellation there. Once you know the compass directions, if you do not know what time it is at night but can see the moon you can infer the approximate time from there. The lit portion of the moon that we can see, from new moon through the phases to full moon, is actually equivalent to the angle between the moon and sun relative to earth. (I proposed a trigonometric function based on this at 180 degrees in "New Trigonometric Functions", on the Progress Blog). Consider the lit portion of the moon as an arrow pointing toward the unseen sun at night. The sun rises in the east and sets in the west. If you see a crescent moon in the western sky, you know that it is evening because the sun has only recently set. If you see a crescent moon in the east, you know that sunrise is not far away. But if you see a full moon in the east, you know that the sun has only recently set, and a full moon in the west means that the sun is soon to rise.
10) Venus also goes through phases, although they require a telescope to see. Venus is closer to the sun than the earth and so it's distance from us varies greatly from when the two planets are opposite in their orbits to when Venus is on the far side of the sun from earth. But it's brightness does not vary in proportion to it's distance from us. That is because, just as with the moon, Venus is going through phases relative to earth. When Venus is closest to us, it's angular diameter is far greater then when it is on the opposite side of the sun. But this is when we would only see a crescent Venus, similar to a crescent moon, instead of a "full" Venus (although much more distant) when Venus is on the opposite side of the sun.
11) The same side of the moon always faces earth. We cannot see the far side of the moon from earth. It is sometimes called "the dark side of the moon", but this is not correct. The sun shines just as much on the far side of the moon as it does on the close side. The far side of the moon lacks the "seas", the dark areas that we can see on the moon. They are not really seas of water, but dark areas of solidified lava. But the far side is very heavily cratered. This makes perfect sense if we consider that the tidal forces of earth's gravity drew out lava during the early days when the moon was volcanic, in the same way that the moon's gravity creates tides in the earth's oceans. The far side is so heavily cratered because meteors from out in space are more likely to hit that side because it faces outward.
12) The most important scientific fact in the universe is that there are two fundamental electric charges, opposite charges attract and like charges repel. All of the universe is based on these simple rules. There are four basic forces which operate the physical universe. The strong nuclear force binds the positively-charges protons in the nucleus together, against the like-charge repulsion which would otherwise thrust them apart. This is what makes atoms possible. The so-called weak nuclear force acts in the opposite way, by breaking large nuclei apart in the process known as radioactivity. The electromagnetic force is just the results of the rules of the fundamental electric charges. Gravity is the fourth force. It is very weak in comparison with the others, but yet gravity dominates the universe on a large scale. Two large oil tankers or cruise ships docked side-by-side would have only about one pound (0.45 kg) of gravitational force between them. What gravity actually is depends on who you ask. My theory of gravity is that the attractive force between opposite charges is very slightly stronger than the repulsive force between like charges, resulting in a net overall attraction between matter which is very weak in comparison to the other three forces but which dominates the universe on a large scale.
13) A star is born when a vast amount of matter, mostly gas and dust, comes together in space by gravity. The star begins operation when the tremendous gravity at the center is strong enough to overcome electron repulsion between atoms and crunch small atoms together into heavier ones. We know that the sun is a so-called second generation star because it already contains heavy elements. The heavier elements are cooked up in the centers of stars by the tremendous heat and pressure crunching lighter atoms together into heavier atoms. This releases surplus binding energy, which radiates from the star as heat and light. When the universe first formed, there was only the lightest element, hydrogen, some helium and traces of other of the lightest of elements. The reason that we now have heavier matter is that the light atoms have been being crunched together in stars into heavier atoms. Some stars ultimately explode as a supernova and scattered their component matter across space. Gravity then pulls most of the matter back together into a second generation star, and maybe some planets. It requires heavy matter to form planets. This is how our solar system, including the sun, formed. There was a star which exploded, scattering it's matter across space, until much of it was brought back together by gravity to form the solar system. When we use energy that does not come directly or indirectly from the sun, such as tidal or nuclear energy, we are actually using energy from that supernova. Even with all of the crunching of lighter elements into heavier ones that has been going on, the vast majority of the atoms in the universe are still believed to be hydrogen.
14) The sun is a relatively average star. It is a little bit larger than the average star. My understanding is that on a scale of stars from 1 to 10, the sun is about a 6. Some of the stars that you can see in the sky are hundreds of times larger than the sun. The brightest star in the sky, although not as bright as the planets Venus and Jupiter, is Sirius, in the northern hemisphere winter sky. Sirius is about twice as large as the sun, but is relatively close by, about eight light-years. If the giant red star Antares, in the northern summer constellation Scorpio, was put in the place of the sun, you would not be reading this because the earth would be well within the star. Usually by random chance, the stars appear to form patterns called constellations, which ancient people gave names to. The stars in a constellation may actually be nowhere near each other, but just happen to be in the same line of sight from earth. Just as a faint star may actually be very bright, but be a great distance away or an exceptionally bright star may just happen to be close to us.
15) Stars, such as our sun, which are the product of more than one generation of a star forming, cooking up heavier elements by the fusion of light elements together, and then scattering it's component matter across space when it explodes in a supernova until much of the material is drawn back together by gravity to form another star which already contains a significant portion of heavy elements due to the previous star, are known as Population 1 stars. But there are still some older stars, from much further back in time, which contain little in the way of heavier elements, these are considered as a glimpse back to the earlier universe and are known as Population 2 stars. Obviously, planets can only form with Population 1 stars because they require debris of heavy elements floating in space to form.
16) The distance that light travels in one year is used to measure distances in the universe and is known as a light-year. It is equivalent to nearly six trillion miles or about 9.5 trillion km. Our galaxy is spiral in form and is about 100,000 light years in diameter. The earth and the solar system are nowhere near the center, but are about 30,000 light years out in one of the spiral arms. When you look at the dense band of stars across the sky, known as the Milky Way, you are looking at the plane of the galaxy. The earth faces a different direction in space during each of the four seasons as it revolves around the sun. The center of the galaxy is toward the southern stars of the northern hemisphere summer. The stars of the northern hemisphere winter are looking outward toward the spiral arms of the galaxy. The reason that much of modern astronomy revolves around the nation of Chile is that the vast majority of the earth's people live in the northern hemisphere, and this has resulted in the northern stars being much better studied than the southern stars.
17) The planets in our solar system generally increase in size as we go outward from the sun. This is due to the sun's gravity and also the sun's heat. Much of the debris blasted outward by the supernova of the sun's predecessor star fell back together to form the sun and planets. The entire mass of debris began a mutual orbit from all directions until, due to collisions and mutual gravitational attraction, one geometric plane of orbit predominated. This debris in orbit around the newly-forming sun coalesced at periodic intervals to form the planets. Close to the sun, the new planets had more competition from the gravity of the sun for debris that was still floating in space. The heat of the sun prevented the massive oceans of liquid methane and ammonia from forming around the planets closest to it, but not the larger planets further out, Jupiter, Saturn, Uranus and, Neptune. Outside of this we find the Kuiper Belt, of many rocky and icy objects in orbit around the distant sun, and beyond that the Oort Cloud of comets.
18) According to conventional physics, the reason that large bodies, such as stars and planets, tend to be spherical in shape is that a sphere is the most compact three-dimensional form in that it has the lowest surface area per volume, and thus the lowest state of potential energy of position and the universe tends to seek the lowest energy state.
19) As a planet, such as the earth, rotates, the spin produces an outward force which counteracts gravity and is known as centrifugal force. The effect of this is known as the Coriolis Force, and Wikipedia gives it as 1/289 that of gravity on earth. On the large planets of the outer solar system, which rotate very rapidly, this outward force has a much greater effect than it does on earth. The lateral bands of different shades and colors that we can see on Jupiter, Saturn, Uranus and, Neptune are the result of the Coriolis Force affecting the formation of clouds as those planets spin. The reason for the banding is that the spin, and thus the Coriolis Force, is the greatest in the equatorial regions of those planets because the spin of rotation is the fastest there. As we move away from the equator, the spin is less and this affects the formation of clouds resulting in the lateral banding seen from earth.
20) The asteroids, mostly between Mars and Jupiter and Saturn rings give us a glimpse into the early solar system. The asteroids would have coalesced by gravity into a planet, if they were not prevented from doing so by the powerful gravity of Jupiter. A vast cloud of debris from a supernova of a star coalesced by gravity into the sun and planets. We know that the sun is at least a "second-generation" star because it already contains heavy elements which it could not yet have synthesized. The pieces of debris went into a mutual orbit until one geometric plane predominated by collisions and gravitational attractions among the pieces. This is the plane in which the planets and asteroids orbit the sun today. Most of the other debris has coalesced into planets by gravity, but the asteroids remain as they are due to the gravity of Jupiter. Saturn's ring system, composed of debris and pieces of ice in orbit around the planet, were similarly prevented by Saturn's gravity from coalescing into one or more moons. But the rings are remarkable for how they have aligned in one orbital plane, which cannot even be seen from earth when the rings are aligned edge-on to us.
21) All energy in the universe ultimately comes from the Big Bang, which began the universe. Remember that energy can never be created or destroyed, but only changed in form. The energy which crunches smaller atoms together into larger ones in the centers of stars comes from the force of the matter falling together by gravity to create the star, which was thrown out across space by the Big Bang in the first place. The sun and planets of our solar system are composed of material that was once part of a star that exploded. The energy that we use today which is not directly from the sun, such as nuclear and tidal energy, originates from the energy within this former star. Nuclear energy of fission, and also of radioactive decay, is from the energy that went into crunching smaller atoms together within that star. Tidal energy from the earth's oceans comes from the kinetic energy of position of the moon and earth relative to the sun, which comes from the matter within that star being thrown outward when it exploded in a supernova from nuclear forces within.
22) Just as an exercise in grasping the magnitude of infinity, suppose that the universe of matter and space that we see is infinite. This means simply that it never ends, there is always more matter and space. If this were correct, there would have to be an infinite number of solar systems just like ours. The chances of another solar system forming that is just like ours, down to the exact atom, is exceedingly slight. But infinity is such that if it is divided by exceedingly slight, no matter how exceedingly slight, it is still infinity. This means an infinite number of earths exactly like ours, with all the islands and continents in exactly the same places with every mountain just as high and every spot in the ocean just as deep. This concept of infinity comes up against religion because it would not include an infinity of person identical to you because the one and only you would be a creation of God.
23) One dimension is a straight line. Two dimensions is a flat sheet. Three dimensions is a cube or sphere. Dimensions are the number of pieces of information that it requires to describe an equilateral area. A living being, of a given number of dimensions, cannot be aware of or move in space outside of those dimensions. If a two-dimensional being existed within a sheet, it would be utterly unaware if a higher-dimensional being bent the sheet so that it now actually occupied three dimensions. No matter how much force we apply to an object, it can never move the object outside of the dimensions to which the force is applied. If the Big Bang threw the matter which composes our bodies across a given number of dimensions, we can never see or move in or be affected by what happens in dimensions beyond these. If we could see matter composed of five dimensions, we would still only see three of those dimensions.
24) Atoms consist of a nucleus, composed of positively-charged protons and electrically neutral neutrons, with negatively-charged electrons in orbitals around the nucleus. In ordinary atoms the number of positively-charged protons and negatively-charged electrons are equal so that the overall electrical charge of the atom balances out to zero. An atom is defined as one or another element, such as hydrogen, oxygen, iron, uranium, etc. There are over 100 known elements. Which element an atom is defined as depends on the number of protons in the nucleus. Hydrogen has 1 proton in the nucleus, carbon has 6, oxygen has 8 and lead has 82, for example. Atoms of the same element can have different numbers of neutrons in the nucleus. Atoms, of the same element with the same number of protons but with different numbers of neutrons, are known as isotopes. The atomic number of an element is simply the number of protons in the nucleus. The atomic weight (or mass) of an element might be about double the atomic number and is the number of total nucleons (protons and neutrons) in the nucleus. If there are different isotopes of the element, these are averaged together to get the atomic weight. Electrons are so light that they do not even count in atomic weight. A proton is 1,836 the mass of an electron. An atom with one or two more or less electrons than protons, so that it has a net negative or positive charge, is known as an ion.
25) You may have heard of heavy water. It really is water that is about 10% heavier than ordinary water. The difference is that the two hydrogen atoms in a molecule of H2O heavy water are not ordinary hydrogen, but isotopes of hydrogen with a neutron as well as the one proton in the nucleus of the hydrogen atoms. Ordinary water usually cannot be used as a moderator in a nuclear reactor, in which a carefully controlled nuclear reaction produces useful heat by boiling water, because it will absorb the high-speed neutrons. If such absorption of neutrons is part of the design, it is called a "light water reactor". But heavy water has the two hydrogen atoms in the molecule already with neutrons in the nucleus, so that it does not absorb more neutrons. Heavy water thus accomplishes the purpose of slowing the neutrons down, but not absorbing them. The extra neutrons in the hydrogen atoms of molecules of heavy water make it useful in nuclear fusion, in which smaller atoms are crunched together by tremendous heat and pressure into larger atoms, and the leftover binding energy is released as heat. A hydrogen bomb is an ordinary atomic bomb that is surrounded by a layer of heavy water whose atoms will fuse together and release their excess binding energy when the bomb is detonated.
26) Atoms seem to be solid particles to us, but are actually by far mostly empty space. The reason that atoms do not just merge into one another is that the electrons of adjacent atoms are all negatively-charged, and we know that like charges repel while opposite charges attract. The outermost electrons in adjacent atoms mutually repel one another when they are in close proximity, this is known as electron repulsion and it is what prevents atoms from merging together. It takes the tremendous heat and pressure at the centers of stars to overcome this repulsion and crunch smaller atoms together into larger ones. The pressure of tectonic collisions on earth or the pressure in the depths of the oceans is nowhere remotely enough to overcome electron repulsion in atoms.
27) Quarks were first theorized in 1964 and are the particles of which nucleons, protons and neutrons are composed. There are several different quarks, but all ordinary matter is composed of up and down quarks. An up quark has an electric charge of +2/3, and a down quark has a charge of -1/3. This means that two up quarks and one down quark form a proton, with a charge of +1, and two down quarks and one up quark form a neutron, with a charge of zero. Particles which are such composites of quarks are known as hadrons. Protons and neutrons are hadrons known as baryons, as opposed to other hadrons known as mesons. The electrons in orbitals around the nucleus are classified as leptons, and are not composed of quarks.
28) Antimatter is matter but with the electrical charges reversed in the atom. Instead of positively-charged protons, antimatter has negatively-charged anti-protons in the nucleus with positively-charged positrons in the orbitals, instead of the electrons in ordinary matter. Due to these opposite charges, matter and antimatter brought into contact will mutually annihilate in a spectacular burst of energy far greater than an equivalent nuclear explosion. I imagine that neutrons would be the same overall in either matter or antimatter, since they have no electric charge. As far as I know, we could not tell a galaxy made of antimatter apart from one composed of ordinary matter just by observation, since both would handle light in the same way.
29) Lighter atoms are crunched together by the tremendous heat and pressure in the centers of stars to form heavier elements. The reason that some elements are much more abundant than others can be explained by a simple factor tree. An atom with one proton is hydrogen. Some of the atoms created in the beginning were helium, with two protons, and two hydrogen atoms can also be crunched together to form helium. The reason that elements like carbon and oxygen are common can be explained as multiples of helium. Three atoms of helium crunched together forms an atom of carbon, and four atoms of helium crunched together forms an atom of oxygen. The reason that gold is rare is that, with 79 protons, it's atom does not fit as readily into the factor tree. This process of crunching lighter atoms together into heavier ones by the tremendous heat and pressure in the centers of stars only goes so far, elements heavier than nickel and iron require the input of energy when a large star ultimately explodes as a supernova to form. This is why these heavier elements are so much less common than the lighter elements.
30) Nuclear energy comes in two forms, fusion and fission. Both are based on the release of excess binding energy as either smaller atoms are crunched together or larger ones are split apart. The nuclear energy which produced heat and light in the sun and stars is all from fusion, and the nuclear energy used by people is (so far) all from fission. Fusion has been achieved by using lasers to push atoms together, but is still in the experimental stage. The energy which holds the positively-charged protons in the nucleus together, which should mutually repel, is known as binding energy. The basis of fusion and the reason that the sun and stars release heat and light is that there is more total binding energy in the smaller atoms which are crunched together than there is in the larger atom that they are crunched together into. When this happens, the leftover energy is released and we see the star shining. The basis of fission is that there is more total binding energy in the atom of plutonium or 235 isotope of uranium then there is in the "daughter atoms" which result when it is split by a high-speed neutron. The leftover energy is released as heat in an atomic bomb or nuclear reactor. In fission, the splitting of the large atom sends out several neutrons at high speed, which can split other atoms to repeat the process and start a chain reaction. The reason that the neutrons are released is that the resulting smaller "daughter atoms" contain fewer total neutrons than the original large atom. When atoms are larger, they tend to have more neutrons per proton to hold the atom together against the mutual electrical repulsion of the positively-charged protons. This is why the weights of elements do not increase at a steady rate as we move to heavier atoms. The only elements that can undergo fission (splitting) is plutonium, an entirely man-made element, and the 235 isotope of uranium. The atoms of any elements can be crunched together, by enough heat and pressure, to form heavier elements.
31) What is known as the binding energy curve seems to preclude the release of binding energy from either the fusing of light atoms or the fission of heavy atoms, since the binding energy per nucleon in the nucleus (a nucleon is a member of the nucleus, either a proton or neutron) increases from the lightest elements onward, then levels off, then decreases as we get to the heaviest elements. This seems to mean that crunching light elements together into heavier ones, or splitting the heaviest elements into lighter ones, should require an input of energy rather than giving off energy. The reason that this is not the case is that as the elements get heavier, there tends to be more neutrons relative to protons in the nucleus. This means that neutrons are formed by the combination of an electron and proton in fusion of lighter elements, and neutrons are released by the fission (splitting) of certain heavy elements. So that even though there is less binding energy per nucleon in the heavier elements which will undergo fission, several neutrons are released when the atom is split so that there is binding energy to be released based on the fact that there are fewer nucleons in the resulting "daughter" atoms. The heavier elements require an input of energy to form the atoms and this comes only with the energy released by a star that explodes as a supernova.
32) The atoms of some very heavy elements, and some isotopes of lighter elements, are somewhat unstable and give off particles or energy as they decay to a more stable atom. This is known as radioactivity. There are three types of radioactivity; alpha, beta and, gamma. Alpha particles that are given off is essentially the same as a helium nucleus, two protons and two neutrons. Beta particles are electrons and gamma rays are electromagnetic waves. All are given off as an unstable atom seeks stability by decaying into something more stable. Alpha particles are given off as an atom of one element changes into another element. Beta particles are given off as a neutron changes into a proton. One unstable isotope of a light element that is particularly useful is that of carbon-14, which means carbon with it's six protons and also eight neutrons. Carbon-14 is less stable after the plant dies and the rate at which it decays, or it's half-life, is known. This makes carbon-14 analysis very useful for archaeological dating. Atoms of heavier elements in dust in space, the debris of a supernova or exploding star, can also be split back into smaller atoms by cosmic rays, in a process known as spallation.
33) Atoms can join together to form molecules, which can have completely different characteristics then their component atoms. The bonds between atoms which form molecules are known as chemical bonds. these bonds fall into two basic categories, ionic and covalent bonds. Ionic bonds are formed when one atom loses an electron to another, giving one an overall negative charge and the other an overall positive charge which causes opposite charge attraction to bind them together. Ionic bonds tend to be brittle. Covalent bonds are where two atoms share one or more electrons, thus binding them together. The molecular bonds in living things tend to be covalent bonds.
34) The chemical behavior of atoms depends only on the electrons in the outer orbital shell. Atoms with only a few electrons in the outer shell relative to the size of the atoms, such as one or two or three electrons, tend to lose those electrons to atoms with a mostly-full outer shell. This forms an ionic bond between the two atoms. Atoms with a medium number of atoms in the outer shell, maybe four or five, tend to share these electrons with other atoms to form covalent bonds. Atoms with completely full outermost electron shells tend to be unreactive chemically. These include argon, neon and, xenon. The periodic table of the elements arranges elements by their atomic number, which is the number of protons in the nucleus, and places the elements in columns with other elements with the same number of electrons in the outermost shell,and thus similar chemical characteristics just as the columns on a calendar represent the same day of the week.
35) Metals are different from non-metals in that large numbers of atoms in metals share some of their electrons among themselves. These community electrons are known as delocalized electrons. This is what causes metals to appear different from non-metals. It is also why metals tend to conduct electricity, a voltage pressure can cause these delocalized electrons to move in one direction. There can also be electricity in non-metals, but this is static electricity caused by friction that knocks electrons out of the outer electron orbitals of one of the materials involved.
36) Burning is a chemical reaction which releases a considerable amount of energy. Burning does not involve or affect the nucleus of the atom, it only involves the electrons in the outermost orbital. this makes burning different from nuclear reactions in that simple burning cannot make one element out of another. There is energy in the molecular bonds holding atoms together in a way similar to the far greater energy holding the nucleus of the atom together. But it also requires energy to break those molecular bonds. Basically, a substance will burn if there is more energy in the molecular bonds than it takes to break those bonds. Oxygen, or some other oxidizer, is vital to burning because it combines chemically with the loose atoms, which have already had their bonds broken, so that they do not smother the further burning process. Most compounds that readily burn (a material formed of molecules composed of different atoms is known as a compound, rather than an element of one type of atom) contain carbon in their molecules and were formerly a part of living things which had their molecules put together by solar energy, such as wood or oil. When these materials are burned, the original solar energy that went into creating the bonds is released as heat and light. Remember that energy can never be created or destroyed, but only changed in form. A flame consists mainly of glowing particles of carbon, which later appear black. The stomach also breaks chemical bonds to release their energy, but it does it by way of acid
37) The structures of living things are based on carbon. It is a small atom which can form extremely complex molecular structures, based on covalent bonds, that are assembled with energy from the sun. Carbon can form something like 35 times as many molecular compounds as all other elements combined. The energy of sunlight on the leaves of plants is used by the plant to split molecules of carbon dioxide in the air. The carbon in the molecule is used by the plant to build it's structure, the two atoms of oxygen in the molecule of carbon dioxide are released back into the air. This means we can say that plants, including the largest trees, literally appear out of thin air. When humans or animals eat the plants, or the plants are burned, the original energy of the sun, which was stored in the molecular bonds of the plant, is released.
38) Entropy is a principle which comes into play when we deal with two different levels of complexity. A simple example of entropy which is often used involves an open bottle of ink placed in a container of water. Osmosis will cause the ink to leave the bottle and disperse throughout the water. Entropy is that it is a lot easier for the ink to leave the bottle and disperse throughout the water than it is for the ink to return to the bottle. But my understanding of entropy is that there are no meaningful examples of it outside of living things, and the implements such as the bottle of ink that are made by living things. Entropy results because we are more complex than out surrounding environment. It is much easier for a situation to proceed from a state of higher complexity to a state of lower complexity then vice-versa. The bottle of ink, produced by complex human beings even though it seems simple, becomes affected by the lower level of complexity around it and the information present in the higher level of complexity cannot readily be found by the lower complexity of that environment.
39) The electrons in orbit around the nucleus of an atom exert electromagnetic force. We usually do not notice this force because it tends to be balanced out in all directions, and thus dissipated. Magnetism refers to the force that is exerted if these electron orbitals can all be lined up. We usually think of iron as being potentially magnetic, although it is not the only material that can be magnetized. A piece of soft iron can be magnetized temporarily as an electromagnet by having the electrons in orbitals lines up by the influence of a nearby electric current. The effect of magnetism can also be used as an electrical transformer. Magnetism is a far more powerful force than gravity. When a magnet lifts a piece of iron or steel, it is overpowering the gravity of the entire earth on the metal.
40) The earth is continuously bombarded by particles from outer space, known as cosmic rays. The term is actually a misnomer from the days when cosmic rays were believed to be electromagnetic waves. They are actually particles, mostly positively-charged such as protons and alpha particles (which is essentially a helium nucleus of two protons and two neutrons). There are electrons among cosmic rays, which are negatively-charged. Cosmic rays move at nearly the speed of light. They do not come from the sun, nor seem to even come from within our galaxy. Cosmic rays impact atoms and molecules high in the atmosphere to produce what are known as secondary cosmic rays, which bombard the earth. There are particles with electric charge which come from the sun and are referred to as the solar wind. These particles could be harmful except that we are shielded by the earth's magnetic field. This is why there is an occasional spectacular display of light around the earth's magnetic pole, known as the northern lights or aurora borealis. I saw the northern lights once, from the top of a parking ramp. It looked like a glowing green curtain. There are two zones around the earth, extending well into space, within which high-energy particles such as protons and electrons are trapped. These zones are linked to the earth's magnetic poles and are known as the Van Allen Belts. The charged particles within the belts can be hazardous to the electronics in spacecraft and satellites.
41) The spectrum of visible light that we can see is only a small portion of the electromagnetic spectrum. Electromagnetic waves move through space at the speed of light, 186,282 miles per second or 300 million meters per second. If these waves have a shorter wavelength, they will have a higher frequency in waves per second. A longer wavelength will mean a lower frequency. The color that we can see with the longest wavelength, and thus the lowest frequency, is red. From there, as frequency gets higher, we come to orange, yellow, green, blue and, violet. We see an absence of light as black, a mix of all colors as white, a mix of black and white as gray, and a mix of certain colors as brown. The longest electromagnetic waves in terms of wavelength are radio waves. As we get higher in frequency, we come to microwaves, infrared (heat), visible light from red to violet, ultraviolet (which causes sunburn), X-rays and finally gamma rays.
42) All waves, including electromagnetic waves, have an interrelated wavelength, a frequency and, also an amplitude. The wavelength is simply the distance from a point on one wave to the corresponding point on the next wave, such as crest to crest or trough to trough. Frequency depends on the velocity of the wave and is usually expressed as waves per second. In radio waves, a hertz is one wave per second and a megahertz is a million waves per second. The amplitude is the strength of the wave as the distance of the crest above and the trough below the origination line. Electromagnetic waves can be polarized, meaning that they can be confined to moving in one geometric plane, rather from all planes like the hands on a clock. When a source of waves is moving, it produces what is known as the Doppler Effect. This is a crowding together of the waves in the direction of motion, to produce an effectively higher-frequency, and a spreading out of the waves in the direction away from the motion to produce an effectively lower-frequency. The best example is the whistle of a train coming toward you. The whistle seems high-pitched, but then suddenly drops in pitch as the train passes. The Doppler Effect is especially useful in astronomy, the so-called "red shift" of distant galaxies moving away from us. The further away galaxies were, the more red-shifted was their light, meaning that distant galaxies were moving away faster, and this is what led to the conclusion that the universe was expanding. Sound is waves also, but is takes on a different form as compression and rarefaction, rather than as peaks and troughs. This means that the atoms or molecules of the medium undergo alternating compression and rarefaction (spreading out) by the energy of the sound wave. In machines, sound takes the form of cyclical vibrations. It is important to design rockets, for example, so that vibrations cancel one another out rather than amplifying each other. This means that rather than having two compression or two rarefaction together, we want the two to cancel each other out to zero.
43) The reason that we see the range of wavelengths that we do is explained by the complexity of the structure in our eyes necessary to receive and process electromagnetic radiation, which is composed of atoms, and the size of those component atoms relative to the wavelengths of the electromagnetic radiation. It is impossible for an eye to see gamma rays because the wavelength of the rays is too short relative to the size of the atoms which would have to form the very complex structure in the eye to see the waves. An eye that was large enough to see in radio waves wouldn't make sense because it could be so extremely complex with it's size made of atoms that the little bit of information which could be conveyed by the long-wavelength radio waves would be nowhere near worth it.
44) Electromagnetic waves come in different wavelengths, from very long-wavelength radio waves to gamma rays at the other end of the spectrum. All electromagnetic waves can be reflected by matter, but are reflected by objects which are roughly comparable in size to the wavelength. This is why, if you have the radio on while driving, long waves (such as AM in North America) will fade when you go under a bridge, but shorter waves (such as FM in North America) will not fade. This is because the longer waves have a wavelength roughly comparable to the size of the bridge, and so are reflected away by it. But the shorter waves can be reflected off the road and the underside of the bridge so that they are reflected underneath the bridge and do not fade. The properties of the extremely short wavelength x-rays and gamma rays to be able to actually penetrate matter is simply due to their wavelengths being short enough to pass between atoms.
45) We certainly did not learn all we know about the universe from visual observation using optical telescopes. There are separate studies of the stars and the universe going on in all sections of the electromagnetic spectrum. Giant antennas collect radio waves from space for radio astronomy. Satellites and telescopes look at the universe in infrared, ultraviolet and X-rays. The great advantage of instruments like the Hubble Space Telescope is that it sees from above the earth's atmosphere, which would otherwise interfere with much of the electromagnetic information from space. A primary goal of astronomy is to avoid atmospheric interference as much as possible which is why the ideal place for an astronomical observatory is on a mountain in the desert. Visible light is useful in another way than direct observation. Light can be broken down into it's component colors using a spectroscope with a prism. Incandescent chemical elements give off or absorb certain wavelengths of visible light, and we can thus focus on a distant star and find out what elements are present in the star by the patterns of it's spectrum. Helium is named for the sun (helios) because it was discovered in the sun, by spectroscopy, before it was found on earth.
46) Transparency to light results when the atoms of a material are lined up in such perfect rows that light can pass between the atoms. Water molecules are polar, meaning that one side of the molecule is more negatively-charged and the other side more positively-charged. This causes the molecules to line up in rows, positive-to-negative, because opposite charges attract. Light can thus pass between the molecules. Refraction in a transparent medium is when light enters the medium at and angle to the alignment of the rows and bounces off the rows of atoms so that it is bent. This affects the component colors of light differently so that they end up separated from one another as they exit the medium on the opposite side. This is the basis of a prism. Longer wavelengths naturally have a more difficult time squeezing between the rows of molecules, which is why red light is absorbed by water first. In underwater photographs, you may notice that there is nothing red below about 9 meters (30 feet). The reason that deep water appears blue is that it is only the shorter wavelengths of light that can pass far enough through water without being absorbed to be refracted back to the surface. The reason that matter reflects light at all is that the electrons in atomic orbitals are electrically-charged and light is an electromagnetic wave so that it is diverted by the charges of the electrons.
47) Matter will reflect light, and other electromagnetic radiation. But to get a mirror image, the reflecting surface must be smooth down to the wavelength of the light. Water molecules are small and the smooth surface of water will give such a mirror image. The surfaces of most materials will reflect light, but will not give a mirror image because the surface may seem smooth from our level but is not smooth relative to the wavelength of light. If we could see in other wavelengths of the electromagnetic spectrum, the surfaces which give mirror images would be different.
48) We do not actually see objects, what we see is the light that shines from or is reflected by the objects. This means that there are some optical illusions. The classic optical illusion is a rainbow. The light of the sun from behind us is refracting through spherical drops of water back to us. The drops act as prisms because it handles the different wavelengths of the colors of light differently so that we see them individually. The blue of the sky is also an optical illusion. The size of dust particles that are small enough to remain airborne is such that they reflect the shorter wavelength blue light so that the blue light from the sun is scattered around the sky. We cannot see water vapor, but it does have an effect on light in that it does the opposite of the prism by blending all colors together into white. Notice that the sky over a desert tends to appear as a deeper blue due to the absence of water vapor. This is also why clouds and snow appear white. Remember that white is a mix of all colors and black is a complete absence of light. Sunrise and sunset may appear red or orange due to an optical illusion based on simple geometry. The sun is shining through a thicker section of the atmosphere so that light at the longer wavelength end of the visible spectrum is scattered out as well. Color is actually an optical illusion as well since it does not really exist, but is the result of how our brains interpret different wavelengths of visible light.
49) States of matter are the three major ways in which matter can exist. A solid has a definite shape and a definite volume. A liquid has a definite volume, but no definite shape. A liquid adapts the shape of the container holding it. A gas has no definite shape or definite volume, it tends to fill a container that is holding it. Liquids tend to be exceedingly rare relative to gases and solids. The state of matter depends on temperature, and only two of the more than one hundred chemical elements are liquid at room temperature, mercury and bromine. In science laboratory terminology, standard temperature and pressure refers to a pressure of one atmosphere at 25 degrees Celsius. The three states of matter are based on the existence of atoms, there are two other possible states of matter in the universe. Plasma is found where it is so hot that even the atoms break apart into their component particles. A neutron star and a black hole are extreme environments in which the force of gravity is so great that even the structures of atoms have collapsed so that state of matter is meaningless. A black hole in space is where the components of matter have been so completely crushed together from the extreme gravity that not even light can escape, hence the name.
50) There is a way to calculate approximately how many atoms or molecules are in a pure sample of a substance that can be weighed. Every element has an atomic mass and every compound a molecular mass. There is a number known as Avogadro's number, which is 6.02 x 10 raised to the 23rd power. If you have a sample of matter weighing as many grams as it's atomic or molecular weight, this is how many atoms or molecules are in the sample. This number of atoms or molecules is referred to as a mole. This concept is very useful in that it can be easily calculated just how much of one chemical will undergo a complete reaction with just how much of another, in terms of the chemical formula and how many moles of each.
51) Heat is simply the kinetic energy of moving atoms and molecules. This means that there must be a temperature at which all molecular movement stops and it cannot get any colder. This temperature is known as absolute zero and temperature measured from it is known as absolute temperature. Absolute zero is -459 degrees Fahrenheit or -273.16 degrees Celsius. Measuring in Celsius degrees but starting from absolute zero, instead of the freezing point of water, is known as the Kelvin scale. This means that on a scorching hot summer day, there is only about 15% more heat than there is on an extremely frigid winter day. The reason that living things are temperature-sensitive is that we depend on and our bodies contain so much water, which is sensitive to temperature. Due to the vastness of empty space, the universe is overall a very cold place. Despite the tremendous heat given off by stars, the average temperature across the universe is only a few degrees above absolute zero. Heat tends to accelerate chemical reactions and when heat is a measured component of a reaction, it must use the Kelvin scale because it is a measure of absolute heat. Refrigeration is used to preserve food because it slows the reactions by which the food decays.
52) There are three ways in which heat can move. These are conduction, convection and, radiation. Conduction takes place when a colder object is brought in physical contact with a hotter object. Some of the kinetic energy of the moving atoms and molecules in the hot object goes to strike the atoms and molecules in the cooler object. If the two are kept together long enough, they should reach the same temperature. Convection is a form of conduction except that it involves a fluid, such as water or air to pick up and then carry away heat. Radiation is where heat energy is carried away from a source by infrared radiation, examples are when you feel the heat from a heating element in an oven or get a sun tan from solar radiation. Heat by radiation is different from conduction in that there is no direct contact, and from convection in that there is no movement of matter.
53) It is Newton's Laws of Motion which govern classic physics. Force in a moving object is equal to mass x acceleration. An object either at rest or in motion stays that way until acted upon by an outside force. Every action must result in an equal and opposite reaction.
54) All machines, no matter how complex, can be broken down into several simple machines. There are varying definitions of what the fundamental simple machines are. My understanding is that there are only three simple machines. These are the lever, the wheel (including pulleys) and the inclined plane. The lever changes the direction of a force, as well as exchanging distance for force. The wheel converts between circular motion and linear, and can also act as a lever by using gears and wheels of different sizes. The inclined plane (or wedge) converts force in one direction to force in a perpendicular direction. All machines and tools are composed of these fundamental simple machines.
55) The vast majority of atoms in the air are nitrogen, about 78%. Nitrogen does not combine chemically with other elements easily, but when it does the bonds contain a lot of energy which is why nitrogen-based fertilizer is so explosive. Air is about 21% oxygen, up to a few percent carbon dioxide and up to several percent water vapor (vapour). These gases in the air are not combined chemically with one another, so that air is a mixture but not a compound.
56) The basis of weather is that water is actually lighter than air by molecule. This is why it evaporates and also a reading on the barometer of low pressure usually means that a storm is pending because wet air is less dense than dry air. But a water molecule is polar, meaning that it is more positively-charged on one side and more negatively-charged on the other side. this is known as hydrogen bonding and causes the water molecules to line up negative-to-positive. This brings the water molecules very close together so that, at sea level, water is actually 800 times as heavy as air. Evaporated water condenses around particles of dust high in the air to form clouds. Warm air can hold more water than cold air so that when there is a drop in temperature, more water condenses so that it must begin to fall as precipitation. Water is pulled upward by the fact that the surface of the air heats unevenly so that there is an updraft over the warmer places and an downdraft over the cooler places. Water vapor condenses at altitude because the sun does not directly warm the air, but warms the earth which then warms the air, so that temperature decreases with altitude. This is what forms those fluffy cumulus clouds. Clouds that form in strata, without the updrafts, are called stratus cloud. A stratus cloud can form at ground level, where it is known as fog. It gets very cold higher up in the sky, and the high wispy clouds that form there are composed of ice crystals rather than water vapor and are known as cirrus and tend to be aligned along the direction of the high winds at that altitude. Weather over the sea may not seem to be important, but that is what dissolves oxygen in water so that fish can breathe. Water has a tremendous heat capacity, which is why it moderates the climate or nearby land and why deep lakes take longer to freeze over in winter, or may not freeze over at all. Shallow Lake Erie is the only one of North America's Great Lakes which regularly freezes over in winter. This polarity of water molecules is what makes it so useful as a solvent and for washing, molecules of dirt and other foreign molecules are held by the polarity of the water molecules.
57) An important factor in weather is fronts, either warm air pushing cold air or vice-versa. The front is shaped like a wedge, meaning that the front boundary is not perpendicular to the ground but is at a sharp angle with the lighter warm air above the denser cold air. A front usually means a storm because it is where warm air and cold air come into contact. Warm air can hold more water, which means that it loses some of it's ability to hold water when it comes in contact with cold air and this excess water falls as precipitation. The fact that a front is shaped like a wedge means that it's approach can often be seen by high cirrus clouds in the sky, followed by lower clouds as the front approaches.
58) Hurricane formation depends on dust being brought out over the sea by the wind. Circular storms, such as hurricanes, in the western hemisphere are fed by dust from north Africa, and those in the eastern hemisphere by dust from Australia. The dust acts as condensation nuclei so that the air can hold more water, condensed on the particles of dust, as cloud droplets. The dust as condensation nuclei also concentrates water which falls as rain, but it does not mean that more water evaporates then would otherwise. This concentration of rain can be seen in how India gets Monsoon rains, but downwind Arabia is very dry.
59) The basis of global warming is that as the sun warms the earth, the earth absorbs the wavelengths of radiation coming from the sun and then re-radiates the energy back into space but at different wavelengths. The trouble is that there are certain greenhouse gases in the earth's atmosphere which allow the wavelengths from the sun to get through but then block the wavelengths being radiated back into space so that the earth gets warmer. This operates just like a greenhouse, the glass or plastic allows solar radiation in but blocks the re-radiated energy from getting out. A couple of degrees warmer may not seem like much to most people but if average global temperatures rise even a couple of degrees it will mean that mush more ice will melt and much more water will evaporate. When ice melts, it exposes darker ground beneath and since dark colors absorb heat which was reflected away by the white snow on the melted ice, we now have a warming spiral started. Carbon dioxide is the most important gas, although methane and even water vapor are actually greenhouse gases too. The earth used to have a lot more carbon in the air but it was buried as the structures of plants and became coal and oil over millions of years. When we burn fossil fuels, we are releasing this carbon back into the air. Global warming will exaggerate what is already there. The average storm is more ferocious, wet areas will be even wetter while dry areas are even dryer.
60) Water will evaporate as vapor in the air. Water usually only evaporates from the surface of water, but if we apply enough heat to the water then evaporation will take place from throughout the volume of water. this condition is known as boiling. The temperature at which boiling will take place is not fixed like the freezing point, but depends on atmospheric pressure. The lower the atmospheric pressure, the lower the temperature at which water will boil. Atmospheric pressure naturally decreases with altitude so that it takes less heat to boil water in Denver than it does in Miami. When heat is applied to water that is already boiling, it will not cause the water to get hotter but will cause it to evaporate faster. A pressure cooker operates by sealing in the steam so that it increases the pressure on the boiling water in order to raise it's boiling point and so cook the food faster. The heat and pressure in steam comes from the energy that it took to vaporize it in the first place. Steam is different from vapor in that steam consists of tiny droplets of condensed vapor. The heat released by steam when it condenses is the heat that it took to vaporize it in the first place. It takes energy to evaporate water and this is the principle behind sweating, it absorbs heat energy and so results in cooling.
61) Glaciers are vast sheets of ice that are found in the polar regions, and sometimes on mountains where it is very cold. Glaciers form especially during ice ages. They begin when the temperature is cold enough that the snow of one winter has not fully melted when the snow of the following winter begins to fall. Snow piles up year after year, decade after decade, and century after century. The weight of the snow above compresses that below into ice. Glaciers can reach about 2 km in thickness, depending on the altitude of the weather which brings the snowfall. When an object is large enough, such as the glacial ice sheets, it is affected by the rotation of the earth and is pulled toward the equator and also somewhat eastward by the momentum of the earth's rotation. This is why ice, which usually covers about 10 percent of the earth's surface, covers about 30 percent during the ice ages. This means that a significant portion of the earth's total water becomes locked up in glaciers during the ice ages so that sea level drops and the area of dry land increases. This formed a land connection between Siberia and Alaska, and this is how the ancestors of Indians in the western hemisphere got there. It is also how people settled Japan. ( It must also mean there was a significant increase in the salinity of the sea during the ice ages).
62) The surface of the earth is about 72% water. The southern hemisphere is only about 10% land. The northern hemisphere is nearly half land. Ice usually covers about 10% of the earth's surface, but during ice ages that increases to about 30%. The average depth of the world's oceans is about 5 km (3.25 miles). There are often wide areas of relatively shallow water off the coasts, known as continental shelves.
63) Rock is so abundant on earth because it is a compound of silicon and oxygen, both of which are abundant. Rock formed by volcanic heat, such as the very hard granite and basalt, are known as igneous rock. There is a rock cycle, which takes place over very long periods of time. Igneous rock is gradually worn down into grains by the waves on water. These grains collect on the seafloor until they are compressed back into solid rock. But now, the rock is a weaker sedimentary rock such as sandstone, slate or, shale. Limestone is a sedimentary rock which is formed by the skeletons of countless microscopic creatures. The rock may be forced upward or compressed by tectonic movement, or various other factors, which changes the sedimentary rock into another form. Now it is known as metamorphic rock, which means change. An example of a metamorphic rock is marble, which is formed when limestone is placed under extreme tectonic pressure. As one example marble was formed when Italy, of volcanic origin, was pushed into Europe by movement of the African Tectonic Plate. This is the collision which formed the Alps in Europe. The movement also forced limestone seafloor upward, and then transformed it into marble when the collision with Europe took place.
64) The bodies of microscopic creatures in warm shallow seas build up on the bottom of the sea over millions of years. This is compressed into rock to form limestone. This limestone on the seafloor may later be forced upward by tectonic movement. Limestone can be dissolved by flowing water, and this is what forms caves. Limestone also has the property of recrystallizing after it has dissolved. This is what forms the stalactite and stalagmite formations on the floors and ceilings of caves. Limestone terrain is also vulnerable to sinkholes, in which a cavern is formed by flowing water underground until the roof suddenly collapses. This property of limestone to recrystallize is used to make cement, which is a mixture of limestone and clay.
65) There are two basic types of island in the oceans. There are volcanic islands where the solidified magma from an eruption breaks the surface of the water to form an island. If such a volcano does not break the surface of the water, but is close enough that sunlight can reach it, coral may build up on it over millions of years until that breaks the surface of the water to form a coral atoll.
66) Beneath the continents and oceans, there are about twenty tectonic plates in the world. It is along the boundaries of these plates that the most powerful earthquakes occur. The earth's rotation causes a jostling of the plates. A tsunami occurs when there is an earthquake under the sea. It is so deadly because the powerful wave created by the pressure can barely be seen out at sea until it reaches the shallow water adjacent to land. It is in the gaps between the plates that magma from below tends to emerge and solidify to form volcanic islands. Although both earthquakes and volcanoes can occur that are not on plate boundaries. Mountains can be of volcanic origin or can be formed when continental land masses, moving tectonically by the earth's rotation, collide with other land masses or plate boundaries.
67) The heat released by volcanic activity is left over from the early days of the planet, and also comes from the radioactive decay of certain elements within the earth. This heat will not last forever. The moon was once volcanic, but it is no longer. The same was probably also true of Mars. The reason is that the moon and Mars are smaller than the earth so that they have a greater surface area per volume so that not only can they potentially hold less heat than earth, but their heat had a greater relative surface area from which to escape into space.
68) The earth is tilted on it's axis 23 1/2 degrees relative to the plane of it's orbit around the sun. This is what produces seasons. Each of the four seasons represents a quadrant, a quarter of a circle, in the earth's orbital path. The Tropic of Cancer is the line 23 1/2 degrees north of the equator and the Tropic of Capricorn is 23 1/2 degrees south of the equator. The tropics, in between these two lines, represent the area in which the sun is directly overhead at some point during the year. The equinoxes are the two days in the year when the sun is directly overhead at the equator. The Vernal Equinox is the first day of spring in the northern hemisphere and the first day of autumn in the southern hemisphere, the Autumnal Equinox is the opposite. The solstices are the two days in the year when the sun is directly overhead at one of the tropics. In the northern hemisphere, the Summer Solstice is the first day of summer when the sun is directly overhead at the Tropic of Cancer and the Winter Solstice is when the sun is directly overhead at the Tropic of Capricorn as the first day of winter, in the southern hemisphere it is the opposite. The Arctic Circle is 23 1/2 degrees from the north pole at 66 1/2 degrees north latitude because the north pole is 90 degrees north, the Antarctic Circle is the corresponding line in the southern hemisphere. From the equator to the Arctic or Antarctic Circles, the sun always rises and sets once every 24 hours. But above the Arctic or Antarctic Circles, that may not be the case. There are very long days in summer and very long nights in winter. The sun is lower in the sky during the day in winter, due to the earth's tilt on it's axis, but this means that the moon is higher in the sky at night.
69) The reason that it is hot at the equator and cold at the poles is due to the simple geometry of a sphere. The higher the latitude, away from the equator, the wider the area that the sun's light is spread over. At the equator, the earth's surface is essentially perpendicular to the incoming solar radiation so that it catches the most. As we go to higher latitudes this changes, until at the poles the surface is essentially parallel to the incoming radiation and catches little of it. The radiation is proportional to the cosine of the latitude. It is like shining a flashlight directly at a wall so that the light is concentrated, in comparison with shining it angle so that the light is dispersed over a wider area. The reason that there is not more extreme temperature difference than there are is that heat is redistributed across the world by winds and ocean currents. Altitude is also a factor, the sun does not warm the air but warms the earth which then warms the air. This is why temperature drops with altitude, there can be high mountains with snow on the peak at the equator. This is also why Antarctica is the coldest place on earth, not only is it at the south pole but is the highest continent in elevation.
70) The Milky Way that can be seen across the sky in rural areas on dark nights is the plane of our galaxy. The Milky Way is not the same as the celestial equator because the plane of the earth's orbit around the sun is tilted about 60 degrees to the plane of the galaxy. The celestial equator, a reflection of the earth's equator in the sky, is not the same as the apparent path that the sun and the planets move across the sky because of the earth's 23 1/2 degree tilt on it's axis. This path of the planets and sun across the sky, due to the plane of the planets' orbits around the sun, is known as the ecliptic and it stretches across the constellations of the zodiac. The earth faces different directions in space as it revolves around the sun over the course of a year. This is why each season brings it's own set of stars.
71) The proportion of the stars that can be seen over the course of a year, in comparison with the total number of stars which can be seen from earth, varies with latitude. The only place where all of the stars can potentially be seen is at the equator. At the north or south pole, only half of all the stars can ever be seen. In between, there is a zone of circumpolar stars which are always seen at all times of the year. These circumpolar stars replace the ones that are out of view due to the latitude. At one of the poles, all stars visible are circumpolar, while at the equator none are circumpolar and completely different stars are seen as the earth moves around the sun. In the northern hemisphere, the Big Dipper is the best known of the circumpolar constellations. The zone of circumpolar stars, in degrees from the celestial north or south pole, is the same as the observer's latitude in degrees.
72) Suppose you wanted to calculate where celestial north would be if the earth were not tilted the 23 1/2 degrees on it's axis. I figure that it would be 23 1/2 degrees south of the North Star, and the corresponding point in the southern hemisphere, on the vernal or autumnal equinox (the first day of spring or autumn when the sun is directly overhead at the equator) in the exact middle of the night, halfway between sunset and sunrise. This point in the sky would be exactly on a flat horizon as seen over a flat horizon from the Tropic of Cancer at vernal equinox, or the Tropic of Capricorn at autumnal equinox. Remember that the tropics are the lines of altitude 23 1/2 degrees north and south of the equator, which are the furthest extent away from the equator that the sun is ever directly overhead. This is because 23 1/2 degrees is the angle at which the earth is tilted on it's axis. If there was no atmosphere to disperse light, so that we could see stars during the day as it is on the moon, you would see the same set of stars exactly six months and twelve hours apart. The other planets do not have the same axial tilt as the earth. The North Star on Mars, for example, is Deneb in the constellation Cygnus.
73) In the days of sailing ships, it was a simple matter to take a reading of the ship's latitude. All that was necessary was the measure the angular distance of the north star above a flat horizon. The earth is a 360 degree sphere and latitude is the angular distance north or south of the equator, with the equator being 0 degrees, the north pole being 90 degrees north (the north star would be directly overhead at the north pole or at 90 degrees), and the south pole being 90 degrees south. Lines of latitude thus run parallel to the equator. But measuring the ship's longitude was more difficult. The solution came when John Harrison invented a very accurate clock that did not rely on the motion of a pendulum. A pendulum-based clock was considered as unreliable at sea because the pitching and rolling of the ship in rough water might affect the timing of the pendulum. The ship's clock could be set to what became known as Greenwich Mean Time, GMT or the time at 0 degrees longitude which was designated as the Prime Meridian. Local solar time could be determined on ship by devices such as a sundial, and this revealed the location of the ship relative to the Prime Meridian. Each hour ahead of GMT represented 15 degrees latitude east, and each hour behind GMT represented 15 degrees west. 360 degrees in a circle divided by 24 hours in a day equals 15 degrees.
74) You may have heard of a nautical mile. This is a seafaring unit of distance based on the circumference of the earth, because that made it easier to measure by the position of the sun. A circle is divided into 360 degrees and each degree is then divided into 60 minutes. A nautical mile is one minute of arc on a circumference of the earth. It is equivalent to 1.16 conventional miles. The speed of a ship, expressed in knots, means a nautical mile per hour. A similar concept to that of the nautical mile is used as a unit of distance in the universe. There are 360 degrees in a circle. Each degree can be divided into 60 minutes and each minute into 60 seconds. This does not apply to time on earth because the earth rotates 15 degrees (1/24 circle) per hour. This means that there are 1,296,000 seconds in a complete circle. Far out into space, we would come to a point at which the distance between the earth and sun would occupy only one second of arc in the sky. That great distance is known as a parallax-second or parsec. A parsec is equivalent to 3.26 light years.
75) We measure time of day by the rotation of the earth relative to the sun. But it is not quite as simple as it may seem. The sun is actually closer to the earth in the northern hemisphere winter, in January. This means that the earth moves faster in it's orbit during this time than it does in June. Although the earth is rotating at the same rate, this would cause the sun to appear to move across the sky faster in January. In order to maintain the same number of hours in a day, we use what we call mean solar time. This is simply an average of the apparent motion of the sun throughout the year. Finally, the fact that the earth is revolving around the sun while it is rotating has an effect on apparent time. We can also measure time by the stars, and this is known as sidereal time. If we measure time by the stars, a year is actually about 20 minutes longer than if we measure it by the sun. Time zones across the world, where it is the same time by hour, are artificial units that are established by convention. The truth is that it is not exactly the same solar time in any two places unless they are on exactly the same longitude. The international date line, where one day ends and the next day begins, is also an artificial creation. It is intended to be at 180 degrees longitude, opposite the Prime Meridian, but it is curved in places so that it does not cross any land. This is so that no one would have to live in one day but go to work or school in another day.
76) The density of something that floats, relative to the water which it floats in, can easily be seen in what proportion of it submerges in the water as it floats. If 90% of a floating log is submerged, then it's density is 90% that of water. If you see a floating iceberg, we know that ice is about .9 as dense as water and the reciprocal of .9 is 1.11 so you are seeing only about 11% of the total iceberg. An object that floats on water displaces water equal to it's weight. You can weigh something if you can put it in a bowl on water and measure how much water it displaces, in comparison with the empty bowl. If an object is lighter than water, it floats and displaces it's weight in water. If an object is heavier then water, it sinks and displaces it's volume in water. Remember that water cannot be compressed.
77) When forces from more than one direction are acting on an object, we calculate the resulting force with a vector diagram. Draw a line representing each force in the direction of the force and with the length of the line being proportional to the force. Then draw a parallelogram with the two lines of the forces being half of the parallelogram. The diagonal across the parallelogram, moving away from the object, will be the resultant vector. If there are more than two forces acting on the object, then simply repeat the process with the resultant vector.
78) When you do a calculation involving data, it is important to keep significant figures (sigfigs) in mind. Suppose you take one data reading of 4.28, and then another reading of 6.36142. These two numbers cannot be added, subtracted or multiplied together because there is a mismatch in the number of significant figures. The only way to relate the numbers together is to reduce to the least number of significant figures. The second number will have to be reduced to 6.36. It does not make sense to combine an accurate reading with a less accurate reading. There are cases where significant figures are not an issue, such as if you have six apples. If a reading is known to have more accuracy than it's significant figures indicates, then add zeroes onto the end. The two numbers would have an equal number of significant figures if the first number was 4.28000. The zeroes tell us that these places can be considered as significant figures.
79) Modern mathematics would not be possible without a simple concept, that of zero. Humans have used counting devices such as the abacus and counting boards for thousands of years. But really complex calculations are impossible without an understanding of zero. It once occurred to me that, while geometry and the movement of astronomical bodies was quite advanced, there is not a single complex calculation referred to anywhere in the Bible. I believe that his is what held back the development of technology for so long ("The Zero Hypothesis" on the Progress Blog). The Arabic numerals that we use today were part of the solution to breaking out of ancient times, but the simple but vital concept of zero almost certainly came from India. The circle on the Indian flag is a Buddhist symbol. Buddhism originated in India, even though today it is very much a minority faith there. But the circle can also stand for zero, because the modern world would not have been possible without it.
80) Algebra is the branch of mathematics dealing with variables. The primary principle of algebra is that we can do anything to an equation, and it will still be an equation, as long as we do the same thing to both sides of the equation. An equation is a mathematical statement with two sides balanced by an equal sign. Variables are usually represented by letters, often x and y. Suppose that we have the equation 6y = 3x, and we want to redefine the equation in terms of x. We can do whatever we like to the equation, as long as we do the same to both sides. So, if we divide both sides by 3 we get 6y/3 = 3x/3. On the 3x/3 side, the 3s can simply cancel out to leave x alone becaise there is a 3 both above and below the bar. On the 6y/3 side, we can divide 6 by 3 to give us 2. This brings us to x = 2y so that we now have the equation defined in terms of x.
81) One of the most important concepts in mathematics is that of pi. The name comes from the letter of the Greek alphabet. Pi is simply the ratio of the circumference of a circle to the diameter of the circle. Pi cannot be described perfectly by numbers. In decimal, it can supposedly be calculated to an infinite number of decimal places. I can remember it as far as 3.1415927. It is often expressed simply as 3.14 or the fraction 22/7. While this is not perfectly accurate, it is good enough for most calculations involving pi. The concept of pi appears in all manner of calculations involving lines and circles or semi-circles. Suppose we want to calculate how fast the earth is moving in it's orbit around the sun. Just take the average distance from the earth to the sun. Then multiply it by two because we have to reach the earth's orbit on the other side of the sun to get the diameter of the earth's orbit . Then multiply that by pi to find the circumference of the earth's orbit. Then divide that by the number of days in a year to find how far the earth travels per day. Then divide that by 24 to find how far the earth travels per hour.
82) Any formula for calculating the area, volume or, circumference of a circle or sphere will necessarily involve pi. The circumference of a circle is given by pi x the diameter or twice the radius of the circle. The area of a circle is given by pi x the radius of the circle squared. The surface area of a sphere, such as the earth, is given by 4 pi x the radius squared. The volume of a sphere is given by 4/3 pi x the radius of the circle cubed. Raising a number to a power means to multiply it that many times by itself. Something raised to the second power, or multiplied by itself once, is referred to as squared. Something raised to the third power, or multiplied by itself twice, is referred to as cubed.
83) The geometry that we use today is known as euclidean geometry, after the ancient Greek by that name. There have been other systems developed, known as non-euclidean geometries, but the common geometry taught in school is traced back ti Euclid. In euclidean geometry, postulates and theorems are proven by linking to previously proven geometric facts. But this results in a logic structure which ultimately rests on something which cannot be actually proven in the same way, but must be presumed to be true. The basis of euclidean geometry, without which we could never have built the modern world, comes down to a simple but vital assumption. The basis of the geometry is the belief (we could call it faith) that if we have a straight line, and one point outside that line, there will be one and only one line that can include the outside point and will be parallel to the existing line.
84) A number that shows up in many scientific and financial formulae is e. This is equivalent to 2.718... and on to an infinite number of decimal places. The number e is defined as ( 1 + 1/x ) raised to the x power, where x is any large number. The formula to calculate interest earned on a given principal amount is remembered by the acronym "pert". P is the original principal amount, multiplied by e, and then the product of these two numbers raised to the power rt, or interest rate multiplied by time. This gives the total amount of money at the end of that time. The time of the interest rate and the term is almost always expressed in years.
85) When you have a loan at a certain interest rate, and you make a payment on it, how much have you paid in interest and how much have you actually paid down the balance? Take the balance of the loan and multiply it by the interest rate. The interest rate is almost always by the year and payments made by the month. Don't multiply the balance by the interest rate as a percentage, use a decimal instead. If the interest rate is 5%, multiply the balance by .05. Then divide that figure by 12 months. That is how much of your payment is going to pay interest, the rest is paying down the balance.
86) A very important concept in mathematics is the Pythagorean Theorem. This involves the diagonal of a right triangle. Suppose that you are constructing a fence or wall and want to be sure that it forms a perfect right angle. The Pythagorean Theorem is that C squared = A squared + B squared. A and B are the perpendicular sides of the right triangle, and C is the diagonal. We know that 3 squared = 9 and 4 squared = 16 and 5 squared = 25, and also that 9 + 16 = 25. So, all that we would have to do to get a perfect right angle is to measure 3 units from the end of the wall on one of the perpendicular sides, and 4 units from the end of the wall on the other perpendicular side. It does not matter which units we use, meters, yards or feet, or just the length of some improvised unit. But the longer the unit used, the more accurate it will be. The final step is to simply position the two perpendicular sides so that the distance between the two measured points on the two perpendicular sides is 5 units. The two perpendicular sides will then form a perfect right angle.
87) Another very important concept, that I have written quite a bit about getting more out of, is the Inverse Square Law. This basically states that if a light is three times as distant, it will be 1/9 as bright. This also applies to gravity, as well as to so many other things. If we are twice as far away from the center of a planet, it's gravitational force will be 1/4. This is because 9 is the square of 3 and 4 is the square of 2. I also noticed that it solves what is known as Galileo's Paradox of Perfect Squares. The paradox is that every number must have a perfect square, the number multiplied by itself, but few numbers are perfect squares. This basically proves that numbers must be infinite because, while this is clearly true, it cannot be true of any finite set of numbers. I found that as we get to higher numbers, the proportion of perfect squares decreases according to the Inverse Square Law. More obvious applications include the loudness of sound and the strength of radio signals and the apparent visual size of objects with distance. The force of gravity also diminishes with distance in accord with the Inverse Square Law.
88) The Inverse Square Law is related to pi in the circumference of a circle at a given radius. The circumference of a circle is always 2 pi. The distance to an object can be considered as the radius of a circle. If the object is twice as far away, it will appear as having only one-quarter the angular size. This is because we are dealing with two circumferences sharing as radius, which is the distance from the observer to the object. One circle is horizontal and the other is vertical. Both circles pass through the distant object. It is twice as far away on both circles, therefore it will appear only one-quarter as large. The Inverse Square Law is a quirk of our multidimensional space.
89) If there is an Inverse Square Law, then there must be some type of "Square Law" for it to be an inverse of. Put simple, a smaller sphere will have more surface area per volume than a larger sphere and a smaller circle will have more circumference per area than a larger circle, all expressible as squares. The earth is 4 times the diameter of the moon. The earth has 16 times the surface area of the moon and 64 times the volume of the moon. This is because diameter is one-dimensional, surface area is two-dimensional and volume is three-dimensional. The four times the moon's diameter squared is sixteen times the surface area and and the four times cubed is sixty-four times the volume.
90) I would like to just review fractions because I consider them as so important. Even if fractions are somewhat awkward to use, so much of how we express in numbers is really fractions. A percentage is a fraction with an agreed-upon denominator of 100. All expressions of ratio and proportions are fractions. Angular degrees are fractions of a complete circle, 360 degrees. Trigonometric functions are fractions. But we try to express everything in decimal because of the perceived awkwardness of fractions. Multiplying fractions is easy, just multiply across: 2/5 x 2/3 = 4/15. To divide fractions, just invert and multiply: 2/3 divided by 2/5 is the same as 2/3 x 5/2 = 10/6, which can be reduced to 5/3. But to add or subtract fractions, a common denominator is required. 1/3 + 1/4 must be converted to a common denominator such as 4/12 + 3/12 = 7/12.
91) The three angles of a triangle always add up to 180 degrees, and the four angles of a rectangle, square or, parallelogram always add up to 360 degrees. Remember that a complete circle is 360 degrees. Knowing this is very useful in calculations involving geometry. In a parallelogram, which is formed by the intersection of two sets of parallel lines which may not be perpendicular to each other, remember that opposite angles must be equal to one another.
92) If we have a graph with a horizontal x axis and a vertical y axis, with both axes having numbers spaced at intervals, we can express a line on the graph with an equation. The standard form of such an equation is y = mx + b. The y is the location relative to the y-axis of a point and the x is the location relative to the x-axis of the same point. This is similar to finding a street on the map of a city by use of coordinates on the grid of the map. The set of points which meet the condition of the equation will form a straight line. m will be the slope of the line and b will be the offset location of the line. If we let m = 3 so that y = 3x, we will have a steep line on the graph such that where x = 1, y = 3, where x = 2, y = 6. This concept of a graphed line being equivalent to an equation can also be used to graph a curve when we use squared variables, such as y = x squared. This is the foundation of calculus.
93) A very important concept in spatial mathematics is the trigonometric functions known as sine, cosine and, tangent. These are often abbreviated as sin, cos and, tan. These represent the relationships between angles and the proportions of lengths of adjacent sides. Suppose we have a horizontal X-axis and a vertical Y-axis that is perpendicular (forming a right angle) to the X-axis. From the point where these two axes intersect, suppose we have a line at some angle between the two that we refer to as the radius, or R. The radius extends to the opposite corner of a rectangle of which two sides are the X- and Y-axis. The relative lengths of the X- and Y-axis will depend on the angle that the radius between them is set at. The X-and Y-axis lines will be equal in length only if the radius is set at 45 degrees, which is exactly half of the right angle at which the X- and Y-axis lines are set. If the angle at which the radius is set at is lower than 45 degrees, the X-axis will be longer than the Y-axis. If the angle at which the radius is set is higher than 45 degrees, the vertical Y-axis will be longer than the horizontal X-axis. The sine of the angle that the radius is set as is defined as sine = Y/R. In the same way, the cosine of the angle of the radius relative to the two axes is defined as cosine = X/R. The tangent of any angle to which the radius has been set is tangent = Y/X. The tangent is thus the ration between the lengths of the axes and does not involve the length of the radius line. Both sine and cosine range from 0 to 1. If the radius is at the lowest possible angle, zero degrees, so that it is one and the same as the X-axis, the cosine would be 1 and the sine would be 0. If the radius is at it's highest possible angle so that it is one and the same with the vertical Y-axis, the sine would be 1 and the cosine would be 0. The tangent can be any number because it is defined as the ratio of the vertical Y-axis to the horizontal X-axis. The tangent is less than 1 only when the angle of the radius, which defines the lengths of the X- and Y-axes, is less than 45 degrees. As the angle of the radius increases or decreases, the sine and cosine do not change at a steady rate but change at the rate of the edge of a quadrant (a quarter of a circle) inscribed on a right angle so that the center of the circle is at the intersection of the X- and Y-axes and the radius. This means that if the angle at which the radius is set is exactly halfway between the X- and Y-axes, at 45 degrees, the values of the sine and cosine will not be .5, but .707. The importance of the trigonometric functions can be seen in how this value of .707, both the sine and cosine at 45 degrees, also represents the overall value of an alternating electric current. The alternating current forms what is known as a sine wave as it reaches a peak in one direction, drops momentarily to zero, reaches a peak flowing in the opposite direction, then back to zero, and so on. There are actually six trigonometric functions overall because the sine, cosine and tangent are rations of the three values (X- and Y-axes and radius) that can be reversed. The inverse of the sine is the cosecant, the inverse of the cosine is the secant and, the inverse of the tangent is the cotangent. If the name of a trigonometric function begins with co-, it means that it's value decreases as the angle increases. The cotangent is valuable in parallax, the measurement of the distance to an object by the change in the angle at which it appears as we change position over a known distance.
94) The distances to stars can be measured by trigonometry, using a technique known as parallax. The earth is on opposite sides of it's orbit around the sun six months apart. This distance is used as a baseline to measure the shift in the angles at which certain stars are seen against the background of more distant stars, and this is used to calculate their distance. But this method is not accurate for stars that are very distant. Fortunately, there are stars known as cepheid variables. These stars undergo a cycle during which they vary in brightness. It was found that the length of the cycle of these stars is in direct proportion to the actual brightness of the star. So, all we have to do is to measure the length of the cycle of a cepheid variable and then we can discern it's distance by comparing it's actual brightness with it's apparent brightness as seen from earth.
95) There is a very important number with regard to circles and waves. We know from the Pythagorean Theorem that if there is a square, with sides 1 unit in length, the diagonal will have a length of the square root of 2, or 1.414. The reciprocal of 1.414, as well as half of it, is the very special number of .707. If the diagonal of a square has a length of 1 unit, each of the sides will each have a length of .707. This is why, in the trigonometric functions, both the sine and the cosine of 45 degrees, at which the sides will form an equal square is .707. This number really comes into play with waves. Take an alternating electric current, for example. The voltage starts at 0, increases to 1, drops to 0 again, increases to 1 in the opposite direction, drops to zero, and then starts the cycle over. The mystery is what the actual strength of such a current would be, direct current is much simpler to deal with. The overall strength of such a wave, with a peak of 1, is .707. A wave like this is known as a sine wave, and remember that it is 45 degrees where the two sides would form an equal-sides square, rather than a rectangle, and the sine of 45 degrees, the proportion of the sides to the diagonal is .707. The top of a wave is known as a peak, and the mirror image opposite is known as a trough. Peaks and troughs average out to zero. I used an alternating electric current as an example here, but this same concept of .707 applies to any such wave.
96) The same concept of 1.414 and .707 applies to the range of artillery relative to maximum altitude of the shot. The acceleration due to gravity on earth is 32 feet (9.75 meters) per second squared. I noticed that this means that objects fall in units of 16 feet, and I named this unit a grav (for gravity). It is extremely useful, in anything to do with altitude, to remember that 1 second = 16 feet. An object starts to fall with a velocity of zero and reaches the 32 feet per second at the end of the first second. This means that it's average velocity during the first second was half of that, or 16 feet per swcond, during the first second so that it fell 16 feet. During the second second, it's velocity started at 32 and went to 64 feet per second, meaning that the average velocity during the second second was 48 feet per second and this is how far it fell during the second second of fall. 48 is three times 16. So, the falling distance for a compound object is one grav during the first second, three gravs during the second second, five during the third second, and so on. This means that, to find out how far something will fall in a given number of seconds, simply square the number of seconds and multiply by 16 feet. To find out how long in seconds an object will take to fall a given distance, barring air resistance, divide the altitude by 16 feet and then find the square root of it. This concept of a grav being 16 feet can also be used to find the muzzle velocity of a gun. Fire the gun straight upward and time until the splash of the returning bullet in still water. Then multiply this time by 16 feet to get the initial velocity of the bullet or shell. If the splash took 18 seconds, the initial velocity was 288 feet per second and it reached a height of 81 gravs, or 1296 feet. This is because the bullet must have been on it's way back down for half of it's time in the air, 9 seconds, and 9 consecutive odd numbers add to 81, just as 9 squared equals 81. The bullet's initial and final fall velocities would be the same, half of the time that the bullet was in the air would be spent ascending and half falling. If we can use the grav to measure the maximum altitude of an artillery shot, we can use the .707 and 1.414 to determine artillery range because the trajectory of a shell is essentially half of a wave. The maximum range of the shell is when it is aimed at an upward angle of 45 degrees. It's range would be 1.414 times it's maximum altitude when fired vertically, at that angle an altitude of .707 the maximum vertical altitude would be reached (when the gun is fired straight upward as measured by timing the splash). To find the maximum altitude that a shell will reached when fired at a given angle, just multiply the sine of that angle by the maximum altitude reached when fired straight upward. Here is another example of creating your own mathematics when necessary, remember in "New Trigonometric Functions" on the progress blog we saw that maximum range of artillery is given by multiplying the sine of an angle by the cosine of the angle and then multiplying by two. This gives us a peak of 1 at 45 degrees. The required angle of aim to hit a target at a given distance within the maximum range can be calculated according to this or, better yet, a chart drawn up beforehand. If multiple guns are firing at the target, and you want to be sure of which shot is from your gun, just find the maximum altitude that your shell will reach, multiply it by four and then divide that distance by the initial velocity of the shell. This will give the time it will take to reach the target. We multiply the maximum altitude by four because the shell climbs to the maximum altitude with an average velocity of only half the initial velocity, and then descends from that velocity at again only half of the initial velocity of the shell.
97) One useful geometric fact for telecommunications and space travel is that the intersection of the radius of a sphere, such as a planet, and a line tangent to the surface of the sphere must form a right angle. Looking down at the sphere from some altitude can be seen as two right triangles with the line between the triangles being the line of our vision perpendicular to the surface of the sphere. The closer to the sphere we are, the lower our altitude, the greater the angle between the tangent and the central line of our vision perpendicular to the surface of the sphere. This means that the third angle in each of the right triangles must be less when we are lower in altitude. When we are at an essentially infinite distance from the sphere, this internal angle within the sphere approaches 90 degrees so that it points to the edge of the sphere nearly perpendicular to our line of vision so that we can see essentially half of the sphere. When we move closer to the sphere, this third angle moves, like a hand on a clock, closer and closer to us until it reaches us as we land on the surface of the planet and the visible horizon shrinks to a minimal sphere. The intersection of this third angle with the surface of the sphere gives us the outer edge of the visible horizon from any given altitude above the sphere. Just remember that the tangent line where our line of vision intersects the edge of the sphere must form a right angle with the radius of the sphere.
98) Being proficient with mathematics means making up your own, when necessary. Learn to pick out the patterns in things and apply numbers and geometric shapes to it. This is different from doing pre-prepared problems out of a textbook. There may be a number of ways to find a solution. I once wanted to add up all the numbers up to a certain number. I thought that there must be a quick way to do it, but I did not know the way. However, after a few minutes of trial, I noticed that if you divide a number in half, add 1/2 to it and then multiply it by the original number, we get the answer. This means that the numbers from 1 to 10 should add up to 55, and they do. Some mathematical problems can be solved by representing them as geometric shapes, and getting the answer by the area of the shape.
99) It is very useful to have a sense of odds. Suppose that there are ten gloves mixed in a drawer. There are five right gloves, and five left ones. Without looking, we reach in and take two gloves. What are the odds that we have a matching left and right pair? Your first reaction may be to answer fifty percent, but that is actually incorrect. The odds that we have a matching pair by randomly taking two gloves are 5/9. When we take the first glove, whichever it is, that leaves four that would not form a matching pair and five that would. It would only be if we had an infinite number of gloves that the odds would be 50/50.
100) The number of possible orders of something is known as permutations. It is represented by an exclamation point in mathematics "!" and means multiplying all numbers up to a certain number. 5! means 1 x 2 x 3 x 4 x 5, or 120. This means that, if you have five coins and want to arrange them in a line on a table, there are 120 possible permutations. There will be 5 possibilities for the first coin, multiplied by four for the second coin, three for the third, two for the fourth, leaving one for the fifth. With multiple permutations, remember to add the permutations rather than multiply. If the permutations of the coins can also include which of the two sides of the coin faces up, we would multiply two possibilities by itself five time to get 32, and then add that to the 120 to get 152 possible permutations.
101) We tend to run into errors by treating something as if it were infinite, when it isn't. One example which I wrote about is the center of gravity of a mass such as a planet. Textbooks tend to presume that the center of gravity of the planet is the same as it's center of mass. But this cannot be correct. If we are approaching a planet in a spacecraft, the closer side of the planet will have more gravitational influence on us than the far side. This means that the center of gravity must be closer to us than the planet's actual center of mass. The closer we are to the planet, the greater the difference between the center of mass and the effective center of gravity. It is only when we are at an infinite distance from the planet that the center of gravity and the center of mass are the same. (I believe that this is why the orbits of satellites in low orbit tend to be unstable, not just because of possible friction with the upper reaches of the atmosphere. The center of gravity is continuously changing, with greater relative proportion than would be if the satellite were in a higher orbit).
102) Here is another example of how textbooks tend to give simplified examples of concepts. To get really accurate calculations, it is essential to discern whether what you are dealing with is infinite, finite or, infinitesimal. Textbooks tend to have simple examples using the infinite and the infinitesimal, while situations in the "real world" tend toward the finite, and this leads to calculations which may be fairly close but not as accurate as they could be. In textbook illustrations of the sun shining on the earth, for example, day and night are portrayed as equal in length. Yet this is highly unlikely. The sun is not really an infinitesimal point radiating light toward the earth. The sun has a certain angular diameter in the sky, and this means that day should actually be a little bit longer than the night. I calculate that day on earth is about 4 minutes longer than night due to the angular diameter of the sun. The sun averages .532 degrees angular diameter from earth, and the earth rotates a degree every four minutes, with the addition of extra day taking place both at sunset and sunrise. But there is an opposite factor making for a longer night. Unless the sun is at an infinite distance from the earth, which it isn't, a portion of the earth's surface should be hidden from the sun by the curvature of the earth so that the sun will actually shine on slightly less than half the earth at a time. However, I calculate that this shortens the day by only about 4.4 seconds so that it is nowhere near the extension of the day due to the angular diameter of the sun. This also means that we can see slightly less than half of the moon's surface from any given point on earth at any given moment, due to the curvature of the moon, but more than half of the moon overall because of the width of the earth. This kind of thinking is necessary for really accurate calculations. Also consider that the earth is closer to the sun in the southern hemisphere summer (January), this means that the sun looms larger in the sky so that the southern hemisphere has longer days in summer but shorter days in winter then the northern hemisphere.
103) To calculate the odds of something, it is vital to understand the wording. Suppose that three random events just happen to take place on the same day of the week. The first event sets the day of the week. The odds of the second event occurring on the same day of the week are 1/7, and the odds of the third event being on the same day are another 1/7, so that the total odds are 1/7 x 1/7 or 1/49. Notice that we do not multiply by the first event because the wording was only on the same day of the week, but not on any particular day of the week. But if the wording was the odds of the three events taking place on a Wednesday, then we would multiply the 1/7 three times because the first event couldn't take place on just any day. The odds of all three events falling on a Wednesday would be 1/343. The wording determines whether the odds of the first one are counted, or whether it merely sets the pace.
104) To calculate multiple odds, multiply the fractions. If the odds of one event occurring is 1/2 and that of an unrelated event 1/3 (make sure that one event doesn't affect the other or it will be a different calculation altogether), the odds of both occurring are 1/2 x 1/3 or, 1/6. What if you play a game where there is a given odds of winning and you play a given number of times? We would add the odds of the remainder. If the odds were 1/2, and you wanted the odds of winning once, the first time you played the odds would be 1/2. The second time you played, you would add 1/2 of the remaining one-half so that the overall odds of winning would be 3/4. If you played a game with odds of 1/2 three times, the overall odds of winning would be 7/8 (87.5%). You keep adding the odds (1/2 in this case) to the remainder for each time played. Remember that in any uncertain event or game of chance, the odds of winning can never equal 1, which would be certainty. I have found that if there is a game with a certain odds of winning, or an event with a certain chance of taking place, and we play the game that number of times, the overall odds of winning are a shade over 1/2 for any large number. The lower the number, the higher the chances of winning. If the odds were 1/3, and you played 3 times, your odds of winning are greater than if the odds were 1/100, and you played 100 times. If the odds were 1/1000, and you played 1000 times, your overall chances of winning once would be a shade over 1/2. If you played 2000 times, at 1/1000 odds, the odds of winning once are 3/4. If you played 3000 times, at 1/1000 odds each time, the overall chance of winning once is 7/8 (87.5%). This can also be used for applications like calculating critical mass in nuclear physics. If the nucleus of an atom represents about one-millionth of the atom's cross-section, a high-speed neutron that passes through the atom has a millionth of a chance of hitting the nucleus, and splitting the atom.
Thoroughly learn everything here, as well as the posting on this blog about everyday technology "The Way Things Work" and the posting about perspective "Scientific Perspectives And Facts" on the physics and astronomy blog, www.markmeekphysics.blogspot.com , and you will really have a good background in science.
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