How to have a wx party
So I was kidding
Measure it!
Chemistry. World of atoms
Organic chemistry.
Biology and the study of life.
The ultimate life form.
The deadly Hydra...
Wrong again!
How to square a square.
When things don't add up.
E = MC2
Computer aided design
Aboriginal Science
The development of a scientist
Fundamental Waves
BILLION The cardinal number equal to 10^9 or 1000 million
British define it as 10^12 or one million million.
So be very careful placing large bets in Britain!.
Party balloons are cheap so no problem there. What to fill them with........ I knew how to make Hydrogen which most first year chemistry students would also. Not wanting to blow myself up again, I elected to get a party kit from Wal-Mart for $18.50. The kit contained 30 latex party balloons and a small disposable tank of Helium to fill them with. Now I had to justify why to spend $20 for something so ..useless. My wife suggested they might be good for the Grandkids birthdays if nothing else so my conscience was appeased.
The next step was to glean something useful from each 67 cent launch so I elected to justify each launch with a few estimates of weather data. I inflated the balloon to a diameter of 6.5 inches, tied it off and released it from the floor. the balloon took about 2.5 seconds to reach the 8 foot ceiling so my ascent rate was around 3 feet per second. It was launch time!
I failed to mention a stopwatch would be handy here, which I happen to have built into my Casio wrist watch. Start the watch and release the balloon! Its amazing how fast a 6.5 inch balloon disappears from view, especially if its white and the background is clouds. I tracked it for 21 seconds. Its altitude at that point should have been A=V*T or A=3*21=63 feet high.
To be honest, I fashioned one more device to use in the great launch. For around $1.50 you can purchase 2 ea. 1/2 circle protractors, the kind used to measure angles at school. Enlarge one hole and push a screw through and thread it into the hole in the other protractor. This makes a pretty good sextant. Now you sight the balloon along the edge of the fixed screw protractor while the loose protractor swings free, seeking its gravity level. Clamp the two protractors together when you have your sighting and you now have the elevation angle of the balloon. Incidently, you can use this device to estimate how high a satellite will pass overhead.
Now if you're as lucky as I am, you may also own a scientific calculator. Now it will be possible to solve the triangle formed by you, the balloon and the ground directly beneath the balloon. In my case, I measured the balloon after 21 seconds at an angle of 30 degrees to the South East. That meant the wind was blowing oppositely from the North West. Since my
balloon was 63 feet high at last sighting, the calculator gives the ground distance as 109 feet. This distance was covered in 21 seconds so that gives a wind speed of 109/21 = 5.2 ft/sec or 3.5 miles per hour. Don't tell anyone but the calculator uses (uggg!) trigonometry to solve the triangle.
So thats the end of the great balloon experience......Now lets find out how fast a mouse can run!
(DJS)
Ok, so I decided to conserve helium for balloons because of the expense and try Hydrogen.
(Some things never change. The weather bureau did the same thing.) The next question was how to hook up a hydrogen generator that would be reasonably safe. Kids, don't try this at home without parental guidance. Even with guidance it may not be safe.
The Weather bureau generates hydrogen by the reaction of sodium hydroxide on Aluminum metal. The reaction is: 2Al + 2H2O + 2NaOH = 2NaAlO2 + 3H2.
The Sodium hydroxide is cheaply provided by lye and Aluminum by heavy duty Aluminum foil. The reaction vessel is a 2 liter plastic pop bottle. Glass pressure vessels are not particularly safe. Also keep a plastic bucket handy with several inches of warm water in the bottom. Wearing eye protection such as glasses or goggles is also an excellent idea.
Put 2.5 inches of distilled or aged tap water in the pop bottle. Slowly add 10 grams of lye through a funnel or with a small plastic spoon. Rotate the bottle to dissolve. The solution will get warm. Remember to add acids or bases to water. Never mix water to bases or acids or boiling splatter could result.
Next, tear off a 12" by 24" piece of aluminum foil and cut it into 2" by 12" strips with scissors. You should end up with around 12 strips. Fold each strip until you get a little block of aluminum 2" by 1/2 inch square. Try to keep the folding tight. Take your equipment and materials outside and place in a shady spot away from children and pets. Set it where it can be seen through a window so you can monitor the reaction. Now you're ready to go.
Put all 12 Aluminum sticks in the plastic reactor bottle and squeeze the bottle till it dents. Place the neck of the balloon all the way over the threads of the bottle and wait. The chemical reaction is exothermic. That is, the solution will get hot as the hydrogen is seen bubbling off the Aluminum sticks. In fact you could observe "thermal runaway" where the solution boils and the balloon fills quickly with steam and water vapor. This spoils the lift and will require a new start.
The solution should never get too hot to touch the plastic bottle. on the other hand, Hydrogen generation depends on solution temperature; warm but not boiling. If the solution gets too hot, cool it down by setting the bottle in the bucket with 2.5 inches of warm water. If your solution gets too cold, it may take all day to fill. Hot water in the bucket will help stabilize the reacting solutions temperature by acting as a constant temperature heat sink.
When the balloon diameter reaches 6.5 to 7 inches in diameter, it will probably float and you're ready for a launch. Pinch the nozzle and remove from the reactor bottle. Tie a knot in the nozzle and launch just as initially described for Helium. Be sure to clean up the reactor bottle by filling it with cold tap water and dumping it down the drain. Catch any unused Aluminum in a strainer to be reused. The liquid could still be caustic so wash your hands after cleanup. If you get some on you, wipe with white vinegar. Now you see why Helium is so much easier and safer. One last word.. Don't use Hydrogen balloons for the Grandkids birthday parties, especially near the birthday cake! (DJS)
It has been said that if you have not or cannot make measurements about a reaction, then your knowledge of that reaction is of an unsatisfactory kind. (Or something similar to that.) In other words, knowledge comes to those who can measure results. We measure everything we experience in life.. only the degree of accuracy varies. Even ON/OFF is a measurement albeit of a limited binary form.
All fundamental units in nature can be measured and combined to make derived units. The most used system worldwide is the MKS system or Meter, Kilogram, Second units of measure. Of less use now is the CGS or Centimeter, Gram, Second system and the old English FPS or Foot, pound, Second system. The usefulness of the MKS system is that we can use decimal multipliers pr prefixes to extend the range of our measurements to cover the expected values, like Kilometers, Microseconds, milligrams, and every scientist in the world will know what you're talking about.
The symbols M, L, and T are usually reserved for Mass, Length, and Time respectively. Area, for example, is a derived unit which is Length times Length or L*L or L^2. The formulas that described the derived units are called dimensional formulae. Examples are L^3 for volume and L/T or L*T^-1 for speed. (^ means raise T to the power of -1 which is 1/T) (DJS)
In our macro world, all matter is composed of elements which are the smallest constituents that retain unique properties representative of that element. Elements can join together to form compounds that have their own chemical and physical properties. Atoms can connect together with others of their kind or different atoms to form a group called a molecule. Sometimes they even group in right or left handed formations but they always group in whole numbers or integers. You won't find a half of an oxygen atom. Sometimes the atoms lose or gain electrons and become ions, but the atoms nucleus remains intact unless it really gets blasted by something.
One atom of carbon can match up with two atoms of oxygen to form a stable molecule of carbon dioxide or CO2. The numeric subscript is the number of each of the elements that are hanging out together; one carbon C and two oxygen's O.
Sometimes even the molecules will combine together to form larger molecules, but the ratio of the number of atoms will remain the same. A copy of the periodic table of the elements is a necessity for solving the next problem. The periodic table shows the symbol for each element, its atomic number and its atomic weight. The atomic number shows how many protons are in the nucleus. The atomic weight of an atom shows the total of protons and neutrons in the nucleus and represents the true weight of an atom when compared to the atomic weights of other atoms. From a physics point of view, it would probably be better to call it atomic MASS than weight since an atom has less weight on the moon than on the earth but the mass stays the same everywhere.
The name of the book I am using is called CHEMISTRY, second edition by Steven S. Zumdahl at the University of Illinois. Start with page 113, problem 37. Determine the molecular formula to which the following empirical formula and molecular weight pertain. (a) SNH, 188.32.
Answer: The three elements sulfur (S), Nitrogen (N) and hydrogen (H) occur together in the lowest whole numbers of each element.. one of each. This could have been S2N2H2. Since SNH is the lowest whole number form, its called the empirical formula. (S1N1H1 means the same thing.) The S2N2H2 is called the molecular formula and may be actually how the molecule is put together. Now we have to find the atomic weight of the empirical formula. We do this by adding the atomic weights of each element multiplied by the number of atoms of each element. Sulfur is 32.06 times 1, Nitrogen is 14.01 times 1, and H is 1.008 times 1. This totals to a molecular mass of 47.078. Since the molecular mass was given as 188.32, we need to find how big this is compared to our empirical form. 188.32 / 47.078 = 4.000. So each of our elements in our empirical formula must be multiplied by 4 giving a molecular formula of S4N4H4. Using the same technique, try solving (b) where the empirical form and molecular mass is N1P1CL2, 347.66.
By the way, if you're wondering why the atomic mass of an element in the periodic table is not a whole number, its because each element is a mixture of the element and some of its isotopes where an isotope is the same element with one or more extra neutrons in its nucleus. This is how nature made our elements. Hydrogen, for example, has two isotopes, deuterium or heavy hydrogen, and Tritium or heavier hydrogen, famous as being a major component in hydrogen bombs and fusion reactors. (DJS)
(How handy...)
One eye from a turtle and one eye from a female wood duck..
Each animal knows the danger sensing capabilities of the other and they make use of it by staying together. Make sense?
I don't think I'll ever understand an Amoeba. Wheres its brain? what makes it go where it goes? I found this critter in a sample of Boise pond water. It was obviously going somewhere at a speed equal to a snail in granny gear. First all the stuff inside would start to flow out from the side, then stop and begin to flow out from some other part. I've never seen one leave any part of its body behind, perhaps because the whole body has an enclosing membrane.
The small sphere in the middle is the nucleus I think and maybe thats the "brains" of the outfit. Wherever it goes, thats where the amoeba goes. I'm not sure what all the other pieces are for- food vacuoles maybe. Does it feel? does it have a nose? Who knows...
Around 1965 I was teaching electronics at Boise Junior College. I was looking for some research projects for my students to work on. I had recently read an article on the persistence of vision using different light blink rates between left and right eyes. I began thinking about hearing and whether the ears could hear interfering beats between two sounds as they got close together in frequency.
The human ear can tune pianos and other instruments by listening for the zero beat between the string and a standard pitch fork. In this case the two sound sources strike both ears with similar beats. As the soundwaves move in and out of phase, the characteristic beats can be heard. I wondered if one of the sounds could be channeled in the left ear and the other in the right ear and whether the brain would do the mixing and produce a beat. I hooked up a pair of earphones and two audio generators; a separate sound source for each ear. I felt sure the brain would not mix the two signals.
After doing the test, I found no beats and concluded that there was no common pathway for the signals to mix in the auditory nerves or the brain. I wrote a paper and submitted it to a magazine, (I'm not sure but think it was Scientific American letters to the editor.) A few months later, this same magazine published an article saying you COULD hear a beat! Flabbergasted, I tried the experiment again and there it was. A beat that somehow had eluded me on the first try showed up on the second. It would be interesting to try it again some 40 years later. Perhaps there is an external bone conduction path through the skull that provides the signal mixing. Maybe some of my teachers were wrong.. there IS something between my ears after all. (DJS)
For many centuries the Pythagorean theorm has been taught in schools the world over. It's one useful beast and has had many proofs to validate its claim to fame. Here is another that I probably rediscovered thinking it may be original. (Ed. It was not)
Start by cutting a 7 inch square out of a piece of paper. Then trim the 7 inch square down to a 5 inch square. Try to keep the right angles as accurate as you can. Now place the 5 inch square in the 7 inch square hole and rotate the square till the corners lock against the sides of the 7 inch hole. Guess what! You end up with 4 triangular holes. In this case, believe it or not, the 4 triangles will be right triangles with the sides 3, 4, and 5 inches long.
Whats the area of the big 7 inch square hole? Right, 49 square inches. How about the 5 inch square? Right again! 25 square inches. Now whats the total area of the 4 right triangles? 49 - 25 = 24 square inches or 6 square inches each.
Now the big hole has an area of (A + B) times (A + B)
Multiply this out and you have A*A + 2*A*B + B*B (Where * stands for "times.")
The inside smaller square has an area of C * C.
By now you may notice the area of all four triangles is 1/2 * A * B * 4 or 2*A*B
If we take away the area of the 4 triangles from the big square hole, we see it equals the area of the small inside square C * C. Lets write it out..
A * A + 2*A*B + B * B - 2 *A*B = C*C
The 2*A*B terms cancel to give: A^2 + B^2 = C^2 which is the Pythagorean theorm.
(The ^ symbol in most calculators stands for raising to a power or A*A = A^2 )
This is the 101st year of the revolutionary publications made by a one, Albert Einstein. In 1905 these publications changed the world as we know it. No longer was our knowledge of the physical world just a conglomerate of unconnected ideas for which nothing matched.
Basically the problems began many years before when Galileo did some physical experiments on falling bodies and noted how velocities of objects seemed to add. Basically, if we run down the road at 5 miles per hour and throw a ball that leaves our hand with a speed of 10 miles per hour, we would expect to find the ball moving ahead with a speed of 15 mph. This is the addition of velocities rule that Galileo came up with. Excuse me for not defining the direction our runner threw the ball. Indeed an observer standing on the road would closely measure just this condition... but not quite. Some years before Einstein but after Galileo, Michael Faraday made some electrical charge measurements and came up with some formulas on how charges act when they move, and how the conductors carrying the charges act when THEY move. A Genius by the name of James C. Maxwell took those equations, added a few of his own, and stated that the movement of the electrical charges could result in an electromagnetic wave that would travel through space with a velocity C which is close to 186,000 miles per second. Maxwell also showed that regardless of the velocity of the emitting antenna, the velocity was still always C. He surmised correctly that light was probably this E-M radiation. A fellow by the name of H. Hertz shortly generated radio waves that we now use for our communications needs and proved Maxwell's mathematical work. Basically, there was nothing in Maxwells equations that showed the velocity of light to be anything else but C. They didn't even follow Galileo's addition of velocities rule. C depended only on two constants of nature called permativity and permeability of space.. how space reacts to electric and magnetic fields. Enter Einstein. He asked HOW CAN THAT BE? Using nothing more than high school algebra, Einstein showed that our measurements of distance and/or time didn't follow Galileo's ideas very well. Another scientist by the name of Lorentz had already come up with the relativity equations that Einstein derived, but for the wrong reasons. Even the famous Newton didn't depart from the ideas of Galileo, so his mathematical treatment of the movement of bodies was in error when the body velocities were near the velocity of light.
If you want to come up with the relativity equations that Einstein did, then solve the following problem: Two Motorboats have a water speed of C miles per second. They decide to have a race on a river where the downstream flow is V miles per second. One motorboat goes directly downstream for a measured shoreline distance of one mile and returns upstream to the starting gate. The other motorboat leaves at the same time as the first boat but moves directly across the stream for one mile and then comes back to the starting gate. The question.. which boat will win? According to Galileo, there will be a winner. But suppose we found that they finished in a dead heat.. This race was actually performed by a pair of scientists named Michelson and Morley, only they used light beams instead of motorboats. The race DID finish in a dead heat and only Einstein came up with a satisfactory answer.. Time and space kind of melted together. His writings indicate he knew the outcome of the light race, but had already made up his mind from the findings of Faraday and Maxwell. Now that Einstein has solved all this, even his famous E = MC^2 equation makes sense. If you push a mass, it will move faster. Push it long enough and its speed will increase till it begins to get close to light speed. Now Einstein used the maximum speed of anything is the speed of light as one of his postulates. If we keep pushing, but the speed doesn't increase much, then where does all this pushing energy go? This is exactly what would happen if the mass of the object got greater and greater. It now appears as if the mass of the object has increased as the speed approaches light speed. Thus it is this equation that ties mass and energy together, that is, the magical properties of C, the speed of light.
Lets calculate the cross current time out and back and label it T90 for Time at a direction 90 degrees with the current moving with a velocity V. On the way out across the current, the boat must crab upstream a bit to stay on course. Using a bit of vector algebra and the old faithful pythagorean theorm, we find the actual velocity over the 1 mile course is not C, but sqr(C^2-V^2) where sqr means square root of the quantity within the parenthesis. Now time out and back equals total distance divided by the velocity which is 2 miles divided by actual velocity, or T90 = 2/sqr(C^2 - V^2).
Ok, lets calculate the time for the down and upstream part of the course. Label it T0 (T sub zero). This is for zero degrees with respect to the direction of the current that is moving with a velocity V. Time down = 1 mile divided by velocity down. Time back = 1 mile divided by velocity back. T0 = 1 / (C+V) + 1 / (C-V). With a bit of algebra, this simplifies to T0 = 2*C / (C^2 - V^2).
Now find the ratio between T0 and T90 to see which one is longer.
T0 / T90 = [2*C / (C^2 - V^2)] / [2 / sqr(C^2 - V^2)] simplifies to 1 / sqr( 1 - V^2/C^2). This is Lorenz and Einstein's equation for special relativity. Now if there were no current V, the time ratio is 1/1 or the race ends in a dead heat. However if V is not zero, then the value becomes greater than one meaning that T0 will be greater than T90 and that means the boat going cross stream and back would be the winner.
The relevance of this race is that in order for it to come out in a tie, either the path for the cross stream racer was a bit longer than one mile, or the clock beating out the time for the cross stream racer was a bit faster. The paradox was that when the race was performed with light rays instead of motorboats, the cross current time equaled the down current time, just as if there were no current at all.....
Shortly after publishing the paper on special relativity, Einstein quickly realized another outcome of his postulate that the velocity of light was the upper speed limit for all motion of masses. If a mass is moving near the speed of light, it will require much more energy to make it move any faster than Newtons laws would give. It's like the mass of a body was not a constant when compared to the same mass traveling at a different velocity. In fact the mass of a body must have the relativistic correction applied to it. That factor is: sqr(c^2/(c^2 -v^2)). (See the derivation above.) So the mass appears to be rest mass times the factor.So the momentum of a body is not mass times velocity anymore, but mass times velocity times the relativity factor. Now Newton and Leibnitz invented a form of math called calculus which would allow one to sum up a series of small changes to calculate the total change. If we do that with the momentum of a body with respect to its velocity, we find the total energy of the moving body is: E = M* C*sqr(C^2 - V^2) or if the mass is at rest, (V=0), then E = M*C^2 where E is kinetic energy of the mass in Joules, M is the rest mass in kilograms, and C is the velocity of light in meters per second, which turns out to be a HUGE number.
(C = 300,000,000 meters/second nearly.) Thus even a mass at rest was shown to have an energy content unknown before Einstein. Mass and energy were equivalent and interchangeable.
This discovery immediately solved the problem of where stars got the energy to keep on shining for billions of years beyond their gravitational lifetimes. Gravity heated the protostar to ignition and then E = MC2 took over. Let there be light!
The C's and R's in the top row are the values for the capacitors and resistors shown in the schematic at the bottom. For example, C=.01*10-6 means the value for the capacitor is .01 microfarads. The R's are given in ohms of resistance and f is the frequency in hertz. The great thing about CAD programs is that once the mathematical model is constructed, the value of any component can be changed and the program will instantly calculate the new output and even draw a response curve saving many hours of recalculation.
(From an original essay published by the SOCIETY OF AMATEUR SCIENTISTS)
I
I would like to explain my feelings about the values of an informal science background and the evolution of those feelings into a life long interest.
In my youth I was not a particularly good student mostly because I was impatient and argumenative. One day I noticed a rather large telescope prominently displayed through a neighbors picture window. My curiosity grew and I soon gained the courage to ring the doorbell and meet the telescopes owner. The gentleman was happy to explain the workings of this mysterious device and invited me to look at the stars. This meeting instigated an interest in astronomy that I carry to this day, some 70 years later.
The Buhl planetarium in Pittsburgh Pa was a huge and wonderous place with star shows, science fairs, exhibits, and everything to ignite the imagination of a young boy. Astronomy was my reason for starting regular visits to the planetarium. They had amateur astronomer meetings, Shops to grind your own telescope mirrors and lots of knowledgeable and helpful people.
I soon found that the science of astronomy was not the only science in the world. After all, the planetarium also had beautiful exhibits in a number of physical and life sciences. I soon found myself wanting to learn all of it. Perhaps my reason at the time was more selfish. I wanted to know more than my school chums about these complicated devices and ideas. I embarked on a regimen of self study which gave me a spotty but gradually expanding mastery of math and general science. The self study habit has never left me and when I graduated from high school, the first jobs I held were in radio-TV service and at the Allegheny observatory as a computer.
Through my working years my accumulated knowledge of electronics, science and math continued to open doors to technical jobs even though I had no formal college training. Eventually, with the help of military schooling I began teaching electronics to others. This launched my teaching career. I taught electronics in the military, then private tech schools, and finally in college level vocational-technical programs.
I'm retired now with a circle of friends that include a chemist, a physicist, an AV design engineer, and an electrical engineer, all retired and working in planetarium-like places on a volunteer basis. We all love to get together and share thoughts and ideas on anything of a scientific or technological bent. We are all aging now, and our families are beginning to bog down with age related medical problems, but as long as we can talk and do science, we're all young again. Our passion for science and technology is like a flame - small and searing when young, warm and comfortable in age.
I found I enjoy seeing the spark of interest and understanding spread across the faces of young people just as it must have shown on my own. I began going to grade schools in the capacity of the government AMERICA READS program. My job was to help the kids learn to read better. I soon found that analyzing their stories, they became more interested. As an incentive, I would make or buy some device, usually from a science supply house like Edmund Scientific. The kids were thrilled to see these demos in static electricity, micro chemistry, rocket balloons, optical illusions, all the things I remembered getting excited about years ago. Soon the teachers wanted me to give the demos to entire classes, which brings me to the present. I now have a set of science demos for 4th to 6th grade and although the expense was considerable, it also fits in with my own lifestyle. Even us old-timers love to see old science with a new twist!
In our Universe there are some things so common, so ubiquitous that they are seen everywhere. One we have known from birth was probably when we were rocked in our Mothers arms. It is a motion that has a periocity to it. The periocity is measured in time. It may be a second or day or week, or any time you like. It is as natural as we are. It is a product of life...of stored energy. The other natural wave is one more akin to death. It begins large an vigorous and simply drops its energy level until after much time has past, we cannot see how little it has become. I'll write about these two waves types in the next few days. They have both been a large part of my life.
To illustrate the periodic wave, I'll give you a problem. You're at the top of a very tall building. The building is swaying very slowly. You make some measurements and find at noon on the first day, the top of the building has moved 1.17365 meters since noon the day before. The next day at noon, the building has swayed to a distance of 1.19081 meters. One more day at noon and the building is at 1.20791 meters deflection. The last day at noon finds the building to be deflected to 1.22495 meters. We will make the a-priori assumption the building is swaying in a transcendental motion called a sine curve. Now describe the motion of the top of the building, How many days it will take to sway one full cycle, the offset distance the building was at before it began moving, and whether the wave was actually a true, pure sine wave. In the next few days, I'll try to answer this problem and why it had a real world use when I solved the general form.
Now I can tell you a bit more about it. I was working for a company that made very low frequency receivers....20 to 60 kilohertz range. At these frequencies, radio waves can follow the Earth surface and go worldwide, even underwater I've been told. Purpose? to transmit extremely accurate time standards to submarines, remote locations. The site usually had a very accurate oscillator. They just needed to lock it in to a common standard oscillator. So the time standard was sent by VLF radio, and the local oscillator or clock was controlled by the standard. Here are two sine wave oscillators, one controlling the other. What do you get when you beat two sine waves together? Another very low frequency sine wave or beat. When they lock, the beat goes constant. My mathematical contribution was to measure 4 equal timed beat signals and calculate the period of the time difference between the signals. I will tell you now..I learned a lot of math trying to solve 4 simultaneous trig equations. Recently, using a CAD program, I solved it in about 25 minutes. CAD programs weren't available to many in 1967.
The 4 equations look like the following..
e1 = E*sin (2*pi* F* T +0) + C e2 = E*sin (2*pi* F* (T+dt))+C e3 = E*sin (2*pi*F*(T+2*dt))+ C
e4 = E*sin (2*pi* F* (T+3*dt) + C Where e1,e2,e3, and e4 are the 4 measured signals of the beat. E is the peak that the signal rises to, F is the frequency of the beat and T is the zero starting time, and dt is the common time interval between e measurements. C is the common displacement e might have from zero.
There's your 4 equations. Knowns are e1, e2, e3, e4, dt and of course, pi. Unknowns are:
E, F, T, and C. The numbers I gave you at first are the 4 e values. dt is one day. Next time I will give you the derived solution. By the way, the solution was published in the engineering magazine FREQUENCY in 1969. Good luck figuring that one out long hand!
Its time for a little enlightenment! The actual incremental angle da (radians) that the sine wave steps through in time dt is.... da = acos[1/2*((e3-e0)/(e2-e1))-1] (acos is arc or inverse)
From this solution, all the other unknowns may be solved. Incidently, there will be times when no solution will be found. It doesn't mean the process has failed. It simply means there was no sine wave that would fit the points, or the wave was not a single sine if it was periodic. The angular velocity capital omega is the angle da moved in time dt or W = da/dt. This should be enough info to get an interested party started, perhaps even carry it much further than I have.
The other curve that succumbs to a similar analysis is the natural logarithmic curve. four known points also defines a definite solution for the equation: Y =Ln(x) + C
If one of the curves doesn't fit, try the other. Maybe you'll get lucky...(DJS)