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| [[File:Kater pendulum use.png|thumb|250px|Kater's original pendulum, showing use, from Kater's 1818 paper. The pendulum's [[Frequency|period]] was timed by comparing its swing with the pendulum in the precision clock behind it. The sight ''(left)'' was used to avoid [[parallax error]].]]
| | Msvcr71.dll is an important file which assists help Windows process different components of the program including important files. Specifically, the file is utilized to aid run corresponding files in the "Virtual C Runtime Library". These files are important in accessing any settings which support the different applications plus programs inside the system. The msvcr71.dll file fulfills numerous important functions; though it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer might have a difficult time processing plus reading components of the program. However, users require not panic considering this problem can be solved by following several procedures. And I usually show you certain strategies about Msvcr71.dll.<br><br>Google Chrome crashes on Windows 7 by the corrupted cache contents and problems with all the stored browsing data. Delete the browsing information plus obvious the contents of the cache to solve this issue.<br><br>The Windows registry is a system database of info. Windows plus alternative software store a lot of settings plus alternative info inside it, plus retrieve such information within the registry all of the time. The registry is furthermore a bottleneck inside which considering it's the heart of the running system, any problems with it may result mistakes plus bring the running system down.<br><br>Fixing tcpip.sys blue screen is easy to do with registry repair software.Trying to fix windows blue screen error on your will be challenging because in the event you remove or damage the registry it can result severe damage to a computer. The registry must be cleaned and all erroneous and incomplete info removed to stop blue screen errors from occurring.The advantage of registry repair software is not limited to simply getting rid of the blue screen on startup.We may be amazed at the better plus more improved speed plus performance of your computer system after registry cleaning is completed. Registry cleaning can definitely develop your computer's working abilities, particularly when we choose a certain registry repair software that is extremely efficient.<br><br>Another thing you need to check is whether the [http://bestregistrycleanerfix.com/system-mechanic iolo system mechanic] system you are considering has the ability to identify files and programs which are wise. One of the registry cleaner programs you might try is RegCure. It is helpful for speeding up and cleaning up difficulties on the computer.<br><br>The system is designed plus built for the purpose of helping we accomplish tasks and not be pestered by windows XP error messages. When there are errors, what do you do? Some folks pull their hair plus cry, while those sane ones have their PC repaired, while those actually wise ones research to have the mistakes fixed themselves. No, these errors were not equally designed to rob you off a cash and time. There are aspects which you can do to really avoid this from happening.<br><br>Perfect Optimizer is a advantageous Registry Product, updates consistently and has lots of features. Despite its price, there are which the update are actually worthwhile. They offer a lot of support through phone, send and forums. You might want to head over to the free trial to check it out for yourself.<br><br>A registry cleaner is a program that cleans the registry. The Windows registry usually gets flooded with junk information, info that has not been removed from uninstalled programs, erroneous file organization and other computer-misplaced entries. These neat small program software tools are quite prevalent today and you are able to find very a few good ones on the Internet. The superior ones give you option to maintain, clean, update, backup, and scan the System Registry. Whenever it finds supposedly unwelcome elements in it, the registry cleaner lists them and recommends the consumer to delete or repair these orphaned entries and corrupt keys. |
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| A '''Kater's pendulum''' is a reversible freeswinging [[pendulum]] invented by British physicist and army captain [[Henry Kater]] in 1817<ref>{{cite journal
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| | last = Kater
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| | first = Henry
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| | authorlink =
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| | coauthors =
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| | title = An account of experiments for determining the length of the pendulum vibrating seconds in the latitude of London
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| | journal = Phil. Trans. R. Soc.
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| | volume = 104
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| | issue = 33
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| | pages = p.109
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| | publisher =
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| | location = London
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| | year = 1818
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| | url = http://books.google.com/?id=uaQOAAAAIAAJ&pg=PA83&lpg=PA83&dq=%22Henry+Kater%22+kater+pendulum
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| | doi =
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| | accessdate = 2008-11-25}}</ref> for use as a [[gravimeter]] instrument to measure the local [[acceleration of gravity]]. Its advantage is that, unlike previous pendulum gravimeters, the pendulum's [[centre of gravity]] and [[center of oscillation]] don't have to be determined, allowing greater accuracy. For about a century, until the 1930s, Kater's pendulum and its various refinements remained the standard method for measuring the strength of the Earth's gravity during [[Geodetics|geodetic]] surveys. It is now used only for demonstrating pendulum principles.
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| ==Description==
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| The pendulum consists of a rigid metal bar with two pivot points, one near each end of the bar. It can be suspended from either pivot and swung. It also has either an adjustable weight that can be moved up and down the bar, or one adjustable pivot, to adjust the periods of swing. In use, it is swung from one pivot, and the [[Frequency|period]] timed, and then turned upside down and swung from the other pivot, and the period timed. The movable weight (or pivot) is adjusted until the two periods are equal. At this point the period is equal to the period of an 'ideal' simple pendulum of length equal to the distance between the pivots. From the period and the measured distance between the pivots, the acceleration of gravity can be calculated with great precision from the periodicity equation for a simple pendulum, (1) below. | |
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| ==Gravity measurement with pendulums==
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| A pendulum can be used to measure the [[acceleration of gravity]] ''g'' because its [[Frequency|period]] of swing ''T'' depends only on ''g'' and its length ''L'':<ref>{{cite web
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| | last = Nave
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| | first = C. R.
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| | authorlink =
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| | coauthors =
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| | title = Simple Pendulum
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| | work = Hyperphysics
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| | publisher = Dept. of Physics and Astronomy, Georgia State Univ.
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| | year = 2005
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| | url = http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html
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| | doi =
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| | accessdate = 2009-02-20}}</ref>
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| :<math>T = 2 \pi \sqrt { \frac{L}{g} } \qquad \qquad \qquad (1)\,</math>
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| So by measuring the length ''L'' and period ''T'' of a pendulum, ''g'' can be calculated. The first person to discover that gravity varied over the Earth's surface was French scientist [[Jean Richer]], who in 1671 was sent on an expedition to [[Cayenne]], [[French Guiana]] by the French [[Académie des Sciences]], assigned the task of making measurements with a [[pendulum clock]]. Through the observations he made in the following year, Richer determined that the clock was 2½ minutes per day slower than at Paris, or equivalently the length of a pendulum with a swing of one second there was 1¼ Paris ''lines'', or 2.6 mm, shorter than at Paris.<ref>{{cite book
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| | last = Poynting
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| | first = John Henry
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| | authorlink =
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| | coauthors = Joseph John Thompson
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| | title = A Textbook of Physics, 4th Ed.
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| | publisher = Charles Griffin & Co.
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| | year = 1907
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| | location = London
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| | pages = 20
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| | url = http://books.google.com/?id=TL4KAAAAIAAJ&pg=PA20#
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| | doi =
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| | isbn = }}</ref><ref>{{cite conference
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| | first = Lenzen
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| | last = Victor F.
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| | authorlink =
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| | coauthors = Robert P. Multauf
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| | title = Paper 44: Development of gravity pendulums in the 19th century
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| | booktitle = United States National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
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| | pages = 307
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| | publisher = Smithsonian Institution Press
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| | year = 1964
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| | location = Washington
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| | url = http://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA307
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| | doi =
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|
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| | accessdate = 2009-01-28}}</ref> It was realized by the scientists of the day, and proven by [[Isaac Newton]] in 1687, that this was due to the fact that the Earth was not a perfect sphere but slightly [[Oblate spheroid|oblate]]; it was thicker at the equator because of the Earth's rotation. Since the surface was farther from the Earth's center at Cayenne than at Paris, gravity was weaker there. Since that time pendulums began to be used as precision [[gravimeter]]s, taken on voyages to different parts of the world to measure the local gravitational acceleration. The accumulation of geographical gravity data resulted in more and more accurate models of the overall shape of the Earth.
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| Pendulums were so universally used to measure gravity that, in Kater's time, the local strength of gravity was usually expressed not by the value of the acceleration ''g'' which is now used, but by the length at that location of the ''[[seconds pendulum]]'', a pendulum with a period of two seconds, so each swing takes one second. It can be seen from equation (1) that for a seconds pendulum, the length is simply proportional to ''g'':
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| :<math>g = \pi^2 L \,</math>
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| ==Inaccuracy of gravimeter pendulums==
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| In Kater's time, the period ''T'' of pendulums could be measured very precisely by timing them with precision clocks set by the passage of stars overhead. Prior to Kater's discovery, the accuracy of ''g'' measurements was limited by the difficulty of measuring the other factor ''L'', the length of the pendulum, accurately. ''L'' in equation (1) above was the length of an ideal mathematical 'simple pendulum' consisting of a point mass swinging on the end of a massless cord. However the 'length' of a real pendulum, a swinging rigid body, known in mechanics as a [[compound pendulum]], is more difficult to define. In 1673 Dutch scientist [[Christiaan Huygens]] in his mathematical analysis of pendulums, ''Horologium Oscillatorium'', showed that a real pendulum had the same period as a simple pendulum with a length equal to the distance between the pivot point and a point called the ''[[center of oscillation]]'', which is located under the pendulum's [[center of gravity]] and depends on the mass distribution along the length of the pendulum. The problem was there was no way to find the location of the center of oscillation in a real pendulum accurately. It could theoretically be calculated from the shape of the pendulum if the metal parts had uniform density, but the metallurgical quality and mathematical abilities of the time didn't allow the calculation to be made accurately.
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| To get around this problem, most early gravity researchers, such as [[Jean Picard]] (1669), [[Charles Marie de la Condamine]] (1735), and [[Jean-Charles de Borda]] (1792) approximated a simple pendulum by using a metal sphere suspended by a light wire. If the wire had negligible mass, the center of oscillation was close to the center of gravity of the sphere. But even finding the center of gravity of the sphere accurately was difficult. In addition, this type of pendulum inherently wasn't very accurate. The sphere and wire didn't swing back and forth as a rigid unit, because the sphere acquired a slight [[angular momentum]] during each swing. Also the wire stretched elastically during the pendulum's swing, changing ''L'' slightly during the cycle.
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| ==Kater's solution==
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| However, in ''Horologium Oscillatorium'', Huygens had also proved that the pivot point and the center of oscillation were interchangeable. That is, if any pendulum is suspended upside down from its center of oscillation, it has the same period of swing, and the new center of oscillation is the old pivot point. The distance between these two conjugate points was equal to the length of a simple pendulum with the same period.
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| As part of a committee appointed by the [[Royal Society]] in 1816 to reform British measures, Kater had been contracted by the House of Commons to determine accurately the length of the seconds pendulum in London.<ref>{{cite book
| |
| | last = Zupko
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| | first = Ronald Edward
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| | authorlink = Ronald Edward Zupko
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| | coauthors =
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| | title = Revolution in Measurement: Western European Weights and Measures since the Age of Science
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| | publisher = Diane Publishing
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| | year = 1990
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| | location = New York
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| | pages = 107–110
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| | url = http://books.google.com/?id=twUNAAAAIAAJ&pg=PA110
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| | doi =
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| | isbn = 0-87169-186-8}}</ref> He realized Huygens principle could be used to find the center of oscillation, and so the length ''L'', of a rigid (compound) pendulum. If a pendulum were hung upside down from a second pivot point which could be adjusted up and down on the pendulum's rod, and the second pivot were adjusted until the pendulum had the same period as it did when swinging right side up from the first pivot, the second pivot would be at the center of oscillation, and the distance between the two pivot points would be ''L''.
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| Kater wasn't the first to have this idea.<ref>[http://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA315 Lenzen & Multauf 1964], p.315</ref>
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| <ref>[http://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA12# Poynting & Thompson 1907], p.12</ref> French mathematician [[Gaspard de Prony]] first proposed a reversible pendulum in 1800, but his work was not published till 1889. In 1811 [[Bohnenberger|Friedrich Bohnenberger]] again discovered it, but Kater independently invented it and was first to put it in practice.
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| [[File:Kater pendulum.png|thumb|400px|Drawing of Kater's pendulum <br>''<span style="color:red;">(a)</span>'' opposing knife edge pivots from which pendulum is suspended <br>''<span style="color:red;">(b)</span>'' fine adjustment weight moved by adjusting screw <br>''<span style="color:red;">(c)</span>'' coarse adjustment weight clamped to rod by setscrew<br>''<span style="color:red;">(d)</span>'' bob <br>''<span style="color:red;">(e)</span>'' pointers for reading]]
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| ==The pendulum==
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| Kater built a pendulum consisting of a brass rod about 2 meters long, 1½ inches wide and one-eighth inch thick, with a weight ''<span style="color:red;">(d)</span>'' on one end.<ref>{{cite book
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| | last = Elias Loomis
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| | first =
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| | authorlink =
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| | coauthors =
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| | title = Elements of Natural Philosophy, 4th Ed.
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| | publisher = Harper & Brothers
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| | year = 1864
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| | location = New York
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| | pages = 109
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| | url = http://books.google.com/?id=-jwJAAAAIAAJ&pg=PA109
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| | doi =
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| | isbn = }}</ref> For a low friction pivot he used a pair of short triangular 'knife' blades attached to the rod. In use the pendulum was hung from a bracket on the wall, supported by the edges of the knife blades resting on flat agate plates. The pendulum had two of these knife blade pivots ''<span style="color:red;">(a)</span>'', facing one another, about a meter (40 in) apart, so that a swing of the pendulum took approximately one second when hung from each pivot.
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| Kater found that making one of the pivots adjustable caused inaccuracies, making it hard to keep the axis of both pivots precisely parallel. Instead he permanently attached the knife blades to the rod, and adjusted the periods of the pendulum by a small movable weight ''<span style="color:red;">(b,c)</span>'' on the pendulum shaft. Since gravity only varies by a maximum of 0.5% over the Earth, and in most locations much less than that, the weight only had to be adjusted slightly. Moving the weight toward one of the pivots decreased the period when hung from that pivot, and increased the period when hung from the other pivot. This also had the advantage that the precision measurement of the separation between the pivots only had to be made once.
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| ==Experimental procedure==
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| To use, the pendulum was hung from a bracket on a wall, with the knife blade pivots supported on two small horizontal agate plates, in front of a precision pendulum clock to time the period. It was swung first from one pivot, and the oscillations timed, then turned upside down and swung from the other pivot, and the oscillations timed again. The small weight ''<span style="color:red;">(c)</span>'' was adjusted with the adjusting screw, and the process repeated until the pendulum had the same period when swung from each pivot. By putting the measured period ''T'', and the measured distance between the pivot blades ''L'', into the period equation (1), ''g'' could be calculated very accurately.
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| Kater performed 12 trials. He measured the period of his pendulum very accurately using the clock pendulum by the ''method of coincidences''; timing the interval between the ''coincidences'' when the two pendulums were swinging in synchronism. He measured the distance between the pivot blades with a microscope comparator, to an accuracy of 10<sup>−4</sup> in. (2.5 μm). As with other pendulum gravity measurements, he had to apply small corrections to the result for a number of variable factors:
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| *the finite width of the pendulum's swing, which increased the period
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| *temperature, which caused the length of the rod to vary due to [[thermal expansion]]
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| *atmospheric pressure, which reduced the effective mass of the pendulum by the buoyancy of the displaced air, increasing the period
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| *altitude, which reduced the gravitational force with distance from the center of the Earth. Gravity measurements are always referenced to [[sea level]].
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| He gave his result as the length of the [[seconds pendulum]]. After corrections, he found that the mean length of the solar seconds pendulum at London, at sea level, at 62°F, swinging in vacuum, was 39.1386 inches. This is equivalent to a gravitational acceleration of 9.81158 m/s<sup>2</sup>. The largest variation of his results from the mean was {{convert|0.00028|in|um}}. This represented a precision of gravity measurement of 7(10<sup>−6</sup>) (7 [[milligal]]s).
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| In 1824, the British Parliament made Kater's measurement of the seconds pendulum the official standard of length for defining the [[yard (unit of length)|yard]].
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| ==Use==
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| The large increase in gravity measurement accuracy made possible by Kater's pendulum established [[gravimetry]] as a regular part of [[geodesy]]. To be useful, it was necessary to find the exact location (latitude and longitude) of the 'station' where a gravity measurement was taken, so pendulum measurements became part of [[surveying]]. Kater's pendulums were taken on the great historic [[Geodetic surveying|geodetic surveys]] of much of the world that were being done during the 19th century. In particular, Kater's pendulums were used in the [[Great Trigonometric Survey]] of India.
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| Reversible pendulums remained the standard method used for absolute gravity measurements until they were superseded by free-fall [[gravimeter]]s in the 1950s.<ref name="Torge">{{cite book
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| | last = Torge
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| | first = Wolfgang
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| | authorlink =
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| | coauthors =
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| | title = Geodesy: An Introduction
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| | publisher = Walter de Gruyter
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| | year = 2001
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| | location =
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| | pages = 177
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| | url = http://books.google.com/?id=pFO6VB_czRYC&pg=PA177&lpg=PA177
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| | doi =
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| | isbn = 3-11-017072-8}}</ref>
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| ===Repsold–Bessel pendulum===
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| [[File:Repsold pendulum.png|thumb|80px|Repsold pendulum.]]
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| Repeatedly timing each period of a Kater pendulum, and adjusting the weights until they were equal, was time consuming and error-prone. [[Friedrich Bessel]] showed in 1826 that this was unnecessary. As long as the periods measured from each pivot, T<sub>1</sub> and T<sub>2</sub>, are close in value, the period ''T'' of the equivalent simple pendulum can be calculated from them:<ref>[http://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA15# Poynting & Thompson 1907], p.15</ref>
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| :<math>T^2 = \frac{T_1^2 + T_2^2}{2} + \frac{T_1^2 - T_2^2}{2} \left ( \frac {h_1 + h_2}{h_1-h_2} \right ) \, \qquad \qquad \qquad (2)</math>
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| Here <math>h_1\,</math> and <math>h_2\,</math> are the distances of the two pivots from the pendulum's center of gravity, so <math>h_1 + h_2 = L\,</math>. <math>h_1\,</math> and <math>h_2\,</math> cannot be measured with great accuracy. They are found by balancing the pendulum on a knife edge to find its center of gravity, and measuring the distances of each of the pivots from the center of gravity. However, because <math>T_1^2 - T_2^2\,</math> is so much smaller than <math>T_1^2 + T_2^2\,</math>, the second term on the right in the above equation is small compared to the first, so <math>h_1 - h_2\,</math> doesn't have to be determined with high accuracy, and the balancing procedure described above is sufficient to give accurate results.
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| Therefore the pendulum doesn't have to be adjustable at all, it can simply be a rod with two pivots. As long as each pivot is close to the [[center of oscillation]] of the other, so the two periods are close, the period ''T'' of the equivalent simple pendulum can be calculated with equation (2), and the gravity can be calculated from ''T'' and ''L'' with (1).
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| In addition, Bessel showed that if the pendulum was made with a symmetrical shape, but internally weighted on one end, the error caused by effects of air resistance would cancel out. Also, another error caused by the finite diameter of the pivot knife edges could be made to cancel out by interchanging the knife edges.
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| Bessel didn't construct such a pendulum, but in 1864 Adolf Repsold, under contract to the Swiss Geodetic Commission, developed a symmetric pendulum 56 cm long with interchangeable pivot blades, with a period of about 3/4 second. The Repsold pendulum was used extensively by the Swiss and Russian Geodetic agencies, and in the [[Great Trigonometric Survey|Survey of India]]. Other widely used pendulums of this design were made by [[Charles Sanders Peirce|Charles Peirce]] and C. Defforges.
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.shef.ac.uk/polopoly_fs/1.14262!/file/G1_Katers_script_V2.1.pdf The Accurate Measurement of ''g'' using Kater's pendulum, U. of Sheffield] Has derivation of equations
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| *[http://books.google.com/books?id=LWEJAAAAQAAJ&pg=PA407&dq=kater+pendulum&lr=&as_brr=1 Kater, Henry (June 1818) An Account of the Experiments for determining the length of the pendulum vibrating seconds in the latitude of London, The Edinburgh Review, Vol. 30, p.407] Has detailed account of experiment, description of pendulum, value determined, interest of French scientists
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| {{DEFAULTSORT:Kater's Pendulum}}
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| [[Category:Pendulums]]
| |
Msvcr71.dll is an important file which assists help Windows process different components of the program including important files. Specifically, the file is utilized to aid run corresponding files in the "Virtual C Runtime Library". These files are important in accessing any settings which support the different applications plus programs inside the system. The msvcr71.dll file fulfills numerous important functions; though it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer might have a difficult time processing plus reading components of the program. However, users require not panic considering this problem can be solved by following several procedures. And I usually show you certain strategies about Msvcr71.dll.
Google Chrome crashes on Windows 7 by the corrupted cache contents and problems with all the stored browsing data. Delete the browsing information plus obvious the contents of the cache to solve this issue.
The Windows registry is a system database of info. Windows plus alternative software store a lot of settings plus alternative info inside it, plus retrieve such information within the registry all of the time. The registry is furthermore a bottleneck inside which considering it's the heart of the running system, any problems with it may result mistakes plus bring the running system down.
Fixing tcpip.sys blue screen is easy to do with registry repair software.Trying to fix windows blue screen error on your will be challenging because in the event you remove or damage the registry it can result severe damage to a computer. The registry must be cleaned and all erroneous and incomplete info removed to stop blue screen errors from occurring.The advantage of registry repair software is not limited to simply getting rid of the blue screen on startup.We may be amazed at the better plus more improved speed plus performance of your computer system after registry cleaning is completed. Registry cleaning can definitely develop your computer's working abilities, particularly when we choose a certain registry repair software that is extremely efficient.
Another thing you need to check is whether the iolo system mechanic system you are considering has the ability to identify files and programs which are wise. One of the registry cleaner programs you might try is RegCure. It is helpful for speeding up and cleaning up difficulties on the computer.
The system is designed plus built for the purpose of helping we accomplish tasks and not be pestered by windows XP error messages. When there are errors, what do you do? Some folks pull their hair plus cry, while those sane ones have their PC repaired, while those actually wise ones research to have the mistakes fixed themselves. No, these errors were not equally designed to rob you off a cash and time. There are aspects which you can do to really avoid this from happening.
Perfect Optimizer is a advantageous Registry Product, updates consistently and has lots of features. Despite its price, there are which the update are actually worthwhile. They offer a lot of support through phone, send and forums. You might want to head over to the free trial to check it out for yourself.
A registry cleaner is a program that cleans the registry. The Windows registry usually gets flooded with junk information, info that has not been removed from uninstalled programs, erroneous file organization and other computer-misplaced entries. These neat small program software tools are quite prevalent today and you are able to find very a few good ones on the Internet. The superior ones give you option to maintain, clean, update, backup, and scan the System Registry. Whenever it finds supposedly unwelcome elements in it, the registry cleaner lists them and recommends the consumer to delete or repair these orphaned entries and corrupt keys.