title: David Hume

text: However, in works such as "Of Superstition and Enthusiasm", Hume specifically seems to support the standard religious views of his time and place. This still meant that he could be very critical of the Catholic Church, dismissing it with the standard Protestant accusations of superstition and idolatry, as well as dismissing as idolatry what his compatriots saw as uncivilised beliefs. He also considered extreme Protestant sects, the members of which he called "enthusiasts", to be corrupters of religion. By contrast, in "The Natural History of Religion", Hume presents arguments suggesting that polytheism had much to commend it over monotheism. Additionally, when mentioning religion as a factor in his "History of England", Hume uses it to show the deleterious effect it has on human progress. In his "Treatise of Human Nature", Hume wrote: "Generally speaking, the errors in religions are dangerous; those in philosophy only ridiculous."


title: David Hume

text: Lou Reich (1998) argues that Hume was a religious naturalist and rejects interpretations of Hume as an atheist. Paul Russell (2008) writes that Hume was plainly sceptical about religious belief, although perhaps not to the extent of complete atheism. He suggests that Hume's position is best characterised by the term "irreligion," while philosopher David O'Connor (2013) argues that Hume's final position was "weakly deistic". For O'Connor, Hume's "position is deeply ironic. This is because, while inclining towards a weak form of deism, he seriously doubts that we can ever find a sufficiently favourable balance of evidence to justify accepting any religious position." He adds that Hume "did not believe in the God of standard theism ... but he did not rule out all concepts of deity", and that "ambiguity suited his purposes, and this creates difficulty in definitively pinning down his final position on religion".


title: David Hume

text: One of the traditional topics of natural theology is that of the existence of God, and one of the "a posteriori" arguments for this is the "argument from design" or the teleological argument. The argument is that the existence of God can be proved by the design that is obvious in the complexity of the world, which "Encyclopædia Britannica" states is "the most popular", because it is:…the most accessible of the theistic arguments ... which identifies evidences of design in nature, inferring from them a divine designer ... The fact that the universe as a whole is a coherent and efficiently functioning system likewise, in this view, indicates a divine intelligence behind it.In "An Enquiry Concerning Human Understanding", Hume wrote that the design argument seems to depend upon our experience, and its proponents "always suppose the universe, an effect quite singular and unparalleled, to be the proof of a Deity, a cause no less singular and unparalleled". Philosopher Louise E. Loeb (2010) notes that Hume is saying that only experience and observation can be our guide to making inferences about the conjunction between events. However, according to Hume: We observe neither God nor other universes, and hence no conjunction involving them. There is no observed conjunction to ground an inference either to extended objects or to God, as unobserved causes.Hume also criticised the argument in his "Dialogues Concerning Natural Religion" (1779). In this, he suggested that, even if the world is a more or less smoothly functioning system, this may only be a result of the "chance permutations of particles falling into a temporary or permanent self-sustaining order, which thus has the appearance of design".


title: David Hume

text: A century later, the idea of order without design was rendered more plausible by Charles Darwin's discovery that the adaptations of the forms of life result from the natural selection of inherited characteristics. For philosopher James D. Madden, it is "Hume, rivaled only by Darwin, [who] has done the most to undermine in principle our confidence in arguments from design among all figures in the Western intellectual tradition".


title: David Hume

text: Finally, Hume discussed a version of the anthropic principle, which is the idea that theories of the universe are constrained by the need to allow for man's existence in it as an observer. Hume has his sceptical mouthpiece Philo suggest that there may have been many worlds, produced by an incompetent designer, whom he called a "stupid mechanic". In his "Dialogues Concerning Natural Religion", Hume wrote:


title: David Hume

text: Many worlds might have been botched and bungled throughout an eternity, ere this system was struck out: much labour lost: many fruitless trials made: and a slow, but continued improvement carried on during infinite ages in the art of world-making.


title: David Hume

text: American philosopher Daniel Dennett has suggested that this mechanical explanation of teleology, although "obviously ... an amusing philosophical fantasy", anticipated the notion of natural selection, the 'continued improvement' being like "any Darwinian selection algorithm".


title: David Hume

text: In his discussion of miracles, Hume argues that we should not believe miracles have occurred and that they do not therefore provide us with any reason to think God exists. In "An Enquiry Concerning Human Understanding" (Section 10), Hume defines a miracle as "a transgression of a law of nature by a particular volition of the Deity, or by the interposition of some invisible agent". Hume says we believe an event that has frequently occurred is likely to occur again, but we also take into account those instances where the event did not occur:


title: David Hume

text: A wise man ... considers which side is supported by the greater number of experiments. ... A hundred instances or experiments on one side, and fifty on another, afford a doubtful expectation of any event; though a hundred uniform experiments, with only one that is contradictory, reasonably beget a pretty strong degree of assurance. In all cases, we must balance the opposite experiments ... and deduct the smaller number from the greater, in order to know the exact force of the superior evidence.


title: David Hume

text: Hume discusses the testimony of those who report miracles. He wrote that testimony might be doubted even from some great authority in case the facts themselves are not credible: "[T]he evidence, resulting from the testimony, admits of a diminution, greater or less, in proportion as the fact is more or less unusual."


title: David Hume

text: Although Hume leaves open the possibility for miracles to occur and be reported, he offers various arguments against this ever having happened in history. He points out that people often lie, and they have good reasons to lie about miracles occurring either because they believe they are doing so for the benefit of their religion or because of the fame that results. Furthermore, people by nature enjoy relating miracles they have heard without caring for their veracity and thus miracles are easily transmitted even when false. Also, Hume notes that miracles seem to occur mostly in "ignorant and barbarous nations" and times, and the reason they do not occur in the civilised societies is such societies are not awed by what they know to be natural events. Finally, the miracles of each religion argue against all other religions and their miracles, and so even if a proportion of all reported miracles across the world fit Hume's requirement for belief, the miracles of each religion make the other less likely.


title: David Hume

text: Hume was extremely pleased with his argument against miracles in his "Enquiry". He states "I flatter myself, that I have discovered an argument of a like nature, which, if just, will, with the wise and learned, be an everlasting check to all kinds of superstitious delusion, and consequently, will be useful as long as the world endures." Thus, Hume's argument against miracles had a more abstract basis founded upon the scrutiny, not just primarily of miracles, but of all forms of belief systems. It is a common sense notion of veracity based upon epistemological evidence, and founded on a principle of rationality, proportionality and reasonability.


title: David Hume

text: The criterion for assessing Hume's belief system is based on the balance of probability whether something is more likely than not to have occurred. Since the weight of empirical experience contradicts the notion for the existence of miracles, such accounts should be treated with scepticism. Further, the myriad of accounts of miracles contradict one another, as some people who receive miracles will aim to prove the authority of Jesus, whereas others will aim to prove the authority of Muhammad or some other religious prophet or deity. These various differing accounts weaken the overall evidential power of miracles.


title: David Hume

text: Despite all this, Hume observes that belief in miracles is popular, and that "the gazing populace… receive greedily, without examination, whatever soothes superstition, and promotes wonder."


title: David Hume

text: Critics have argued that Hume's position assumes the character of miracles and natural laws prior to any specific examination of miracle claims, thus it amounts to a subtle form of begging the question. To assume that testimony is a homogeneous reference group seems unwise- to compare private miracles with public miracles, unintellectual observers with intellectual observers and those who have little to gain and much to lose with those with much to gain and little to lose is not convincing to many. Indeed, many have argued that miracles not only do not contradict the laws of nature, but require the laws of nature to be intelligible as miraculous, and thus subverting the law of nature. For example, William Adams remarks that "there must be an ordinary course of nature before anything can be extraordinary. There must be a stream before anything can be interrupted." They have also noted that it requires an appeal to inductive inference, as none have observed every part of nature nor examined every possible miracle claim, for instance those in the future. This, in Hume's philosophy, was especially problematic.


title: David Hume

text: Little appreciated is the voluminous literature either foreshadowing Hume, in the likes of Thomas Sherlock or directly responding to and engaging with Hume—from William Paley, William Adams, John Douglas, John Leland, and George Campbell, among others. Regarding the latter, it is rumoured that, having read Campbell's Dissertation, Hume remarked that "the Scotch theologue had beaten him."


title: David Hume

text: Hume's main argument concerning miracles is that miracles by definition are singular events that differ from the established laws of nature. Such natural laws are codified as a result of past experiences. Therefore, a miracle is a violation of all prior experience and thus incapable on this basis of reasonable belief. However, the probability that something has occurred in contradiction of all past experience should always be judged to be less than the probability that either ones senses have deceived one, or the person recounting the miraculous occurrence is lying or mistaken, Hume would say, all of which he had past experience of. For Hume, this refusal to grant credence does not guarantee correctness. He offers the example of an Indian Prince, who, having grown up in a hot country, refuses to believe that water has frozen. By Hume's lights, this refusal is not wrong and the Prince "reasoned justly;" it is presumably only when he has had extensive experience of the freezing of water that he has warrant to believe that the event could occur.


title: David Hume

text: So for Hume, either the miraculous event will become a recurrent event or else it will never be rational to believe it occurred. The connection to religious belief is left unexplained throughout, except for the close of his discussion where Hume notes the reliance of Christianity upon testimony of miraculous occurrences. He makes an ironic remark that anyone who "is moved by faith to assent" to revealed testimony "is conscious of a continued miracle in his own person, which subverts all principles of his understanding, and gives him a determination to believe what is most contrary to custom and experience." Hume writes that "All the testimony whichever was really given for any miracle, or ever will be given, is a subject of derision."


title: David Hume

text: From 1754 to 1762 Hume published "The History of England", a six-volume work, that extends (according to its subtitle) "From the Invasion of Julius Caesar to the Revolution in 1688." Inspired by Voltaire's sense of the breadth of history, Hume widened the focus of the field away from merely kings, parliaments, and armies, to literature and science as well. He argued that the quest for liberty was the highest standard for judging the past, and concluded that after considerable fluctuation, England at the time of his writing had achieved "the most entire system of liberty that was ever known amongst mankind". It "must be regarded as an event of cultural importance. In its own day, moreover, it was an innovation, soaring high above its very few predecessors." Hume's "History of England" made him famous as a historian before he was ever considered a serious philosopher. In this work, Hume uses history to tell the story of the rise of England and what led to its greatness and the disastrous effects that religion has had on its progress. For Hume, the history of England's rise may give a template for others who would also like to rise to its current greatness.


title: David Hume

text: Hume's "The History of England" was profoundly impacted by his Scottish background. The science of sociology, which is rooted in Scottish thinking of the eighteenth century, had never before been applied to British philosophical history. Because of his Scottish background, Hume was able to bring an outsider's lens to English history that the insulated English whigs lacked.


title: David Hume

text: Hume's coverage of the political upheavals of the 17th century relied in large part on the Earl of Clarendon's "History of the Rebellion and Civil Wars in England" (1646–69). Generally, Hume took a moderate royalist position and considered revolution unnecessary to achieve necessary reform. Hume was considered a Tory historian, and emphasised religious differences more than constitutional issues. Laird Okie explains that "Hume preached the virtues of political moderation, but ... it was moderation with an anti-Whig, pro-royalist coloring." For "Hume shared the ... Tory belief that the Stuarts were no more high-handed than their Tudor predecessors". "Even though Hume wrote with an anti-Whig animus, it is, paradoxically, correct to regard the "History" as an establishment work, one which implicitly endorsed the ruling oligarchy".


title: David Hume

text: Historians have debated whether Hume posited a universal unchanging human nature, or allowed for evolution and development.


title: David Hume

text: The debate between Tory and the Whig historians can be seen in the initial reception to Hume's "History of England". The whig-dominated world of 1754 overwhelmingly disapproved of Hume's take on English history. In later editions of the book, Hume worked to "soften or expunge many villainous whig strokes which had crept into it."


title: David Hume

text: Hume did not consider himself a pure Tory. Before 1745, he was more akin to an "independent whig." In 1748, he described himself as "a whig, though a very skeptical one." This description of himself as in between whiggism and toryism, helps one understand that his "History of England" should be read as his attempt to work out his own philosophy of history.


title: David Hume

text: Robert Roth argues that Hume's histories display his biases against Presbyterians and Puritans. Roth says his anti-Whig pro-monarchy position diminished the influence of his work, and that his emphasis on politics and religion led to a neglect of social and economic history.


title: David Hume

text: Hume was an early cultural historian of science. His short biographies of leading scientists explored the process of scientific change. He developed new ways of seeing scientists in the context of their times by looking at how they interacted with society and each other. He covers over forty scientists, with special attention paid to Francis Bacon, Robert Boyle, and Isaac Newton. Hume particularly praised William Harvey, writing about his treatise of the circulation of the blood: "Harvey is entitled to the glory of having made, by reasoning alone, without any mixture of accident, a capital discovery in one of the most important branches of science."


title: David Hume

text: The "History" became a best-seller and made Hume a wealthy man who no longer had to take up salaried work for others. It was influential for nearly a century, despite competition from imitations by Smollett (1757), Goldsmith (1771) and others. By 1894, there were at least 50 editions as well as abridgements for students, and illustrated pocket editions, probably produced specifically for women.


title: David Hume

text: Hume's writings have been described as largely seminal to conservative theory, and he is considered a founding father of conservatism. In contrast, many of his ideas, such as limited government, private property when there is scarcity, and constitutionalism, are first principles of liberalism. Thomas Jefferson banned the "History" from University of Virginia, feeling that it had "spread universal toryism over the land." By comparison, Samuel Johnson thought Hume to be "a Tory by chance [...] for he has no principle. If he is anything, he is a Hobbist." A major concern of Hume's political philosophy is the importance of the rule of law. He also stresses throughout his political essays the importance of moderation in politics, public spirit, and regard to the community.


title: David Hume

text: Throughout the period of the American Revolution, Hume had varying views. For instance, in 1768 he encouraged total revolt on the part of the Americans. In 1775, he became certain that a revolution would take place and said that he believed in the American principle and wished the British government would let them be. Hume's influence on some of the Founders can be seen in Benjamin Franklin's suggestion at the Philadelphia Convention of 1787 that no high office in any branch of government should receive a salary, which is a suggestion Hume had made in his emendation of James Harrington's "Oceana".


title: David Hume

text: The legacy of religious civil war in 18th-century Scotland, combined with the relatively recent memory of the 1715 and 1745 Jacobite risings, had fostered in Hume a distaste for enthusiasm and factionalism. These appeared to him to threaten the fragile and nascent political and social stability of a country that was deeply politically and religiously divided. Hume thought that society is best governed by a general and impartial system of laws; he is less concerned about the form of government that administers these laws, so long as it does so fairly. However, he also clarified that a republic must produce laws, while "monarchy, when absolute, contains even something repugnant to law."


title: David Hume

text: Hume expressed suspicion of attempts to reform society in ways that departed from long-established custom, and he counselled peoples not to resist their governments except in cases of the most egregious tyranny. However, he resisted aligning himself with either of Britain's two political parties, the Whigs and the Tories:


title: David Hume

text: My views of "things" are more conformable to Whig principles; my representations of "persons" to Tory prejudices.


title: David Hume

text: Canadian philosopher Neil McArthur writes that Hume believed that we should try to balance our demands for liberty with the need for strong authority, without sacrificing either. McArthur characterises Hume as a "precautionary conservative," whose actions would have been "determined by prudential concerns about the consequences of change, which often demand we ignore our own principles about what is ideal or even legitimate." Hume supported the liberty of the press, and was sympathetic to democracy, when suitably constrained. American historian Douglass Adair has argued that Hume was a major inspiration for James Madison's writings, and the essay "Federalist No. 10" in particular.


title: David Hume

text: Hume offered his view on the best type of society in an essay titled "Idea of a Perfect Commonwealth", which lays out what he thought was the best form of government. He hoped that, "in some future age, an opportunity might be afforded of reducing the theory to practice, either by a dissolution of some old government, or by the combination of men to form a new one, in some distant part of the world". He defended a strict separation of powers, decentralisation, extending the franchise to anyone who held property of value and limiting the power of the clergy. The system of the Swiss militia was proposed as the best form of protection. Elections were to take place on an annual basis and representatives were to be unpaid. Political philosophers Leo Strauss and Joseph Cropsey, writing of Hume's thoughts about "the wise statesman", note that he "will bear a reverence to what carries the marks of age." Also, if he wishes to improve a constitution, his innovations will take account of the "ancient fabric", in order not to disturb society.


title: David Hume

text: In the political analysis of philosopher George Holland Sabine, the scepticism of Hume extended to the doctrine of government by consent. He notes that "allegiance is a habit enforced by education and consequently as much a part of human nature as any other motive."


title: David Hume

text: In the 1770s, Hume was critical of British policies toward the American colonies and advocated for American independence. He wrote in 1771 that "our union with America…in the nature of things, cannot long subsist."


title: David Hume

text: Hume expressed his economic views in his "Political Discourses", which were incorporated in "Essays and Treatises" as Part II of "Essays, Moral and Political". To what extent he was influenced by Adam Smith is difficult to assess; however, both of them had similar principles supported from historical events. At the same time Hume did not demonstrate concrete system of economic theory which could be observed in Smith's "Wealth of Nations". However, he introduced several new ideas around which the "classical economics" of the 18th century was built. Through his discussions on politics, Hume developed many ideas that are prevalent in the field of economics. This includes ideas on private property, inflation, and foreign trade. Referring to his essay "Of the Balance of Trade", economist Paul Krugman (2012) has remarked that "David Hume created what I consider the first true economic model."


title: David Hume

text: In contrast to Locke, Hume believes that private property is not a natural right. Hume argues it is justified, because resources are limited. Private property would be an unjustified, "idle ceremonial," if all goods were unlimited and available freely. Hume also believed in an unequal distribution of property, because perfect equality would destroy the ideas of thrift and industry. Perfect equality would thus lead to impoverishment.


title: David Hume

text: David Hume anticipated modern monetarism. First, Hume contributed to the theory of quantity and of interest rate. Hume has been credited with being the first to prove that, on an abstract level, there is no quantifiable amount of nominal money that a country needs to thrive. He understood that there was a difference between "nominal" and "real" money.


title: David Hume

text: Second, Hume has a theory of causation which fits in with the Chicago-school "black box" approach. According to Hume, cause and effect are related only through correlation. Hume shared the belief with modern monetarists that changes in the supply of money can affect consumption and investment.


title: David Hume

text: Lastly, Hume was a vocal advocate of a stable private sector, though also having some non-monetarist aspects to his economic philosophy. Having a stated preference for rising prices, for instance, Hume considered government debt to be a sort of substitute for actual money, referring to such debt as "a kind of paper credit." He also believed in heavy taxation, believing that it increases effort. Hume's economic approach evidently resembles his other philosophies, in that he does not choose one side indefinitely, but sees gray in the situation


title: David Hume

text: Due to Hume's vast influence on contemporary philosophy, a large number of approaches in contemporary philosophy and cognitive science are today called "Humean."


title: David Hume

text: The writings of Thomas Reid, a Scottish philosopher and contemporary of Hume, were often critical of Hume's scepticism. Reid formulated his "common sense" philosophy, in part, as a reaction against Hume's views.


title: David Hume

text: Hume influenced, and was influenced by, the Christian philosopher Joseph Butler. Hume was impressed by Butler's way of thinking about religion, and Butler may well have been influenced by Hume's writings.


title: David Hume

text: Attention to Hume's philosophical works grew after the German philosopher Immanuel Kant, in his "Prolegomena to Any Future Metaphysics" (1783), credited Hume with awakening him from his "dogmatic slumber."


title: David Hume

text: According to Arthur Schopenhauer, "there is more to be learned from each page of David Hume than from the collected philosophical works of Hegel, Herbart and Schleiermacher taken together."


title: David Hume

text: A. J. Ayer, while introducing his classic exposition of logical positivism in 1936, claimed:The views which are put forward in this treatise derive from…doctrines…which are themselves the logical outcome of the empiricism of Berkeley and David Hume.Albert Einstein, in 1915, wrote that he was inspired by Hume's positivism when formulating his theory of special relativity.


title: David Hume

text: Hume's problem of induction was also of fundamental importance to the philosophy of Karl Popper. In his autobiography, "Unended Quest", he wrote: "Knowledge ... is "objective"; and it is hypothetical or conjectural. This way of looking at the problem made it possible for me to reformulate Hume's "problem of induction"." This insight resulted in Popper's major work "The Logic of Scientific Discovery". In his "Conjectures and Refutations", he wrote:


title: David Hume

text: I approached the problem of induction through Hume. Hume, I felt, was perfectly right in pointing out that induction cannot be logically justified.


title: David Hume

text: Hume's rationalism in religious subjects influenced, via German-Scottish theologian Johann Joachim Spalding, the German neology school and rational theology, and contributed to the transformation of German theology in the age of enlightenment. Hume pioneered a comparative history of religion, tried to explain various rites and traditions as being based on deception and challenged various aspects of rational and natural theology, such as the argument from design.


title: David Hume

text: Danish theologian and philosopher Søren Kierkegaard adopted "Hume's suggestion that the role of reason is not to make us wise but to reveal our ignorance," though taking it as a reason for the necessity of religious faith, or "fideism". The "fact that Christianity is contrary to reason…is the necessary precondition for true faith." Political theorist Isaiah Berlin, who has also pointed out the similarities between the arguments of Hume and Kierkegaard against rational theology, has written about Hume's influence on what Berlin calls the "counter-Enlightenment" and on German anti-rationalism. Berlin has also once said of Hume that "no man has influenced the history of philosophy to a deeper or more disturbing degree."


title: David Hume

text: In 2003 philosopher Jerry Fodor, described Hume's "Treatise" as "the founding document of cognitive science."


title: David Hume

text: Hume engaged with contemporary intellectuals including Jean-Jacques Rousseau, James Boswell, and Adam Smith (who acknowledged Hume's influence on his economics and political philosophy).


title: David Hume

text: Morris and Brown (2019) write that Hume is "generally regarded as one of the most important philosophers to write in English."


title: David Hume

text: In September 2020, the David Hume Tower, a University of Edinburgh building, was renamed to 40 George Square; this was following a campaign led by students of the university to rename it, in objection to Hume's writings related to race.


title: Chernobyl Nuclear Power Plant

text: The Chernobyl Nuclear Power Plant (ChNPP; ; ), is a nuclear power plant undergoing decommissioning. ChNPP is located near the abandoned city of Pripyat in northern Ukraine northwest of the city of Chernobyl, from the Belarus–Ukraine border, and about north of Kyiv. The plant was cooled by an engineered pond, fed by the Pripyat River about northwest from its juncture with the Dnieper.


title: Chernobyl Nuclear Power Plant

text: ChNPP was commissioned in phases with the four reactors entering commercial operation between 1978 and 1984. In 1986, reactor No. 4 was the site of the Chernobyl disaster; as a result of this, the power plant is now within a large restricted area known as the Chernobyl Exclusion Zone. Both the zone and the power plant are administered by the State Agency of Ukraine on Exclusion Zone Management. The three other reactors remained operational post-accident maintaining a capacity factor between 60 and 70%. In total, units 1 and 3 had supplied 98 terawatt-hours of electricity each, with unit 2 slightly behind at 75 TWh. In 1991, unit 2 was placed into a permanent shutdown state by the plant's operator due to complications resulting from a turbine fire. This was followed by unit 1 in 1996 and unit 3 in 2000. Their closures were largely attributed to foreign pressures. In 2013, the plant's operator announced that units 1-3 were fully defueled, and in 2015 entered the decommissioning phase, during which equipment contaminated during the operational period of the power station will be removed. This process is expected to take until 2065 according to the plant's operator. Although the reactors have all ceased generation, Chernobyl maintains a large workforce as the ongoing decommissioning process requires constant management.


title: Chernobyl Nuclear Power Plant

text: From 24 February to 31 March 2022, Russian troops occupied the plant as part of their invasion of Ukraine.


title: Chernobyl Nuclear Power Plant

text: The Chernobyl Nuclear Power Plant consisted of four RBMK-1000 reactors, each capable of producing 1,000 megawatts (MW) of electric power (3,200 MW of thermal power), and the four together produced about 10% of Ukraine's electricity. Like other sites which housed multiple RBMK reactors such as Kursk, the construction of the plant was also accompanied by the construction of a nearby city to house workers and their families. In the case of the ChNPP, the new city was Pripyat. Construction of the station concluded in the late 1970s, with reactor No. 1 being commissioned in 1977. It was the third Soviet RBMK nuclear power plant, after the Leningrad Nuclear Power Plant and the Kursk Nuclear Power Plant, and the first plant on Ukrainian soil.


title: Chernobyl Nuclear Power Plant

text: The completion of the first reactor in 1977 was followed by reactor No. 2 in 1978, No. 3 in 1981, and No. 4 in 1983. Two more blocks, numbered five and six, of more or less the same reactor design, were planned at a site roughly a kilometer from the contiguous buildings of the four older blocks. This is similar to the layout of units 5 and 6 at Kursk and shows the similarity in design between the RBMK sites. Reactor No. 5 was around 70% complete at the time of Reactor 4's explosion and was scheduled to come online approximately seven months later, on November 7, 1986. In the aftermath of the disaster, construction on No. 5 and 6 was suspended, and eventually cancelled in April 1989, days before the third anniversary of the 1986 explosion. At one point six other reactors were planned on the other side of the river, bringing the total to twelve.


title: Chernobyl Nuclear Power Plant

text: Reactors No. 3 and 4 were second-generation units, whereas No. 1 and 2 were first-generation units, like those in operation at the Kursk power plant. Second-generation RBMK designs were fitted with a more secure containment structure visible in photos of the facility.


title: Chernobyl Nuclear Power Plant

text: The power plant is connected to the 330 kV and 750 kV electrical grid. The block has two electrical generators connected to the 750 kV grid by a single generator transformer. The generators are connected to their common transformer by two switches in series. Between them, the unit transformers are connected to supply power to the power plant's own systems; each generator can therefore be connected to the unit transformer to power the plant, or to the unit transformer and the generator transformer to also feed power to the grid.


title: Chernobyl Nuclear Power Plant

text: The 330 kV line was normally not used, and served as an external power supply, connected to a station's transformer – meaning to the power plant's electrical systems. The plant was powered by its own generators, or at any event got power from the 750 kV national grid through the main grid backup feed in transformer, or from the 330 kV level feed in grid transformer 2, or from the other power plant blocks via two reserve busbars. In case of total external power loss, the essential systems could be powered by diesel generators. Each unit's transformer is therefore connected to two 6 kV main power line switchboards, A and B (e.g., 7A, 7B, 8A, 8B for generators 7 and 8), powering principal essential systems and connected to even another transformer at 4 kV, which is backed up twice (4 kV reserve busbar).


title: Chernobyl Nuclear Power Plant

text: The 7A, 7B, and 8B boards are also connected to the three essential power lines (for the coolant pumps), each also having its own diesel generator. In case of a coolant circuit failure with simultaneous loss of external power, the essential power can be supplied by the spinning down turbogenerators for about 45 to 50 seconds, during which time the diesel generators should start up. The generators were started automatically within 15 seconds at loss of off-site power.


title: Chernobyl Nuclear Power Plant

text: Electrical energy was generated by a pair of 2x500 MW hydrogen-cooled turbo generators per unit. These are located in the -long machine hall, adjacent to the reactor building. The turbines—the venerable five-cylinder K-500-65/3000—are supplied by the Kharkiv turbine plant; the electrical generators are the TBB-500. The turbine and the generator rotors are mounted on the same shaft; the combined weight of the rotors is almost and their speed is 3,000 revolutions per minute.


title: Chernobyl Nuclear Power Plant

text: The turbo generator is long and its total weight is . The coolant flow for each turbine is 82,880 t/h. The generator produces 20 kV 50 Hz AC power. The generator's stator is cooled by water while its rotor is cooled by hydrogen. The hydrogen for the generators is manufactured on-site by electrolysis. The design and reliability of the turbines earned them the State Prize of Ukraine for 1979.


title: Chernobyl Nuclear Power Plant

text: The Kharkiv turbine plant later developed a new version of the turbine, K-500-65/3000-2, in an attempt to reduce use of valuable metal. The Chernobyl plant was equipped with both types of turbines; block 4 had the newer ones. The newer turbines, however, turned out to be more sensitive to their operating parameters, and their bearings had frequent problems with vibrations.


title: Chernobyl Nuclear Power Plant

text: The construction of two partially completed reactors, No. 5 and 6, were suspended immediately after the accident at reactor No. 4, and eventually cancelled in 1989. Reactors No. 1 and 3 continued to operate after the disaster. Reactor No. 2 was permanently shut down in 1991 after a fire broke out due to a faulty switch in a turbine. Reactors No. 1 and 3 were to be eventually closed due to a 1995 agreement Ukraine made with the EU.


title: Chernobyl Nuclear Power Plant

text: Ukraine agreed to close the remaining units in exchange for EU assistance in modernizing the shelter over reactor No. 4 and improving the energy sector of the country, including the completion of two new nuclear reactors, Khmelnitski 2 and Rovno 4. Reactor No. 1 was shut down in 1996 with No. 3 following in 2000.


title: Chernobyl Nuclear Power Plant

text: SKALA (Russian: СКАЛА, система контроля аппарата Ленинградской Атомной; "sistema kontrolya apparata Leningradskoj Atomnoj", “Control system of the devices of the Leningrad Nuclear Power Plant”) was the process computer for the RBMK nuclear reactor at the Chernobyl nuclear power plant prior to October 1995. Dating back to the 1960s, it used magnetic-core memory, magnetic tape data storage, and punched tape for loading software.


title: Chernobyl Nuclear Power Plant

text: SKALA monitored and recorded reactor conditions and control board inputs. It was wired to accept 7200 analog signals and 6500 digital signals. The system continuously monitored the plant and displayed this information to operators. Additionally, a program called PRIZMA (Russian: ПРИЗМА, программа измерения мощности аппарата; "programma izmereniya moshchnosti apparata", “Device power measurement program”) processed plant conditions and made recommendations to guide plant operators. This program took 5 to 10 minutes to run, and could not directly control the reactor.


title: Chernobyl Nuclear Power Plant

text: On 9 September 1982, a partial core meltdown occurred in reactor No. 1 due to a faulty cooling valve remaining closed following maintenance. Once the reactor came online, the uranium in the tank overheated and ruptured. The extent of the damage was comparatively minor, and no one was killed during the accident. However, due to the negligence of the operators, the accident was not noticed until several hours later, resulting in significant release of radiation in the form of fragments of uranium oxide and several other radioactive isotopes escaping with steam from the reactor via the ventilation stack. The accident was not made public until several years later, despite cleanups taking place in and around the power station and Pripyat. The reactor was repaired and put back into operation after eight months.


title: Chernobyl Nuclear Power Plant

text: According to KGB documents, declassified in Ukraine on 26 April 2021, serious incidents occurred in the third and fourth reactors in 1984. According to the same documents, the central government in Moscow knew as early as 1983 that the powerplant was "one of the most dangerous nuclear powerplants in the USSR".


title: Chernobyl Nuclear Power Plant

text: On 26 April 1986, the Chernobyl disaster occurred at reactor No. 4, caused by a catastrophic power increase resulting in core explosions and open-air fires. This caused large quantities of radioactive materials and airborne isotopes to disperse in the atmosphere and surrounding land. The disaster is regarded as the worst accident in the history of nuclear power. As a result, Reactor No. 4 was destroyed entirely, and therefore enclosed in a concrete and lead sarcophagus, followed more recently by a large steel confinement shelter to prevent further escape of radioactivity. Large areas of Europe were affected by the accident. The radioactive cloud spread as far away as Norway.


title: Chernobyl Nuclear Power Plant

text: Reactor No. 2 was permanently shut down shortly after October 1991 when a fire broke out due to a faulty switch in a turbine.


title: Chernobyl Nuclear Power Plant

text: On 11 October 1991, a fire broke out in the turbine hall of reactor No. 2. The fire began in reactor No. 2's fourth turbine, while the turbine was being idled for repairs. A faulty switch caused a surge of current to the generator, igniting insulating material on some electrical wiring. This subsequently led to hydrogen, used as a coolant in the synchronous generator, being leaked into the turbine hall "which apparently created the conditions for fire to start in the roof and for one of the trusses supporting the roof to collapse." The adjacent reactor hall and reactor were unaffected, but due to the political climate it was decided to shut down this reactor permanently after this incident.


title: Chernobyl Nuclear Power Plant

text: The 2017 Petya cyberattack affected the radiation monitoring system and took down the power plant's official website, which hosts information about the incident and the area.


title: Chernobyl Nuclear Power Plant

text: The Chernobyl Exclusion Zone was the site of fighting between Russian and Ukrainian forces during the Battle of Chernobyl as part of the Russian invasion of Ukraine. On 24 February 2022, Russian forces captured the plant. The resulting activity reportedly led to a 20-fold increase of radiation levels in the area due to disruption of contaminated soil. The increase in radiation levels was published on an online map maintained by a Ukrainian NGO, but the corresponding sensors at the plant itself exhibited no increase.


title: Chernobyl Nuclear Power Plant

text: On 9 March 2022, there was a power cut at the plant itself. No radiation leaks were reported as of 9 March 2022. However, Ukrainian authorities reported that there was a risk of a radiation leak due to spent fuel coolant being unable to circulate properly.


title: Chernobyl Nuclear Power Plant

text: On 31 March 2022, Russian forces formally handed control of the plant back to its employees, and most occupying forces withdrew. Ukrainian National Guard personnel were moved to Belarus as prisoners of war. On 2 April 2022, Ukrainian media reported that the flag of Ukraine was raised at the plant.


title: Chernobyl Nuclear Power Plant

text: Chernobyl operator Energoatom claimed that Russian troops had dug trenches in the most contaminated part of the Chernobyl exclusion zone, receiving "significant doses" of radiation. "BBC News" reported unconfirmed reports that some were being treated in Belarus.


title: Chernobyl Nuclear Power Plant

text: After the explosion at reactor No. 4 and construction of the Shelter Object, the remaining three reactors were re-launched and continued to operate until the post-Soviet period. The Chernobyl New Safe Confinement is equipped with 2 overhead main cranes, which will be used to remove unstable parts of the original sarcophagus. The majority of the external gamma radiation emissions at the site are from the isotope caesium-137, which has a half-life of 30.17 years. , the radiation exposure from that radionuclide has declined by half since the 1986 accident.


title: Chernobyl Nuclear Power Plant

text: In October 1991, reactor No. 2 caught fire, and was subsequently shut down. Ukraine's 1991 independence from the Soviet Union generated further discussion on the Chernobyl topic, because the Verkhovna Rada, Ukraine's new parliament, was composed largely of young reformers. Discussions about the future of nuclear energy in Ukraine ultimately moved the government toward a decision to decommission reactor No. 2.


title: Chernobyl Nuclear Power Plant

text: In November 1996, following pressure from foreign governments, reactor No. 1 was shut down. Removal of uncontaminated equipment has begun at reactor No. 1 and this work could be complete by 2020–2022. In December 2000, reactor No. 3 was shut down after operating briefly since March 1999 following 5 months of repairs, and the plant as a whole ceased producing electricity. In April 2015, units 1 through 3 entered the decommissioning phase.


title: Chernobyl Nuclear Power Plant

text: In 2013, the pump lifting river water into the cooling reservoir adjacent to the facility was powered down, with the thermal sink expected to slowly evaporate.


title: Chernobyl Nuclear Power Plant

text: Originally announced in June 2003, a new steel containment structure named the New Safe Confinement was built to replace the aging and hastily built sarcophagus that protected reactor No. 4. Though the project's development had been delayed several times, construction officially began in September 2010. The New Safe Confinement was financed by an international fund managed by the European Bank for Reconstruction and Development and was designed and built by the French-led consortium Novarka.


title: Chernobyl Nuclear Power Plant

text: In February 2013, a portion of the roof and wall adjacent to the covered part of the turbine hall collapsed into the entombed area of the turbine hall. The collapse did not affect any other part of the Object Shelter or the New Safe Confinement. No variances in radiation levels as a result of the incident were detected. The collapsed roof was built after the Chernobyl disaster and was later repaired.


title: Chernobyl Nuclear Power Plant

text: Novarka built a large arch-shaped structure out of steel wide, high, and long to cover the old crumbling concrete dome that was in use at the time. The structure was built in two segments which were joined in August 2015. In November 2016, the completed arch was placed over the existing sarcophagus. This steel casing project was expected to cost $1.4 billion, and was completed in 2017. The casing also meets the definition of a nuclear entombment device.


title: Chernobyl Nuclear Power Plant

text: A separate deal has been made with the American firm Holtec International to build a storage facility within the exclusion zone for nuclear waste produced by Chernobyl. This facility, named the Interim Storage Facility 2, will have storage for the 21,297 Spent Fuel assemblies currently at the power plant, which will be loaded into approximately 231 waste canisters, and stored in the ISF-2 for 100 years. In 2020, the storage facility was completed, and on November 18, 2020, the first canister of nuclear waste was loaded into the storage area.


title: Max Minghella

text: Max Giorgio Choa Minghella (born 16 September 1985) is an English actor, film producer, director, and screenwriter. He is known for his roles in the films "Syriana" (2005), "Art School Confidential" (2006), "Elvis and Anabelle" (2007), "The Social Network" (2010), "The Ides of March" (2011), "The Internship" (2013), "Horns" (2013), and "Spiral" (2021), as well as his role as Nick Blaine in the television series "The Handmaid's Tale" (2017–present), which earned him a Primetime Emmy Award nomination in 2021.


title: Max Minghella

text: Minghella was born in Hampstead, London, the son of director Anthony Minghella (1954–2008) and dancer and choreographer Carolyn Jane Chao. His father was born in Ryde, Isle of Wight, and was of Italian descent, and his mother was born in Hong Kong.


title: Max Minghella

text: While Minghella was growing up he spent time on his father's film sets. He has said that he has "fond memories" of them and that he felt "no pressure" from his father to succeed in the entertainment industry. He was educated at St Anthony's Preparatory School and University College School in Hampstead. He then attended Columbia University, which he considered his "first priority", and studied history, graduating in 2009. He was a resident of John Jay Hall during his studies at Columbia. He typically worked on films only during his summer breaks. He has said that he felt like "an English boy at an American school", that he kept to himself, and that most of his fellow students did not know that he was an actor.


title: Max Minghella

text: Minghella's father was born on the Isle of Wight, and was of Italian descent. His mother, who was from Hong Kong, is from a family of multiple heritage. His maternal grandfather was of three-quarters Chinese and one-quarter Jewish descent, and his maternal grandmother was of Indian Parsi, English, Irish, Swedish and Chinese ancestry.


title: Max Minghella

text: Sir Robert Kotewall is his great-grandfather. Olympic swimmer Robyn Lamsam is his second cousin through the Kotewalls.


title: Max Minghella

text: Minghella was inspired to become an actor during his young years, after seeing a production of the play "This Is Our Youth" in London's West End; he subsequently dropped out of the University College School to pursue an acting career, and attended the National Youth Theatre. He did not always want to become an actor. He said: "I thought it was sort of embarrassing to say you wanted to be an actor it was, like, uncool. And growing up I was massively concerned with seeming cool. I thought up until about age 16 that I was going to be a music video director."


title: Max Minghella

text: Minghella had an uncredited appearance as an extra in his father's film "Cold Mountain". His first professional role was in "Bee Season", playing the son of a dysfunctional Jewish American family. The film was released in November 2005 to mostly mixed reviews and low box office, grossing only $1 million in its limited release, although critics remarked that Minghella was "a talented young actor to watch, delivering a strong performance".


title: Max Minghella

text: Minghella's other November 2005 role was the political thriller "Syriana" in which he played the son of George Clooney's CIA agent character. In 2006, he starred in the Daniel Clowes adaptation "Art School Confidential", a comedy directed by Terry Zwigoff. He got the part after meeting Zwigoff when he visited the set of "Bee Season". His film "Elvis and Anabelle", a dark romantic drama in which he plays an undertaker's son, premiered at the South by Southwest film festival in 2007. He described it as "a really sweet film". He was to play Art Bechstein in the film version of writer Michael Chabon's novel "The Mysteries of Pittsburgh", but dropped out of the project due to his university schedule.


title: Max Minghella

text: The press reported in 2007 that Minghella was cast in Beeban Kidron's "Hippie Hippie Shake", a film about writer Richard Neville set in 1960s London. In March 2008, it was announced that he would star in Alejandro Amenábar's second English language film, "Agora". The film is set in 4th century Egypt and revolves around Hypatia of Alexandria. Later in 2008, Minghella played a pompous film director in "How to Lose Friends & Alienate People", and was seen in a segment of the dark comedy "Brief Interviews with Hideous Men".


title: Max Minghella

text: In David Fincher's "The Social Network", the 2010 film about the origins of Facebook, he plays Divya Narendra, one of the Harvard upperclassmen who sues Mark Zuckerberg for stealing the idea behind Facebook. In June 2010, Minghella was cast in the Russian science-fiction film "The Darkest Hour", released in December 2011. He appeared in the ensemble dramedy "10 Years" (2012). The following year, he had a supporting role as Graham Hawtrey in the comedy "The Internship".


title: Max Minghella

text: In November 2013, it was announced that Minghella would be playing Richie Castellano in the second season of the Fox sitcom "The Mindy Project". In 2013, he appeared in the music video for "Shot at the Night" by The Killers.


title: Max Minghella

text: Minghella has been playing the role of Nick Blaine, an Eye in the Republic of Gilead and the love interest of June Osborne, in the Hulu dystopian series "The Handmaid's Tale" since 2017. He was nominated for a Primetime Emmy Award for Outstanding Supporting Actor in a Drama Series for the role in 2021.


title: Ayan Mukerji

text: Ayan Mukerji (born 15 August 1983) is an Indian film director, screenwriter and producer who works in Hindi films. Mukerji made his directorial debut at the age of 26 with the coming-of-age comedy "Wake Up Sid" (2009). The film opened to strong reviews and box-office returns. His next directorial, the 2013 romantic comedy-drama, "Yeh Jawaani Hai Deewani", ranks among the highest-grossing Hindi films. He then directed the fantasy action-adventure "", which is the first in a planned trilogy of a cinematic universe named the "Astraverse".


title: Ayan Mukerji

text: Mukerjee was born in a Bengali Hindu family in Kolkata, India, the son of Bengali film actor Deb Mukherjee. His family was steeped in the Indian film industry since the 1930, when his grandfather, Sashadhar Mukherjee, first ventured by chance into the field. Sashadhar Mukherjee was a pioneering film producer who was one of the founding partners of Filmistan Studio in Mumbai, and produced films like "Dil Deke Dekho" (1959), "Love in Simla" (1960), "Ek Musafir Ek Hasina" (1962) and "Leader" (1964). The studio is still run by Ayan's extended family.


title: Ayan Mukerji

text: Ayan's grandmother, Satidevi Mukherjee (née Ganguly), wife of Sashadhar Mukherji, was the sister of pioneer actor Ashok Kumar, Anup Kumar and singer Kishore Kumar. Ayan's grandfather's brothers were film producer Prabodh Mukherjee, director Subodh Mukherjee and Ravindra Mohan Mukherjee, grandfather of actress Rani Mukherjee. Ayan's father's brothers were actors Joy Mukherjee, Subir Mukherjee and Shomu Mukherjee, who is the husband of actress Tanuja and he made 3 movies and father of actresses Kajol and Tanisha Mukherjee.


title: Ayan Mukerji

text: Mukerji studied at Jamnabai Narsee School in Vile Parle, Mumbai. He later attended Rajiv Gandhi Institute of Technology but dropped out after the first year to assist Ashutosh Gowariker as clapper boy on "Swades" (2004).


title: Ayan Mukerji

text: Mukerji started his career as an assistant to Ashutosh Gowariker, his brother-in-law, on "Swades," and later to Karan Johar on "Kabhi Alvida Naa Kehna" (2006). After taking a short break from filmmaking, Mukerji wrote the screenplay for "Wake Up Sid" and directed it. The film released in 2009, starring Ranbir Kapoor and Konkona Sen Sharma in the lead roles and went on to become a critical and commercial success. The film earned him a Best Debut Director at Filmfare.


title: Ayan Mukerji

text: Mukerji's second film, also under Johar's banner Dharma Productions, was "Yeh Jawaani Hai Deewani", with Ranbir Kapoor and Deepika Padukone in lead roles. It was a huge commercial success. The movie grossed Rs. 1 billion at the box-office in just 7 days. It became the third highest-grossing Indian film of all time. The film earned him a second Best Director nomination at Filmfare.


title: Ayan Mukerji

text: Mukerji's third film, "", starring Ranbir Kapoor, Amitabh Bachchan, Alia Bhatt, Mouni Roy and Nagarjuna Akkineni released on 9th September 2022 under Johar's banner Dharma Productions. It is the first in a planned trilogy known as the "Astraverse".


title: Ayan Mukerji

text: Mukerji is all set to direct the next in this trilogy, "Brahmа̄stra: Part Two – Dev" and "Brahmāstra: Part Three".


title: Crayfish

text: Crayfish are freshwater crustaceans belonging to the clade Astacidea, which also contains lobsters. In some locations, they are also known as crawfish, craydids, crawdaddies, crawdads, freshwater lobsters, mountain lobsters, rock lobsters, mudbugs, baybugs or yabbies. Taxonomically, they are members of the superfamilies Astacoidea and Parastacoidea. They breathe through feather-like gills. Some species are found in brooks and streams, where fresh water is running, while others thrive in swamps, ditches, and paddy fields. Most crayfish cannot tolerate polluted water, although some species, such as "Procambarus clarkii", are hardier. Crayfish feed on animals and plants, either living or decomposing, and detritus.


title: Crayfish

text: The name "crayfish" comes from the Old French word ' (Modern French '). The word has been modified to "crayfish" by association with "fish" (folk etymology). The largely American variant "crawfish" is similarly derived.


title: Crayfish

text: Some kinds of crayfish are known locally as lobsters, crawdads, mudbugs, and yabbies. In the Eastern United States, "crayfish" is more common in the north, while "crawdad" is heard more in central and southwestern regions, and "crawfish" farther south, although considerable overlaps exist.


title: Crayfish

text: The body of a decapod crustacean, such as a crab, lobster, or prawn (shrimp), is made up of twenty body segments grouped into two main body parts, the cephalothorax and the abdomen. Each segment may possess one pair of appendages, although in various groups, these may be reduced or missing. On average, crayfish grow to in length. Walking legs have a small claw at the end.


title: Crayfish

text: Crayfish are closely related to lobsters, and together they belong to the infraorder Astacidea. Their phylogeny can be shown in the simplified cladogram below:


title: Crayfish

text: Four extant (living) families of crayfish are described, three in the Northern Hemisphere and one in the Southern Hemisphere. The Southern Hemisphere (Gondwana-distributed) family Parastacidae, with 14 extant genera and two extinct genera, live(d) in South America, Madagascar, and Australasia. They are distinguished by the absence of the first pair of pleopods. Of the other three Northern Hemisphere families (grouped in the superfamily Astacoidea), the four genera of the family Astacidae live in western Eurasia and western North America, the 15 genera of the family Cambaridae live in eastern North America, and the single genus of Cambaroididae live in eastern Asia.


title: Crayfish

text: The greatest diversity of crayfish species is found in southeastern North America, with over 330 species in 15 genera, all in the family Cambaridae. A further genus of astacid crayfish is found in the Pacific Northwest and the headwaters of some rivers east of the Continental Divide. Many crayfish are also found in lowland areas where the water is abundant in calcium, and oxygen rises from underground springs.


title: Crayfish

text: In 1983, Louisiana designated the crayfish, or crawfish as they are commonly called, as its official state crustacean. Louisiana produces of crawfish per year with the red swamp and white river crawfish being the main species harvested. Crawfish are a part of Cajun culture dating back hundreds of years. A variety of cottage industries have developed as a result of commercialized crawfish iconography. Their products include crawfish attached to wooden plaques, T-shirts with crawfish logos, and crawfish pendants, earrings, and necklaces made of gold or silver.


title: Crayfish

text: Australia has over 100 species in a dozen genera. It is home to the world's three largest freshwater crayfish:


title: Crayfish

text: Many of the better-known Australian crayfish are of the genus "Cherax", and include the common yabby ("C. destructor"), western yabby ("C. preissii"), and red-claw crayfish ("C. quadricarinatus").


title: Crayfish

text: The marron species "C. tenuimanus" is critically endangered, while other large Australasian crayfish are threatened or endangered.


title: Crayfish

text: In Australia, New Zealand, and South Africa, the term "crayfish" or "cray" generally refers to a saltwater spiny lobster, of the genus "Jasus" that is indigenous to much of southern Oceania, while the freshwater species are usually called "yabbies" or "", from the indigenous Australian and Māori names for the animal, respectively, or by other names specific to each species. Exceptions include western rock lobster (of the Palinuridae family) found on the west coast of Australia (it is a spiny lobster, but not of "Jasus)"; the Tasmanian giant freshwater crayfish (from the Parastacidae family and therefore a true crayfish) found only in Tasmania; and the Murray crayfish found along Australia's Murray River.


title: Crayfish

text: In Singapore, the term crayfish typically refers to "Thenus orientalis", a seawater crustacean from the slipper lobster family. True crayfish are not native to Singapore, but are commonly found as pets, or as an invasive species ("Cherax quadricarinatus") in the many water catchment areas, and are alternatively known as "freshwater lobsters".


title: Crayfish

text: In England and Ireland, the terms crayfish or crawfish commonly refer to the European spiny lobster, a saltwater species found in much of the East Atlantic and Mediterranean. The only true crayfish species native to the British Isles is the endangered white clawed crayfish.


title: Crayfish

text: Fossil records of crayfish older than 30 million years are rare, but fossilised burrows have been found from strata as old as the late Palaeozoic or early Mesozoic. The oldest records of the Parastacidae are in Australia, and are 115 million years old and the crayfish "Palaeocambarus" and "Cricoidoscelosus" from the Yixian Formation of China are likely around 120 million years old, making them both one of, if not, the oldest known crayfish to date.


title: Crayfish

text: Crayfish are susceptible to infections such as crayfish plague and to environmental stressors including acidification. In Europe, they are particularly threatened by crayfish plague, which is caused by the North American water mold "Aphanomyces astaci." This water mold was transmitted to Europe when North American species of crayfish were introduced. Species of the genus "Astacus" are particularly susceptible to infection, allowing the plague-coevolved signal crayfish (native to western North America) to invade parts of Europe.


title: Crayfish

text: Acid rain can cause problems for crayfish across the world. In whole-ecosystem experiments simulating acid rain at the Experimental Lakes Area in Ontario, Canada, crayfish populations crashed – probably because their exoskeletons are weaker in acidified environments.


title: Crayfish

text: In several countries, particularly in Europe, native species of crayfish are under threat by imported species, particularly the signal crayfish ("Pacifastacus leniusculus"). Crayfish are also considered an invasive predatory species, endangering native European species such as the Italian agile frog.


title: Crayfish

text: Crayfish are eaten worldwide. Like other edible crustaceans, only a small portion of the body of a crayfish is eaten. In most prepared dishes, such as soups, bisques and étouffées, only the tail portion is served. At crawfish boils or other meals where the entire body of the crayfish is presented, other portions, such as the claw meat, may be eaten.


title: Crayfish

text: Global crayfish production is centered in Asia, primarily China. In 2018, Asian production accounted for 95% of the world's crawfish supply.


title: Crayfish

text: Crayfish is part of Swedish cuisine and is usually eaten in August at special crayfish parties (Swedish "Kräftskiva"). Documentation of the consumption of crayfish dates to at least the 16th century. On the Swedish west coast, Nephrops norvegicus (Swedish "Havskräfta", "sea crayfish") is more commonly eaten while various freshwater crayfish are consumed in the rest of the country. Prior to the 1960s, crayfish was largely inaccessible to the urban population in Sweden and consumption was largely limited to the upper classes or farmers holding fishing rights in fresh water lakes. With the introduction of import of frozen crayfish the crayfish party is now practiced widely practiced across all spheres in Sweden and among the Swedish-speaking population of Finland.


title: Crayfish

text: In the United States, crayfish production is strongly centered in Louisiana, with 93% of crayfish farms located in the state as of 2018. In 1987, Louisiana produced 90% of the crayfish harvested in the world, 70% of which were consumed locally. In 2007, the Louisiana crayfish harvest was about 54,800 tons, almost all of it from aquaculture. About 70–80% of crayfish produced in Louisiana are "Procambarus clarkii" (red swamp crawfish), with the remaining 20–30% being "Procambarus zonangulus" (white river crawfish). Optimum dietary nutritional requirement of freshwater crayfish, or crayfish nutrient specifications are now available for aquaculture feed producers


title: Crayfish

text: Like all crustaceans, crayfish are not kosher because they are aquatic animals that do not have both fins and scales. They are therefore not eaten by observant Jews.


title: Crayfish

text: Crayfish are preyed upon by a variety of ray-finned fishes, and are commonly used as bait, either live or with only the tail meat. They are a popular bait for catching catfish, largemouth bass, smallmouth bass, striped bass, perch, pike and muskie. When using live crayfish as bait, anglers prefer to hook them between the eyes, piercing through their hard, pointed beak which causes them no harm; therefore, they remain more active.


title: Crayfish

text: When using crayfish as bait, it is important to fish in the same environment where they were caught. An Illinois State University report that focused on studies conducted on the Fox River and Des Plaines River watershed stated that rusty crayfish, initially caught as bait in a different environment, were dumped into the water and "outcompeted the native clearwater crayfish". Other studies confirmed that transporting crayfish to different environments has led to various ecological problems, including the elimination of native species. Transporting crayfish as live bait has also contributed to the spread of zebra mussels in various waterways throughout Europe and North America, as they are known to attach themselves to exoskeleton of crayfishes.


title: Crayfish

text: Crayfish are kept as pets in freshwater aquariums. They prefer foods like shrimp pellets or various vegetables, but will also eat tropical fish food, regular fish food, algae wafers, and small fish that can be captured with their claws. A report by the National Park Service as well as video and anecdotal reports by aquarium owners indicate that crayfish will eat their moulted exoskeleton "to recover the calcium and phosphates contained in it." As omnivores, crayfish will eat almost anything; therefore, they may explore the edibility of aquarium plants in a fish tank. However, most species of dwarf crayfish, such as "Cambarellus patzcuarensis", will not destructively dig or eat live aquarium plants.


title: Crayfish

text: In some nations, such as the United Kingdom, United States, Australia, and New Zealand, imported alien crayfish are a danger to local rivers. The three most widespread American species invasive in Europe are "Faxonius limosus", "Pacifastacus leniusculus" and "Procambarus clarkii". Crayfish may spread into different bodies of water because specimens captured for pets in one river are often released into a different catchment. There is a potential for ecological damage when crayfish are introduced into non-native bodies of water: e.g., crayfish plague in Europe, or the introduction of the common yabby ("Cherax destructor") into drainages east of the Great Dividing Range in Australia.


title: Crayfish

text: The Protivin brewery in the Czech Republic uses crayfish outfitted with sensors to detect any changes in their bodies or pulse activity in order to monitor the purity of the water used in their product. The creatures are kept in a fish tank that is fed with the same local natural source water used in their brewing. If three or more of the crayfish have changes to their pulses, employees know there is a change in the water and examine the parameters.


title: Crayfish

text: Scientists also monitor crayfish in the wild in natural bodies of water to study the levels of pollutants there.


title: Moment of inertia

text: The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. The moment of inertia is a standard mechanical quantity. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.


title: Moment of inertia

text: It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.


title: Moment of inertia

text: For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.


title: Moment of inertia

text: When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moments of inertia may be expressed in units of kilogram metre squared (kg·m) in SI units and pound-foot-second squared (lbf·ft·s) in imperial or US units.


title: Moment of inertia

text: The moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by formula_1, where formula_2 is the distance of the point from the axis, and formula_3 is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.


title: Moment of inertia

text: In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum. The term "moment of inertia" was introduced by Leonhard Euler in his book "Theoria motus corporum solidorum seu rigidorum" in 1765, and it is incorporated into Euler's second law.


title: Moment of inertia

text: The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.


title: Moment of inertia

text: The moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.


title: Moment of inertia

text: The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.


title: Moment of inertia

text: The moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section.


title: Moment of inertia

text: The moment of inertia is also defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is


title: Moment of inertia

text: If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive.


title: Moment of inertia

text: If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque on a body to the angular acceleration around a principal axis, that is


title: Moment of inertia

text: For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as,


title: Moment of inertia

text: Thus, the moment of inertia of the pendulum depends on both the mass of a body and its geometry, or shape, as defined by the distance to the axis of rotation.


title: Moment of inertia

text: This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.


title: Moment of inertia

text: In general, given an object of mass , an effective radius can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is


title: Moment of inertia

text: Mathematically, the moment of inertia of a simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum this is found to be the product of the mass of the particle formula_3 with the square of its distance formula_2 to the pivot, that is


title: Moment of inertia

text: This can be shown as follows: The force of gravity on the mass of a simple pendulum generates a torque formula_11 around the axis perpendicular to the plane of the pendulum movement. Here formula_12 is the distance vector from the torque axis to the pendulum center of mass, and formula_13 is the net force on the mass. Associated with this torque is an angular acceleration, formula_14, of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is formula_15. Since formula_16 the torque equation becomes:


title: Moment of inertia

text: where formula_18 is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the vector triple product expansion with the perpendicularity of formula_14 and formula_12.) The quantity formula_21 is the "moment of inertia" of this single mass around the pivot point.


title: Moment of inertia

text: The quantity formula_21 also appears in the angular momentum of a simple pendulum, which is calculated from the velocity formula_23 of the pendulum mass around the pivot, where formula_24 is the angular velocity of the mass about the pivot point. This angular momentum is given by


title: Moment of inertia

text: Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield


title: Moment of inertia

text: This shows that the quantity formula_21 is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values formula_1 for all of the elements of mass in the body.


title: Moment of inertia

text: A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The natural frequency (formula_29) of a compound pendulum depends on its moment of inertia, formula_30,


title: Moment of inertia

text: where formula_3 is the mass of the object, formula_33 is local acceleration of gravity, and formula_2 is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.


title: Moment of inertia

text: Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point formula_35 so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (formula_36), to obtain


title: Moment of inertia

text: A simple pendulum that has the same natural frequency as a compound pendulum defines the length formula_39 from the pivot to a point called the center of oscillation of the compound pendulum. This point also corresponds to the center of percussion. The length formula_39 is determined from the formula,


title: Moment of inertia

text: The seconds pendulum, which provides the "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency of formula_43 for the pendulum. In this case, the distance to the center of oscillation, formula_39, can be computed to be


title: Moment of inertia

text: Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. Kater's pendulum is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a gravimeter.


title: Moment of inertia

text: The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.


title: Moment of inertia

text: These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the x-axis formula_46 and horizontal moment of the y-axis formula_47.<br>


title: Moment of inertia

text: Height ("h") and breadth ("b") are the linear measures, except for circles, which are effectively half-breadth derived, formula_2


title: Moment of inertia

text: The moment of inertia about an axis of a body is calculated by summing formula_1 for every particle in the body, where formula_2 is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia.)


title: Moment of inertia

text: Consider the kinetic energy of an assembly of formula_56 masses formula_57 that lie at the distances formula_58 from the pivot point formula_35, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses,


title: Moment of inertia

text: Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.


title: Moment of inertia

text: The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:


title: Moment of inertia

text: Here, the function formula_65 gives the mass density at each point formula_66, formula_12 is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point formula_66 in the solid, and the integration is evaluated over the volume formula_69 of the body formula_70. The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.


title: Moment of inertia

text: Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the formula_71-axis perpendicular to the cross-section, weighted by its density. This is also called the "polar moment of the area", and is the sum of the second moments about the formula_72- and formula_73-axes. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the formula_72-axis or formula_73-axis depending on the load.


title: Moment of inertia

text: The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass.


title: Moment of inertia

text: A list of moments of inertia formulas for standard body shapes provides a way to obtain the moment of inertia of a complex body as an assembly of simpler shaped bodies. The parallel axis theorem is used to shift the reference point of the individual bodies to the reference point of the assembly.


title: Moment of inertia

text: As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that can form the sphere whose centers are along the axis chosen for consideration. If the surface of the ball is defined by the equation


title: Moment of inertia

text: then the square of the radius formula_2 of the disc at the cross-section formula_71 along the formula_71-axis is


title: Moment of inertia

text: Therefore, the moment of inertia of the ball is the sum of the moments of inertia of the discs along the formula_71-axis,


title: Moment of inertia

text: If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis formula_18 perpendicular to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the "polar moment of inertia". The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles.


title: Moment of inertia

text: If a system of formula_101 particles, formula_102, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point formula_103, and absolute velocities formula_104:


title: Moment of inertia

text: where formula_24 is the angular velocity of the system and formula_107 is the velocity of formula_103.


title: Moment of inertia

text: For planar movement the angular velocity vector is directed along the unit vector formula_109 which is perpendicular to the plane of movement. Introduce the unit vectors formula_110 from the reference point formula_103 to a point formula_112, and the unit vector formula_113, so


title: Moment of inertia

text: This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.


title: Moment of inertia

text: Note on the cross product: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.


title: Moment of inertia

text: The moment of inertia formula_118 about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the "polar moment of inertia". Specifically, it is the second moment of mass with respect to the orthogonal distance from an axis (or pole).


title: Moment of inertia

text: For a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Figure skaters can change their moment of inertia by pulling in their arms. Thus, the angular velocity achieved by a skater with outstretched arms results in a greater angular velocity when the arms are pulled in, because of the reduced moment of inertia. A figure skater is not, however, a rigid body.


title: Moment of inertia

text: Let the reference point be the center of mass formula_116 of the system so the second term becomes zero, and introduce the moment of inertia formula_118 so the kinetic energy is given by


title: Moment of inertia

text: Newton's laws for a rigid system of formula_101 particles, formula_102, can be written in terms of a resultant force and torque at a reference point formula_103, to yield


title: Moment of inertia

text: The kinematics of a rigid body yields the formula for the acceleration of the particle formula_132 in terms of the position formula_103 and acceleration formula_134 of the reference particle as well as the angular velocity vector formula_24 and angular acceleration vector formula_14 of the rigid system of particles as,


title: Moment of inertia

text: For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along formula_18 perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors formula_139 from the reference point formula_103 to a point formula_112 and the unit vectors formula_113, so


title: Moment of inertia

text: where formula_145, and formula_146 is the unit vector perpendicular to the plane for all of the particles formula_132.


title: Moment of inertia

text: Use the center of mass formula_116 as the reference point and define the moment of inertia relative to the center of mass formula_118, then the equation for the resultant torque simplifies to


title: Moment of inertia

text: The scalar moments of inertia appear as elements in a matrix when a system of particles is assembled into a rigid body that moves in three-dimensional space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque of the rigid system of particles.


title: Moment of inertia

text: Let the system of formula_101 particles, formula_102 be located at the coordinates formula_112 with velocities formula_104 relative to a fixed reference frame. For a (possibly moving) reference point formula_103, the relative positions are


title: Moment of inertia

text: where formula_24 is the angular velocity of the system, and formula_159 is the velocity of formula_103.


title: Moment of inertia

text: Note that the cross product can be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix, formula_161, constructed from the components of formula_162:


title: Moment of inertia

text: The inertia matrix is constructed by considering the angular momentum, with the reference point formula_103 of the body chosen to be the center of mass formula_116:


title: Moment of inertia

text: where the terms containing formula_159 (formula_168) sum to zero by the definition of center of mass.


title: Moment of inertia

text: Then, the skew-symmetric matrix formula_169 obtained from the relative position vector formula_170, can be used to define,


title: Moment of inertia

text: is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass formula_116.


title: Moment of inertia

text: The kinetic energy of a rigid system of particles can be formulated in terms of the center of mass and a matrix of mass moments of inertia of the system. Let the system of formula_101 particles formula_102 be located at the coordinates formula_112 with velocities formula_104, then the kinetic energy is


title: Moment of inertia

text: , the second term in this equation is zero. Introduce the skew-symmetric matrix formula_169 so the kinetic energy becomes


title: Moment of inertia

text: where formula_172 is the inertia matrix relative to the center of mass and formula_187 is the total mass.


title: Moment of inertia

text: The inertia matrix appears in the application of Newton's second law to a rigid assembly of particles. The resultant torque on this system is,


title: Moment of inertia

text: where formula_189 is the acceleration of the particle formula_132. The kinematics of a rigid body yields the formula for the acceleration of the particle formula_132 in terms of the position formula_103 and acceleration formula_193 of the reference point, as well as the angular velocity vector formula_24 and angular acceleration vector formula_14 of the rigid system as,


title: Moment of inertia

text: Use the center of mass formula_116 as the reference point, and introduce the skew-symmetric matrix formula_198 to represent the cross product formula_199, to obtain


title: Moment of inertia

text: The inertia matrix of a body depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the center of mass formula_116 and the inertia matrix relative to another point formula_103. This relationship is called the parallel axis theorem.


title: Moment of inertia

text: Consider the inertia matrix formula_206 obtained for a rigid system of particles measured relative to a reference point formula_103, given by


title: Moment of inertia

text: where formula_211 is the vector from the center of mass formula_116 to the reference point formula_103. Use this equation to compute the inertia matrix,


title: Moment of inertia

text: The first term is the inertia matrix formula_172 relative to the center of mass. The second and third terms are zero by definition of the center of mass formula_116. And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix formula_218 constructed from formula_211.


title: Moment of inertia

text: where formula_211 is the vector from the center of mass formula_116 to the reference point formula_103.


title: Moment of inertia

text: Note on the minus sign: By using the skew symmetric matrix of position vectors relative to the reference point, the inertia matrix of each particle has the form formula_224, which is similar to the formula_1 that appears in planar movement. However, to make this to work out correctly a minus sign is needed. This minus sign can be absorbed into the term formula_226, if desired, by using the skew-symmetry property of formula_227.


title: Moment of inertia

text: The scalar moment of inertia, formula_228, of a body about a specified axis whose direction is specified by the unit vector formula_18 and passes through the body at a point formula_103 is as follows:


title: Moment of inertia

text: where formula_206 is the moment of inertia matrix of the system relative to the reference point formula_103, and formula_169 is the skew symmetric matrix obtained from the vector formula_235.


title: Moment of inertia

text: This is derived as follows. Let a rigid assembly of formula_101 particles, formula_102, have coordinates formula_112. Choose formula_103 as a reference point and compute the moment of inertia around a line L defined by the unit vector formula_18 through the reference point formula_103, formula_242. The perpendicular vector from this line to the particle formula_132 is obtained from formula_244 by removing the component that projects onto formula_18.


title: Moment of inertia

text: where formula_247 is the identity matrix, so as to avoid confusion with the inertia matrix, and formula_248 is the outer product matrix formed from the unit vector formula_18 along the line formula_39.


title: Moment of inertia

text: To relate this scalar moment of inertia to the inertia matrix of the body, introduce the skew-symmetric matrix formula_251 such that formula_252, then we have the identity


title: Moment of inertia

text: where the dot and the cross products have been interchanged. Exchanging products, and simplifying by noting that formula_244 and formula_18 are orthogonal:


title: Moment of inertia

text: Thus, the moment of inertia around the line formula_39 through formula_103 in the direction formula_18 is obtained from the calculation


title: Moment of inertia

text: where formula_206 is the moment of inertia matrix of the system relative to the reference point formula_103.


title: Moment of inertia

text: This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body.


title: Moment of inertia

text: For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The moment of inertia tensor is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.


title: Moment of inertia

text: Here formula_46 denotes the moment of inertia around the formula_72-axis when the objects are rotated around the x-axis, formula_283 denotes the moment of inertia around the formula_73-axis when the objects are rotated around the formula_72-axis, and so on.


title: Moment of inertia

text: These quantities can be generalized to an object with distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has


title: Moment of inertia

text: where formula_287 is their outer product, E is the 3×3 identity matrix, and "V" is a region of space completely containing the object.


title: Moment of inertia

text: The inertia tensor can be used in the same way as the inertia matrix to compute the scalar moment of inertia about an arbitrary axis in the direction formula_290,


title: Moment of inertia

text: where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such as formula_292 is obtained by the computation


title: Moment of inertia

text: and can be interpreted as the moment of inertia around the formula_72-axis when the object rotates around the formula_73-axis.


title: Moment of inertia

text: The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,


title: Moment of inertia

text: It is common in rigid body mechanics to use notation that explicitly identifies the formula_72, formula_73, and formula_71-axes, such as formula_46 and formula_283, for the components of the inertia tensor.


title: Moment of inertia

text: There are some CAD and CAE applications such as SolidWorks, Unigraphics NX/Siemens NX and MSC Adams that use an alternate convention for the products of inertia. According to this convention, the minus sign is removed from the product of inertia formulas and instead inserted in the inertia matrix:


title: Moment of inertia

text: If one has the inertia data formula_303 without knowing which inertia convention that has been used, it can be determined if one also has the principal axes. With the principal axes method, one makes inertia matrices from the following two assumptions:


title: Moment of inertia

text: Next, one calculates the eigenvectors for the two matrices. The matrix whose eigenvectors are parallel to the principal axes corresponds to the inertia convention that has been used.


title: Moment of inertia

text: The distance formula_2 of a particle at formula_307 from the axis of rotation passing through the origin in the formula_308 direction is formula_309, where formula_308 is unit vector. The moment of inertia on the axis is


title: Moment of inertia

text: For multiple particles, we need only recall that the moment of inertia is additive in order to see that this formula is correct.


title: Moment of inertia

text: Let formula_314 be the inertia tensor of a body calculated at its center of mass, and formula_103 be the displacement vector of the body. The inertia tensor of the translated body respect to its original center of mass is given by:


title: Moment of inertia

text: where formula_3 is the body's mass, E is the 3 × 3 identity matrix, and formula_318 is the outer product.


title: Moment of inertia

text: Let formula_103 be the matrix that represents a body's rotation. The inertial tensor of the rotated body is given by:


title: Moment of inertia

text: The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame. This means that as the body moves the components of the inertia matrix change with time. In contrast, the components of the inertia matrix measured in a body-fixed frame are constant.


title: Moment of inertia

text: Let the body frame inertia matrix relative to the center of mass be denoted formula_321, and define the orientation of the body frame relative to the inertial frame by the rotation matrix formula_134, such that,