## От редакцииМы продолжаем публикации шнобелевских лауреатов. На этот раз Роберт Мэтьюз. Он получил шнобелевскую премию по физике в 1996 г. за изучение физической стороны законов Мерфи применительно к бутерброду, падающему, как известно, маслом вниз. Результат этого исследования, предлагаемый вниманию читателей, был опубликован в 1995 г. в Eur. J. Phys. ## Robert A.J. Matthews. TUMBLING TOAST, MURPHY'S LAW AND THE FUNDAMENTAL CONSTANTS## AbstractWe investigate the dynamics of toast tumbling from a table to the floor. Popular opinion is that the final state is usually butter-side down, and constitutes prima facie evidence of Murphy’s Law. The orthodox view, in contrast, is that the phenomenon is essentially random, with a ## 1. IntroductionThe term Murphy’s Law has its origins in dynamical experiments conducted by the US Air Force in the late 1940s involving an eponymous USAF captain The phenomenon of toast falling from a table to land butter-side down on the floor is popularly held to be empirical proof of the existence of Murphy’s Law. Furthermore, there is a widespread belief that it is the result of a genuine physical effect, often ascribed to a dynamical asymmetry induced by one side of the toast being buttered. Quite apart from whether or not the basic observation is true, this explanation cannot be correct. The mass of butter added to toast (~ 4 ## 2. Dynamics of falling toast.In what follows we model the tumbling toast problem as an example of a rigid, rough, homogeneous rectangular lamina, mass m, side 2a, falling from a rigid platform set a height h above the ground. We consider the dynamics of the toast from an initial state where its centre of gravity overhangs the table by a distance δ0 as shown in fig. 1. Initially, we ignore the process by which the toast arrives at this state, and also assume that it has zero horizontal velocity; the important effect of a non-zero horizontal velocity is addressed later. Finally we assume a perfectly inelastic impact with the floor with zero rebound. With these assumptions, the dynamics of the lamina are determined by the forces shown in fig. 1: Fig. 1. The initial orientation of the rotating toast✓ the weight, ✓ the frictional force, ✓ the reaction of the table, The resulting angular velocity about the point of contact,
where k is the appropriate radius of gyration, such that for the rectangular lamina considered here. Multiplying (3) by 2ω and integrating from the initial conditions ω = 0 at θ = 0 leads to:
where we have used δ ≡ ηa, with η (0 < η < 1) being the overhang parameter. Equation (4) is the central equation of the tumbling toast problem, as it gives the rate of rotation of the toast once it has detached from the table from a specific state of overhang. Unless the toast can complete sufficient rotation on its descent to the floor to bring the buttered side facing upwards, the toast will land buttered side down. Thus if the toast begins its descent at an angle φ to the horizontal, then for it to land butter-side up again we must have
where ω
The frictional force acting on the lamina will prevent detachment until the lamina has rotated through at least an angle φ at which point slipping occurs. This minimum value of φ follows from the usual condition F = μR, where μ is the coefficient of static friction between the lamina and the table edge. From (l), (2) and (4) we find
To calculate the free-falling angular rotation rate ω
where the value of the critical overhang parameter η
where and . For conventional tables and slices of toasts, we have h ~ 75 cm, 2α ~ 10 cm leading to R ~ 15, α ~ 0.06 and thus a lower limit on the critical overhang parameter of
if the toast is to complete sufficient rotation to avoid a butter-side down final state. ## 3. Experimental results and implications.An experimental determination of qo holds the key to establishing whether or not the fall of toast constitutes a manifestation of Murphy's Law. Tests were carried out using a lamina derived from a standard white loaf. The lamina was cut into a rectangle of 10 cm × 7.3 cm × 1.5 cm, and placed on a rigid flat and level platform of kitchen Contiboard, used to model the surface of a clean, uncovered table. Measurements of the value of the coefficient of static friction μ between the lamina and the platform were made by measuring the angle of the platform at which sliding just began; the tangent of this angle is then equal to μ. Test were carried out on both bread and toast, leading to
Measurements of the value of the critical overhang parameter η This was found to be
Both bread and toast are thus relatively unstable to tumbling from overhanging positions. Crucially, neither can sustain overhangs anywhere near as large as the critical value given in (10). This implies that laminae with either composition do not have sufficient angular rotation to land butter-side up following free-fall from a table-top. In other words, the material properties of slices of toast and bread and their size relative to the height of the typical table are such that, in the absence of any rebound phenomena, they lead to a distinct bias towards a butter-side down landing. But before this can be taken as confirmation of popular belief, however, some practical issues must be addressed. ## 4. The effects of non-zero horizontal velocity.So far, we have ignored the means by which the toast comes to be in the overhang condition shown in fig. 1. This is clearly of practical importance, however, as the toast will typically leave the table as the result of sliding off a tilted plate, or being struck by a hand or arm. The consequent horizontal velocity may dominate the dynamics if the gravitational torque has insufficient time to induce significant rotation. In this case, the toast will behave like a simple projectile off the edge of the table, keeping its butter-side up throughout the flight. This raises the possibility that, while dynamically valid, the butter-side down phenomenon may only be witnessed for an in feasibly small range of horizontal velocities. To investigate this range, we first note that the time for an initially horizontal lamina of overhang parameter η to acquire inclination ψ follows from (8):
where
If the lamina has a horizontal velocity V
At speeds considerably below this value (below, say, the torque-induced rotation should still dominate the dynamics of the falling toast, and the butter-side down phenomenon should still be observed. This conclusion is supported by observation. Furthermore, the relatively high value of V ## 5. Tumbling toast and the fundamental interactions.We have seen that the outcome of the fall of toast from a table is dictated by two parameters: the surface properties of the toast, which determine η
where is the fall velocity, f is the fraction of kinetic energy that goes into breaking N polymeric bonds of binding energy E
where m
A simple Bohr-atom model shows that
where α is the electronic fine structure constant, m
where μ is the radius of the polymeric atoms in units of the Bohr radius α
where R
_{H} gives, alter some reduction,
where Inserting the various values, we find that this first-principles argument leads to a maximum safe height for human of around 3 m. Although the estimate of L
and inserting the observed value η ~ 0.015 given in (12) leads to R ~ 60 and h ~ 3 m. The limit (23) thus implies that all human-like organisms are doomed to experience tumbling toast landing butter-side down. ## 6. Conclusions.Our principal conclusion is a surprising one, given the apparently quotidian nature of the original phenomenon: all human-like organisms are destined to experience the tumbling toast manifestation of Murphy’s Law because of the values of the fundamental constants in our universe. As such, we have probably confirmed the suspicions of many regarding the innate cussedness of the universe. We therefore feel we must conclude this investigation on a more optimistic note. What can human-like-and thus presumably intelligent-organisms do to avoid toast landing butter-side down? Building tables of the ~ 3 m height demanded by (24) is clearly impracticable. Reducing the size of toast is dynamically equivalent, but the required reduction in size is also nnsatisfactory. The best approach is somewhat counter-intuitive: toast seen heading off the table should be given a smart swipe forward with the hand. Similarly, a plate off which toast is sliding should be moved swiftly downwards and backwards, disconnecting the toast from the plate. Both actions have the effect of minimising the amount of time the toast is exposed to the gravitationally-induced toque, either by giving the toast a large horizontal velocity or by sudden disconnection of the point of contact. In both cases, the toast will descend to the floor keeping the butter side uppermost. We end by noting that, according to Einstein, God is subtle, but He is not malicious. That may be so, but His inhence on falling toast clearly leaves much to be desired. ## Acknowledgements.It is a pleasure to thank Professor Ian Fells and Robin Bootle for providing background on Murphy’s Law.## References.[l] Bootle R. and Fells I. 1991 QED: Murphy’s Law (London, BBC).
[2] Bootle R. 1995 personal communication.
[3] Press W.H. 1980 Am. J. Phys. 48 597-8.
[4] Davies P.C.W. 1982 The Accidental Universe. (Cambridge: Cambridge University Press) 44-9.## Invitation to Reading Discuss |
## Из wikipedia.org
Robert A.J. Matthews (born 1959) is a British physicist, mathematician, computer scientist, and journalist. Байесовская вероятность, интерпретация понятия вероятности, используемая в байесовской теории. Robert Pershing Wadlow (1918 – 1940) also known as the Alton Giant and the Giant of Illinois, is the tallest person in history for whom there is irrefutable evidence. Ian Fells is Emeritus Professor of Energy Conversion at the University of Newcastle upon Tyne, and former chairman of the "New and Renewable Energy Centre" at Blyth, Northumberland, England. William H. Press (born 1948) is an astrophysicist, theoretical physicist, and computational biologist. Paul Charles William Davies (born 1946) is an English physicist, writer and broadcaster, a professor at Arizona State University as well as the Director of BEYOND: Center for Fundamental Concepts in Science. ## Компьютеры (продолжение)Пенти - ум, а Интер - нет
Перед тем, как процессор сгорит, в его памяти проносятся все операции, которые он когда-либо совершил.
Плохому сайту всегда баннеры мешают.
Повестка в военкомат - это бесплатный купон на двухгодичное отключение от Интернета.
Поломал Вася-хакер своего провайдера - и сидит, как дурак, без интернета...
Послать по e-мейлу к е-матери…
Почему книги Гейтса выходят в двух томах?- Второй том - patСh к первому...
Правда, что Билл Гейтс самый знаменитый человек в мире? - Нет. Самый знаменитый - его мать.
Предложение по усовершенствованию Windоws: пусть система создаёт не только папки Му dосumеnts, Му рiсturеs и Му musiс, но и Му роrnо.
При звуках коннекта теряет волю
При использовании трекбола надевайте коврик на палец
Прикладник является приложением к компьютеру, операционной системе и тому начальнику, который говорит ему, что делать.
Приложение совершило несовершаемую ошибку и будет закрыто
Программа MiСrоsоft не совершила никакой ошибки, но по привычке будет закрыта
Программист - это человек, который сначала долго думает, чтобы потом ничего не делать.
Программисты не пишут нелогичных программ, бывают нелогичные пользователи.
Программисты не умирают - они теряют память...
Продаётся модем, подключённый к Интернет
Продается стиральная машина "Windows-95". Стирает все
Пропала собака, особые приметы: Shift+2
Протокол IPX/SPX является быстрым маршрутизируемым протоколом для небольших сетей, но у него есть один недостаток - он разработан фирмой Novell... (С) «Секреты Windows NT 4.0 Server»
Пусти "Осла" в сеть, так он, осёл, весь траффик съест!
Пьяный русский хакер практически непобедим!
Ракета - она не дура. У неё внутри, извиняюсь за выражение, компьютер.
Раньше микропроцессоры были большими и занимали целые комнаты...
Российские хакеры взломали бортовой компьютер российского истребителя СУ-27. Теперь боекомплект самолёта нескончаемый.
С тех пор, как к интернету подключились женщины, нельзя доверять ни одному рейтингу или голосованию...
С точки зрения программиста пользователь - это периферийное устройство, вводящее набор символов в ответ на команду READ
Самый злобный вирус, который ничем нельзя вывести - это справка от Микрософт.
Самый страшный вирус всегда сидит перед компьютером.
Сегодня пробовал секс с женщиной. Жалкая пародия на установку Windows. (Из дневника ламера)
Секрет долговечности батареек Энерджайзерс - они работают на операционках фирмы MiСrosoft и повисают на определённом уровне зарядки!!!
Семь бед - один Reset
Сервер - не суетись под клиентом!
Сидим, как в деревне! Ни радио, ни телевизора - один Интернет остался!
Скажи мне какая у тебя стартовая страница, и я скажу кто ты.
Скажи мне, кто Билл Гейтс, и я скажу кто ты
Скажи мне, что ты ищешь в Интернете, и я скажу, кто ты.
Скачанного не воротишь.
Сначала было слово. И слово было два байта...
Совместим с гибким дисководом.
Спать на работе - грех, не для того вам там дан бесплатный интернет!
Супер акция!!! Загляни под крышку винчестера и выиграй приз!!!
Там, на неведомых дорожках китайских ломанных СD...
Тамбовский волк тебе провайдер...
Тарантино снял фильм "Убить Билла" после трехчасовой установки Windows XP. |