Nonextensive statistical mechanics - Applications to nuclear and high energy physics

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Nonextensive statistical mechanics -Applications to nuclear and high energy physics ?Constantino Tsallis Centro Brasileiro de Pesquisas F′?sicas Rua Xavier Sigaud 150,22290-180Rio de Janeiro,RJ,Brazil Ernesto P.Borges Escola Politecnica,Universidade Federal da Bahia,Rua Aristides Novis 240210-630Salvador-BA,Brazil ?(February 2,2008)Abstract A variety of phenomena in nuclear and high energy physics seemingly do not satisfy the basic hypothesis for possible stationary states to be of the type covered by Boltzmann-Gibbs (BG)statistical mechanics.More speci?-cally,the system appears to relax,along time,on macroscopic states which violate the ergodic assumption.Some of these phenomena appear to follow,instead,the prescriptions of nonextensive statistical mechanics.In the same manner that the BG formalism is based on the entropy S BG =?k i p i ln p i ,the nonextensive one is based on the form S q =k (1? i p q i )/(q ?1)(with

S1=S BG).Typically,the systems following the rules derived from the former

exhibit an exponential relaxation with time toward a stationary state char-

acterized by an exponential dependence on the energy(thermal equilibrium),

whereas those following the rules derived from the latter are characterized by

(asymptotic)power-laws(both the typical time dependences,and the energy

distribution at the stationary state).A brief review of this theory is given here,

as well as of some of its applications,such as electron-positron annihilation

producing hadronic jets,collisions involving heavy nuclei,the solar neutrino

problem,anomalous di?usion of a quark in a quark-gluon plasma,and?ux of

cosmic rays on Earth.In addition to these points,very recent developments

generalizing nonextensive statistical mechanics itself are mentioned.

I.INTRODUCTION

The foundation of statistical mechanics comes from mechanics(classical,quantum,rel-ativistic,or any other elementary dynamical theory).Consistently,in our opinion,the expression of entropy to be adopted for formulating statistical mechanics(and ultimately thermodynamics)depends on the particular type of occupancy of phase space(or Hilbert space,or Fock space,or analogous space)that the microscopic dynamics of the system under study collectively favors.In other words,it appears to be nowadays strong evidence that statistical mechanics is larger than Boltzmann-Gibbs(BG)statistical mechanics,that the concept of physical entropy can be larger than

S BG=?k

W

i

p i ln p i(1)

(hence S BG=k ln W for equal probabilities),pretty much as geometry is known today to be larger than Euclidean geometry,since Euclid’s celebrated parallel postulate can be properly generalized in mathematically and physically very interesting manners.

Let us remind the words of A.Einstein[1]expressing his understanding of Eq.(1):

Usually W is put equal to the number of complexions[...].In order to calculate W,one needs a complete(molecular-mechanical)theory of the system under consideration.Therefore it is dubious whether the Boltzmann principle has any meaning without a complete molecular-mechanical theory or some other theory which describes the elementary processes.S= R

(q∈R;S1=S BG).(2)

q?1

It is well known that for microscopic dynamics which relax on an ergodic occupation of phase space,the adequate entropic form to be used is that of Eq.(1).Such assumption is ubiquitously satis?ed,and constitutes the physical basis for the great success of BG thermostatistics since over a century.We believe that a variety of more complex occupations of phase space may be handled with more complex entropies.In particular,it seems that Eq.

(2),associated with an index q which is dictated by the microscopic dynamics(and which generically di?ers from unity),is adequate for a vast class of stationary states ubiquitously found in Nature.In recent papers,E.G.D.Cohen[3]and M.Baranger[4]also have addressed this question.

A signi?cant amount of systems,e.g.,turbulent?uids([5,6]and references therein), electron-positron annihilation[7,8],collisions of heavy nuclei[9–11],solar neutrinos[12,13], quark-gluon plasma[14],cosmic rays[15],self-gravitating systems[17],peculiar velocities of galaxy clusters[18],cosmology[19],chemical reactions[20],economics[21–23],motion of

Hydra viridissima[24],theory of anomalous kinetics[25],classical chaos[26],quantum chaos [27],quantum entanglement[28],anomalous di?usion[29],long-range-interacting many-body classical Hamiltonian systems([30]and references therein),internet dynamics[31], and others,are known nowadays which in no trivial way accommodate within BG statistical mechanical concepts.Systems like these have been handled with the functions and concepts which naturally emerge within nonextensive statistical mechanics[2,32,33].

We may think of q as a biasing parameter:q<1privileges rare events,while q>1privi-leges common events.Indeed,p<1raised to a power q<1yields a value larger than p,and the relative increase p q/p=p q?1is a decreasing function of p,i.e.,values of p closer to0(rare events)are bene?ted.Correspondingly,for q>1,values of p closer to1(common events) are privileged.Therefore,the BG theory(i.e.,q=1)is the unbiased statistics.A concrete consequence of this is that the BG formalism yields exponential equilibrium distributions (and time behavior of typical relaxation functions),whereas nonextensive statistics yields (asymptotic)power-law distributions(and relaxation functions).Since the BG exponential is recovered as a limiting case,we are talking of a generalization,not an alternative.

To obtain the probability distribution associated with the relevant stationary state(ther-mal equilibrium or metaequilibrium)of our system we must optimize the entropic form(2) under the following constraints[2,32]:

p i=1,(3)

i

and

i p q i E i

,(5)

Z q

where

Z q≡ j[1?(1?q)βq(E j?U q)]1/(1?q),(6) and

β

βq≡

S will be optimized by the same distribution.Nevertheless,only a very restricted class of such entropic forms can be considered as serious candidates for constructing a full thermo-statistical theory,eventually connected with thermodynamics.In particular,one expects the correct entropy to be concave and stable.Such is the case[35]of S q as well as of the generalized entropy recently proposed[36,37]for the just mentioned superstatistics[34].We brie?y address these questions in this Section.

Let us?rst consider the following di?erential equation:

dy

=e?t/τ.(12)

O(0)?O(∞)

(iii)We may refer to the energy distribution at thermal equilibrium of a Hamiltonian system,and consider x≡E i,where E i is the energy of the i-th microscopic state,y=

Z p(E i),where p is the energy probability and Z the partition function,and?a≡βis the inverse temperature.In this case Eq.(10)reads in the familiar BG form:

p(E i)=

e?βE i

dx

=ay q(y(0)=1).(14) The solution is given by

y=e ax q≡[1+(1?q)ax]1/(1?q),(15) e x q being from now on referred to as the q-exponential function(e x1=e x).The three above possible physical interpretations of such solution now become

(i)For the sensitivity to the initial conditions,

ξ(t)=eλq t

q

=[1+(1?q)λq t]1/(1?q),(16) whereλq=0is the generalized Lyapunov coe?cient(see[26]),and,at the edge of chaos,λq>0and q<1.

(ii)For the relaxation,

O(t)?O(∞)

[1+(q?1)τq t]1/(q?1)

,(17) whereτq>0is a generalized relaxation time,and typically q≥1[38].

(iii)For the energy distribution,we get the form which emerges in nonextensive statistical mechanics,namely[2,32]

p(E i)=e?β′q E i

q

[1+(q?1)β′q E i]1/(q?1)

(Z′q≡ j e?β′q E j q),(18)

where usually,but not necessarily,β′q>0and q≥1.This distribution is the one that optimizes the entropy S q under appropriate constraints for the canonical ensemble.

Let us next unify Eqs.(9)and(14)as follows:

dy

+1 1

a1

Hagedorn.This scenario is now con?rmed.The ingredients for a microscopic model within this approach have also been proposed[8].

Heavy nuclei collisions:

A variety of high-energy collisions have been discussed in terms of the present nonexten-sive formalism.Examples include proton-proton,central Pb-Pb and other nuclear collisions [9,10].Along related lines,entropic inequalities applied to pion-nucleon experimental phase shifts have provided strong evidence of nonextensive quantum statistics[11].

Solar neutrino problem:

The solar plasma is believed to produce large amounts of neutrinos through a variety of mechanisms(e.g.,the proton-proton chain).The calculation done using the so called Solar Standard Model(SSM)results in a neutrino?ux over the Earth,which is roughly the double of what is measured.This is sometimes referred to as the neutrino problem or the neutrino enigma.There is by no means proof that this neutrino?ux defect is due to

a single cause.It has recently been veri?ed that neutrino oscillations do seem to exist(

[12]and references therein),which would account for part of the de?cit.But it is not at all clear that it would account for the entire discrepancy.Quarati and collaborators[13] argue that part of it–even,perhaps,an appreciable part of it–could be due to the fact that BG statistical mechanics is used within the SSM.The solar plasma involves turbulence, long-range interactions,possibly long-range memory processes,all of them phenomena that could easily defy the applicability of the BG formalism.Then they show[13]in great detail how the modi?cation of the“tail”of the energy distribution could considerably modify the neutrino?ux to be expected.Consequently,small departures from q=1(e.g.,|q?1|of the order of0.1)would be enough to produce as much as50%di?erence in the?ux.This is due to the fact that most of the neutrino?ux occurs at what is called the Gamow peak. This peak occurs at energies quite above the temperature,i.e.,at energies in the tail of the distribution.

Quark di?usion:

The anomalous di?usion of a charm quark in a quark-gluon plasma has been analyzed by

Walton and Rafelski[14]through both nonextensive statistical mechanical arguments and quantum chromodynamics.The results coincide,as they should,only for q=1.114. Cosmic rays:

The?uxΦof cosmic rays arriving on Earth is a quantity whose measured range is among the widest experimentally known(33decades in fact).This distribution refers to a range of energies E which also is impressive(13decades).This distribution is very far from exponential:See Figs.1and2.This basically indicates that no BG thermal equilibrium is achieved,but some other(either stationary,or relatively slow varying)state,characterized in fact by a power law.If the distribution is analyzed with more detail,one veri?es that two,and not one,power-law regimes are involved,separated by what is called the“knee”(slightly below1016eV).At very high energies,the power-law seems to be interrupted by what is called the“ankle”(close to1019eV)and perhaps a cuto?.The momenta

M l≡ (E? E )l =[ E cutof f

0dEΦ(E)(E? E )l]/[ E cutof f

dEΦ(E)](l=1,2,3)as functions

of the cuto?energy E cutoff(assumed to be abrupt for simplicity)are calculated as well:See Figs.3,4and5.At high cuto?energies, E saturates at2.48944...×109eV[15],a value which is over ten times larger than the Hagedorn temperature(close to1.8×108eV[8]).In the same limit,we obtain for the speci?c-heat-like quantity M2? E2 ?6.29×1021(eV)2. Finally,M3? E3 diverges with increasing E cutoff.This is of course due to the fact that, in the high energy limit,Φ∝1/E1

dE i

=?b′p q′i?bp q i.(21)

This di?erential equation has remarkable particular cases.The most famous one is(q′,q)= (1,2),since it precisely corresponds to the di?erential equation which enabled Planck,in his October1900paper,to(essentially)guess the black-body radiation distribution,thus opening(together with his December1900paper)the road to quantum mechanics.The more general case q′=1and arbitrary q is a simple particular instance of the Bernoulli equation,and,as such,has a simple explicit solution(Eq.(20)with a1=?b′and a q=?b). This solution has proved its e?ciency in a variety of problems,including in generalizing the Zipf-Mandelbrot law for quantitative linguistics(for a review,see Montemurro’s article in the Gell-Mann–Tsallis volume[33]).Finally,the generic case q>q′>1also has an explicit solution(though not particularly simple,but in terms of two hypergeometric functions; see[38])and produces,taking also into account the ultra-relativistic ideal gas density of states,the above mentioned quite good agreement with the observed?uxes.Indeed,if we assume0

For possible e?ects of a slightly nonextensive black-body radiation on cosmic rays see [40].Finally,other aspects related to cosmic rays have been shown to exhibit?ngerprints of nonextensivity[41].

IV.CONCLUSIONS

In nuclear and high energy physics,there is a considerable amount of anomalous phe-nomena that bene?t from a thermostatistical treatment which exceeds the usual capabilities of Boltzmann-Gibbs statistical mechanics.This fact is due to the relevance of long-range forces,as well as to a variety of dynamical nonlinear dynamical aspects,possibly leading to nonmarkovian processes,i.e.,long-term microscopic memory.Some of these phenomena

appear to be tractable within nonextensive statistical mechanics,and we have illustrated this with a few typical examples.For the particular case of cosmic rays,we have indi-cated their average energy E ?2.48944...×109eV,and the speci?c-heat-like quantity E2 ? E 2?6.29×1021(eV)2,with the hope that they might be usefully compared to related astrophysical quantities,either already available in the literature,or to be studied. Along the same vein we have also presented the dependence of such momenta on a possibly existing high-energy cuto?.

In addition to this,we have sketched the possible generalization of nonextensive statistical mechanics on the basis of a recently introduced entropic form[36],whose stationary state is the Beck-Cohen superstatistics[34].The metaequilibrium distribution associated with a crossover between q-statistics and q′-statistics can be seen as a particular case of this generalized nonextensive statistical mechanics.

It is worthy to mention at this point that the present attempts for further generalization of BG statistical mechanics are to be understood on physical grounds,and by no means as informational quantities that can be freely introduced to deal with speci?c tasks,and which can in principle depend on as many free(or?tting)parameters as one wishes.Examples of such informational quantities are the Renyi entropy(depending on one parameter and being usefully applied in the multifractal characterization of chaos),Sharma-Mittal entropy (which contains both Renyi entropy and S q as particular cases),and very many others that are available in the literature.The precise criteria for an entropic form to be considered a possible physical entropy are yet not fully understood,although it is already clear that it must have a microscopic dynamical foundation.It seems however reasonable to exclude,at this stage,those forms which,in contrast with S BG and S q,(i)are not concave(or convex),since this would seriously damage the capability for thermodynamical stability and for satisfactory thermalization of di?erent systems put in thermodynamical contact,or(ii)are not stable, since this would imply that the associated quantity would not be robust under experimental measurements.These crucial points,and several others(probably equally important,such as the?nite entropy production per unit time),have been disregarded by Luzzi et al[42]in their

recent criticism of nonextensive statistical mechanics.Indeed,Renyi entropy,Sharma-Mittal entropy(that Luzzi et al mention without any justi?cation at the same epistemological level as S q)are neither concave nor stable for arbitrary values of their parameters.These and other information measures(most of them not concave and/or not stable)have been freely introduced,along various decades,as optimizing tools for speci?c tasks.They can certainly be useful for various purposes,which do not necessarily include the speci?c one we are addressing here:a thermodynamically meaningful generalization of the Boltzmann-Gibbs physical entropy.

ACKNOWLEDGMENTS

We are indebted to T.Kodama,G.Wilk,I.Bediaga,E.G.D.Cohen,M.Baranger and J. Anjos for useful remarks that we have received along the years.One of us(CT)is grateful to M.Gell-Mann for many and invaluable discussions on this subject.This work has been partially supported by PRONEX/MCT,CNPq,and FAPERJ(Brazilian agencies).

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Mechanics

Mechanics is an area of science concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes\ (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo, Kepler, and especially Newton, laid the foundation for what is now known as classical mechanics. It is a branch of classical physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as a branch of science which deals with the motion of and forces on objects. In the undergraduate, we Theoretical Mechanics Statics kinematics (运动学) 动力学（Dynamics） Strength of materials, also called mechanics of materials, is a subject which deals with the behavior of solid objects subject to stresses and strains. Dynamics is a branch of physics (specifically classical mechanics) concerned with the study of forces and torques（力矩） and their effect on motion, as opposed to kinematics, which studies the motion of objects without reference to its causes. Isaac Newton defined the fundamental physical laws which govern dynamics in physics, especially his second law of motion. Generally speaking, researchers involved in dynamics study how a physical system might develop or alter over time and study the causes of those changes. In addition, Newton established the fundamental physical laws which govern dynamics in physics. By studying his system of mechanics, dynamics can be understood. In particular, dynamics is mostly related to Newton's second law of motion. However, all three laws of motion are taken into account because these are interrelated in any given observation or experiment.

Mechanics of Writing 英文写作过程

CHAPTER1 Mechanics of Writing Error-free writing requires more than just using good grammar.You must also use correct mechanics of writing in your documents.The mechanics of writing speci?es the established conventions for words that you use in your documentation.Grammar re?ects the forms of words and their relationships within a sentence.For instance,if you put an apostrophe in a plural word(“Create two?le’s”),you have made a mistake in the mechanics of writing,not grammar. The mechanics of writing guidelines in this chapter work well for computer documentation,but other style guides might suggest different rules that are equally effective.In most cases,which rules you follow doesn’t matter as long as you are consistent within your document or documentation set.See Chapter2for options related to the use of text and graphical elements,such as section headings,tables,and cross-references. This chapter discusses the following topics: I“Capitalization”on page2 I“Contractions”on page4 I“Gerunds and Participles”on page5 I“Numbers and Numerals”on page6 I“Pronouns”on page10 I“Technical Abbreviations,Acronyms,and Units of Measurement”on page11 I“Punctuation”on page14 1

01-Mechanics of Vibration

振动力学Mechanics of Vibration

张亚辉Email: zhangyh@https://www.doczj.com/doc/4e14544070.html, 张辉 Offi h S d 00000 Office: 综合实验楼1号楼505 Office hours: Saturday 10:00am ‐11:00am Meetings by appointment are also available 研究方向 ?结构动力学（土木、机械、航空航天、车辆）?复杂结构随机振动及其工程应用?大跨度结构抗震?结构抗风 ?车辆与桥梁动力相互作用? 噪声与振动控制

Textbook and optional references Required textbook 11.张亚辉, 林家浩. 结构动力学基础. 大连理工大学出版社, 2007 Reference textbooks 1.Thomson W T, Dillon D M. Theory of Vibration with Applications. New Jersey: Prentice Hall, 1998 2.Clough R W, Penzien J. Dynamics of Structures (Third Edition). Computers & Structures Inc., 2003（克拉夫R W等著, 王光远等译. 结构动力学. 科学出版社, 1985, 2006） 3.Meirovitch L. Elements of Vibration Analysis (2nd Edition). McGraw‐ Hill inc., inc1986

AP Physics C Mechanics Practice Tests

The questions contained in this AP ? Physics C: Mechanics Practice Exam are written to the content specifications of AP Exams for this subject. Taking this practice exam should provide students with an idea of their general areas of strengths and weaknesses in preparing for the actual AP Exam. Because this AP Physics C: Mechanics Practice Exam has never been administered as an operational AP Exam, statistical data are not available for calculating potential raw scores or conversions into AP grades.This AP Physics C: Mechanics Practice Exam is provided by the College Board for AP Exam preparation. Teachers are permitted to download the materials and make copies to use with their students in a class-room setting only. To maintain the security of this exam, teachers should collect all materials after their administration and keep them in a secure location. Teachers may not redistribute the files electronically for any reason. ? 2008 The College Board. All rights reserved. College Board, Advanced Placement Program, AP , AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trade-mark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web: https://www.doczj.com/doc/4e14544070.html,. Practice Exam Advanced Placement Program AP ? Physics C: Mechanics

泛珠力学基础试试题及答案-PanPhO2008_Mechanics

香港物理奥林匹克委员会主办、中国教育学会物理教学专业委员会和台北市资优教育发展协会协办 第四届泛珠三角及中华名校物理奥林匹克邀请赛 力 学 基 础 试 赛 题 （2008年2月14日9:00-12:00） 一、选择题 (4分×10 = 40分) 1. 由地面竖直上抛一小球，若小球受到的阻力的大小与小球运动的速率成正比，以a 1表示小球抛出时加速度的大小，a 2表示小球升到最高点时加速度的大小及a 3表示小球刚要落回到地面时加速度的大小，则在小球往返过程中， A. a 1a 2>a 3 C. a 2a 3>a 1 E. a 3a 1>a 2 2. 某半径为R =1,600km 的行星有A 和B 二个卫星绕其做圆周运动，它们距行星表面的高度分别为 h A =81km 和h B = 164km ，则卫星A 和B 的速度之比B A v v 和周期之比B A T 为 A.41/42和423/413 B.413/423和42/41 C.423/413和41/42 D.42/41和413/423 E.422/412和412/422 3. 质量为M 及倾角为α的三角形木块位于光滑水平桌面上，而质量为 m 的小滑块位于木块的粗糙斜面上。木块受到水平外力F 作用而运动， 在运动过程中滑块相对木块始终保持相对静止。滑块所受摩擦力的方向 与外力F 的大小的关系是 A. αsin )(g m M F +>时摩擦力沿斜面向下 B. αsin )(g m M F +=时不存在摩擦力 C. αcos )(g m M F +>时摩擦力沿斜面向下 D. αcos )(g m M F +=时不存在摩擦力 E. αtan )(g m M F +>时摩擦力沿斜面向下 F. αtan )(g m M F +>时摩擦力沿斜面向上 一个滑块以初速度 5m/s 在光滑水平桌面上滑行，之后它平滑地滑上 一座有光滑斜面的三角形木块。木块可在桌面上自由滑行。木块质量 是滑块质量的 4倍。 4. 滑块到达斜面最高点时木块的速度和该最高点的高度为 A. 1m/s 和1m B. ?1m/s 和1m C. 2m/s 和1m D. ?2m/s 和1m E. 1m/s 和2m F. ?1m/s 和2m 5. 滑块在斜面滑行到最高点后再沿三角形木块滑下。滑块離开木块时，木块和滑块的速度分别为 A.1m/s 和2m/s B.1m/s 和?2m/s C.2m/s 和2m/s D.2m/s 和?2m/s E. 2m/s 和3m/s F. 2m/s 和?3m/s 6. AB 为一光滑水平直轨道，BCD 是以O 为圆心及半径R =2m 的半 圆形光滑轨道，水平直轨道与半圆轨道连接在一起。在水平轨道上 有两个小球，P 球的质量为Q 球的2倍，P 球与静止的Q 球碰撞后 速度变为原来的1/3。使小球Q 刚能通过轨道的最高点D ，P 球去 碰撞Q 球所需的最小速度是 A. 2.5m/s B. 5m/s C. 7.5m/s D. 10m/s E. 10m/s F. 12.5m/s 7. 在离桌面高25m 处手持一上一下很接近的两个小球，下球的质量是上球的质量4倍。现将两球同时释放，设此后发生的下球与桌面的碰撞以及两小球之间的碰撞都是完全弹性碰撞，而且碰撞时间极短。两小球碰撞后，上球能升到离桌面的高度约是 A. 81m B. 100m C. 121m D. 144m E. 169m F. 196m 8. 一光滑圆圈用细绳垂直挂在天花板上。兩个质量相同的小圆环从圈顶由静止开 始同时向兩边下滑。已知小圆环质量是圆圈质量的2倍。当细绳张力为零时小圆环 的位置θ = A. 300 B. 400 C. 500 D. 600 E. 700 F. 800

力学词汇mechanics

力学词汇(mechanics) dynamic similarity 动力相似[性] plane flow 平面流 potential 势 potential flow 势流 velocity potential 速度势 complex potential 复势 complex velocity 复速度 stream function 流函数 source 源 sink 汇

velocity head 速度[水]头 corner flow 拐角流 cavity flow 空泡流曾用名“空腔流”。 supercavity 超空泡 supercavity flow 超空泡流 aerodynamics 空气动力学 low-speed aerodynamics 低速空气动力学 high-speed aerodynamics 高速空气动力学 aerothermodynamics 气动热力学 subsonic flow 亚声速流[动]又称“亚音速流[动]”。 transonic flow 跨声速流[动]又称“跨音速流[动]”。 supersonic flow 超声速流[动]又称“超音速流[动]”。 hypersonic flow 高超声速流[动]又称“高超音速流[动]”。

conical flow 锥形流 wedge flow 楔流 cascade flow 叶栅流 non-equilibrium flow 非平衡流[动] slender body 细长体 slenderness 细长度 bluff body 钝头体 blunt body 钝体 airfoil 翼型 chord 翼弦 thin-airfoil theory 薄翼理论 configuration 构型

Deformation (mechanics)

The deformation of a thin straight rod into a closed loop. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. In this particular case of bending,displacements associated with rigid translations and rotations of material elements in the rod are much greater than displacements associated with straining. Deformation (mechanics) From Wikipedia, the free encyclopedia Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.[1] A configuration is a set containing the positions of all particles of the body. A deformation may be caused by external loads,[2] body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. Strain is a description of deformation in terms of relative displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered.In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations,e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed are called elastic deformations . In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is plastic deformation , which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is viscous deformation ,which is the irreversible part of viscoelastic deformation. In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material. Contents 1 Strain 1.1 Strain measures

《Mechanics 1历年考试真题分类汇编》

Section A Velocity and Acceleration （Chapter 1） 1． A man runs in a straight line. He passes through a fixed point A with constant velocity 7ms_1at time t = 0. At time t s his velocity is v ms_1. The diagram shows the graph of v against t for the period 0 ≤t ≤ 40. （02w） (i) Show that the man runs more than 154m in the first 24 s. [2] (ii)Given that the man runs 20m in the interval 20 ≤t ≤ 24, find how far he is from A when t = 40.[2] 2 The diagram shows the velocity-time graphs for the motion of two cyclists P and Q, who travel in the same direction along a straight path. Both cyclists start from rest at the same point O and both accelerate at 2ms?2 up to a speed of 10ms?1. Both then continue at a constant speed of 10ms?1.(03s) Q starts his journey T seconds after P. (i) Show in a sketch of the diagram the region whose area represents the displacement of P, from O,at the instant when Q starts. [1]

力学词汇(mechanics)

力学词汇(mechanics)1 mechanics 力学 Newtonian mechanics 牛顿力学 classical mechanics 经典力学 statics 静力学 kinematics 运动学 dynamics 动力学 kinetics 动理学 macroscopic mechanics，macromechanics 宏观力学 mesomechanics 细观力学尺度约为0.01——100μm . microscopic mechanics，micromechanics 微观力学 general mechanics 一般力学 solid mechanics 固体力学 fluid mechanics 流体力学 theoretical mechanics 理论力学 applied mechanics 应用力学 engineering mechanics 工程力学experimental mechanics 实验力学computational mechanics 计算力学rational mechanics 理性力学 physical mechanics 物理力学geodynamics 地球动力学 force 力 point of action 作用点 line of action 作用线 system of forces 力系 reduction of force system 力系的简化又称“力系的约化”。 equivalent force system 等效力系 rigid body 刚体 transmissibility of force 力的可传性parallelogram rule 平行四边形定则 force triangle 力三角形 force polygon 力多边形 null-force system 零力系 equilibrium 平衡 equilibrium of forces 力的平衡 equilibrium condition 平衡条件equilibrium position 平衡位置 equilibrium state 平衡态component force 分力 resultant force 合力 resolution of force 力的分解 composition of forces 力的合成 couple 力偶 arm of couple 力偶臂 system of couples 力偶系 resultant couple 合力偶 moment arm of force 力臂 moment of force 力矩 moment of couple 力偶矩 moment of area 面矩 center of moment 矩心 moment vector 矩矢 moment vector of couple 力偶矩矢principal vector 主矢 principal moment 主矩 torque 转矩 force screw 力螺旋 acting force 作用力 reacting force 反作用力 reaction at support 支座反力 friction force 摩擦力 kinetic friction 动摩擦 rolling friction 滚动摩擦 coefficient of rolling friction 滚动摩擦系数sliding friction 滑动摩擦 coefficient of sliding friction 滑动摩擦系数static friction 静摩擦 coefficient of maximum static friction 最大静摩擦系数 angle of friction 摩擦角 Coulomb law of friction 库仑摩擦定律center of reduction 简化中心又称“约化中心”。 internal force 内力 external force 外力 elastic force 弹性力 distributed force 分布力 concurrent force 汇交力 forces acting at the same point 共点力coplanar force 共面力 constraint 约束 constraint force 约束力

泛珠力学基础试试题及答案-PanPhO2011_Mechanics

香港物理奧林匹克委員會主辦 中國教育學會物理教學專業委員會協辦 第七屆泛珠三角物理奧林匹克暨中華名校邀請賽 力學基礎試賽題 (2011年2月10日9:00-12:00) **有需要时,如无说明,取g =10m/s 2; G =6.67×10?11Nm 2/kg 2. ** *** 选择题1至16(48分,答案唯一)和简答题17至20(52分),做在答题纸上.*** 1. 一位观察者站在静止的列车第一节车厢的前端。当列车以等加速度开动时，第一节车厢经过其旁需5s ，则第十节车厢经过其旁的时间大约是 A. 1.18s B. 1.07s C. 0.98s D. 0.91s E. 0.86s F. 0.81s 2. 如图所示，两个用轻线相连的位于光滑水平面上的物块，质量分 别为m l 和m 2，拉力F 1和F 2方向相反，与刚度系数为k 的轻线沿同一 水平直线运动，且F 1>F 2。在两个物块运动过程中轻线的伸长x 为 A.212211m m k m F m F ?+ B.)(212211m m k m F m F ++ C.)(212211m m k m F m F +? D.)(211221m m k m F m F +? E.)(211221m m k m F m F ++ F.2 11221m m k m F m F ?+ 3. 如图所示，一质量为M 的三角形木块放在水平桌面上，它的顶角为900， 两底角为α和β，两个质量均为m 的小木块位于両侧斜面上。己知所有接触 面都是光滑的。现发现两小木块沿斜面下滑，而模形木块静止不动，这时三 角形木块对水平桌面的压力等于 A. Mg B. 2mg C. Mg +2mg D. Mg +mg E. Mg +mg (βαsin sin +) F. Mg +mg (βαcos cos +) (题4-5) 一个倾角为α的固定斜面上，在斜面底部有一小物块。现给物块一个沿斜面初速率v 1与使它沿斜面向上滑动，经过时间t 1到达最高点，之后它又自动滑回到底部，此时物体的末速率为v 2，所用时间为t 2。若物体与斜面间的滑动摩擦系数为μ，则 4. t 1: t 2为 5. v 1: v 2为 A.αμααμαcos sin cos sin ?+ B.αμααμαcos sin cos sin ?+ C.αμααμαcos sin cos sin +? D.αμααμαcos sin cos sin +? E.αμαcos sin F.α μαsin cos 6. 质量m =10kg 的物体在水平推力F =200N 作用下，从不光滑斜面的底端由静止开始沿固定不动的斜面运动，斜面的倾角为θ 且sin θ=0.6及cos θ=0.8。力F 作用2s 时间后撤去，物体在斜面上继续滑行1.25s 时间后，速度减为零。此后物体下滑回斜面底端所用的时间t 约为 A. 2.85s B. 2.70s C. 2.55s D. 2.40s E. 2.25s F.2.10s 7. 已知地球质量M =6.4×1024kg 和半径R = 6,400km ，则同步通信卫星在赤道上方的高度(km) 约为 A. 43,200 B. 36,800 C. 30,400 D. 10,720 E.4,320 F. 3,680 8. 太阳光线在春分日与地球赤道平面平行。某颗同步卫星正下方的地球表面上有一观察者，在春分这一天用天文望远镜观察到被太阳光照射的该卫星。不考虑大气对光的折射作用，则观察者从日落起直到恰好看不见卫星，所经历的时间(小时hour)大约为 A. 3.21 B. 4.32 C. 5.43 D. 6.54 E. 7.65 F. 8.76 (题9-10) 在光滑水平面上放置一个质量为2m 和半径为R =0.5m 的光滑半球形 碗。在半球形碗上搁置一根质量为m 的竖直杆，此杆只能沿竖直方向运动，如 图所示。开始时竖直杆下端正好和碗的半球面上边缘接触，然后从静止释放杆， 杆和碗开始运动。若某瞬时杆的位置θ=600，则