Material Space Motion Time Phenomenon of Kinetic Energy and Inertia of Material Bodies

Size: px
Start display at page:

Download "Material Space Motion Time Phenomenon of Kinetic Energy and Inertia of Material Bodies"

Transcription

1 AASCIT Journl of Physics 15; 1(4): 9-96 Published online July, 15 ( Mteril Spce Motion Time Phenomenon of Kinetic Energy nd Inerti of Mteril Bodies F F Mende B Verkin Institute for Low Temperture Physics nd Engineering, NAS Ukrine, Khrkov, Ukrine Emil ddress mende_fedor@milru, fmende@milru Keywords Mteril, Spce, Motion, Time, Kinetic Energy, Sclr-Vector Potentil Received: My 8, 15 Revised: June 6, 15 Accepted: June 7 15 Cittion F F Mende Mteril Spce Motion Time Phenomenon of Kinetic Energy nd Inerti of Mteril Bodies AASCIT Journl of Physics Vol 1, No 4, 15, pp 9-96 Abstrct In the rticle re exmined the problems, connected with the determintion of such concepts s spce, mteril, time Is introduced the concept of the point of united plce, which is only in the spce The concept of clustering spce, which presents the spce in the form is introduced of clusters with the intended sizes It is shown tht the time is not primry vlue, which re the length, mss nd force, but it depends on the prmeters indicted Is introduced the new system of units, in which the time is expressed through the mss nd the length It is well known tht for ccelerting the mteril bodies it is necessry to spend energy, in this cse the executed work psses into the kinetic energy With the brking the body returns this energy to the surrounding bodies, for which required the forces, the reverse fcts, which ccelerted body This is the phenomenon of the inerti However, nture of this phenomenon ws up to now not cler In the rticle it is shown tht the inerti nd kinetic energy of mteril bodies re the consequence of the dependence of the sclr potentil of chrge on the speed 1 Introduction The universe is filled with different forms of mteril, nd this mteril is found in the continuous motion Specificlly, with the motion of mtter is connected the introduction of the concept of time But in the existing scientific literture there is no cler physicl definition of such concepts s mteril nd spce There is no cler determintion nd concept of time Moreover, mny reserchers re inclined to consider tht this time the sme physicl substnce s mteril nd spce; therefore during the geometriztion of spce the time consider s one of the coordintes, whose scle cn depend on the speed of the motion of frme of reference The specil theory of reltivity is built on these principles In connection with this pproch the time is introduced s the primry physicl prmeter, which enters into ll existing systems of units In this cse is not considered the fct tht for mesuring the time necessrily in the required order the ttrction of other physicl quntities, such s mss, the length nd force Therefore nturl question rises, nd it is not possible whether to express time through physicl the quntities indicted Actully, the concept of length is inseprbly connected with the spce metrics cn be introduced s the distnce between two mteril objects, or s wvelength, which is extended in the free spce The mterility of the mteril objects surrounding us does not cuse doubt The concept of force we lso encounter in our dily life, undergoing grvity force, nd lso when we observe the ccelertion of bodies, under the ction of force This rticle is dedicted to the exmintion of mtters under discussion

2 9 F F Mende: Mteril Spce Motion Time Phenomenon of Kinetic Energy nd Inerti of Mteril Bodies Mteril nd the Spce In order to introduce wht or concept necessry to, first of ll, describe its properties of the within the frmework existing concepts In nture we observe two forms of mteril First of ll, these re the toms, of which consist the bodies, which is conventionlly designted s mteril Into the structure of toms enter the smller formtions, which is conventionlly designted s nucleons Atoms possess sizes, nd one of the bsic its of spce consists in the fct tht t its one nd the sme point it is not possible to simultneously plce two toms By this is determined the uniqueness of the point indicted, which in the spce is only in its kind We will cll this property of the point of spce the property of united plce From this property escpes the concept of time, which indictes tht t the point of united plce cnnot simultneously be locted two different mteril bodies, but to occupy this plce they cn only in the specific sequence This sequence introduces the concept of time [1,] There is nother form of mteril, which includes different fields nd electromgnetic wves The properties of this mteril re differed from the lredy exmined mteril tel Of this mteril is inherent the property of the interference, with which the fields interfere (they re dded) ccording to the specific lws Wve fields re chrcterized by tht property, tht they re found in the constnt motion, but interfering, they lso cn form structures fixed in the spce, which re the stnding wves Permnent fields lso interfere ccording to the specific lws nd cn destroy (to compenste) ech other An exmple is point, which is found on the line, which connects two similr dwn or two identicl mteril bodies If point is exist equidistntlyed from both objects, then fields t this point re red lso on the object, locted t the point indicted, they do not ct Introducing the concept of united plce, we re thus chieved clustering spce, mking with his discrete, nd this discretion hs its prmeters By the unitry unit, which determines clustering spce, we will cll unitry cluster Prcticl of the reliztion of clustering is determined by the technicl cpbilities of mesuring the sizes of mteril objects nd by the shortest wvelengths, which re known to us A clssicl rdius of electron composes 8x1-15 m In proton rdius still less composes 9x1-16 m The spectrum of the wvelengths of X-rdition vries in the limits of m Simplest is the cubic clustering, when unitry clusters re represented in the form the cube, whose edge they correspond to the lengths indicted With this pproch the point of united plce is the center of the cube indicted The minimlly resolvble volume of this cluster nd the minimlly resolvble distnce between the points of united plce, is determined by the resolution of the mens of the mesurement of length We will consider one of the possible lines, which contin the minimum number of such points, the distnce between two points of united plce We will cll this line stright line For the introduction of the concept of plne it is necessry to hve three points of the united plce, connected by stright lines We will cll the surfce, stretched between such lines nd which hs smllest possible quntity of points of united plce flt surfce For the introduction of the concept of plne it is necessry to hve three points of the united plce, connected by stright lines If we put on stright lines the plnes indicted, then they will limit the volume (we will cll this volume geometric figure), in which there will be locted specific quntity of points of united plce The summry volume of the clusters, entering the limited volume will represent the volume geometric of the figure in question For the ide of figures with the complex geometric form it is necessry to isolte more thn four points of united plce in the spce For the introduction of the concept of ngle let us introduce the concept of circle We will round count the line, locted on the plne, whose points re exist equidistnt from the selected point of united plce We will cll this point the center of circle We will consider tht two not coinciding stright lines, which emnte of one point of united plce form ngle If these lines proceed from the center of circle nd re locted in the plne, on which the circle is locted, then these stright lines interesect the line of circle We will cll distnce between centers of circle nd points of intersection of lines with the circle rdius of circle The length of the section of circle, locted between the points of intersection, expressed in the lengths of rdius, we will consider the mesure of the ngle, locted between rdii Rdin is the unit of the mesurement of ngles This is tht cse, when the length of the section between the points of intersection is equl to the length of rdius Motion nd the Time We will cll the displcement of body with the miniml sizes between the points of united plce simple motion This motion cn be rectiliner, if body is moved between the points of united plce, locted on the stright line nd by curviliner, if the condition indicted is not stisfied The motion of body cnnot rise spontneously, nd stisfction of the specified conditions is required for the ppernce of this motion So tht the body would begin to move to it necessry the ction from the side of other bodies, which is clled force In the existing systems of units the eqution of motion is written s follows F = m, where F is force, which cts on the body, m is the mss of body, is the ccelertion of body In this cse the ccelertion is defined s the second derivtive of wy by the time d x = dt In order to use this reltionship, the system of units, which introduces mss length nd time, is used s the primry units

3 AASCIT Journl of Physics 15; 1(4): But, if for the introduction of length nd mss there re physicl bses, since these re the ctully observed physicl quntities, there re no such bses for the introduction to time, since the time is not ctully observed The ctully observed physicl quntities re mss length nd force; therefore let us ttempt to express time through these vlues But for this it is necessry to select the units of length, force nd mss As the units of length nd mss we cn select the lredy existing units (in the system SI this is meter nd kilogrm)for the selection of the unit of force we will use the lw of universl grvittion We will consider tht there re two identicl msses M, whose centers re locted t distnce R In this cse we will consider tht the liner dimensions of the msses considerbly less thn the distnce between them In ccordnce with the lw of universl grvittion the force, which cts between such msses, will compose F = M 4R (1) In this reltionship we specilly lowered the grvittionl constnt, which is introduced in other systems of units, nd which contins second, since we ttempt to build the new system of units, which does not contin time The time, which will be required by the mss of M in order under the ction of the force indicted to cover distnce R, it is determined by the reltionship [ 1,] t = ± R M () This vlue let us ccept for the unit of time Its vlue is determined by specific physicl quntities nd their properties tking into ccount the lw of universl grvittion Exponentil is the fct tht in the dt the time cn be both positive nd negtive vlueis known tht time reversl, ie, sign chnge of time does not chnge the form of equtions of motion This mens tht for ny possible motion of system cn be chieved the time-reversed motion, when system consecutively psses to the reverse order of the sttes, symmetricl to sttes, pssed in the previous motion In this posing of the question nturlly to ssume tht, when in the system it occurs no chnges, then time for this system not t ll flows When in the system some reversible chnges occur, ie, it fter certin evolution returns reversibly to its initil stte, the time flows first in one, nd then in other direction, reversing the sign since in this cse the concept of time used to in ppliction to this concrete system, it is possible to introduce the proper time of system, ie to ssume tht in ech seprtely undertken system there is its proper time Sttes symmetricl on the time re chrcterized by opposite directions of the speeds (pulses) of prticles nd mgnetic field Temporry invrince leds to specific rtios between the probbilities of direct nd reverse rections, to the prohibition of some sttes of the polriztion of prticles in the rections, to the equlity to zero electricl dipole moment of elementry prticles nd T d It follows from the generl principles of the quntum field theory tht ll processes in nture re symmetricl reltive to the work of three opertions: the time reversl, three-dimensionl inversion nd chrge conjugtion Using the exmined method of introduction to time it is possible to build hours If re locted two identicl msses M, locted t distnce R, then, in ccordnce with the lw of universl grvittion, the force of their ttrction determines the dependence: If the msses indicted revolve round the overll center of msses nd cts the principle of the equivlence of grvittionl nd inert mss, then the equlity will be crried out: R T = ±4π () M where T is period of revolution of msses round the overll center It is evident tht this reltionship is differed from reltionship () only in terms of coefficient in order to trnsfer this vlue into seconds, should be used by coefficient For its obtining the lw of universl grvittion (1) nd the vlues, entering reltionship () one should write down in one of the systems of units In the system SI the vluet, clculted in seconds, will be equl R T = ±4π GM Consequently, conversion fctor will be equl T H K = = T H G (4) where grvittionl constnt G is determined in the system SI If we clculte the coefficient K, then it will be evident tht the newly introduced unit of time is pproximtely five orders more thn second This, of course, is not very convenient, but in order to void these inconveniences, it is possible to introduce dimensionless coefficient (4) into reltionship () Then the time, mesured by the hours exmined will be clculted in seconds Since time now hs its own dimensionlity, pssge to the electricl systems of units lso does not compose lbor, simply into the pproprite dimensionlity of ones it is necessry to put the new dimensionlity of time with the selected dimensionless conversion fctor If we for mesuring the electricl units use to Guss system nd to express in it time in the units of mss nd length, then ll electricl nd mgnetic units will be lso expressed in the units of mss nd length It is well known tht for ccelerting the mteril bodies it is necessry to spend energy, in this cse the executed work psses into the kinetic energy With the brking the body

4 95 F F Mende: Mteril Spce Motion Time Phenomenon of Kinetic Energy nd Inerti of Mteril Bodies returns this energy to the surrounding bodies, for which be required the forces, the reverse fcts, which ccelerted body This is the phenomenon of the inerti It is cler tht in the process of ccelertion the body ccumultes some form of energy, which returns then to the environment with its brking But none of the existing t present theories gives nswer to question, tht this fter energy nd how it is ccumulted nd returns Chrged the bodies nd in the chrges re hd the electricl field, possessing of the energy It is possible to expect tht the dependence of these pour on from the speed it cn shed light to this question In the specil theory of reltivity (SR) the electric fields of chrges depend on speed, nd, it would seem, this theory hd to give nswer to the presented question Butinto SR the chrge is the invrint of the speed Its fields lthough chnge in the process of ccelertion, these chnges occur in such wy tht to n increse pour on norml to the direction of motion it is compensted by the decrese of longitudinl pour on, nd the flow of the electric field through the surfce, which surrounds chrge remins constnt In works [-1] it is shown tht within the frmework the Glileo conversions the sclr potentil of chrge depends on speed In this cse the electric fields, norml to the direction of its motion, increse, while longitudinl fields they remin constnt Similr of the pproch gives of the possibility to explin of the phenomenon of the kinetic energy nd the inerti of the mteril bodies 4 Kinetic Energy of the Electrified Bodies The electron hs the electricl fields, energy which esy to clculte Specific energy of the electricl fields is written w = 1 ε E The tension of the electricl fields of the electron is determined by the equlity e E = 4πε r Using the element the volume 4π r dr, we obtin the energy of the fields on resting of the electron: e dr W = = 8πε 8 r e πε, where e is the chrge, is rdius of the electron If electron moves with the speed v, then, ccording to the concept of sclr-vector potentil, its electric fields, norml to the direction of motion, increse [6-1] v 1v E = Ech E 1+ c c Let us write down the electric fields, norml to the direction of motion in the coordinte system, represented in Fig 1 Fig 1 The element of the volume, utilized for the clcultion of the energy fields moving of the electron Then the energy of the fields moving electron will be written down W 1v e sin qdqdr = 1+ v c 8πε r The integrtion to the ngle gives π π sin (1 cos ) (cos ) cos cos q 4 Therefore qdq = q d q = q + = 4 4 1v e dr 4 v 1v e Wv = 1+ = c 8 πε r c c πε For of the speeds is considerble smller of the speed of the 1v light the term cn be disregrded, therefore 4c 4 v e Wv = 1 + c 8πε The connection between by energy of the fields nd with the mss of the rest of the electron is given by the equlity [11]: 4 e W = = mc 8πε Consequently, dditionl energy of electron, connected with the fct tht of its field they depend on speed, to be determined by the reltionship W = mv v This is the kinetic energy moving of the electron It is differed from the conventionl vlue in terms of the coefficient 1, but this indictes only tht the fct tht the

5 AASCIT Journl of Physics 15; 1(4): officilly tken vlue of the mss of electron must be decresed two times Thus, we estblished the physicl cuse for the presence in the moving electrified bodies of kinetic energy, nd, therefore, lso their inerti properties These the property re connected with the dependence of the sclr potentil of chrges on the speed, nd since ll mteril bodies consist of the free or bound chrges, this rule is universl 5 Conclusion In the rticle re exmined the problems, connected with the determintion of such concepts s spce, mteril, time Is introduced the concept of the point of united plce, which is only in the spce The concept of clustering spce, which presents the spce in the form is introduced of clusters with the intended sizes It is shown tht the time is not primry vlue, which re the length, mss nd force, but it depends on the prmeters indicted Is introduced the new system of units, in which the time is expressed through the mss nd the length The concept of the globl of counting nd globl time is introduced Are given the conversions of electromgnetic pour on upon trnsfer from the globl of counting into the inertil system It is well known tht for ccelerting the mteril bodies it is necessry to spend energy, in this cse the executed work psses into the kinetic energy With the brking the body returns this energy to the surrounding bodies, for which be required the forces, the reverse fcts, which ccelerted body This is the phenomenon of the inerti However, nture of this phenomenon ws up to now not cler In the rticle it is shown tht the inerti nd kinetic energy of mteril bodies re the consequence of the dependence of the sclr potentil of chrge on the speed The reltionships, which reflect this lw, re obtined References [1] F F Mende, The problem of contemporry physics nd method of their solution, LAP LAMBERT Acdemic Publishing, 1 [] F F Mende, Problems of modern physics nd their solutions, PALMARIUM Acdemic Publishing, 1 [] F F Mende, On refinement of equtions of electromgnetic induction, Khrkov, deposited in VINITI, No 774 B88 Dep, 1988 [4] F F Mende, Are there errors in modern physics Khrkov, Constnt, [5] FF Mende, On refinement of certin lws of clssicl electrodynmics, rxivorg/bs/physics/484 [6] F F Mende, Conception of the sclr-vector potentil in contemporry electrodynmics, rxivorg/bs/physics/568 [7] F F Mende, Gret misconceptions nd errors physicists XIX-XX centuries Revolution in modern physics, Khrkov, NTMT, 1 [8] F F Mende, New electrodynmics, revolution in the modern physics Khrkov, NTMT, 1 [9] F F Mende, New pproches in contemporry clssicl electrodynmics Prt II, Engineering Physics,, 1, p -17 [1] F F Mende, Considertion nd the Refinement of Some Lws nd Concepts of Clssicl Electrodynmics nd New Ides in Modern Electrodynmics, Interntionl Journl of Physics, 14, Vol, No 8, 1-6 [11] R Feynmn, R Leighton, M Sends, Feynmn lectures on physics, М Mir, Vol6, 1977

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Classical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011

Classical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011 Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,

More information

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if

More information

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

More information

Set up Invariable Axiom of Force Equilibrium and Solve Problems about Transformation of Force and Gravitational Mass

Set up Invariable Axiom of Force Equilibrium and Solve Problems about Transformation of Force and Gravitational Mass Applied Physics Reserch; Vol. 5, No. 1; 013 ISSN 1916-9639 E-ISSN 1916-9647 Published by Cndin Center of Science nd Eduction Set up Invrible Axiom of orce Equilibrium nd Solve Problems bout Trnsformtion

More information

Summary of equations chapters 7. To make current flow you have to push on the charges. For most materials:

Summary of equations chapters 7. To make current flow you have to push on the charges. For most materials: Summry of equtions chpters 7. To mke current flow you hve to push on the chrges. For most mterils: J E E [] The resistivity is prmeter tht vries more thn 4 orders of mgnitude between silver (.6E-8 Ohm.m)

More information

This final is a three hour open book, open notes exam. Do all four problems.

This final is a three hour open book, open notes exam. Do all four problems. Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

More information

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which

More information

Name Solutions to Test 3 November 8, 2017

Name Solutions to Test 3 November 8, 2017 Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

INTERACTION BETWEEN THE NUCLEONS IN THE ATOMIC NUCLEUS. By Nesho Kolev Neshev

INTERACTION BETWEEN THE NUCLEONS IN THE ATOMIC NUCLEUS. By Nesho Kolev Neshev INTERACTION BETWEEN THE NUCLEONS IN THE ATOMIC NUCLEUS By Nesho Kolev Neshev It is known tht between the nucleons in the tomic nucleus there re forces with fr greter mgnitude in comprison to the electrosttic

More information

INTRODUCTION. The three general approaches to the solution of kinetics problems are:

INTRODUCTION. The three general approaches to the solution of kinetics problems are: INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The

More information

Week 10: Line Integrals

Week 10: Line Integrals Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles

Method of Localisation and Controlled Ejection of Swarms of Likely Charged Particles Method of Loclistion nd Controlled Ejection of Swrms of Likely Chrged Prticles I. N. Tukev July 3, 17 Astrct This work considers Coulom forces cting on chrged point prticle locted etween the two coxil,

More information

Intro to Nuclear and Particle Physics (5110)

Intro to Nuclear and Particle Physics (5110) Intro to Nucler nd Prticle Physics (5110) Feb, 009 The Nucler Mss Spectrum The Liquid Drop Model //009 1 E(MeV) n n(n-1)/ E/[ n(n-1)/] (MeV/pir) 1 C 16 O 0 Ne 4 Mg 7.7 14.44 19.17 8.48 4 5 6 6 10 15.4.41

More information

ragsdale (zdr82) HW2 ditmire (58335) 1

ragsdale (zdr82) HW2 ditmire (58335) 1 rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

More information

ME 141. Lecture 10: Kinetics of particles: Newton s 2 nd Law

ME 141. Lecture 10: Kinetics of particles: Newton s 2 nd Law ME 141 Engineering Mechnics Lecture 10: Kinetics of prticles: Newton s nd Lw Ahmd Shhedi Shkil Lecturer, Dept. of Mechnicl Engg, BUET E-mil: sshkil@me.buet.c.bd, shkil6791@gmil.com Website: techer.buet.c.bd/sshkil

More information

DIRECT CURRENT CIRCUITS

DIRECT CURRENT CIRCUITS DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

More information

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.

- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students. - 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting

More information

Candidates must show on each answer book the type of calculator used.

Candidates must show on each answer book the type of calculator used. UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

More information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Measuring Electron Work Function in Metal

Measuring Electron Work Function in Metal n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd

More information

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015 Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, 15 1. (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be

More information

MAT187H1F Lec0101 Burbulla

MAT187H1F Lec0101 Burbulla Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

CAPACITORS AND DIELECTRICS

CAPACITORS AND DIELECTRICS Importnt Definitions nd Units Cpcitnce: CAPACITORS AND DIELECTRICS The property of system of electricl conductors nd insultors which enbles it to store electric chrge when potentil difference exists between

More information

Physics 1402: Lecture 7 Today s Agenda

Physics 1402: Lecture 7 Today s Agenda 1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:

More information

Period #2 Notes: Electronic Structure of Atoms

Period #2 Notes: Electronic Structure of Atoms Period # Notes: Electronic Structure of Atoms The logicl plce (for civil engineers) to begin in describing mterils is t the tomic scle. The bsic elements of the tom re the proton, the neutron, nd the electron:

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Physics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018

Physics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018 Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.

More information

Electric Potential. Concepts and Principles. An Alternative Approach. A Gravitational Analogy

Electric Potential. Concepts and Principles. An Alternative Approach. A Gravitational Analogy . Electric Potentil Concepts nd Principles An Alterntive Approch The electric field surrounding electric chrges nd the mgnetic field surrounding moving electric chrges cn both be conceptulized s informtion

More information

General Relativity 05/12/2008. Lecture 15 1

General Relativity 05/12/2008. Lecture 15 1 So Fr, We Hve Generl Reltivity Einstein Upsets the Applecrt Decided tht constnt velocity is the nturl stte of things Devised nturl philosophy in which ccelertion is the result of forces Unified terrestril

More information

1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum?

1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum? Which of the following summrises the chnge in wve chrcteristics on going from infr-red to ultrviolet in the electromgnetic spectrum? frequency speed (in vcuum) decreses decreses decreses remins constnt

More information

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B

Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere

More information

Physics 2135 Exam 3 April 21, 2015

Physics 2135 Exam 3 April 21, 2015 Em Totl hysics 2135 Em 3 April 21, 2015 Key rinted Nme: 200 / 200 N/A Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. 1. C Two long stright

More information

Lecture XVII. Vector functions, vector and scalar fields Definition 1 A vector-valued function is a map associating vectors to real numbers, that is

Lecture XVII. Vector functions, vector and scalar fields Definition 1 A vector-valued function is a map associating vectors to real numbers, that is Lecture XVII Abstrct We introduce the concepts of vector functions, sclr nd vector fields nd stress their relevnce in pplied sciences. We study curves in three-dimensionl Eucliden spce nd introduce the

More information

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5.

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5. PHY1 Electricity Topic 5 (Lectures 7 & 8) pcitors nd Dielectrics In this topic, we will cover: 1) pcitors nd pcitnce ) omintions of pcitors Series nd Prllel 3) The energy stored in cpcitor 4) Dielectrics

More information

Motion of Electrons in Electric and Magnetic Fields & Measurement of the Charge to Mass Ratio of Electrons

Motion of Electrons in Electric and Magnetic Fields & Measurement of the Charge to Mass Ratio of Electrons n eperiment of the Electron topic Motion of Electrons in Electric nd Mgnetic Fields & Mesurement of the Chrge to Mss Rtio of Electrons Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1.

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

Conservation Law. Chapter Goal. 5.2 Theory

Conservation Law. Chapter Goal. 5.2 Theory Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very

More information

Kepler's Three LAWS. Universal Gravitation Chapter 12. Heliocentric Model. Geocentric Model. Other Models. Johannes Kepler

Kepler's Three LAWS. Universal Gravitation Chapter 12. Heliocentric Model. Geocentric Model. Other Models. Johannes Kepler Universl Grvittion Chpter 1 Johnnes Kepler Johnnes Kepler ws Germn mthemticin, stronomer nd strologer, nd key figure in the 17th century Scientific revolution. He is best known for his lws of plnetry motion,

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

and that at t = 0 the object is at position 5. Find the position of the object at t = 2. 7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

Problems for HW X. C. Gwinn. November 30, 2009

Problems for HW X. C. Gwinn. November 30, 2009 Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object

More information

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions: Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGI OIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription

More information

Vector potential quantization and the photon wave-particle representation

Vector potential quantization and the photon wave-particle representation Vector potentil quntiztion nd the photon wve-prticle representtion Constntin Meis, Pierre-Richrd Dhoo To cite this version: Constntin Meis, Pierre-Richrd Dhoo. Vector potentil quntiztion nd the photon

More information

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s 4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The

More information

Density of Energy Stored in the Electric Field

Density of Energy Stored in the Electric Field Density of Energy Stored in the Electric Field Deprtment of Physics, Cornell University c Tomás A. Aris October 14, 01 Figure 1: Digrm of Crtesin vortices from René Descrtes Principi philosophie, published

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah 1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,

More information

Recitation 3: More Applications of the Derivative

Recitation 3: More Applications of the Derivative Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

More information

2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm

2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm 2.57/2.570 Midterm Exm No. 1 Mrch 31, 2010 11:00 m -12:30 pm Instructions: (1) 2.57 students: try ll problems (2) 2.570 students: Problem 1 plus one of two long problems. You cn lso do both long problems,

More information

Lesson 8. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER)

Lesson 8. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER) Lesson 8 Thermomechnicl Mesurements for Energy Systems (MEN) Mesurements for Mechnicl Systems nd Production (MME) A.Y. 205-6 Zccri (ino ) Del Prete Mesurement of Mechnicl STAIN Strin mesurements re perhps

More information

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016

Physics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016 Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric

More information

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS 1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

More information

CHAPTER 4 MULTIPLE INTEGRALS

CHAPTER 4 MULTIPLE INTEGRALS CHAPTE 4 MULTIPLE INTEGAL The objects of this chpter re five-fold. They re: (1 Discuss when sclr-vlued functions f cn be integrted over closed rectngulr boxes in n ; simply put, f is integrble over iff

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

E S dition event Vector Mechanics for Engineers: Dynamics h Due, next Wednesday, 07/19/2006! 1-30

E S dition event Vector Mechanics for Engineers: Dynamics h Due, next Wednesday, 07/19/2006! 1-30 Vector Mechnics for Engineers: Dynmics nnouncement Reminders Wednesdy s clss will strt t 1:00PM. Summry of the chpter 11 ws posted on website nd ws sent you by emil. For the students, who needs hrdcopy,

More information

CHEMICAL KINETICS

CHEMICAL KINETICS CHEMICAL KINETICS Long Answer Questions: 1. Explin the following terms with suitble exmples ) Averge rte of Rection b) Slow nd Fst Rections c) Order of Rection d) Moleculrity of Rection e) Activtion Energy

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

Chapter 14. Matrix Representations of Linear Transformations

Chapter 14. Matrix Representations of Linear Transformations Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

More information

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation 1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview

More information

Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc

Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc Physics 170 Summry of Results from Lecture Kinemticl Vribles The position vector ~r(t) cn be resolved into its Crtesin components: ~r(t) =x(t)^i + y(t)^j + z(t)^k. Rtes of Chnge Velocity ~v(t) = d~r(t)=

More information

PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS

PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?

More information

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

+ x 2 dω 2 = c 2 dt 2 +a(t) [ 2 dr 2 + S 1 κx 2 /R0

+ x 2 dω 2 = c 2 dt 2 +a(t) [ 2 dr 2 + S 1 κx 2 /R0 Notes for Cosmology course, fll 2005 Cosmic Dynmics Prelude [ ds 2 = c 2 dt 2 +(t) 2 dx 2 ] + x 2 dω 2 = c 2 dt 2 +(t) [ 2 dr 2 + S 1 κx 2 /R0 2 κ (r) 2 dω 2] nd x = S κ (r) = r, R 0 sin(r/r 0 ), R 0 sinh(r/r

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

west (mrw3223) HW 24 lyle (16001) 1

west (mrw3223) HW 24 lyle (16001) 1 west (mrw3223) HW 24 lyle (16001) 1 This print-out should hve 30 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Reding ssignment: Hecht, sections

More information

Lecture 3: Curves in Calculus. Table of contents

Lecture 3: Curves in Calculus. Table of contents Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up

More information

Physics 202, Lecture 14

Physics 202, Lecture 14 Physics 202, Lecture 14 Tody s Topics Sources of the Mgnetic Field (Ch. 28) Biot-Svrt Lw Ampere s Lw Mgnetism in Mtter Mxwell s Equtions Homework #7: due Tues 3/11 t 11 PM (4th problem optionl) Mgnetic

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

10 Vector Integral Calculus

10 Vector Integral Calculus Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

More information

13: Diffusion in 2 Energy Groups

13: Diffusion in 2 Energy Groups 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

More information

Energy creation in a moving solenoid? Abstract

Energy creation in a moving solenoid? Abstract Energy cretion in moving solenoid? Nelson R. F. Brg nd Rnieri V. Nery Instituto de Físic, Universidde Federl do Rio de Jneiro, Cix Postl 68528, RJ 21941-972 Brzil Abstrct The electromgnetic energy U em

More information

UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3

UNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Light and Optics Propagation of light Electromagnetic waves (light) in vacuum and matter Reflection and refraction of light Huygens principle

Light and Optics Propagation of light Electromagnetic waves (light) in vacuum and matter Reflection and refraction of light Huygens principle Light nd Optics Propgtion of light Electromgnetic wves (light) in vcuum nd mtter Reflection nd refrction of light Huygens principle Polristion of light Geometric optics Plne nd curved mirrors Thin lenses

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

Conducting Ellipsoid and Circular Disk

Conducting Ellipsoid and Circular Disk 1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information