Experimental evaluation of the process of decohesion of adhesive joints with polymer films

Size: px
Start display at page:

Download "Experimental evaluation of the process of decohesion of adhesive joints with polymer films"

Transcription

1 olimey, No , 2, p. 82 Expeimentl evlution of the pocess of decohesion of dhesive oints with polyme films M. Zenkiewicz nd J. Dzwonkowski Tnsltion submitted by J.E. Bke Selected fom Intentionl olyme Science nd Technology, 28, No. 1, 21, efeence T /11-12/82; tnsl. seil no Accute detemintion of the dhesive stength of oints in which one of the elements is plstics film is quite difficult. Selection of the method of investigting dhesive stength, nd pope intepettion of the esults obtined, e both vey impotnt; these esults seve not only fo evluting the popeties of the oint unde investigtion, but e lso the bsis fo selecting the conditions of ctivtion of the plstics film. The methods of investigting the dhesive stength of oints with plstics film cn be divided into two min goups (ef. 1). The methods in the fist goup e chcteised in tht oint filue duing the investigtions occus simultneously nd instntneously ove its whole sufce, whees the chcteistic fetue of the methods in the second goup is tht oint filue poceeds gdully, ove longe peiod of time, genelly fom few seconds to sevel tens of seconds. In investigtions of dhesive oints of the plstics film/dhesive/ plstics film type o oints of the igid plte (most often of metl o plstics)/dhesive/plstics film type, methods fom the second goup e the most impotnt, especilly methods using peeling nd teing-off (efs. 2 5). The method of teing-off t n ngle of 18 (ef. 3) is employed univeslly in investigtions of the dhesive stength of oints of the igid plte/dhesive/plstics film type. An essentil condition fo its use is tht neithe ccking no fctue should occu in the flexible lye of the dhesive oint tht is pulled wy (i.e. in the plstics film), which is submitted to bending t n ngle of 18. Using this method, it is possible to investigte the dhesion between the dhesive nd The uthos e fom The 'Metlchem' Institute of lstics ocessing, Toun the plstics film, povided it is less thn the dhesion between the dhesive nd the igid plte nd less thn the decohesion of this dhesive in the solid stte. In this ticle we pesent the esults of investigtions of dhesive oints of the igid luminium plte/cylic dhesive/polyethylene film type. The investigtions included detemintion of the dhesive stength of dhesive/film oints in eltion to film ctivtion conditions, nd detemintion of the wok of dhesion nd the wok of defomtion of the film. ANALYSIS OF THE ROBLEM When nlysing the test esults obtined using the method of teing off t n ngle of 18, we must tke into ccount the fundmentl diffeences tht ise between mutul dhesion of dhesive nd plstics film nd the dhesive stength of the oint unde investigtion. A numeicl mesue of dhesion is the wok of dhesion ( ), which is the wok pefomed ginst the foces of dhesion (being the sum of ll the intemolecul intections), equied fo foming unit boundy sufce of two phses in equilibium duing evesible isotheml pocess. Moeove, mesue of the dhesive stength of given oint, detemined by mens of teing off, is the wok of teing off ( ), being the sum of the wok of dhesion, the wok of defomtion of the test-piece duing teing-off, nd wok connected with ovecoming the foces of intection of the electic chges developing in the oute lyes of the septed mteils nd the foces of fiction tht ises duing teing-off. The fist two components e the most impotnt hee, nd the contibution of the othe components cn be neglected in the moity of technicl nlyses. Intentionl olyme Science nd Technology, Vol. 28, No. 5, 21 T/95

2 Theefoe the teing-off wok pe unit sufce e of the test oint, clled the unit wok of teing-off ( ), is: 1 = = sl sl 2L + L F () l dl (2) Fig. 1 A scheme pesenting the teing-off test cied out though n ngle of 18 : 1 te teste's immobile holde nd foce senso; 2 te teste's mobile holde; 3 two-side dhesive tpe; 4 film being ton off; 5 igid plte; L ton-off length, m length of fixing the fee end of film (fo detils see min text) Mesuements of dhesive stength by the method of teing-off t n ngle of 18 should be cied out in the mnne depicted in Fig. 1. The test oints, compising igid plte (5) to which is glued two-sided dhesive tpe (3), oined dhesively ove section of length L to the film (4) tht is ton off, e clmped in the fixed gip (1) of the stength testing mchine, which is equipped with system fo mesuing the teing-off foce (F ). The fee end of the film, of length L + m, is clmped ove section of length m in the moving gip (2) of the stength testing mchine. In the stting position, the uppe plten of this gip is in contct with the bottom edge of the igid plte. When the mchine is stted, this gip moves downwds t constnt speed, cting upon the fee end of the film with foce F. The foce F cuses the film to be ton wy fom the two-sided dhesive tpe, with consequent filue of the dhesive oint. This pocess comes to n end when the distnce between the moving gip nd the bottom edge of the igid plte is 2 L + L, whee L is the incese in length of the section of film, tking plce unde the ction of the foce F. Theefoe the wok cn be clculted on the bsis of the eltion: whee l is the film teing-off pth ( l 2 L + L). F 2L + l L = F () l dl m (1) whee s is the width of the test-piece. ope peption of plstics film pio to pinting, gluing, metllising o lminting is bsed on effecting physicl nd chemicl chnges in its oute lye such s incese the dheence of the glue o ink to the sid film, theeby incesing the dhesion between them. One of the min methods of peping the oute lye of polyethylene film is its ctivtion using low-tempetue plsm, poduced duing coon dischge in i (ef. 6). Theefoe pt fom the fct tht the wok is mesue of the qulity of n dhesive oint, it lso seves s citeion fo evluting the selection of suitble conditions of film ctivtion. Admittedly thee e no methods fo diect detemintion of the wok duing teing-off test of n dhesive oint of the dhesive/plstics film type, but the vlue of this wok cn be detemined to n ppoximtion tht is dequte fo technologicl needs on the bsis of the esults of tests of teing-off nd stetching (ef. 7). This ppoximtion is bsed on the ssumption tht the wok is the sum of ust the wok nd the wok of stetching ( ) of the specimen by length L. Then, on the bsis of the esults of film teing-off nd stetching tests, the wok cn be found fom the eltion: (3) Fo the pupose of comping the esults of tests on diffeent dhesive oints, we should evlute the vlues of the wok of teing-off, stetching nd dhesion pe unit sufce e of the test oint, o the coesponding unit wok vlues. These vlues e clculted in the mnne descibed by eltion (2). Then the unit wok of dhesion ( ) is clculted fom the eltion: whee is the unit wok of stetching. The pecentge contibution (u) of the wok to the wok is defined by the fomul: Film stetching duing the teing-off test tkes plce in diffeent conditions thn duing the usul stetching test. One of the esons fo this is tht the ton-off film is submitted to uw ( %) = 1 = 1 (4) (5) T/96 Intentionl olyme Science nd Technology, Vol. 28, No. 5, 21

3 bending, nd the length of the stetched section inceses s teing-off of the film poceeds. Owing to the smll dimensions of the test-piece, nd especilly the smll coss-sectionl e nd the low vlue of the coefficient of longitudinl elsticity of polyethylene (ef. 8), the vlue of the foce bending the test-piece duing teing-off is insignificnt, not exceeding 1 5 N. This is the bsis fo neglecting the influence of this foce on the vlue of the foce F. To simulte the film stetching conditions duing the teingoff test, specil equipment ws designed (ef. 9), pemitting continuous chnge in length of the specimen of film duing the stetching test. TEST ROCEDURE AND SCOE The method of investigting the dhesive oints is depicted in Fig. 1. The igid plte (5) (25 mm long, 25 mm wide nd 5 mm thick) ws mde fom luminium sheet. The two-sided dhesive tpe (3), type Z 2139, is composed of fomed polyethylene copolyme, coveed with lye of cylic dhesive. The film (4) ws extuded fom polyethylene of type 'Blen FB 8-64', poduced by the compny Slovnft Btislv. Joints with film of thickness 5 µm, 7 µm nd 15 µm wee investigted. This film ws ctivted by coon dischge t the test bench using n ctivto AGT-2, designed nd built t the 'Metlchem' Institute of lstics ocessing in Toun. The specimens of film wee ctivted with diffeent unit enegy of ctivtion (E ), the uppe vlue of which ws 7.5 kj/m 2. Two kinds of studies wee cied out: teing-off tests pefomed in ccodnce with the stndd (ef. 3) nd stetching tests ccoding to the pocedue descibed in the stndd (ef. 1). A 'Titest 216' stength testing mchine ws used in these tests. The cylic dhesive/polyethylene film dhesive oints fo the teing-off tests hd length L = 5 mm. The totl length of the section of film stetched duing the teing-off test ws theefoe 2 L = 1 mm. The speed of the moving gip of the testing mchine duing the teing-off test ws 5 mm/s. The clcultions of the wok of teing-off, bsed on eltion (1), used the men teing-off foce (F ), which ws s clculted on the bsis of the gph of F = f (l ) ecoded duing the teing-off test (Fig. 2). The edings of n integto coupled to the plotte ecoding this gph wee used fo clculting the sufce e bounded by the cuve dwn nd the bsciss; this e is diectly popotionl to the vlue of the wok. Dividing the vlue thus obtined by the length of the segment k on the bsciss, which coesponds on cetin scle to the pth of the moving gip of the stength testing mchine duing the teing-off test, we obtin the foce F. The wok of teing-off, defined by eltion (1), is thus s equl to the poduct F (2L + L). The ithmetic men fom s F s F, N k 2L + L Fig. 2 Repesenttive couse of the foce (F ) used to te off egisteed in the test: F s vege foce to te off; k section popotionl to the pthwy of the mobile holde of te teste; s ecode's pthwy five teing-off tests ws tken s the test esult fo the wok of teing-off of n dhesive oint contining film with given level of ctivtion. In ech cse five film specimens diffeing in length: L, 1.5L nd 2L, wee submitted to the stetching tests. Futhe clcultions wee bsed on the ithmetic men of ll these mesuements. To povide conditions of stetching tht e simil to those tht ise duing the teing-off test, the sme time of incese in specimen length by the vlue L ws used in both cses, coesponding to the sme speed of stetching. On the bsis of this condition, the velocity (ν ) of the moving gip of the stength testing mchine duing the stetching test ws clculted s follows: L ν = ν 2L + L (6) whee ν is the speed of teing-off (ν = 5 mm/s), L is the incese in film length duing the teing-off test (ithmetic men of five mesuements). Mintining n identicl speed of film stetching duing the teing-off test nd duing the stetching test is fundmentl condition fo the use of eltion (3). It thus follows tht s the speed of stetching of the polyme film chnges, thee is significnt vition of the wok (ef. 11). The men stetching foce (F ), the wok nd the wok s wee clculted in the sme wy s the foce F, the wok s nd the wok. The wok ws estimted on the bsis of eltion (4). TEST RESULTS AND DISCUSSION The gphs pesented in Figs. 3 5 povide gphicl illusttion of the esults of the mesuements nd clcultions mde duing the teing-off tests. s Intentionl olyme Science nd Technology, Vol. 28, No. 5, 21 T/97

4 F s, N E Fig. 3 The vege foce (F s ) used to te off film smples in eltion to film's specific tetment enegy (E ); film thickness (µm): 1 5; 2 7; 3 15 L, mm, J E Fig. 5 The te-off wok ( ) in eltion to film's specific tetment enegy (E E i Fig. 4 Film's elongtion ( L) in teing test in eltion to film's specific tetment enegy (E ) ; film thickness s in Fig. 3 Regdless of film thickness, with incese in enegy E in the nge < E 2 kj/m 2, the foce F s inceses (Fig. 3). This incese is, ppoximtely, diectly popotionl to the incese in vlue of E, though when E > 2 kj/m 2 futhe incese in the foce F s is much smlle. A chcteistic fetue is tht s the film thickness inceses, the vlue of the foce F inceses. s The pecentge diffeences in the vlues of this foce fo film specimens of thickness 5 µm nd 15 µm (cuves 1 nd 3 espectively) nge fom 45.4% (specimens ctivted in conditions with E = 3 kj/m 2 ) to 99.7% (specimens ctivted in conditions with E =.5 kj/m 2 ). The men vlue of these diffeences is ppox. 65%. Duing teing-off tests of specimens of films 15 µm thick, ctivted in conditions with E = 7.5 kj/m 2, thee ws ccking of the dhesive oints t the phse boundy: two-sided dhesive tpe/luminium plte. Theefoe Figs. 3 5 do not hve the esults of tests on these specimens. The incese in length L duing the teing-off tests lso depends on film thickness (Fig. 4). In the cse of specimens of films with thickness of 5 µm nd 7 µm, this length incese is popotionl to the vlue of E ove the whole nge of unit ctivtion enegy investigted, nd it hs simil vlues fo both kinds of film (cuves 1 nd 2). On the othe hnd, the incese in length of stetched film with thickness of 15 µm is only simil to the pevious vlues in the nge E 1.6 kj/ m 2 (cuve 3), becoming much less when this vlue is exceeded. The eltive elongtions of ll the specimens in the nge E 1.6 kj/m 2 do not exceed 7 8% nd e theefoe t the limit of elstic stins of low-density polyethylene (ef. 8). The couse of the dependence of wok on enegy E (Fig. 5) is simil to the eltions shown in Fig. 3. Thee e meely some diffeences with espect to specimens of films with thickness of 15 µm (cuve 3) in the nge of vlues of E > 2 kj/m 2. This is cused by the much smlle elongtion of these specimens, seen in Fig. 4, which limits the incese in the vlue of. The eltions fo wok s function of enegy E e simil to the eltions shown in Fig. 5. The esults of mesuements nd clcultions bsed on stetching tests on specimens of films e pesented in Fig. 6. These tests elte to five diffeent vlues of E, i.e. smlle numbe of specimens thn in the teing-off tests in Figs Even so, the investigtions coveed the entie nge of E vlues tht ws exmined peviously ( kj/m 2 ). Ove the whole of this nge, egdless of film thickness, s E inceses thee is incese in the wok. The chnges in, J E Fig. 6 The stetching wok ( ) in eltion to film's pecific tetment enegy (E T/98 Intentionl olyme Science nd Technology, Vol. 28, No. 5, 21

5 u, % the vlue of e smllest in the cse of film with thickness of 15 µm (cuve 3), nd these vitions e ppoximtely line. Thee is no simple explntion fo the diffeences in vlues elting to thicknesses of 5 µm nd 7 µm. It cn only be ssumed tht they ise fom the diffeent physicl popeties of these films, connected with diffeent extusion conditions. Fig. 7 is gphicl epesenttion of the esults of clcultions of the pecentge contibution of the wok to the wok, bsed on eltion (5). It cn be seen tht the contibution of to deceses s the unit ctivtion enegy inceses. This is cused by the incese in stength of the dhesive oints nd hence incese in the vlue of (i.e. decese of the numeto in the fction in eltion (5)) duing the teing-off test. Fo ll the films investigted, this eltion hs n ppoximtely line couse, nd in the cse of films with thickness of 5 µm nd 7 µm (cuves 1 nd 2) the eltions follow simil couse, whees the film with thickness of 15 µm (cuve 3) is chcteised by much slowe decese in the contibution of to s E inceses. CONCLUSIONS E Fig. 7 vs (cf. eqn. 5) in eltion to film's specific tetment enegy (E As the unit ctivtion enegy E of polyethylene film inceses, thee is consideble incese in the dhesive stength of oints contining this film. The unit te-off wok of n dhesive oint of ctivted film nd cylic dhesive, in the cse of the mximum vlue tht we investigted E = 7.5 kj/m 2 nd film thickness of 7 µm, is nely twenty-fou times gete thn the te-off wok of coesponding oint contining unctivted film. The incese in vlue of the unit teoff wok is especilly pid in the nge: E 2 kj/m 2. The pecentge incese in this vlue fo E = 2 kj/m 2, in the cse of dhesive oints with film of thickness 5 µm, 7 µm nd 15 µm, is 793%, 1187% nd 1662% espectively. As the unit ctivtion enegy inceses, thee is incese in stetching wok, ssocited with teing-off of the dhesive oints investigted, this incese being of n ppoximtely line chcte in the nge E 3 kj/m 2. The eo in estimting the unit wok of dhesion on the bsis of the esults of the teing-off test inceses with incese in unit ctivtion enegy of the test film. This is connected with incesed mutul dhesion of the polyethylene film nd the cylic dhesive, which cuses n incese in elongtion of specimens of this film duing the teing-off test. In the nge of E vlues used in the pocesses fo peping the oute lye of polyethylene film (E 2 kj/m 2 ), this eo does not exceed 9% nd diffes little fo individul films of diffeent thickness. In the nge of E vlues tht we investigted, the mximum vlue of this eo eches 29% (when E = 7.5 kj/m 2 ). REFERENCES 1. J. Golebiewski, olimey, 43, 1998, p Stndd N ISO Stndd ASTM D Stndd ASTM D Stndd ASTM D M. Zenkiewicz, Zesz. Nuk. ol. Slskie, No. 152, Mechnik, No. 98, Gliwice, M. Zenkiewicz nd J. Golebiewski, Evlution of the enegy of films duing investigtion of dhesive oints. Intentionl Resech Confeence, 'Moden oblems of Mechnics nd Mthemtics', Ukinin Ntionl Acdemy of Science, Lvov, My J. Sechtling, lstics Hndbook, WNT, Wsw, 2 9. olish t (1998) 1. Stndd N-81/C J. Golebiewski nd M. Zenkiewicz, olymes nd stuctul composites. ce Nuk, Ktedy Budowy Mszyn No. 1/98, 23, olitechnik Slsk, Gliwice, 1998 Received Intentionl olyme Science nd Technology, Vol. 28, No. 5, 21 T/99

On the Eötvös effect

On the Eötvös effect On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion

More information

U>, and is negative. Electric Potential Energy

U>, and is negative. Electric Potential Energy Electic Potentil Enegy Think of gvittionl potentil enegy. When the lock is moved veticlly up ginst gvity, the gvittionl foce does negtive wok (you do positive wok), nd the potentil enegy (U) inceses. When

More information

Electric Potential. and Equipotentials

Electric Potential. and Equipotentials Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil

More information

4.2 Boussinesq s Theory. Contents

4.2 Boussinesq s Theory. Contents 00477 Pvement Stuctue 4. Stesses in Flexible vement Contents 4. Intoductions to concet of stess nd stin in continuum mechnics 4. Boussinesq s Theoy 4. Bumiste s Theoy 4.4 Thee Lye System Weekset Sung Chte

More information

SURFACE TENSION. e-edge Education Classes 1 of 7 website: , ,

SURFACE TENSION. e-edge Education Classes 1 of 7 website: , , SURFACE TENSION Definition Sufce tension is popety of liquid by which the fee sufce of liquid behves like stetched elstic membne, hving contctive tendency. The sufce tension is mesued by the foce cting

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 3 Due on Sep. 14, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt

More information

Available online at ScienceDirect. Procedia Engineering 91 (2014 ) 32 36

Available online at   ScienceDirect. Procedia Engineering 91 (2014 ) 32 36 Aville online t wwwsciencediectcom ScienceDiect Pocedi Engineeing 91 (014 ) 3 36 XXIII R-S-P semin Theoeticl Foundtion of Civil Engineeing (3RSP) (TFoCE 014) Stess Stte of Rdil Inhomogeneous Semi Sphee

More information

Fluids & Bernoulli s Equation. Group Problems 9

Fluids & Bernoulli s Equation. Group Problems 9 Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going

More information

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,

More information

9.4 The response of equilibrium to temperature (continued)

9.4 The response of equilibrium to temperature (continued) 9.4 The esponse of equilibium to tempetue (continued) In the lst lectue, we studied how the chemicl equilibium esponds to the vition of pessue nd tempetue. At the end, we deived the vn t off eqution: d

More information

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3 DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl

More information

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept

More information

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system 436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique

More information

Mathematical formulation of the F 0 motor model

Mathematical formulation of the F 0 motor model negy Tnsduction in TP Synthse: Supplement Mthemticl fomultion of the F 0 moto model. Mkov chin model fo the evolution of the oto stte The fou possible potontion sttes of the two oto sp61 sites t the otostto

More information

Important design issues and engineering applications of SDOF system Frequency response Functions

Important design issues and engineering applications of SDOF system Frequency response Functions Impotnt design issues nd engineeing pplictions of SDOF system Fequency esponse Functions The following desciptions show typicl questions elted to the design nd dynmic pefomnce of second-ode mechnicl system

More information

1 Using Integration to Find Arc Lengths and Surface Areas

1 Using Integration to Find Arc Lengths and Surface Areas Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s

More information

Fourier-Bessel Expansions with Arbitrary Radial Boundaries

Fourier-Bessel Expansions with Arbitrary Radial Boundaries Applied Mthemtics,,, - doi:./m.. Pulished Online My (http://www.scirp.og/jounl/m) Astct Fouie-Bessel Expnsions with Aity Rdil Boundies Muhmmd A. Mushef P. O. Box, Jeddh, Sudi Ai E-mil: mmushef@yhoo.co.uk

More information

Qualitative Analysis for Solutions of a Class of. Nonlinear Ordinary Differential Equations

Qualitative Analysis for Solutions of a Class of. Nonlinear Ordinary Differential Equations Adv. Theo. Appl. Mech., Vol. 7, 2014, no. 1, 1-7 HIKARI Ltd, www.m-hiki.com http://dx.doi.og/10.12988/tm.2014.458 Qulittive Anlysis fo Solutions of Clss of Nonline Odiny Diffeentil Equtions Juxin Li *,

More information

Discrete Model Parametrization

Discrete Model Parametrization Poceedings of Intentionl cientific Confeence of FME ession 4: Automtion Contol nd Applied Infomtics Ppe 9 Discete Model Pmetition NOKIEVIČ, Pet Doc,Ing,Cc Deptment of Contol ystems nd Instumenttion, Fculty

More information

Radial geodesics in Schwarzschild spacetime

Radial geodesics in Schwarzschild spacetime Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using

More information

6. Gravitation. 6.1 Newton's law of Gravitation

6. Gravitation. 6.1 Newton's law of Gravitation Gvittion / 1 6.1 Newton's lw of Gvittion 6. Gvittion Newton's lw of gvittion sttes tht evey body in this univese ttcts evey othe body with foce, which is diectly popotionl to the poduct of thei msses nd

More information

NS-IBTS indices calculation procedure

NS-IBTS indices calculation procedure ICES Dt Cente DATRAS 1.1 NS-IBTS indices 2013 DATRAS Pocedue Document NS-IBTS indices clcultion pocedue Contents Genel... 2 I Rw ge dt CA -> Age-length key by RFA fo defined ge nge ALK... 4 II Rw length

More information

Topics for Review for Final Exam in Calculus 16A

Topics for Review for Final Exam in Calculus 16A Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the

More information

RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1

RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1 RELAIVE KINEMAICS he equtions of motion fo point P will be nlyzed in two diffeent efeence systems. One efeence system is inetil, fixed to the gound, the second system is moving in the physicl spce nd the

More information

10 Statistical Distributions Solutions

10 Statistical Distributions Solutions Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques

More information

Answers to test yourself questions

Answers to test yourself questions Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E

More information

CHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD

CHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD ollege Physics Student s Mnul hpte 8 HAPTR 8: LTRI HARG AD LTRI ILD 8. STATI LTRIITY AD HARG: OSRVATIO O HARG. ommon sttic electicity involves chges nging fom nnocoulombs to micocoulombs. () How mny electons

More information

Chapter 6 Thermoelasticity

Chapter 6 Thermoelasticity Chpte 6 Themoelsticity Intoduction When theml enegy is dded to n elstic mteil it expnds. Fo the simple unidimensionl cse of b of length L, initilly t unifom tempetue T 0 which is then heted to nonunifom

More information

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018. SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl

More information

Physics 604 Problem Set 1 Due Sept 16, 2010

Physics 604 Problem Set 1 Due Sept 16, 2010 Physics 64 Polem et 1 Due ept 16 1 1) ) Inside good conducto the electic field is eo (electons in the conducto ecuse they e fee to move move in wy to cncel ny electic field impessed on the conducto inside

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 4 Due on Sep. 1, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt

More information

Algebra Based Physics. Gravitational Force. PSI Honors universal gravitation presentation Update Fall 2016.notebookNovember 10, 2016

Algebra Based Physics. Gravitational Force. PSI Honors universal gravitation presentation Update Fall 2016.notebookNovember 10, 2016 Newton's Lw of Univesl Gvittion Gvittionl Foce lick on the topic to go to tht section Gvittionl Field lgeb sed Physics Newton's Lw of Univesl Gvittion Sufce Gvity Gvittionl Field in Spce Keple's Thid Lw

More information

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.

More information

Electricity & Magnetism Lecture 6: Electric Potential

Electricity & Magnetism Lecture 6: Electric Potential Electicity & Mgnetism Lectue 6: Electic Potentil Tody s Concept: Electic Potenl (Defined in tems of Pth Integl of Electic Field) Electicity & Mgnesm Lectue 6, Slide Stuff you sked bout:! Explin moe why

More information

Friedmannien equations

Friedmannien equations ..6 Fiedmnnien equtions FLRW metic is : ds c The metic intevl is: dt ( t) d ( ) hee f ( ) is function which detemines globl geometic l popety of D spce. f d sin d One cn put it in the Einstein equtions

More information

Work, Potential Energy, Conservation of Energy. the electric forces are conservative: ur r

Work, Potential Energy, Conservation of Energy. the electric forces are conservative: ur r Wok, Potentil Enegy, Consevtion of Enegy the electic foces e consevtive: u Fd = Wok, Potentil Enegy, Consevtion of Enegy b b W = u b b Fdl = F()[ d + $ $ dl ] = F() d u Fdl = the electic foces e consevtive

More information

π,π is the angle FROM a! TO b

π,π is the angle FROM a! TO b Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two

More information

Physics 1502: Lecture 2 Today s Agenda

Physics 1502: Lecture 2 Today s Agenda 1 Lectue 1 Phsics 1502: Lectue 2 Tod s Agend Announcements: Lectues posted on: www.phs.uconn.edu/~cote/ HW ssignments, solutions etc. Homewok #1: On Mstephsics this Fid Homewoks posted on Msteingphsics

More information

Two dimensional polar coordinate system in airy stress functions

Two dimensional polar coordinate system in airy stress functions I J C T A, 9(9), 6, pp. 433-44 Intentionl Science Pess Two dimensionl pol coodinte system in iy stess functions S. Senthil nd P. Sek ABSTRACT Stisfy the given equtions, boundy conditions nd bihmonic eqution.in

More information

Influence of Hydraulic Hysteresis on Effective Stress in Unsaturated Clay

Influence of Hydraulic Hysteresis on Effective Stress in Unsaturated Clay Intentionl Jounl of Envionmentl nd Eth Sciences 1:1 21 Influence of Hydulic Hysteesis on Effective Stess in Unstuted Cly Anuchit Uchipicht Abstct A compehensive pogm of lbotoy testing on compcted kolin

More information

This immediately suggests an inverse-square law for a "piece" of current along the line.

This immediately suggests an inverse-square law for a piece of current along the line. Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line

More information

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI 1 Electomgnetism Alexnde A. Isknd, Ph.D. Physics of Mgnetism nd Photonics Resech Goup Electosttics ELECTRIC PTENTIALS 1 Recll tht we e inteested to clculte the electic field of some chge distiution.

More information

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = =

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = = Chpte 1 nivesl Gvittion 11 *P1. () The un-th distnce is 1.4 nd the th-moon 8 distnce is.84, so the distnce fom the un to the Moon duing sol eclipse is 11 8 11 1.4.84 = 1.4 The mss of the un, th, nd Moon

More information

7.5-Determinants in Two Variables

7.5-Determinants in Two Variables 7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt

More information

Ch 26 - Capacitance! What s Next! Review! Lab this week!

Ch 26 - Capacitance! What s Next! Review! Lab this week! Ch 26 - Cpcitnce! Wht s Next! Cpcitnce" One week unit tht hs oth theoeticl n pcticl pplictions! Cuent & Resistnce" Moving chges, finlly!! Diect Cuent Cicuits! Pcticl pplictions of ll the stuff tht we ve

More information

ELECTRO - MAGNETIC INDUCTION

ELECTRO - MAGNETIC INDUCTION NTRODUCTON LCTRO - MAGNTC NDUCTON Whenee mgnetic flu linked with cicuit chnges, n e.m.f. is induced in the cicuit. f the cicuit is closed, cuent is lso induced in it. The e.m.f. nd cuent poduced lsts s

More information

CHAPTER 2 ELECTROSTATIC POTENTIAL

CHAPTER 2 ELECTROSTATIC POTENTIAL 1 CHAPTER ELECTROSTATIC POTENTIAL 1 Intoduction Imgine tht some egion of spce, such s the oom you e sitting in, is pemeted by n electic field (Pehps thee e ll sots of electiclly chged bodies outside the

More information

Lecture 10. Solution of Nonlinear Equations - II

Lecture 10. Solution of Nonlinear Equations - II Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution

More information

PX3008 Problem Sheet 1

PX3008 Problem Sheet 1 PX38 Poblem Sheet 1 1) A sphee of dius (m) contins chge of unifom density ρ (Cm -3 ). Using Guss' theoem, obtin expessions fo the mgnitude of the electic field (t distnce fom the cente of the sphee) in

More information

NARAYANA I I T / P M T A C A D E M Y. C o m m o n Pr a c t i c e T e s t 0 9 XI-IC SPARK Date: PHYSICS CHEMISTRY MATHEMATICS

NARAYANA I I T / P M T A C A D E M Y. C o m m o n Pr a c t i c e T e s t 0 9 XI-IC SPARK Date: PHYSICS CHEMISTRY MATHEMATICS . (D). (B). (). (). (D). (A) 7. () 8. (B) 9. (B). (). (A). (D). (B). (). (B) NAAYANA I I T / T A A D E Y XIS-I-IIT-SA (..7) o m m o n c t i c e T e s t 9 XI-I SA Dte:..7 ANSWE YSIS EISTY ATEATIS. (B).

More information

dx was area under f ( x ) if ( ) 0

dx was area under f ( x ) if ( ) 0 13. Line Integls Line integls e simil to single integl, f ( x) dx ws e unde f ( x ) if ( ) 0 Insted of integting ove n intevl [, ] (, ) f xy ds f x., we integte ove cuve, (in the xy-plne). **Figue - get

More information

Chapter 21: Electric Charge and Electric Field

Chapter 21: Electric Charge and Electric Field Chpte 1: Electic Chge nd Electic Field Electic Chge Ancient Gees ~ 600 BC Sttic electicit: electic chge vi fiction (see lso fig 1.1) (Attempted) pith bll demonsttion: inds of popeties objects with sme

More information

Week 8. Topic 2 Properties of Logarithms

Week 8. Topic 2 Properties of Logarithms Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e

More information

MAGNETIC EFFECT OF CURRENT & MAGNETISM

MAGNETIC EFFECT OF CURRENT & MAGNETISM TODUCTO MAGETC EFFECT OF CUET & MAGETM The molecul theo of mgnetism ws given b Webe nd modified lte b Ewing. Oested, in 18 obseved tht mgnetic field is ssocited with n electic cuent. ince, cuent is due

More information

Modelling of Low Velocity Impact Damage in Laminated Composites

Modelling of Low Velocity Impact Damage in Laminated Composites Modelling of Lo Velocity Impct mge in Lminted Composites J. Lee*, C. Soutis*, P. T. Cutis nd C. Kong** *Aeospce Engineeing, The Univesity of Sheffield, Sheffield S 3J, UK efence Science nd Technology Lbotoy,

More information

Production Mechanism of Quark Gluon Plasma in Heavy Ion Collision. Ambar Jain And V.Ravishankar

Production Mechanism of Quark Gluon Plasma in Heavy Ion Collision. Ambar Jain And V.Ravishankar Poduction Mechnism of Quk Gluon Plsm in Hevy Ion Collision Amb Jin And V.Rvishnk Pimy im of theoeticlly studying URHIC is to undestnd Poduction of quks nd gluons tht fom the bulk of the plsm ( ) t 0 Thei

More information

Electronic Supplementary Material

Electronic Supplementary Material Electonic Supplementy Mteil On the coevolution of socil esponsiveness nd behvioul consistency Mx Wolf, G Snde vn Doon & Fnz J Weissing Poc R Soc B 78, 440-448; 0 Bsic set-up of the model Conside the model

More information

Physics 11b Lecture #11

Physics 11b Lecture #11 Physics 11b Lectue #11 Mgnetic Fields Souces of the Mgnetic Field S&J Chpte 9, 3 Wht We Did Lst Time Mgnetic fields e simil to electic fields Only diffeence: no single mgnetic pole Loentz foce Moving chge

More information

Chapter 25: Current, Resistance and Electromotive Force. Charge carrier motion in a conductor in two parts

Chapter 25: Current, Resistance and Electromotive Force. Charge carrier motion in a conductor in two parts Chpte 5: Cuent, esistnce nd Electomotive Foce Chge cie motion in conducto in two pts Constnt Acceletion F m qe ndomizing Collisions (momentum, enegy) =>esulting Motion Avege motion = Dift elocity = v d

More information

s c s (b) Hence, show that the entropy for rubber-like materials must have the separable form

s c s (b) Hence, show that the entropy for rubber-like materials must have the separable form EN: Continuum Mechnics Homewok 6: Aliction of continuum mechnics to elstic solids Due Decembe th, School of Engineeing Bown Univesity. Exeiments show tht ubbe-like mteils hve secific intenl enegy ( ) nd

More information

Chapter 25: Current, Resistance and Electromotive Force. ~10-4 m/s Typical speeds ~ 10 6 m/s

Chapter 25: Current, Resistance and Electromotive Force. ~10-4 m/s Typical speeds ~ 10 6 m/s Chpte 5: Cuent, esistnce nd lectomotive Foce Chge cie motion in conducto in two pts Constnt Acceletion F m q ndomizing Collisions (momentum, enegy) >esulting Motion http://phys3p.sl.psu.edu/phys_nim/m/ndom_wlk.vi

More information

Comparative Studies of Law of Gravity and General Relativity. No.1 of Comparative Physics Series Papers

Comparative Studies of Law of Gravity and General Relativity. No.1 of Comparative Physics Series Papers Comptive Studies of Lw of Gvity nd Genel Reltivity No. of Comptive hysics Seies pes Fu Yuhu (CNOOC Resech Institute, E-mil:fuyh945@sin.com) Abstct: As No. of comptive physics seies ppes, this ppe discusses

More information

The influence of the quality of ferrosilicon on the rheology of dense medium and the ability to reach higher densities

The influence of the quality of ferrosilicon on the rheology of dense medium and the ability to reach higher densities The influence of the qulity of feosilicon on the heology of dense medium nd the bility to ech highe densities by H.A. Mybugh* J o u n l Synopsis The dense medium beneficition of hevy metl oxide oes such

More information

Solutions to Midterm Physics 201

Solutions to Midterm Physics 201 Solutions to Midtem Physics. We cn conside this sitution s supeposition of unifomly chged sphee of chge density ρ nd dius R, nd second unifomly chged sphee of chge density ρ nd dius R t the position of

More information

Optimization. x = 22 corresponds to local maximum by second derivative test

Optimization. x = 22 corresponds to local maximum by second derivative test Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible

More information

Chapter 2: Electric Field

Chapter 2: Electric Field P 6 Genel Phsics II Lectue Outline. The Definition of lectic ield. lectic ield Lines 3. The lectic ield Due to Point Chges 4. The lectic ield Due to Continuous Chge Distibutions 5. The oce on Chges in

More information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque

More information

STUDY OF THE UNIFORM MAGNETIC FIELD DOMAINS (3D) IN THE CASE OF THE HELMHOLTZ COILS

STUDY OF THE UNIFORM MAGNETIC FIELD DOMAINS (3D) IN THE CASE OF THE HELMHOLTZ COILS STUDY OF THE UNIFORM MAGNETIC FIED DOMAINS (3D) IN THE CASE OF THE HEMHOTZ COIS FORIN ENACHE, GHEORGHE GAVRIĂ, EMI CAZACU, Key wods: Unifom mgnetic field, Helmholt coils. Helmholt coils e used to estblish

More information

This chapter is about energy associated with electrical interactions. Every

This chapter is about energy associated with electrical interactions. Every 23 ELECTRIC PTENTIAL whee d l is n infinitesiml displcement long the pticle s pth nd f is the ngle etween F nd d l t ech point long the pth. econd, if the foce F is consevtive, s we defined the tem in

More information

GEOMETRY Properties of lines

GEOMETRY Properties of lines www.sscexmtuto.com GEOMETRY Popeties of lines Intesecting Lines nd ngles If two lines intesect t point, ten opposite ngles e clled veticl ngles nd tey ve te sme mesue. Pependicul Lines n ngle tt mesues

More information

Chapter 2. Review of Newton's Laws, Units and Dimensions, and Basic Physics

Chapter 2. Review of Newton's Laws, Units and Dimensions, and Basic Physics Chpte. Review of Newton's Lws, Units nd Diensions, nd Bsic Physics You e ll fili with these ipotnt lws. But which e bsed on expeients nd which e ttes of definition? FIRST LAW n object oves unifoly (o eins

More information

General Physics (PHY 2140)

General Physics (PHY 2140) Genel Physics (PHY 40) Lightning Review Lectue 3 Electosttics Lst lectue:. Flux. Guss s s lw. simplifies computtion of electic fields Q Φ net Ecosθ ε o Electicl enegy potentil diffeence nd electic potentil

More information

Lecture 11: Potential Gradient and Capacitor Review:

Lecture 11: Potential Gradient and Capacitor Review: Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of

More information

Investigations of Boundary Treatments in Incompressible Smoothed Particle Hydrodynamics for Fluid-Structural Interactions

Investigations of Boundary Treatments in Incompressible Smoothed Particle Hydrodynamics for Fluid-Structural Interactions Recent Reseches in Mechnics Investigtions of Boundy Tetments in Incompessile Smoothed Pticle Hydodynmics fo Fluid-Stuctul Intections Fnfn Sun, Mingyi Tn, nd Jing T Xing Astct Two oundy tetment methods

More information

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems.

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems. Physics 55 Fll 5 Midtem Solutions This midtem is two hou open ook, open notes exm. Do ll thee polems. [35 pts] 1. A ectngul ox hs sides of lengths, nd c z x c [1] ) Fo the Diichlet polem in the inteio

More information

3.1 Magnetic Fields. Oersted and Ampere

3.1 Magnetic Fields. Oersted and Ampere 3.1 Mgnetic Fields Oested nd Ampee The definition of mgnetic induction, B Fields of smll loop (dipole) Mgnetic fields in mtte: ) feomgnetism ) mgnetiztion, (M ) c) mgnetic susceptiility, m d) mgnetic field,

More information

Micro-scale adhesive contact of a spherical rigid punch on a. piezoelectric half-space

Micro-scale adhesive contact of a spherical rigid punch on a. piezoelectric half-space http://www.ppe.edu.cn Mico-scle dhesive contct of spheicl igid punch on piezoelectic hlf-spce Z.R. Chen, S.W. Yu * Deptment of Engineeing Mechnics, Tsinghu Univesity, Beijing 84, P.R. Chin Abstct The mico-scle

More information

Chapter Direct Method of Interpolation More Examples Mechanical Engineering

Chapter Direct Method of Interpolation More Examples Mechanical Engineering Chpte 5 iect Method o Intepoltion Moe Exmples Mechnicl Engineeing Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by

More information

ITI Introduction to Computing II

ITI Introduction to Computing II ITI 1121. Intoduction to Computing II Mcel Tucotte School of Electicl Engineeing nd Compute Science Abstct dt type: Stck Stck-bsed lgoithms Vesion of Febuy 2, 2013 Abstct These lectue notes e ment to be

More information

Energy Dissipation Gravitational Potential Energy Power

Energy Dissipation Gravitational Potential Energy Power Lectue 4 Chpte 8 Physics I 0.8.03 negy Dissiption Gvittionl Potentil negy Powe Couse wesite: http://fculty.uml.edu/andiy_dnylov/teching/physicsi Lectue Cptue: http://echo360.uml.edu/dnylov03/physicsfll.html

More information

Continuous Charge Distributions

Continuous Charge Distributions Continuous Chge Distibutions Review Wht if we hve distibution of chge? ˆ Q chge of distibution. Q dq element of chge. d contibution to due to dq. Cn wite dq = ρ dv; ρ is the chge density. = 1 4πε 0 qi

More information

Efficiency of excitation of piezoceramic transducer at antiresonance frequency

Efficiency of excitation of piezoceramic transducer at antiresonance frequency fficiency of excittion of piezocemic tnsduce t ntiesonnce fequency Mezheitsky A.V. fficiency of excittion of piezocemic tnsduce t ntiesonnce fequency dpted fom I Tns. Ultson. Feoelect. Feq. Cont. vol.

More information

The Formulas of Vector Calculus John Cullinan

The Formulas of Vector Calculus John Cullinan The Fomuls of Vecto lculus John ullinn Anlytic Geomety A vecto v is n n-tuple of el numbes: v = (v 1,..., v n ). Given two vectos v, w n, ddition nd multipliction with scl t e defined by Hee is bief list

More information

Elastic scattering of 4 He atoms at the surface of liquid helium

Elastic scattering of 4 He atoms at the surface of liquid helium Indin Jounl of Pue & Applied Physics Vol. 48, Octobe, pp. 743-748 Elstic sctteing of 4 He toms t the sufce of liquid helium P K Toongey, K M Khnn, Y K Ayodo, W T Skw, F G Knyeki, R T Eki, R N Kimengichi

More information

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field

More information

Tests for Correlation on Bivariate Non-Normal Data

Tests for Correlation on Bivariate Non-Normal Data Jounl of Moden Applied Sttisticl Methods Volume 0 Issue Aticle 9 --0 Tests fo Coeltion on Bivite Non-Noml Dt L. Bevesdof Noth Colin Stte Univesity, lounneb@gmil.com Ping S Univesity of Noth Floid, ps@unf.edu

More information

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013 Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo

More information

Mark Scheme (Results) January 2008

Mark Scheme (Results) January 2008 Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question

More information

Wave Generation by Oscillating Wall in Static Media

Wave Generation by Oscillating Wall in Static Media We Genetion by Oscillting Wll in Sttic Medi Hongbin Ju Deptment of Mthemtics Floid Stte Uniesity, Tllhssee, FL.3306 www.eocoustics.info Plese send comments to: hju@mth.fsu.edu Sound, oticity we nd entopy

More information

RELATIONSHIP BETWEEN DESIGN RESPONSE SPECTRA FOR RARE AND FREQUENT EARTHQUAKE LEVELS

RELATIONSHIP BETWEEN DESIGN RESPONSE SPECTRA FOR RARE AND FREQUENT EARTHQUAKE LEVELS th Wold Confeence on Ethquke Engineeing ncouve, B.C., Cnd ugust -6, 00 Ppe No. 9 ELIONHIP BEWEEN EIGN EPONE PEC O E N EQUEN EHQUKE LEEL Yingmin LI Cheng HI Ming LI Ling HN UMMY It is known tht uilding

More information

r = (0.250 m) + (0.250 m) r = m = = ( N m / C )

r = (0.250 m) + (0.250 m) r = m = = ( N m / C ) ELECTIC POTENTIAL IDENTIFY: Apply Eq() to clculte the wok The electic potentil enegy of pi of point chges is given y Eq(9) SET UP: Let the initil position of q e point nd the finl position e point, s shown

More information

ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:

ELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy: LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6

More information

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:

More information

A COMPARISON OF MEMBRANE SHELL THEORIES OF HYBRID ANISOTROPIC MATERIALS ABSTRACT

A COMPARISON OF MEMBRANE SHELL THEORIES OF HYBRID ANISOTROPIC MATERIALS ABSTRACT A COMPARISON OF MEMBRANE SHELL THEORIES OF HYBRID ANISOTROPIC MATERIALS S. W. Chung* School of Achitectue Univesity of Uth Slt Lke City, Uth, USA S.G. Hong Deptment of Achitectue Seoul Ntionl Univesity

More information

Physics 111. Uniform circular motion. Ch 6. v = constant. v constant. Wednesday, 8-9 pm in NSC 128/119 Sunday, 6:30-8 pm in CCLIR 468

Physics 111. Uniform circular motion. Ch 6. v = constant. v constant. Wednesday, 8-9 pm in NSC 128/119 Sunday, 6:30-8 pm in CCLIR 468 ics Announcements dy, embe 28, 2004 Ch 6: Cicul Motion - centipetl cceletion Fiction Tension - the mssless sting Help this week: Wednesdy, 8-9 pm in NSC 128/119 Sundy, 6:30-8 pm in CCLIR 468 Announcements

More information

Dynamically Equivalent Systems. Dynamically Equivalent Systems. Dynamically Equivalent Systems. ME 201 Mechanics of Machines

Dynamically Equivalent Systems. Dynamically Equivalent Systems. Dynamically Equivalent Systems. ME 201 Mechanics of Machines ME 0 Mechnics of Mchines 8//006 Dynmicy Equivent Systems Ex: Connecting od G Dynmicy Equivent Systems. If the mss of the connecting od m G m m B m m m. Moment out cente of gvity shoud e zeo m G m B Theefoe;

More information

Polymer A should have the medium T g. It has a larger sidechain than polymer B, and may also have hydrogen bonding, due the -COOH group.

Polymer A should have the medium T g. It has a larger sidechain than polymer B, and may also have hydrogen bonding, due the -COOH group. MATERIALS 0 INTRODUTION TO STRUTURE AND PROPERTIES WINTER 202 Poblem Set 2 Due: Tuesdy, Jnuy 3, :00 AM. Glss Tnsition Tempetue ) The glss tnsition tempetue, T g, is stongly govened by the bility of the

More information

A Permanent Magnet Device for Measuring the Coercive Force

A Permanent Magnet Device for Measuring the Coercive Force 7th Wold Confeence on Nondestuctive Testing, 5-8 Oct 008, Shnghi, Chin A Penent Mgnet Device fo Mesuing the Coecive Foce Edud S. GORKUNOV, Ve P. TABACHNIK Institute of Engineeing Science, RAS (Ul Bnch)

More information

r a + r b a + ( r b + r c)

r a + r b a + ( r b + r c) AP Phsics C Unit 2 2.1 Nme Vectos Vectos e used to epesent quntities tht e chcteized b mgnitude ( numeicl vlue with ppopite units) nd diection. The usul emple is the displcement vecto. A quntit with onl

More information

EECE 260 Electrical Circuits Prof. Mark Fowler

EECE 260 Electrical Circuits Prof. Mark Fowler EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution

More information