TUTORING TEXTS IN MATHCAD MIROSLAV DOLOZÍILEK and ANNA RYNDOVÁ Faculy of Mechanical Engineering, Brno Universiy of Technology Technická, 616 69 Brno, Czech Republic E-ail: irdo@fyzika.fe.vubr.cz Absrac Mahcad gives users an unique live docuen inerface. The good feaures of his copuing environen for calculaion, daa acquisiion and wriing echnical repors were forerly deonsraed. Soe new feaures of he laes versions of Mahcad - direc Web access, fully ipleened SI unis, iproved -D and 3-D graphing, aniaion - give he good ools also for creaion of uoring exs. This abiliies of Mahcad will be deonsraed on he uoring exs in physics. KEYWORDS: Tuoring ex, Mahcad, Physics Inroducion Mahcad is a rich proble-solving environen ha gives us a wide choice of ools and suppors a variey of analysis and visualisaion echniques. In he os general sense, we can hink of Mahcad as a cobinaion of a powerful echnical copuing environen cenred on real ah noaion, and a flexible, full-feaured echnical word processor. This eans ha wih Mahcad he asks of perforing copuaions and docuening he are inegraed ino one sealess process, resuling in subsanial increases in produciviy. Unlike oher echnical sofware, Mahcad perfors aheaics he sae way we do. Tha is because i looks and works like a pad of paper and pencil. Mahcad's on-screen inerface is a blank workshee on which we ener equaions, graph daa or funcions, and annoae wih ex anywhere on he page. Maheaical expressions in Mahcad look he way you would see he in a ex. The only difference is ha Mahcad's equaions and graphs are live. Change any daa, variable, or equaion, and Mahcad recalculaes your workshee insanly. Apar he good D and 3D graphic environen (vecor field plo, conour plo) we can also creae an aniaion of objec in he graph..
In addiion o he live docuen inerface, anoher iporan feaure ha ses Mahcad apar fro oher echnical copuing ools is is buil-in suppor for unis. Unis are fully inegraed ino all calculaions. Mahcad gives us all he power you need o ge he job done - fro sar o finish. Wih Mahcad we can explore probles, forulae ideas, analyse daa, odel and es scenarios, selec he bes soluion... hen docuen, presen, and counicae he resuls. SarSkech LE allows you o annoae your workshees wih skeches, diagras, and drawings. SarSkech is paraeric so ha your designs and calculaions can drive each oher. We can inser a hyperlink o anoher workshee or a Inerne source. Mahcad is ore han jus a versaile and flexible calculaion and docuenaion environen - i is rapidly evolving as one of he os popular plafors for indusry-specific soluions. There are four ain ways in which Mahcad can be exended in an indusry-specific way: Mahcad Elecronic Books Mahcad Exension Packs User-creaed DLLs (C++ libraries) Ineroperabiliy wih oher applicaions The good feaures of Mahcad for daa acquisiion and wriing a echnical repor fro laboraory easureen as well as for nuerical soluion of exaples in physics were discussed in [1] and []. The iproveens in he versions 6 and 7 and in he las version 000 [3] offer enough cofor ools for creaion also uoring ex for courses of basic physics. These properies of Mahcad environen are deonsraed in he following secion. Inerference of waves We will sudy he inerference of he wo sinusoidal waves. The resul of such superposiion depends on he uual relaion of he apliude, frequency and phase of he boh waves. For wo coheren waves he plane of displaceens and frequency are he sae and he phase difference is consan during he ie. As a resul of he inerference of such waves we can observe an inerference paern - he srucure, which is consan during he ie. There and here, he difference in he pah lenghs of he wo waves arriving a he poin deerines he phase difference of he waves a ha poin. If he pah lengh difference is equal o zero or an ineger uliple of wavelenghs, he arriving waves are exacly in phase and undergo fully consrucive inerference. If he pah lengh difference is odd uliple of half a wavelengh, he arriving waves are exacly ou of phase and undergo fully desrucive inerference. Now we will sudy he inerference of he wo sinusoidal waves fro wo coheren sources Z1 and Z in arbirary poin M (Fig.1) in ore deail. Iediae oscillaion displaceen in he poin M due o wave fro source Z1, Z : U1( A1, ω, k, α1, r1, ) := A1 sin( ω k r1+ α1) (1) A sin( ω k r+ α) U A, ω, k, α, r, := ()
U A, ω, k, α1, α, r1, r, Here A1 resp. A is apliude of he wave; ω is angular frequency; k is angular wave nuber ; r1 resp. r is disance of he poin M and posiion of he source Z1 resp. Z; α1 resp. α is phase consan. In pracice we can find very ofen a siuaion, when he apliudes of he inerference waves are he sae, A1 A A and he plane of displaceens is also he sae. Then he resuling sae of he oscillaions in he poin M is: U1( A, ω, k, α1, r1, ) + U( A, ω, k, α, r, ) := (3) I is a siuaion of superposiion of he haronic oscillaions, when he resuling oscillaions depend on iniial phases α1, α and disances r1, r. Changing he nuerical values of hese paraeers es hese dependencies and find a axiu and iniu of inerference. We define A := 0.001 f := 500 Hz c := 430 s 1 ω := π f r1 r Fig. 1. := 3.86 α1 := 3 α π := λ π := λ = 0.86 4 c := k := f π λ := 0 s, 0.00005 s.. 0.003 s 0.00 U1 U ( A, ω, k, α1, r1, ) ( A, ω, k, α, r, ) U A, ω, k, α1, α, r1, r, 0.001 0.001 0 0.00 0 5. 10 4 0.001 0.0015 0.00 0.005 0.003 s
When we use he relaion sinα + sinβ cos α β sin α β for conversion of he righ side of he equaion (3) A sin ω k r1+ α1 A sin ω k r+ α +, he iediae displaceen can be expressed as a produc of wo funcions Av( r1, r, α1, α) r r1 α1 α := A cos k + (4) sin ω U α1, α, r1, r, U α1, α, r1, r, r1 + r α1 + α := + (5) Av( r1, r, α1, α) U ( α1, α, r1, r, ) := (6) The firs funcion (4), which is independen on he ie, represens he apliude of he resuling oscillaions. The second par, which depends on he ie, represens he frequency and phase of he resuling oscillaions. I is he haronic oscillaions wih angular frequency ω. Noe xz1, yz1, xz, yz he co-ordinaes of he coheren sources Z1, Z, xy, he co-ordinaes of he poin M. Assue, ha he sources are on he y axes and he poins M lay on collinear line in disance d. We will calculae he resul of he inerference in hese poins M. xz1 := 0 xz := 0 d := 10 yz1 := 3 yz := 3 x : = 10 r1() y := x + ( y yz1) r() y := x + ( y yz) Reeber, ha resuling oscillaions on he line here are expressed by Eq. (6) and he apliudes Av by Eq. (4). := y 0 19.95,.. 0 0.00 + Av r1( y), r( y), α1, α 0 0.00 0 10 0 10 0 Observe he resul of he inerference for differen disances d of he line fro he sources Z1, Z. The posiion of he sources is syeric bu he inerference paern is evidenly unsyeric. Is i possible, ha unsyery depends on he phase consans α1, α? y
Apliude Av of he resuling oscillaions depends on difference of he phases α α1, resp. on he arguen of he cosine funcion in Eq. (4). The condiion of he inerference axiu in he poin A can be expressed by eans of uliplicaion of he nuber π: kr ( r1) α1 α + n π (6) Sudy of Eq. (6) shows ha inerference axia of he order n will occur for n 0, 1,.. wih apliude Av A of he resuling oscillaions. The inia in he inerference paern occur when 1 3 5 n,,.. wih Av 0. In any applicaions he difference of he phase consans is zero. So we can ake α1 α 0 and afer subsiuion o Eq. (6) we can find he equivalen relaion r r1 n λ. (7) I eans, ha for he se of poins wih r r1 consan we can find he inerference axia or inia of he specific order n. This condiions are filled by hyperboloids of revoluion. The inerference axia and inia along he line is shown on he previous figure. Now we will show he inerference axia and inia in he plane, which cus he hyperboloids of he revoluion in he 3D space. xz1 := 0 xz := 0 i := 1.. 40 x 0 := 10 x i := x i 1 + yz1 := 3 yz := 3 j := 1.. 40 y 0 := 10 y j := y j 1 + r1 ij ( y j yz1), := x i xz1 + r ij, := x i xz Av ij, := A cos k r ij, r1 ij, α1 α + + ( y j yz) 0.5 0.5 Av
Conclusion All his conribuion is yped in Mahcad environen and is saved in HTML fora. Such a way, every Mahcad docuen can be read as HTML Inerne docuen. For copuing in Mahcad environen we need his docuen in Mahcad fora. The file in he Mahcad fora of his conribuion, which can be calculaed by Mahcad ver. 6 and higher, is available a hp://fyzika.fe.vubr.cz/~irdo/budapes. References 1. M. DOLOŽÍLEK, E.L. LOH.: "MahCAD - Useful Sofware for he Daa Acquisiion of Laboraory Measureens in Physics", Inernaional conference of Copuer Aide Engineering Educaion, Buchares, Sepeber, 1993.. M. DOLOŽÍLEK, E.L. LOH.: "Soe Feaures of Faulus, MahCad, and Spreadshee for Copuer Based Learning in Physics", In.Conference: Copuer Based Learning in Science, Opava, Czech Republic, 1995, s.39-45. 3. Mahcad 000 Professional. MahSof, Inc., 01 Broadway, Cabridge, Massachuses, USA.