Atomic Emission Spectra Using a UV Vis Spectrophotometer and an Optical Fiber Guided Light Source

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Atomic Emission Spectra Using a UV Vis Spectrophotometer and an Optical Fiber Guided Light Source"

Transcription

1 Atomic Emission Spectr Using UV Vis Spectrophotometer nd n Opticl Fiber Guided Light Source Mnuel E. Mins d Piedde Centro de Químic Estruturl, Complexo I, Instituto Superior Técnico, 096 Lisbo Codex, Portugl Mário N. Berbern-Sntos Centro de Químic-Físic Moleculr, Complexo I, Instituto Superior Técnico, 096 Lisbo Codex, Portugl The study of tomic emission spectr in the undergrdute lbortory hs been suggested for mny yers to illustrte quntum mechnicl principles (e.g., the ide tht microscopic mtter exists in quntized sttes) nd the reltion between the electronic structure nd spectrl observtions ( ). Becuse of the simplicity of its spectrum, hydrogen is by fr the element of choice in published experiments in tomic spectroscopy ( 5, 8, 0, ), but experiments involving other elements such s lkli metls hve lso been reported (6, 7, 9). Severl types of pprtus hve been used to record the spectr, including n tomic bsorption spectrophotometer (), spectroscopes ( ), spectrogrphs (5 8) nd UV vis spectrophotometers (9 ). In ll pprtuses of the ltter type, the light source is either prt of the spectrophotometer (deuterium lmp), yielding only few s (), or when not prt, is plced inside the instrument (9, 0) nd is therefore difficult to interchnge. This mkes imprcticl the study of more thn one element in lbortory session. As shown in the present pper, the use of n opticl fiber to guide the light from n externl source onto the monochromtor of the spectrophotometer enbles fst sequentil use of severl light sources, irrespective of their size nd shpe. This setup hs been in use in our undergrdute physicl chemistry lbortory to study the emission spectr of hydrogen, sodium, nd helium in four-hour session. The experiment cn be redily extended to other elements (e.g., Hg, Li, K, Xe, Ne) or modified to obtin moleculr spectr from dischrge tubes nd lmps (e.g., N, D, H ) nd tomic nd moleculr spectr in flmes (e.g., lkli metls, C ), provided the pproprite light sources re vilble. Apprtus The pprtus used consists of Perkin-Elmer Lmbd 5 spectrophotometer, plstic opticl fiber with dimeter of 3.0 mm from Lser Components GmbH (PG-R-FB3000 Plstikfser), nd light source. In the cse of hydrogen nd helium light source from Electro-Technic Products, Inc., including Spectrum tube power supply SP-00 (5 kv, 0 ma) nd n interchngeble gs dischrge lmp ws used. The spectrum of sodium ws recorded using Bellinghm+ Stnley Ltd. sodium lmp unit removed from polrimeter. The opticl fiber is introduced into the spectrophotometer through preexisting hole on the housing bottom nd fixed by clmp support ner the entrnce of the monochromtor. Another clmp support is used to djust the position of the fiber ner the externl light source. The hydrogen nd helium light sources re plced t distnce of c. 0.5 cm from the tip of the opticl fiber in order to void dmge to the fiber by the het generted t the source lmp. In the cse of the much more intense sodium lmp, longer distnce (c. 8 cm) is required to prevent photomultiplier sturtion. Recording the Spectr The Perkin-Elmer Lmbd 5 spectrophotometer is operted in the single-bem mode ccording to the mnufcturer s instructions. A preliminry scn is mde t scn speed of 0 or 0 nm min, to djust the bndpss nd gin so tht the intensity of the stronger s observed in the spectrum does not exceed c. 00. To conveniently detect the weker s it my be necessry to record other spectr with incresed bndpss nd/or gin. In the cse of sodium, for exmple, fter the preliminry scn nd the selection of convenient bndpss nd gin, globl spectrum, without resolved doublets, is recorded. Then, slow scns t 7.5 nm min nd bndpss = 0.5 nm re mde in the ner rnge of ech unresolved detected in the previous spectrum. It is then possible to observe firly resolved doublets (well resolved only for the longer wvelengths, c. λ 568 nm). The typicl wvelength rnges covered in the experiment re nm for hydrogen, nm for sodium, nd nm for helium. The wvelength clibrtion of the pprtus ws checked with mercury lmp from Electro-Technic Products, Inc. Results nd Discussion Using the pprtus nd experimentl procedure described bove, 8 s (prt of the Blmer series) re observed for hydrogen, t lest 8 doublet s re observed in the cse of sodium, nd minimum of 5 s re observed for helium. Becuse of the intrinsic bsorption of the opticl fiber used, the lowest mesurble wvelength is bout 350 nm. No specil precutions re tken to eliminte the contmintion by mbient light. Spectr obtined t high gins nd with wide bndpsses my, however, show extrneous contributions (e.g., the strong mercury s of fluorescent lmps in the lbortory ceiling). Typicl results obtined by the students for hydrogen re nm (H θ ), nm (H η ), nm (H ζ ), nm (H ε ), 0.0 nm (H δ, h ), 33.6 (H γ, G ), 86. nm (H β, F ), (H α, C ) (see ref for ssignments). Those for sodium nd helium re shown in Tbles nd nd in Figures nd. The intensities indicted in Tbles nd re not bsolute, since the efficiencies of the opticl system nd of the photomultiplier of the spectrophotometer re wvelength dependent. The students re sked to complete tsks described in the following sections. JChemEd.chem.wisc.edu Vol. 75 No. 8 August 998 Journl of Chemicl Eduction 03

2 Tsk Assignment of the s in the obtined spectr by comprison with literture dt nd construction of the Grotrin digrm ( ) showing the energy levels nd the observed trnsitions for hydrogen, sodium nd helium (Figs. 3 nd ). Verifiction of the selection rules l = ± nd j = 0, ± for hydrogen nd sodium, nd l = ± nd L = 0, ±, S = 0, J = 0, ± for helium, where l nd L re the orbitl ngulr momentum quntum numbers, j nd J re the totl ngulr momentum quntum numbers for toms tht cn be considered s mono- (e.g., H nd N) nd poly- (e.g., He) electron species, respectively, nd S is the totl spin ngulr momentum quntum number ( ). Tsk Clcultion of the Rydberg constnt, H, nd the ioniztion energy, IE, of hydrogen using the following equtions ( ): λ = H n n () IE = H hc () where λ represents the wvelength, n = (Blmer series), n is n integer lrger thn n, h is the Plnck constnt, nd c is the velocity of light. A lest squres fit of eq to plot of /λ vs /n, using n = 3 0 for λ = nm, leds to H = (.098 ± 0.00) 0 5 cm. (All uncertinties quoted throughout this pper correspond to the stndrd devition multiplied by Student s t prmeter for 95% confidence level.) This result is in good greement with the theoreticl vlue H = 09,677.6 cm (clculted from ): Tble. Wvelengths nd Intensities of Atomic Sodium Emission Lines λ / nm (in ir) Intensity/ O bserved Assignment ( ) Devition rbitrry units (6 D5/,3/ 3 P / ) (6 D5/,3/ 3 P 3 / ) (7 S / 3 P / ) 75. (7 S / 3 P 3 / ) ( 5 D5/,3/ 3 P / ) (5 D5/,3/ 3 P 3 / ) (6 S / 3 P / ) (6 S / 3 P 3 / ) ( D5/,3/ 3 P / ) ( D5/,3/ 3 P 3 / ) ( 3 P 3/ 3 S / ) D ( 3 P / 3 S / D ) ( 5 S / 3 P / ) ( 5 S / 3 P 3 / ) ( 3 D 5 /, 3/ 3 P / ) ( 3 D5/,3/ 3 P 3 / ) Unresolved doublet. b Clculted using the verge doublet wvelength. = µe 8ε 0 h 3 c where e represents the elementry chrge, ε 0 is the vcuum permittivity, h is the Plnck constnt, c is the velocity of light, nd µ is the reduced mss of the nucleus + electron system given by (3) Figure. Sodium emission spectrum (see Tble for ssignments nd reltive intensities). The observed doublet structure recorded t 7.5 nm min is shown for selected s. K is the resonnce doublet from potssium impurity. S/ S/ P/ D5/, 3/ µ = m N m e m N + m e () where m N nd m e re the msses of the nucleus nd of the electron, respectively. From the obtined Rydberg constnt, (.098 ± 0.00) 0 5 cm, nd eq it is possible to conclude tht for hydrogen IE = 3.6 ± 0.03 ev, in good greement with the vlue IE = ev recommended in ref 5. It should be noted tht for more ccurte work the conversion of the experimentl wvelengths, mesured in ir, to the corresponding vlues in vcuum must be tken into ccount (8, ). In our cse, however, this correction is well within the experimentl error nd ws therefore neglected. Tsk 3 Clcultion of the Rydberg constnt nd of the ioniztion energy of sodium. Clcultion of the quntum defect corrections for the shrp (S, ns 3p), principl (P, np 3s), Energy / ev nm 65.3 nm nm 66.0 nm 7.9 nm nm 66.6 nm 98.0 nm nm 89. nm 568. nm 88. nm 6 5 Figure 3. Grotrin digrm for sodium showing the trnsitions observed in the lbortory (see lso Tble ). The trnsitions t 66.6, 7.9, 98.0, nd 55. nm correspond to unresolved doublets. 3 0 Journl of Chemicl Eduction Vol. 75 No. 8 August 998 JChemEd.chem.wisc.edu

3 Tble. Wvelengths nd Intensities of Helium Emission Lines λ / nm (in ir) Intensity/ O bserved Assignment ( ) Devition rbitrry units ( 5 P S 0 ) ( 3 3 P,, 0 3 S ) ( P S 0 ) b ( 5 D 3,, 3 P,, 0) 0. b.9. ( 5 S 3 P,, 0) ( 5 D P ) ( D 3,, 3 P,, 0) ( S 3 P,, 0) 0. b ( D P ) 0. 5 b b ( 3 P S 0 ) 0. 5 b ( S 0 P ) ( 3 3 D 3,, 3 P,, 0 D ( 3 ) 0. 7 b D P ) 0. 3 b ( 3 S 3 P,, 0) ( 3 S 0 P ) b Bndpss = nm, gin = 5. b Bndpss = nm, gin = 5. nd diffuse (D, nd 3p) series of sodium. Anlysis of the doublet structure of the sodium s. The nlogue of eq for sodium is the empiricl eqution ( ) λ = N n δ n l n δ n l (5) where N is the Rydberg constnt for sodium, n = 3,, nd 5 for the P, D, nd S series, respectively, n is n integer equl to or lrger thn n, nd δ n l nd δ n l re the quntum defects or Rydberg corrections. The quntum defect is relted to the interction of the electron whose trnsitions produce the spectrum with the inner core electrons; it is pproximtely constnt for levels of different principl quntum number, n, but with the sme orbitl ngulr momentum quntum number, l (i.e., for the sme series) ( ). The vlue of the quntum defect increses with the degree of penetrtion of the corresponding orbitl. It tkes, therefore, the highest vlues for the S series nd is close to zero for the D series (see below). Eqution 5 ws seprtely fitted to the dt of the diffuse nd shrp series shown in Tble, in the form λ = A n B + C (6) where A, B, nd C re the fitting prmeters, given by A = N (7) B = δ n l (8) Figure. Helium emission spectrum (see Tble for ssignments nd reltive intensities). C = N n δ n l (9) Energy / ev S0 P D 3 S 3 S 3 D3,, 50.5 nm 78.0 nm 36.0 nm 396. nm 50. nm 38.6 nm 9.0 nm nm 706. nm.9 nm 7. nm nm 0. nm 6.9 nm nm Figure. Grotrin digrm for singlet () nd triplet (b) helium showing the trnsitions observed in the lbortory. The corresponding wvelengths re given in Tble. The vlues of these prmeters were found by using the Solver tool of MS Excel 7.0 to minimize the sum of the squres of the devitions between clculted nd experimentl wvelengths (Tble 3). The uncertinties of the prmeters re not determined in the cse of this nonr fit. This cn be done by Monte Crlo simultion (6), but the clcultion is beyond the scope of this lbortory experiment. Comprison of the results obtined for H, N, nd He, with the corresponding (more ccurte) literture vlues, however, llows n posteriori estimtion of the ccurcy of the Rydberg constnts nd quntum defects obtined from eq 6. Accurcies better thn % re found for the Rydberg constnts, nd in the cse of the quntum defects two to three exct deciml plces re observed. From the dt in Tble 3 nd eqs 6 9 it is possible to conclude tht for the diffuse series δ n d = 0.000, δ 3p = 0.876, nd N = 0,5 cm. These results re in good greement with those reported by Stfford nd Wortmn, which ssumed tht δ nd = 0 nd used r djustment of eq 5 to /λ vs. /n plot to derive δ 3p = 0.88 nd N = 0,600 cm (9). Note, however, tht the Rydberg constnt obtined in the present experiment, N = 0,5 cm, is too high when compred to its theoreticl vlue N = 09,73.7 cm cl- JChemEd.chem.wisc.edu Vol. 75 No. 8 August 998 Journl of Chemicl Eduction 05

4 culted from eq 3. In ddition, the quntum defect for the 3p level, δ 3p = 0.876, is lower thn the vlue δ 3p = recommended in (,, 7). For these resons, constrined djustment of eq 5 to the experimentl dt, with N = 09,73.7 cm, ws lso crried out. A slightly worse fit (mximum devition equl to 0. nm) ws obtined, yielding quntum defects δ nd = nd δ 3p = 0.883, in very good greement with the published vlues δ nd = 0.0 nd δ 3p = (,, 7). For the shrp series, the best fit (Tble 3) gve quntum defects δ ns =.36 nd δ 3p = 0.885, nd Rydberg constnt N = 0956 cm ; but imposing gin N = cm yields quntum defects, δ ns =.36 nd δ 3p = 0.883, in better greement with the literture vlues δ ns =.36 nd δ 3p = (,, 7 ). Note, finlly, tht since not ll the doublets of ech sodium were well resolved in the obtined spectrum, the men wvelength vlue of ech doublet in Tble ws selected for the clcultions. For more ccurte clcultion the wvelengths of just one of the two spin-orbit components should be used. The mgnitude of the spin-orbit coupling cn lso be evluted for the resolved doublet s. In the cse of the D, for exmple, spin-orbit coupling of 7.3 cm is obtined from the experimentl dt in Tble. This result is in good greement with the published vlue of 7. cm (). It is lso interesting to note tht very well resolved doublet from potssium impurity is observed in the spectrum of sodium t wvelengths nd nm. This doublet is the equivlent in potssium of the sodium D. In this cse, spin-orbit coupling of 57.6 cm is clculted, in good greement with the published vlue of 57.7 cm (). The strong dependence of the spin-orbit coupling on the tomic number Z it increses with Z ( ) is clerly seen by compring the obtined results for sodium nd potssium indicted bove (7.3 cm nd 57.6 cm, respectively). Another possible topic of discussion is the reltive intensity of the two components of the doublet s. For instnce, in the cse of the D, the theoreticl rtio between the intensities of the P 3/ S / nd P / S / components is :; tht is, tht of the sttisticl weights (j + ) of the corresponding upper sttes (). For most commercil lmps the mesured rtio is frequently lower owing to the rebsorption phenomenon (), which is importnt when the concentrtion of the sodium toms is high. In our cse rtio of.3: is observed (Fig. ). If, however, the D doublet spectrum is recorded immeditely fter turning the lmp on (when the concentrtion of N in the vpor is low) the theoreticl rtio of : is indeed observed. The first ioniztion energy of ground-stte sodium cn lso be clculted from the spectrl dt ( ). In this clcultion it is ssumed tht the ioniztion process cn be decomposed in two steps: in the first one, ground-stte sodium is excited to the 3 P stte (energy required = hν =. ev, where ν is the frequency of the D ); in the second, sodium in the 3 P stte is further excited to the ioniztion limit. The energy required for this second step corresponds to the limit n of eq 5 nd is thus given by the constnt C of eq 6. Conversion of the C vlues in Tble 3 to ev leds to C = 3.0 ev. Thus, IE N = 5.5 ev is derived, in good greement with the recommended vlue, IE N = ev (5). Tble 3. Best-Fit Vlues of A, B, nd C Prmeters (Eq 6) for Sodium D nd S Series Series A/cm Tsk Clcultion of the Rydberg constnt nd the ioniztion energy of helium. Clcultion of the singlet triplet gp for the P sttes. In the cse of helium singlet nd triplet D series the following empiricl eqution is vlid (): λ = He B C/cm D 0, ,505 S 09,56.359,89 n n δ 3p (0) where He is the Rydberg constnt for helium, n = 3,, nd 5, n =, nd δ 3p represents the quntum defect for the singlet or triplet 3P stte. According to eq 0, plot of /λ versus /n should give stright of slope He nd intercept He /( δ 3p ). Lest squre fits to the results in Tble led to He = 0,000 ± 00 cm nd δ 3p = 0.0 for the singlet series nd He = 0,00 ± 600 cm nd δ 3p = for the triplet series. The corresponding vlues in ref re He = 09,7 cm nd δ 3p = nd for the singlet nd triplet series, respectively. The ioniztion energies of the P nd 3 P sttes cn lso be clculted from the intercepts of the lest squre fits referred to bove. The obtined vlues, 3.37 ± 0.0 ev for P nd 3.63 ± 0.0 ev for 3 P sttes, re in excellent greement with the corresponding vlues in ref, ev ( P) nd 3.6 ev ( 3 P), respectively, nd led to singlet triplet gp of 0.6 ev. As for sodium, the ioniztion energy of ground-stte helium cn be clculted by dding the energy of the P S 0 trnsition (.3 ev), which in this cse is tken from the literture (), to the ioniztion energy of the P stte indicted bove (3.37 ev). This leds to IE He =.60 ev, in good greement with the vlue.587 ev recommended in ref 5. Concluding Remrks The experiment described is fst ( four-hour session) nd simple nd yields ccurte results. It illustrtes number of importnt topics in tomic structure nd tomic spectroscopy tht re covered in ll undergrdute physicl chemistry courses but often not studied in the lbortory owing to the lck of pproprite equipment. The required pprtus is ssembled with common instruments (polrimeter or refrctometer sodium lmp nd UV vis spectrophotometer), the dditionl equipment required being only plstic opticl fiber, low-cost hydrogen nd helium dischrge tubes, nd pproprite power supply. It cn lso be used to obtin moleculr spectr from dischrge tubes or lmps (e.g., N, D, H ) nd 06 Journl of Chemicl Eduction Vol. 75 No. 8 August 998 JChemEd.chem.wisc.edu

5 tomic nd moleculr spectr in flmes (e.g., lkli metls, C ). Finlly, the present lbortory experiment cn motivte the discussion of more dvnced topics in tomic spectroscopy, such s the hydrogen fine structure (, 8) nd Rydberg sttes (, 9), nd of the working principles of spectroscopic components, such s dischrge tubes (0), opticl fibers (), monochromtors (8, ), nd photomultiplier detectors (8). Literture Cited. Dougls, J.; von Ngy-Felsobuki, E. I. J. Chem. Educ. 987, 6, 55.. Compnion, A.; Schug, K. J. Chem. Educ. 966, 3, Reiss, E. J. Chem. Educ. 988, 65, 57.. Shields G. C.; Ksh, M. M. J. Chem. Educ. 99, 69, Rppon, M.; Greer, J. M. J. Chem. Educ. 987, 6, McSwiney, H. D.; Peters, D. W.; Griffith, W. B., Jr.; Mthews, C. W. J. Chem. Educ. 989, 66, Miller, K. J. J. Chem. Educ. 97, 5, Shoemker, D. P.; Grlnd, C. W.; Nibler, J. W. Experiments in Physicl Chemistry; McGrw-Hill: New York, Stfford, F. E.; Wortmn, J. H. J. Chem. Educ. 96, 39, Hollenberg, J. L. J. Chem. Educ. 966, 3, 6.. Guiñón, J. L. J. Chem. Educ. 989, 66, Kuhn, H. G. Atomic Spectr; Longmns: London, Softley, T. P. Atomic Spectr; Oxford Chemistry Primers Series; Oxford University Press: New York, 99.. Hken, H.; Wolf, H. C. The Physics of Atoms nd Qunt, 5th ed.; Brewer, W. D., trnsl.; Springer: Berlin, Hndbook of Chemistry nd Physics, 7th ed.; Lide, D. R., Ed.; CRC: Boc Rton, Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flnnery, B. P. Numericl Recipes, nd ed.; Cmbridge University Press: Cmbridge, Slter, J. C. Quntum Theory of Atomic Structure, Vol. I; McGrw- Hill: New York, Powell R. E. J. Chem. Educ. 968, 5, Silvermn, M. P. And Yet it Moves Strnge Systems nd Subtle Questions in Physics; Cmbridge University Press: Cmbridge, Lewin, S. Z. J. Chem. Educ. 965,, A65.. Mellir-Smith, C. M. J. Chem. Educ. 980, 57, 57.. Guiñón, J. L, Grci-Anton, J. J. Chem. Educ. 99, 69, 77. JChemEd.chem.wisc.edu Vol. 75 No. 8 August 998 Journl of Chemicl Eduction 07

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?

1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply? Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Plotting and Graphing

Plotting and Graphing Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

More information

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O The Interntionl Assocition for the Properties of Wter nd Stem Lucerne, Sitzerlnd August 7 Relese on the Ioniztion Constnt of H O 7 The Interntionl Assocition for the Properties of Wter nd Stem Publiction

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Nuclear Magnetic Resonance

Nuclear Magnetic Resonance Nucler Mgnetic Resonnce Chpter 13 1 Rections will often give mixture of products: 2 S 4 + Mjor Minor ow would the chemist determine which product ws formed? Both re cyclopentenes; they re isomers. Spectroscopy

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

, and the number of electrons is -19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow.

, and the number of electrons is -19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow. Prolem 1. f current of 80.0 ma exists in metl wire, how mny electrons flow pst given cross section of the wire in 10.0 min? Sketch the directions of the current nd the electrons motion. Solution: The chrge

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

THERMAL EXPANSION OF TUNGSTEN

THERMAL EXPANSION OF TUNGSTEN . THERMAL EXPANSION OF TUNGSTEN S515 By Peter Hidnert nd W. T. Sweeney ABSTRACT This pper gives the results of n investigtion on the therml expnsion of tungsten (99.98 per cent) over vrious temperture

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

More information

Dispersion in Coaxial Cables

Dispersion in Coaxial Cables Dispersion in Coxil Cbles Steve Ellingson June 1, 2008 Contents 1 Summry 2 2 Theory 2 3 Comprison to Welch s Result 4 4 Findings for RG58 t LWA Frequencies 5 Brdley Dept. of Electricl & Computer Engineering,

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Rate and Activation Energy of the Iodination of Acetone

Rate and Activation Energy of the Iodination of Acetone nd Activtion Energ of the Iodintion of Acetone rl N. eer Dte of Eperiment: //00 Florence F. Ls (prtner) Abstrct: The rte, rte lw nd ctivtion energ of the iodintion of cetone re detered b observing the

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006 dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.

More information

COMPONENTS: COMBINED LOADING

COMPONENTS: COMBINED LOADING LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of

More information

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES Solution to exm in: FYS30, Quntum mechnics Dy of exm: Nov. 30. 05 Permitted mteril: Approved clcultor, D.J. Griffiths: Introduction to Quntum

More information

Project 6 Aircraft static stability and control

Project 6 Aircraft static stability and control Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

PSYCHROMETRICS: HEATING & HUMIDIFYING or COOLING & DEHUMIDIFYING

PSYCHROMETRICS: HEATING & HUMIDIFYING or COOLING & DEHUMIDIFYING PSYCHROMETRICS: HEATING & HUMIDIYING or COOLING & DEHUMIDIYING I) Objective The objective of this experiment is to exmine the stte of moist ir s it enters nd psses through the ir hndling unit. When ether

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING Sung Joon Kim*, Dong-Chul Che Kore Aerospce Reserch Institute, 45 Eoeun-Dong, Youseong-Gu, Dejeon, 35-333, Kore Phone : 82-42-86-231 FAX

More information

Uplift Capacity of K-Series Open Web Steel Joist Seats. Florida, Gainesville, FL 32611; email: psgreen@ce.ufl.edu

Uplift Capacity of K-Series Open Web Steel Joist Seats. Florida, Gainesville, FL 32611; email: psgreen@ce.ufl.edu Uplift Cpcity of K-Series Open Web Steel Joist Sets Perry S. Green, Ph.D, M.ASCE 1 nd Thoms Sputo, Ph.D., P.E., M.ASCE 2 1 Assistnt Professor, Deprtment of Civil nd Costl Engineering, University of Florid,

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:

More information

Applications of X-Ray Photoabsorption Spectroscopy to the Determination of Local Structure in Organometallic Compounds

Applications of X-Ray Photoabsorption Spectroscopy to the Determination of Local Structure in Organometallic Compounds 8059 (28) B. Longto, F. Morndini. nd S. Bresdol, J. Orgnomet. Chem., 88, C7 (33) G. S. Arnold,. L. Klotz,. Hlper, nd M. K. DeArmond, Chem. Phys. (1975). Lett., 19, 546 (1973). (29) M. G. Clerici, S. Di

More information

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule. Text questions, Chpter 5, problems 1-5: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

ORBITAL MANEUVERS USING LOW-THRUST

ORBITAL MANEUVERS USING LOW-THRUST Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ORBIAL MANEUVERS USING LOW-HRUS VIVIAN MARINS GOMES, ANONIO F. B. A. PRADO, HÉLIO KOII KUGA Ntionl Institute

More information

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process

An Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Straight pipe model. Orifice model. * Please refer to Page 3~7 Reference Material for the theoretical formulas used here. Qa/Qw=(ηw/ηa) (P1+P2)/(2 P2)

Straight pipe model. Orifice model. * Please refer to Page 3~7 Reference Material for the theoretical formulas used here. Qa/Qw=(ηw/ηa) (P1+P2)/(2 P2) Air lek test equivlent to IX7nd IX8 In order to perform quntittive tests Fig. shows the reltionship between ir lek mount nd wter lek mount. Wter lek mount cn be converted into ir lek mount. By performing

More information

Mechanics Cycle 1 Chapter 5. Chapter 5

Mechanics Cycle 1 Chapter 5. Chapter 5 Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies

More information

A new algorithm for generating Pythagorean triples

A new algorithm for generating Pythagorean triples A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS 4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Matrix Inverse and Condition

Matrix Inverse and Condition Mtrix Inverse nd Condition Berlin Chen Deprtment of Computer Science & Informtion Engineering Ntionl Tiwn Norml University Reference: 1. Applied Numericl Methods with MATLAB for Engineers, Chpter 11 &

More information

c. Values in statements are broken down by fiscal years; many projects are

c. Values in statements are broken down by fiscal years; many projects are Lecture 18: Finncil Mngement (Continued)/Csh Flow CEE 498 Construction Project Mngement L Schedules A. Schedule.of Contrcts Completed See Attchment # 1 ll. 1. Revenues Erned 2. Cost of Revenues 3. Gross

More information

C-crcs Cognitive - Counselling Research & Conference Services (eissn: 2301-2358)

C-crcs Cognitive - Counselling Research & Conference Services (eissn: 2301-2358) C-crcs Cognitive - Counselling Reserch & Conference Services (eissn: 2301-2358) Volume I Effects of Music Composition Intervention on Elementry School Children b M. Hogenes, B. Vn Oers, R. F. W. Diekstr,

More information

Quick Reference Guide: One-time Account Update

Quick Reference Guide: One-time Account Update Quick Reference Guide: One-time Account Updte How to complete The Quick Reference Guide shows wht existing SingPss users need to do when logging in to the enhnced SingPss service for the first time. 1)

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix. APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Simulation of operation modes of isochronous cyclotron by a new interative method

Simulation of operation modes of isochronous cyclotron by a new interative method NUKLEONIKA 27;52(1):29 34 ORIGINAL PAPER Simultion of opertion modes of isochronous cyclotron y new intertive method Ryszrd Trszkiewicz, Mrek Tlch, Jcek Sulikowski, Henryk Doruch, Tdeusz Norys, Artur Srok,

More information

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

ACID-BASE BASICS Stephen Wright, Ph.D. Dept of Physiology

ACID-BASE BASICS Stephen Wright, Ph.D. Dept of Physiology Foundtions Lerning Objectives for the self-lerning module on ACID-BASE Bsics 1. Define ph. 2. Clculte the hydrogen ion concentrtion from ph nd vice vers. 3. List the mjor body buffer systems. ACID-BASE

More information

The Definite Integral

The Definite Integral Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Applications to Physics and Engineering

Applications to Physics and Engineering Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

2.4 Circular Waveguide

2.4 Circular Waveguide .4 Circulr Wveguide y x Figure.5: A circulr wveguide of rdius. For circulr wveguide of rdius (Fig..5, we cn perform the sme sequence of steps in cylindricl coordintes s we did in rectngulr coordintes to

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

SPARK QUENCHERS EFFECT OF SPARK QUENCHER

SPARK QUENCHERS EFFECT OF SPARK QUENCHER Sprk Quenchers re esily selectble electronic components designed to prevent or substntilly minimize the occurrence of rcing nd noise genertion in rely nd switch contcts. EFFECT OF SPARK QUENCHER Arc suppression

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm

Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting Self-Invertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

Answer, Key Homework 4 David McIntyre Mar 25,

Answer, Key Homework 4 David McIntyre Mar 25, Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his print-out should hve 18 questions. Multiple-choice questions my continue on the next column or pe find ll choices before mkin your selection.

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Final of Math 121, Fall 2014 Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

More information

STATUS OF LAND-BASED WIND ENERGY DEVELOPMENT IN GERMANY

STATUS OF LAND-BASED WIND ENERGY DEVELOPMENT IN GERMANY Yer STATUS OF LAND-BASED WIND ENERGY Deutsche WindGurd GmbH - Oldenburger Strße 65-26316 Vrel - Germny +49 (4451)/9515 - info@windgurd.de - www.windgurd.com Annul Added Cpcity [MW] Cumultive Cpcity [MW]

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

Solar and Lunar Tides

Solar and Lunar Tides Solr nd Lunr Tides evised, Corrected, Simplified Copyriht Steve Olh, Irvine, CA, 009 solh@cox.net Keywords: Tide, Lunr Tide, Solr Tide, Tidl Force Abstrct Solr nd lunr tides were prt of life since there

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

19. The Fermat-Euler Prime Number Theorem

19. The Fermat-Euler Prime Number Theorem 19. The Fermt-Euler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information