Excel Files For Teaching Two Dimensional Motions And Their Curvature
Global Journal of Science Frontier Research: IInterdisciplinaryVolume 16 Issue 3 Version 1.0 Year 2016Type : Double Blind Peer Reviewed International Research JournalPublisher: Global Journals Inc. (USA)Online ISSN: 2249-4626 & Print ISSN: 0975-5896Excel Files for Teaching Two Dimensional Motions and theirCurvatureBy Pavlos MIhasDemocritus University of ThreaceAbstract- Spreadsheets are presented which can help students to understand the the vectorrepresentation of velocity, acceleration and force, and the understanding of the radius ofcurvature\. The examples include: sinusoidal curve, cardioid, Neile’s parabola, motion of acharged body on which act a combination of magnetic and electric field, a body sliding on arotating disk.GJSFR-I Classification: FOR Code: dtheirCurvatureStrictly as per the compliance and regulations of : 2016. Pavlos MIhas. This is a research/review paper, distributed under the terms of the Creative Commons AttributionNoncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use,distribution, and reproduction in any medium, provided the original work is properly cited.
Excel Files for Teaching Two DimensionalMotions and their CurvaturePavlos MIhasII.Introductionn physics education research, student understandingontopics of introductory mechanics has beenthoroughly studiedfor several decades (for exampleClement 1982, Halloun and Hestenes 1985). Thisresearch, has demonstrated that many students retainfundamentalconceptual difficulties, even after instruction(Kim and Pak 2002).Two dimensional motions are examined insecondary education (for example: Motion on a circle,parabolas, elliptic motions of planets, circularpendulums, conical pendulums).Two dimensional motions is a subject whichrequires the understanding of the vector nature ofvelocity, acceleration and force (Mihas & Gemoysakakis2002).We need to present to the students generalprinciples, which they will apply to any kind of twodimensional motions.The principles that students need to learn are:A) the velocity as a vector tangent to the orbit. B) Theacceleration (centripetal acceleration) in case ofconstant speed is perpendicular to the velocity vector.The size of the centripetal acceleration depends on thespeed and the radius of curvature of the orbit. C)Newton’s second law. D) In the case of a tangentialcomponent of the force they have to think of the totalacceleration as composed of two components:tangential acceleration and centripetal acceleration. E)The direction of the force with the direction of velocitycan have an acute, right or obtuse angle if the measureof the velocity increases, remains constant or decreasesand so the kinetic energy increases, remains constant ordecreases. F) The curve can be approximated at acertain point with a circle, which has radius the radius ofAuthor: Democritus University of Threace, Greece.e-mail: pmichas@eled.duth.gr. Theradius of curvature can have values from zero to infinity2(in case of a rectilinear path o if d y 0 . We will seedx 2examples of these cases in the following.In this paper we examine several curves withtheir properties which can help as examples to elucidatethe concepts, and we end up with the relative motion ofa sliding body on a rotating disk which can gives ussome unexpected results.II.Sinusoidal CurveOne curve that can be used to introducestudents to the ideas of radius of curvature and therelation between angle of velocity and acceleration is thesinusoidal curve.The length s on the sinusoidal y B·sinkx isexpressed by use of elliptic integrals of the second kind:s 1 B² k ² cos ² kx ·dx 1 B² k ² E (α , x)whereα B·k1 B² k ²Since elliptic integrals are calculated very fast, itis easy to find solutions of the coordinates as a functionof s. This permits to construct simulations. If we assumea speed that is increasing, with an increase of the speedwhich is proportional to the time, we can use thisexample to show the relation between angle of velocityand acceleration.Usually the students pick up from the simulationsome facts but they do notgenerally have a coherentpicture. So they do not realize that the radius ofcurvature is infinit in the points where the inflectionchanges sides (in the sinusoidal curve at the pointswhere the sign of y changes from plus to minus and viceversa).For example they pick up the information thatfor a constant speed the acceleration should beperpendicular to the velocity, but they generalize it to thelinear parts of the curve where radius of curvature is ρ and so acentripetal v²/ρ 0/ 2016Global Journals Inc. (US)2016curvature which is expressed3/ 2Yearstudents to understand the the vector representation ofvelocity, acceleration and force, and the understanding of theradius of curvature\. The examples include: sinusoidal curve,cardioid, Neile’s parabola, motion of a charged body on whichact a combination of magnetic and electric field, a body slidingon a rotating disk. dx 2 dy 2 ds ds ρ as32 dx d y 2 ds dx231Global Journal of Science Frontier Research ( I ) Volume XVI Issue III Version IAbstract- Spreadsheets are presented which can help
Year2016Excel Files for Teaching Two Dimensional Motions and their CurvatureGlobal Journal of Science Frontier Research ( I ) Volume XVI Issue III Version I24Figure 1: Sinusoidal with vectors of accelerationanalyzed to entripetal and tangentIII.Tutorial CurveThis curve is used by McDermott etal in theTutorials of introductory physics (in Greek p. 285 2011).This curve is a closed one and shows the change ofdirection of the centripetal acceleration but also the totalacceleration the circle that corresponds to the curvatureat the point where the moving body is located. Thestudent can observe that when the body passes throughpoints such as B and C the acceleration is almosttangent to the curve. This curve helps the student tounderstand that the angle of the acceleration to thevelocity is acute when the speed increases and to seethat the centripetal acceleration is greatest in the pointswhere the radius of curvature is smallest. Theworksheets comes in two forms (with macros, whichpermit a continuous change) and without, where thebody moves by sliding a scroll bar (worksheet tutorialcurve.xslx and tutorial curve macros). The points A, Eare points where the student should predict thedirectionof the acceleration and also draw a circle with a radiusequal to the radius of curvature. 2016Global Journals Inc. (US)
Year2016Excel Files for Teaching Two Dimensional Motions and their CurvatureFigure 2: Closed curve used in Tutorials with constanttangential accelerationIV.Neile P arabolaAristotle was the first who was concerned withthe composition of motions (Dugas p. 21). “Let amoving body be simultaneously be actuated by twomotions that are such that the distances traveled in thesame time are in a constant proportion. Then it will moveinto the diagonal of a parallelogram which has as sidestwo lines whose lengths are in this constant relation toeach other” .Much later Galileo (Two New Sciences)analyzedthe motion of projectiles into two componentsand proved that the orbit described by the projectile willhave the shape of a parabola.Parabola was analyzed as performing on thesame time an inertial motion and a motion under theinfluence of gravity. In Neile’s parabola the motion isdydx β: adtdt 0.Fromwork-energytheorem:211 dx 1m·v ² m· m·β ² m·g·y22 dt 2dx a ² 2·g·yβunder the influence of two forces: A horizontal forcearising from the reaction of the curve and a vertical forcearising from the perpendicular component of thereaction and the weight. For a special value of the initialvelocity, its vertical (perpendicular) component remainsconstant. In this case the total force has only ahorizontal component. For larger speed the verticalmotion will be an accelerated motion, and for lesservalues of the speed it will be a decelerated motion. Tofind the curve for the special case of constant verticalcomponent of the speed, letαbe the initial horizontalcomponent of the velocity and β the value of the verticalcomponentof velocity, then the vertical component ofthe reaction is equal to the weight for the case ofconstant β.and sodx dx dy dx · ·β a 2 2 g y :dt dy dt dy3(a ² 2 gy ) 2dy and by integration we get: x y ymax where ymax a²/(2 g)3 gβThis curve was the first one to have its length calculated. This calculation is done easily by using theconservation of energyds v 2x v y2 v02 2 gy , then:dts v02 2 gy ·dt 13 gβ v 3 (v 2 2 gy ) 3 2 0 0 2016Global Journals Inc. (US)Global Journal of Science Frontier Research ( I ) Volume XVI Issue III Version I251
Excel Files for Teaching Two Dimensional Motions and their CurvatureIt is interesting that the radius of curvature ofthis curve is zero for y ymax. If the speed is different fromzero then the centripetal acceleration will be infinite. Theradius of curvature can be expressed as:which for Neile’s curve is expressed as:α 2 2 gy (α 2 β 2 2 gy )ρ β 2gWe can see that for the highest point ρ 0. Incase of finite velocity we get infinite centripetalacceleration. The student can study the Neile’s parabolawith the spreadsheet “Neile curve”3/ 2Year2016.Global Journal of Science Frontier Research ( I ) Volume XVI Issue III Version I26Figure 3: Neile’s Parabola with circle of curvature and accelerations from spreadsheet “Neile Curve”a)CycloidalpendulumIn this pendulum the string is constantly in touchwith a cycloid. For this pendulum the radius of curvatureis studied easily, since it is equal to the length of thestring that does not touch the upper curves. The bob ofthe pendulum describes cycloid. As the bobapproaches the highest point the radius of curvaturebecomes smaller and eventually it becomes zero. In thecase of vibration that starts at this point the velocity iszero and so the centripetal force is zero. This can becontrasted to what happens in Neile curve, At this pointthe total force (weight) is tangent to the curve.The equations of the curve are: x R·(θ-sin(θ)),y -R R·cos(θ) (y 0) where R is the radius of thecircle that produces the cycloid. The length of the stringfor this pendulum is L 4RWe can find that the radius of curvature is:ρ 2 2 yR .For y 0 (at the ends of the cycloid) the radiusof curvature is zero. At the lowest point y -2R and thenρ 4R L string’s length.As it was found in tests and exams the studentscould easily arrange the points 1,2,3 of figure accordingto their radii of curvature but.The cycloidal pendulum has the property of aconstant period T 2π · 2016Global Journals Inc. (US)lthat does not depend on thegamplitude of the oscillations. This can be contrastedwith the simple pendulum for which the period isT 4·K (a )·lwhere a sin(φ/2) (φ amplitude) andgK(a) is the complete elliptic integral.The student can use the spreadsheet “cycloidpendulum” (with macros or not) to study the motion. Thecircle of curvature at the highest point has a minimumvalue which is zero if the bob starts from the cycloid.The student can compare the movement with the simplependulum and see that the period of the cycloidpendulum is independent of the amplitude of theoscillation.
Year2016Excel Files for Teaching Two Dimensional Motions and their CurvatureGlobal Journal of Science Frontier Research ( I ) Volume XVI Issue III Version I271Figure 4: Cycloidal pendulum with curvatur circle and forces actingon the bobb) Motion of a charged body in the combination of acentral force field and a magnetic field – or motion ina rotating frame, Foucault’s pendulum In this case the magnetic force F qv B qB v is the cause of a curving of the orbit. . Withk/m λ, ε q.B/m, The equations of motion are md ²xd²y k·x qBv y m k·y qBv xdt ²dt ²Here k can be either positive or negative.We consider a charged body starting with aninitial velocity v vx0 and located at x0 0, y0 R. Withε qB/m and λ k/m we have two cases: –ε² 4λ 0and–ε² 4λ 0 (which holds for a small repulsive forceand always for attractive central force)In this case we have an oscillatory motion.c) Solution for the equations of the motion of a chargedbody in the combination of a central force field and amagnetic fieldThe equations: d ² x λ·x ε ·v y , d ² y λ ·y ε ·vxdt ²dt ²can be solved with elementary functions. 2016Global Journals Inc. (US)
Year2016Excel Files for Teaching Two Dimensional Motions and their CurvatureGlobal Journal of Science Frontier Research ( I ) Volume XVI Issue III Version I28Figure 5: Movement in a combination of E and B, left with intial speed zero, on the right with asutibable initial horizontal speed so to pass through the origin (in the worksheet kappa -50,B 5, mass 1)By multiplying the second by i 1 andadding to the first we get the complex variable z x iyand the equation of motion:z A·exp(µ ·t )d ²zdz λ ·z iεdt ²dtµ ² iεµ λ 0 µ 1, 2 With iε Α2If –ε² 4λ 0 (λ 0 repulsiveforce) μ1 and μ2 arecomplex,z f1exp((-ie A)t/2) f2exp((-iε-Α)t/2) where f1and f2complex numbers. With initial conditions x 0, y R and(vx0 – εR/2)2/Α Δ we get:f1 M exp(iφ)f2 - Mexp(-iφ)where M ² R ² / 2 και φ arctan( R / ) and Γ Α/2z M [exp(iφ iεt / 2 Γt ) exp( iφ iεt / 2 Γt )]x M [cos(φ εt / 2)·exp(Γt ) cos(φ εt / 2)·exp( Γt )]y M [sin(φ εt / 2)·exp(Γt ) sin(φ εt / 2)·exp( Γt )]If –ε² 4λ 0 (λ 0 orsmall repulsive) we have4λ ε ² Dwe have two imaginary roots μ1μ2µ1 (ε D ) / 2, µ 2 (ε D ) / 2z i 2016( vx0 µ 2 ·R) exp(iµ1t ) ( vx0 µ1 ·R) exp(iµ 2 t )µ1 µ 2Global Journals Inc. (US)x (vx 0 µ2·R) sin( µ1t ) (vx 0 µ1·R) sin( µ2t )µ1 µ2y (vx 0 µ2·R) cos( µ1t ) (vx 0 µ1·R) cos( µ2t )µ1 µ2If vx0-ε·R/2 0,nd the body will pass through the originOr x R·sin(ε·t)·cos(D·t), y R·cos(ε·t)·cos(D·t).We can see that the body will pass through theorigin with a period 2π/D.With a suitable combination of E and B fields wecan have the charge move with very small radius ofcurvature around a circle or much larger radius.d) Similarity with Foucault’s pendulumThis motion is similar to the motion of aFoucault’s pendulum.Actually the equations of motion are verysimilar. In Foucault’s pendulum we have the Coriolis forceF Coriolis 2m ω v ω angular velocitywhich has the same form as the magnetic force. Thiswas mentioned in many papers (Semon & Schmieg1981, Sivardiere 1983, Opat 1990) By comparing thetwo forces we can see that the Magnetic Forcecorresponds to an angular velocity ω qB/(2m) ε/2We propose the following question: for whatcombination of initial speed and magnetic field will theorbit pass through the origin? For finding the answer wecan use the similarity with Foucault’s pendulum. Herewe describe a pendulum located on a turntable. Thebob of the pendulum describes a rotating ellipse. In thelaboratory frame the bob has an initial velocity (v ω R ).
Year2016Excel Files for Teaching Two Dimensional Motions and their CurvatureIf on the other hand the pendulum is hanged from a basis in the laboratory frame then the motion starts withinitial velocity equal to zero in the laboratory frame.Figure 6: The Foucault pendulum in two frames of reference The corresponding velocity in the rotating frame T 4π m /(4λ ε ²) will be v ω R . The motion in the laboratory framewill be confined in a plane. In this case the pendulumwill pass through the origin. The same thing can happenin a magnetic field if vx0 qB/(4m). Then the motion willbe that of a pendulum starting with vx initial 0 in thelaboratory frame. This can help us to calculate the timeneeded for a charge to pass through the origin. For abody moving under a central attractive force F -k·x inthe laboratory frame T 2π m / k . In the case of arotating frame then we should take into account thecentrifugal force mω²r. The constant of the oscillatorymotion is -k effective -k m·ω². This will be the effectiveconstant. Then for the magnetic field -k effective k m·(qB/(2m))² -k ε²/4. With λ -k effective /m andλ0the corresponding value for the laboratory frame, λ0 λ–ε²/4 and so the period is:V.Two Dimensional Motion on aRotating FrameA rotating frame is very interesting from thepoint of didactics of physics. There is a need forintroduction of frames of reference and “inertial” forces.As Galili & Kaplan (1997) pointed out, standardIntroductory Physics Courses do not usually considermore than one observer. He examined the presentationof Energy and Momentum in different frames ofreferences.Another issue that is usually neglected is themass of the base. Galili & Kaplan examined the case ofa ball sliding on base of finite mass and considers theenergy and momentum in the corresponding frame ofreference. 2016Global Journals Inc. (US)291Global Journal of Science Frontier Research ( I ) Volume XVI Issue III Version IFigure 7: Foucault's pendulum
Excel Files for Teaching Two Dimensional Motions and their CurvatureThese issues will be considered here for thecase of a rotating frame of reference. First will beaddressed the rotating frame of reference with constantangular speed ω. In this case the inertia of the base canbe considered infinite. Then there will be a considerationof a base of finite mass.a)Base with constant angular velocityUsually textbooks deal with relative motion inrotating frames of reference in connection with theCentrifugal force and the Coriolis force.Global Journal of Science Frontier Research ( I ) Volume XVI Issue III Version IYear2016b) Absence of frictionThe current paper deals with a particle lying at apoint with coordinates (0, A). The rotating frame has anangular velocityω.One characteristic of the rotating frame is that30 while the angular velocity is the same for every point ofthe rotating frame, the velocity each point is different.c) Considerations of frictionConsidering the effects of friction we obtainsome results, which are not anticipated by the students(or even from physics teachers) if they do not grasp thenotion of relative velocity. For a given location of thesliding body, the relative velocity has to be composedfrom the velocity of the point of the turntable and thevelocity of sliding body.Now the frictional force can be expressed as:Fx µV x , relative Vrelative m.g ,Fx µV y , relative Vrelative m.gThis expresses the fact that the frictional force isdirected opposite to the direction of relative velocity ofthe sliding body to the base.Figure 7: Otbits in different frames of referenceSince the direction of the frictional force isopposite to the direction of the relative velocity, thefriction can act as an accelerating agent in the labframe.The path of a puck sliding on a turntable willbecome a curve. This can be explained by picturing thefriction as dragging the sliding body from its rectilinearmotion to a curved one. In this case the force vector andthe velocity vector in the lab frame will have an acuteangle. In the rotating frame the angle will be 180 .Diagrams of the position of the body, its velocityand acceleration can be generating by using aspreadsheet or a Visual Basic program. In both casesare possible simulations of the motion.The sliding body will get energy from therotating frame. So the energy of the body will increase.d) Energy considerationsThe kinetic energy of the body in the laboratoryframe will be constant in the case of no friction. The 2016Global Journals Inc. (US)kinetic energy will increase in the case of frictional forcesif the angular velocity of the base is constant.A question that may be asked is: ‘In the labframe the friction can act as an accelerating force, whatis its role in the rotating frame?’ In the rotating frame thedirection of the relative velocity is tangential to the path,so the direction of the friction is opposing the directionof velocity. So in the rotating frame the friction willdecelerate the movement as it is anticipated from‘normal’ cases. In the case of the observer on therotating frame then a «centrifugal» potential can beused, to explain the energy increase as due to the actionof the «centrifugal» force on the sliding body. Thisexplanation holds true only if the angular velocity of thebase is constant.On the other hand if the rotating frame is of afinite mass it is expected that the energy
Year2016Excel Files for Teaching Two Dimensional Motions and their CurvatureFigure 8: Energyin a rotating frame which is affected by the frictionof the rotating disk will decrease, due to the interactionof the sliding body (puck) with the rotating body. Thesliding body acts through its frictional force (which is theone part of the action – reaction pair). The moment ofthis force decreases the angular velocity of the base.The decrease of the angular velocity can be calculatedby the use of the conservation of angular momentum.The force vector and the velocity vector in the beginningof the motion will make a 90 angle in the laboratory,later on the angle becomes acute, and later it becomesobtuse. In this case the speed in the laboratory frame isreduced. When the body stops in the rotating frame, theangle becomes 90 in the lab frame.In the case changing angular velocity the workof the friction plus the work of the centrifugal force in therotating frame of reference will not in general be equal tothe change of the kinetic energy. If the angular velocityof the base is affected because of its finite inertia, thenthe complementary acceleration must be taken care of,which gives an additional inertial force. The Coriolisforce of course does not contribute to the change of thekinetic energy. The need to introduce an additionalinertial force can be shown through a computerprogram, which shows to the user the sum of differentterms if it is equal to the change of the kinetic energy inthe rotating frame.The final state of the body for the rotatingobserver will be that the sliding body will be motionless.For the observer in the lab frame the whole system willturn with a reduced angular speed. The body will be willdescribe a circle with the reduced angular speed of thebase.In case of rapid reduction of the angular speed,the graph of velocities in the rotating frame of referenceceases to have the periodic character, which has in thecase where the base is not affected by the sliding body.VI.Applications in ClassThe teaching of two dimensional motions wasdone in the elementary education department ofDemocritus University (Mihas P. and Gemoysakakis T.2007) and there was a special laboratory on friction inwhich except the basic ideas of friction the lesson wasextended for the sliding friction on a rotating disk(Evangelopoulou and Mihas 2011). The results of thiswork were applied with less success to High Schoolstudents (Evangelopoulou A and Mihas P.2012).As was seen by Mihas & Gemoysakakis, thestudents who attended the classes could draw correctlythe vectors of acceleration in different paths, and alsodraw the “circles of curvature”, while for students whodid not attend the classes they did not have anysuccess.The use of the files on friction for a “puck”moving on a rotating disk can be: a) Explanation of thedirection of Friction in a turntable. b) Elucidation of themeaning of relative velocity in the lab frame and in the 2016Global Journals Inc. (US)Global Journal of Science Frontier Research ( I ) Volume XVI Issue III Version I311
Excel Files for Teaching Two Dimensional Motions and their CurvatureYear2016rotating frame of reference. c) The consideration of thedirection of acceleration and the direction of the velocityvectors for the case of increasing, decreasing orconstant speed. d) The application of the Work – Energytheorem in different cases. e) The need to introduce theGlobal Journal of Science Frontier Research ( I ) Volume XVI Issue III Version I32SectionSoftwareSinusoidal CurveTutorial CurveNeile CurveCycloidal pendulumMotion of a charged body in acombination of electric field and magneticfieldFoucault’s PendulumTwo dimensional motion with frictionsinusoidal MAKRO or sinusoidal (without a macro)tutorial curveNeile curvecycloid pendulum with macrosMovement in combination of fields macros or without macrosFOUCAULT PENDULUM MACROSsliding on rotating frame small inertia plux mR2sliding on rotating frame BIG INERTIAAll the files are found at http://kyriakosxolio.gr/w dm curv.htmlReferences Références Referencias1.2.3.4.5.6.7. 2016complementary acceleration in the case of acceleratingframe of reference.A note on the software used in this paperFor each part of this paper there are Excel filesused to draw the figures.E. Kim and S-J Pak (2002) Students do notovercome conceptual difficulties after solving 1000traditional problems Am. J. Phys.70(7) pp. 759-765Dugas Rene A History of Mechanics Editions duGriffon, Neuchatel, Switzerland, 1955 EditionEvangelopoulou A. and Mihas P. (2011) Theinstruction of Galilean relativity and relative rotatingmotion to students of a Department of ElementaryEducation Proceedings of the 6th PanhellenicConference on Science Teaching and ICT inEducation (PP 326-334). Florina, University of WestMacedonia, Greece. The paper is in Greek in theproceedings but there is an English translation inResearch Gate.Evangelopoulou A. and Mihas P. (2012) Perceptionsof students of first grade of Greek High Schoolabout the friction, its laws and its role in the relativetranslational and rotational motion. A Proposal for ateaching intervention in a cooperative constructivestic learning environment. Themes of Sciences andTechnology in Education 5 (1-2), pp. 5-26 (In Greek,An English Translation can be downloaded fromResearchgate).Galili I, Kaplan D (1997) Extending the application , American Journal of Physics, pp 323 335, April 1997.Geoffrey I. Opat.(1990) Coriolis and magneticforces: The gyrocompass and magnetic compass asanalogs, American Journal of Physics 68(12), 11731176.I. A. Halloun and D. Hestenes, "The initialknowledge state of college physics students," Am.J. Phys. 53, 1043 (1985).Global Journals Inc. (US)8.J Sivardiere (1983) On the analogy between inertialand electromagnetic forces, European Journal ofPhysics pp 162-164.9. J.Clement, "Students' preconceptions in introductorymechanics, Am.J.Phys. 50, Jan .(1982).10. Marc Semon, Glenn Schmieg, (1981) Note on theanalogy between Inertial and electromagnetic forces,American Journal of Physics, 49(7) pp 689-690.11. McDermott, L. C., & Shaffer, P. S. (1998). Tutorialsof Introductory Physics. (Greek translation by PavlosMihas (2011). Typothito Publications).12. Mihas P. and Gemoysakakis T, (2007) Difficultiesthat students face with two-dimensional motionPhysics Education 42 (2) 163-169.
It is interesting that the radius of curvature of this curve is zero for y y. max. If the speed is different from zero then the centripetal acceleration will be infiniteThe . radius of curvature can be expressed as: which for Neile's curve is expressed as: ( ) g gy gy. 2 2 2 2 2. 2. 3/ 2. β α α β ρ . We can see that for .
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