Top angle and the maximum speed of falling cones

Transcription

1 Top angle and the maximum speed of falling cones Guiding student investigations J.M. Wooning, A.H. Mooldijk and A.E. van de Valk Cente fo Science and Mathematics Education, Utecht Univesity, Nethelands As a eseach assignment, students investigated the movement of falling pape cones. In thei guiding, teaches could not adequately answe two questions students asked them. One about the point fom what falling distance onwads the speed of the cone is suely at its maximum and one about the elation between the top angle and the maximum speed. To answe these questions, the elation between the top edge and the dag coefficient is needed. In this aticle a fomula is constucted using adapted Newton theoy about the cone. The elation is validated expeimentally using a CBL envionment and elevant coefficients ae detemined. Results ae used fo suggesting teaches how to guide students to make a easonable guess fo the maximum speed distance and to investigate the elation between the top angle and the maximum speed of cones with equal masses. Osbone (1996) has made a plea fo a shift fom pactical wok to eseach and discussion activities. In a ecent cuiculum efom fo uppe seconday education in the Nethelands this shift is pomoted by intoducing a final eseach assignment as a pat of the examination. As pepaing fo it cosses the boundaies of the single subjects (Milla et al. 1994), thee is a need fo coopeation between school depatments. In 1997 ou institute stated the BPS-poject, which aims at guiding the Biology, Chemisty, Physics and Mathematics depatments in schools to coopeate (Boekman and Van de Valk, 1999). The depatments of fou poject schools ageed to conduct a eseach assignment incopoating two subjects. The one on a Physics and Mathematics topic is epoted in this study: Falling Cones. The goal was to have students expeience the eseach pocess fom the stat to the end. The falling cones eseach assignment The falling cone eseach assignment is about the constuction of pape cones and thei movement when they ae eleased. In goups, students exploe the cones and thei motion, oient themselves to elevant vaiables and fomulate a eseach question. They have to plan and do expeiments o mathematics activities in ode to get an answe to that question. The teache stats with the intoduction of the pape cone. Its movement is studied, showing a apidly diminishing acceleation, until its maximum speed is eached. Then pais of cones ae studied, among othes two cones of diffeent sizes, having the same top angle and made of the same pape (see figue 1). To students supise, these two cones fall at equal speed. Students ae challenged to make themselves a couple of cones and study thei movements. Next, woking in goups, they have to fomulate a eseach question, to study some aspect of the cone o of its Figue. Scheme of a cone with vaiables Topangle_00311.doc Page 1 of 7

2 movement in depth, theoetically o expeimentally. Fo this, an investigation design has to be made and implemented. At the end, goups pesent the answe to thei eseach question to each othe on a poste. In figue, a scheme of the cone is dawn and elevant vaiables ae shown. Using Physics and Mathematics theoy, students can undestand how the maximum speed (v max ) of the cone is elated to vaiables Figue 1. Pape cones falling down. such as its mass (m), the adius () of the gound cicle, the ai density (ρ) and the dag coefficient (C): F gav = F dag (1) m g = () v max 1 CAρ v max mg = (3) π Cρ Using these fomulae and the dawings of figue, students can design an expeimental o a theoetical piece of eseach. The teache guides the student goups when they ae tying to fomulate a eseach question, making a eseach design, doing expeiments o mathematical activities. In ode to lean how to conduct a eseach, teaches have to give students many oppotunities to detemine on thei own what to investigate, how to do and why to do it this way. At the same time, teaches have to show students some diection, in ode to get a fuitful investigation. No complete answes should be given, as it would make students own contibutions unnecessay. Instead, the teache can ask fo easons, explanations, can answe questions by asking questions in etun, can offe suggestions etc., in ode to stimulate them to make concete what they want to know and investigate. To povide adequate guidance, teaches have to have a thoough undestanding of the cone issue and how to investigate (Tami 1989). The issue of this study The falling cones investigation assignment was done in fou classes (fom 11). The mateials fo doing the expeiments wee simple: ules, stopwatches, scales etc. No advanced equipment, such as a position senso, was available. Expeiences have shown that students wee challenged to ask questions about the motion, about vaiables being elevant fo the maximum speed, to pobe elationships and to do expeiments. Howeve, students met two content matte poblems that may hinde thei leaning how to conduct a eseach, as the teaches appeaed to have no means fo guiding them to a solution. This study was done to solve this. Fist poblem Expeiments about how the maximum speed of the cone depends on the mass, the shape of the cone, its adius etc. appeaed to be favouite with students. Howeve, tying to measue the maximum speed, a poblem was met: what distance does the cone have to fall fo its speed to become constant? Fom equation (3) it can be seen that one needs to have the value of the dag coefficient C fo calculating that distance. The dag value depends on the shape of the cone, but no fomula is available. So, teaches suggested to make a easonable guess o said: just assume that afte falling1 mete the speed is at its maximum. The students accepted these suggestions, though eluctantly. One goup agued Topangle_00311.doc Page of 7

3 ightly: The stat phase of the fall may be compaable to a cone, that ae found in the liteatue (table 1). Flat cicle 1.11 Open half sphee 0.34 Dop of wate 0.06 Figue 3. Tansciption of the goup s dawing quoted in the text, whee 1 mete is used fo stat phase. dependent on the mass of the cone. In a dawing (see figue 3), they showed what they meant by stat phase. Second poblem One goup studied how the maximum speed depends on the top angle. They found that the bigge the top angle is, the slowe the maximum speed. Fo getting a fomula fo this elation, they studied the theoy and concluded: If you inset all known values into fomula (3), only the dag coefficient and the speed ae left as vaiables. So you can inset v max. Then, the dag coefficient only depends on the shape of the cone, so we ll get thee diffeent values of C. They asked thei teache what fomula to ty. He suggested ( sin v ( γ )) 1 but in fact had no basis max fo it. Both issues can be solved by getting an answe to the question: how does the dag value depend on the top angle? Design of the study The dag coefficient C is dependent on the top angle of a cone. This can be illustated by looking at the C values of some foms Table 1. Dag coefficient of some diffeent foms (Vademecum 1995) The flat cicle equals a cone with a top angle of 180 degees. Cones with shape top angles will have smalle dag coefficients than the flat cicle. The cone C values can be expected to equal the open half sphee at minimum, as the half sphee equals the bottom shape of the ideal wate dop and has an open side, like the cone. So the cone dag coefficients will have values between 0.34 and 1.1. In the theoy pat of the study, the elation between the dag coefficient and the top angle is exploed by using Newton s model of a cone falling though a unifom medium, as it is elaboated by Edwads intenet site (see efeences). This esulted in a fomula fo the dag coefficient, pedicting a sinus squae elation with the top angle. In the expeimental pat, a set of diffeent cones was made, the only vaiable being the top angle. Using a TI position senso and Coach 5 (CMA 000) envionment, the dag coefficient of all cones was detemined and esults wee compaed with the values theoetically pedicted by the Newton/Edwads model. In that way, a elationship was found. The Newton/Edwads model Newton aleady studied the esistance that falling objects expeience when falling though a homogeneous medium. On his website, C.H. Edwads pesents a simplified vesion of the complicated Pincipia Mathematica theoy. Hee, we use the Newton/Edwads model to find a elationship between the cone dag coefficient and the top angle Topangle_00311.doc Page 3 of 7

4 Newton assumed that the medium - ai in ou case - consists of tiny elastic paticles with mass m, unifomly distibuted in space, having no speed. So, Newton (and Figue 4. A paticle colliding elastically with the cone. Edwads) only account fo collisions of paticles on the bottom suface of the cone, which is moving vetically downwad having speed v. If the cone is supposed to be at est, the ai paticles ae moving vetically upwad with speed v. As the diections of the velocities of the actual ai molecules ae unifomly distibuted in space, this model can be applied hee as a fist ode appoximation. It has to be noted that tubulence is not accounted fo. Be N paticles pe unit of volume. All paticles hit the cone suface at angle φ with the suface nomal and bouncing of with the same angle. Fom figue and 4 it can be seen that φ equals π/ - γ/. If a paticle collides elastically with the cone, its momentum p changes by p paticle = mv cos( ϕ ) (4) along the nomal on the outside cone suface. This esults in the opposite change of momentum of the cone: p cone = mv cos( ϕ ) (5) As the hoizontal components of the momentum of the paticles that bounce on the cone ae cancelled out, only the vetical components count. So one collision contibutes: p cone = mv cos ϕ (6) ( ) in the vetically upwad diection. The numbe of paticles that stike a piece da of the cone suface in time t is N da = Nv cos ( ϕ ) da t (7) Using the equality of change of impulse and momentum, ρ being the density of the medium the cone is falling in, with ρ = Nm (8) the vetical foce on the suface piece da fom ai dag is 3 = ρv cos ϕ (9) df dag ( ) da The ai dag on the total cone suface S is given by the suface integal: 3 F = ρ v cos ( ϕ ) da (10) w S This equation can be witten as follows: Fw = ρv R (11) R equalling: R = cos 3 ( ϕ ) da (1) S Accoding to Edwads, you can wite fo the coefficient of esistance: R x = dx (13) ' π 1 + y ( x) 0 with y(x) the function that detemines the shape of the falling body. In the case of the cone, you can wite fo that function y ( x) = ax (14) a being sinϕ a = tan ϕ = (15) cosϕ So y = a. Substitution of the equiement (14) fo the cone into (13) gives: (16) R x x = dx = dx = π 0 1+ y'( x) 0 1+ a 1+ a Inset ing (15) into (16), using ϕ = π γ and A = π one gets: ( π γ ) = sin ( γ ) R = A cos A (17) Topangle_00311.doc Page 4 of 7

5 So we can wite the dag foce: ( γ ) F dag = Aρv sin (18) Assuming that (18) and () ae identical, one finds fo the dag coefficient: ( γ ) = 4 sin ( γ ) C (19) Citical evaluation of this esult: it pedicts that C = 4 when the top angle is 180 and it tends to 0 when the top angle tends to 0. So theoy based on the Newton/Edwads model pedicts 0 < C < 4. Howeve, using expeimental data, it is expected 0.34 < C < 1.1. This shows that the model is not adequate. It only accounts fo dag at the bottom end of the falling cone. At the uppe side tubulence effects ae to be expected, that ae not dependent on the top angle (if the uppe side aea is not too small, conf. the open half sphee). This suggests that the dag coefficient depends on the squae sinus of the half top angle: position of the falling cones at diffeent moments of time. Using an inteface (Coachlab II, CMA 000), the senso signal was sent to the compute and pocessed by the pogam Coach 5 (CMA 000) to poduce distance-time and speedtime gaphs. Fom the gaphs, maximum speed of evey cone was calculated and the aveage of thee measuements was used in the calculations. Fomula (3) was used to calculate the dag coefficients. Results wee plotted in a C, sin (γ/) diagam. Measuement difficulties Befoe becoming a definitive seies of data, seveal poblems had to be ovecome. Fist, it was expeienced that the cone tended to deviate towads the wall when eleased less than 1 mete fom the wall of the oom. This is due to Benouilli effects. Second, the height chosen ( m) appeaed to be too little fo the shape cones to leave a sufficient distance fo measuing ( γ ) = a + b sin ( γ ) C (0) a and b being coefficients. The expeiments To study the elation between the dag coefficient and the top angle, a seies of cones wee constucted. All cones had the same gound cicle aea (A = 150 ± 7 cm ) but diffeent top angles (between 60 o and 10 o ). To keep A constant, the adius of the gound cicle has to be kept constant. Theefoe, an incease of top angle γ means a decease of the length l of the cone. This makes the amount of pape used fo the cones, and thus the mass, vay. The mass of each cone was made g by putting some sand in. A position senso (CBR of Texas Instuments) was used to measue the Figue 5. A diagam fom Coach5 the maximum speed. Due to the height of the laboatoy oom (4,5 m), the falling height was limited to 4 m. Theefoe, no eliable data could be gatheed fo cones with a top angle less than 60 o. Thid, poblems appeaed because of flutteing, in paticula fo the lage top angles. By caeful elease and epetition, fom a lage seies of measuements of one cone, thee movements without flutteing could be selected. This was not possible fo cones with γ > 10 o Topangle_00311.doc Page 5 of 7

6 Results In figue 5, a typical diagam poduced by Coach5 is shown. It illustates that, some time afte elease, the (x, t) gaph shows a staight line, the (v, t) gaph eaching a maximum. The dag coefficient and sin (γ/) ae plotted in figue 7. This diagam confims fomula (0), the coefficients being a = b = Discussion The linea elation between the dag value and sin (γ/) is confimed within the ealm of 60 o < γ < 10 o. The ange of the dag coefficient is: < C < This is within the ange expected (0.34 < C<1.1) detemined by the dag coefficients of a flat cicle and a half open sphee. The Newton/Edwads model does not account fo two phenomena: - paticles not only bouncing on the downwad, outside of the cone, but also on the inne, upwad side - tubulence at the im and the upwad side of the cone. It is likely that these two phenomena ae the impotant factos in explaining the diffeences between the Newton/Edwads model and expeimental findings. Consequences fo guiding students Having found the fomula (0) and the values of its coefficients, the issue of ou eseach has been solved. Now, anothe question aises: how can a teache guide a goup of students that ask one of the two questions that wee at the stat of ou study? The fist poblem descibed in section 3 was: what distance does the cone have to fall fo its speed to become constant? Fist, an appoximation is constucted and checked and then it is discussed how to guide the students. In a fist easonable guess pio knowledge is used: the distance a fee falling body has to fall to each the maximum speed is given by fomula (1): vmax s = (1) g This guess is checked by inseting in (1) the maximum speed fom the figue 5 gaphs, epoduced in figue 7. The esult: Figue 6. Diagam with the dag coefficient as a function of the squae of the sine of the top angle s = 0,3 m, poducing the lowe hoizontal line in figue 7. Using the cossing point of this line with the (x, t) gaph shows that the cone is not yet at its maximum speed as it is not on the linea pat of the gaph. The second guess is suggested by figue 7: double the distance! The uppe line at double distance does coss the linea pat. The same is the case fo the othe cones we used. Ou conclusion: the cone will suely be at its maximum speed having passed distance s max : v s = max max g () When students meet poblem 1, they cannot use fomula (), because they don t know the maximum speed. As a solution, the teache can suggest to guess fom what point onwads the speed is constant. With it, the appoximated maximum speed can be measued. Using Topangle_00311.doc Page 6 of 7

7 that speed, the distance s max can be calculated with fomula (). Only if this distance is about the same o moe than the distance of the guess, the measuement has to be epeated. This pocedue takes less Figue 7. Estimate of distance of falling befoe eaching maximum speed time and is easie to students than the altenative: calculating v max theoetically, using the fomulae (3) and (0)! The second question we stated with was: how does the maximum speed depend on the top angle? Fom a guidance point of view, it is bette to tansfom it into: what gaph is the best to make, to pocess the data gatheed when studying how the maximum speed depends on the top angle? The teache can suggest the following pocedue to the students: - calculate the dag coefficient of all cones fom expeimental data, using fomula (3) - ty a gaph of the dag coefficient and sin(γ/) o sin (γ/). When tying to measue the maximum speed, they will meet poblem 1. As that can be solved now without using fomula 0, they can find the fomula themselves! make a easonable guess using what you aleady know; check the esult and if needed, adapt you guess. This method is an impotant one in the light of the main goal of the investigation assignments: leaning about doing investigations. Refeences Vademecum van de natuukunde (1995), blz. 14, Utecht: Het Spectum, Andeeck B S (1999) measuement of ai esistance on an aitack Ameican Jounal of Physics Boekman H and Valk van de A (1999) Autonomous leaning in the Uppe seconday science and mathematics cuiculum: How to guide the Teaches? Poceedings of the 4 th annual ATEE confeence Leipzig CMA (000) Coach5 and Coachlab II at tml Edwads, C.H. (1997), Newtons s Nose-Cone Poblem; The Mathematica Jounal 7(1), Milla R, Lubben F, Gott R and Duggan S (1994) Investigating in the school science laboatoy: conceptual and pocedual knowledge and thei influence on pefomance. Reseach Papes in Education, 9 () Osbone J (1996) Untying the Godian Knot: diminishing the ole of pactical wok. Phys. Ed. 31 (5), Polya G (1954) Mathematics and plausible easoning Pinceton Pinceton UP Tami P 1989 Taining teaches to teach effectively in the laboatoy Science Education, 73 (1) TI fo infomation about the used motion senso s/featues.html In this study, the elation between the dag coefficient and the top angle of a cone is found: fomula (0). Ways of using this fomula (without giving it to students!) to solve poblems students meet, ae suggested. One impotant, moe geneal aspect of ou suggestions has to be stessed: the Polya method we descibed: Topangle_00311.doc Page 7 of 7