Oscillations due to time-delayed driving of a ball in a water jet -- a challenging problem of the International Physicists' Tournament 2019
OOscillations due to time-delayed driving of a ball ina water jet - a challenging problem of theInternational Physicists’ Tournament 2019
S Michalke , A F¨osel , M Schmiedeberg Department Physik, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Staudtstr.7, 91058 Erlangen, Germany Didaktik der Physik, Department Physik, Friedrich-Alexander-Universit¨atErlangen-N¨urnberg, Staudtstr. 7, 91058 Erlangen, Germany Institut f¨ur Theoretische Physik 1, Friedrich-Alexander-Universit¨atErlangen-N¨urnberg, Staudtstr. 7, 91058 Erlangen, GermanyE-mail: [email protected] april 2020
Abstract.
The
International Physicists’ Tournament (IPT) 2019 dealt with 17challenging problems. In this article, we present experimental as well as theoreticalapproaches exemplarily to one of these tasks. A ball placed on a hard and flat surfacecan show oscillation movement when being hit by a jet of water from above. To explainthe oscillations, a theoretical model is introduced and its predictions are comparedto the experimental measurements. Furthermore, the structure of the IPT itself ischaracterized shortly in this article, as well as the idea of a new and innovative seminarwhich was set up at Erlangen University to prepare and assist students in taking partin International Physicists’ Tournament. Furthermore, the educational relevance ofphysics tournaments like IPT for physics education at university is discussed in detail.
Submitted to:
Eur. J. Phys.
Keywords: physics competitions, International Physicists’ Tournament, non-linearoscillation, time-delayed force, water jet a r X i v : . [ phy s i c s . e d - ph ] A p r scillations of a ball in a water jet
1. Introduction
The
International Physicists’ Tournament (IPT) is a competition where physicsstudents work on experiments, perform simulations or develop theoretical approachesin order to solve open problems. During the tournament the students have to presenttheir solutions and discuss the approaches of other participants. The characteristicsof a (physics) tournament, especially in comparison to a non-tournament (physics)competition, are described in ”International Physicists’ Tournament - the teamcompetition in physics for university students” by Vladimir Vanovskiy, member of theIPT executive committee [1].The German Physicists’ Tournament (GPT) is a corresponding competition inGermany with problems from the upcoming IPT. The winning team of the GPT obtainsthe possibility to qualify for the IPT.In order to support the students who participate in the competitions a new typeof a seminar was developed at the Friedrich-Alexander-University (FAU) Erlangen-N¨urnberg, where students can both work on IPT problems and obtain credits for theirstudies.In this work, we will exemplarily present the experimental as well as theoreticalapproach to one of the problems of the IPT, namely problem number 5 from the 11thIPT that reads [2] ’When a ball lying on a hard and flat surface is hit by a jet of water that fallsperpendicular to the surface, it may start to oscillate. Investigate how the oscillationsdepend on the relevant parameters.’
Figure 1.
A ping pong ball lying on a flat surface being hit by a water jet.
The situation is shown in Fig. 1. The ball, in this case a ping-pong ball, lies on astone slab and is hit by a jet of water. The water jet pulls the ball back into the middle scillations of a ball in a water jet
2. International Physicists’ Tournament, German Physicists’ Tournamentand preparatory seminar at Erlangen University
Since 2009 every year students can compete in the IPT. Within the months of june toaugust in the year before the international tournament, a set of 17 open problems ispublished that teams of up to six students can try to solve with experiments, simulationsor theoretical calculations. The teams are welcome to puzzle over all of the 17 problems.According to the official rules of the international competition IPT [3] a competing teamcan challenge some of the problems and each team has the possibility to reject someof the challenges. However, if a team rejects to many challenges it might be penalized.Therefore, it is recommended to prepare a total of about 13 problems for the IPT.The problems usually deal with physical phenomena or questions that cannot beanswered in a unique way but often allow for many possible ways to be handled. Forexample, the problems of IPT 2019 [2] asked to build a radio with a potato (problem № № №
16) or to study the oscillations of a ball in a water jet (problem № international tournament a team has to be successful in an online qualification process. scillations of a ball in a water jet GermanPhysicists’ Tournament (GPT). It usually takes place in november or december in theyear before the IPT. In december 2018 three teams competed in Frankfurt in order toqualify for the IPT 2019. The GPT 2019, which aimed to be the preselection for findingthe team representing Germany at the IPT 2020, took place at the physics department ofFAU Erlangen-N¨urnberg in december 2019 (for further information, see current websiteof German Physicists’ Tournament [4]). The GPT is based on the same rule set as theIPT with the sole modification that each team has to name the problems it has workedon (at least four per team) and that only these problems can be challenged.While the IPT fosters the autonomy of the students and trains them to deal withcomplex challenges, the work on the problems usually also requires a lot of time whichmight be hard for students that already have to deal with the large workload of theirphysics studies. As a consequence, there are approaches to integrate the work on IPTproblems as elective modules into the curriculum of physics education at the university.At the FAU Erlangen-N¨urnberg we created a new type of seminar, called ’Problems ofthe International Physicists’ Tournament’ at two credit hours and 5 European CreditTransfer System (ECTS) points, corresponding to a workload of about 140 hours.That seminar first took place in the winter term of 2018/2019 (mid of october 2018until mid of february 2019), starting thus significantly after the publication of the IPT2019-problems. Six students that later formed a team at the tournament participated inthe seminar. To each student a supervisor was assigned, e.g. a PhD student interested inthe tournament. The task of the supervisors was to help the students with the theoreticalbackground of the problems, e.g. by finding literature, explaining typical approaches intheoretical modelling, as well as by helping with code development if simulations wereemployed. Furthermore, the supervisors enabled contacts to other researchers at theuniversity that are closer to the field of the problem. Finally, contacts to the practicaltraining courses were supported. The supervisors worked for the seminar as part of theirmandatory teaching load. In addition, they benefited from obtaining insides into howphysical questions can be attacked in various fields of physics, including questions thatare outside of their own field of research.The seminar consisted of two elements: First, the participating students presentedtheir approaches as well as the physical background concerning one IPT problem oftheir choice in talks of 45 minutes. Approaches to a task included various ideas how aproblem can be solved. Not all of these ideas had to be worked out in detail. Usually oneor two those ideas were followed in more detail, e.g. by presenting results of preliminaryexperiments or simulations. However, not a worked-through, single solution in the styleof a presentation in a physics fight had to be presented, but the students had to showthat a problem can be approached from various directions. The talks were followed byan extensive discussion with all members of the audience that not only consisted of the scillations of a ball in a water jet scillations of a ball in a water jet
3. Educational relevance of physics competitions for physics education
Though there don’t exist any studies at all investigating the effectiveness of usingphysics competitions for physics education at university , we definitively think that thereare quite a few arguments for including physics competitions in physics education atuniversity. This is especially true for those so-called physics tournaments which aredealing with creative, innovative and challenging open tasks and which are asking forqualifications and competencies far beyond just only specialised knowledge in physics.Subsequently, we will present and illustrate some of the arguments.Including physics competitions in science education allows for taking into accountthe methodical and didactic ideas of active learning (see e.g. [5], [6]). The followingtext passage may illustrate the main ideas: In 1988, the American theoretical physicist
Richard Feynman gave an account of his interactions with his father in the essay ”How tobecome a Scientist?” published in his book ”What do You care what other people think?Further adventures of a curios character.”[7] Feynman’s father taught him to carefullyobserve phenomena , and he triggered him to ask questions arising from an observation(e.g. ”Why do you think birds pick at their feathers?”). Feynman’s father, he never gavethe answer immediately, but rather he would get Feynman to think of an explanation,to devise an experiment or to provide an explanation - to understand the meaning of thephenomena . So the point was not in giving the answer, but rather in getting Feynman tolearn ”the difference between knowing the name of something and knowing something”([7], p. 14) by being active in learning. Thus active learning in the context of physicstournaments like IPT with all the open problems mostly arising from observations fromdaily live means that students make the observations by themselves, e.g. they carefullylook at the oscillations of the driven ball. They have to design experiments in a self-contained way as well as they have to provide appropriate theories, and they have tofind possible solutions. In the contest itself, they present and defend their own ideasand solutions.(Science) Competitions provide students with an opportunity to evaluate theirperformance with others’ . Student competitions can thus play a crucial role in identifyingtalented students. This is quite a good possibility for students at university to get toknow their own subject-specific competencies (being talented in doing experiments orin making theoretical approaches) as well as to find out something about competenciesconcerning soft-skills (e.g. when getting an award for ”best presenter” or ”bestopponent”). And this is not only a chance for students, but for researchers at university,too: Via competitions they get to know talented and/ or highly motivated studentswhich they can encourage to take part in research studies at their department.The empiricist
Campbell stated, that participation in science competitions helpsstudents become aware of their potential and contributes to their self-confidence (see [8],[9] and [10]). This is a statement about students in schools; it might be true as wellfor students in university, though empirical studies are still missing and without studiesthere is no evidence that one can infer from one population (students in schools) to scillations of a ball in a water jet increase the interestand the motivation of students in schools (see [11], page 7-15, [12] and [13]), at least ofthose students who have been successful.Integrating IPT in physics education pushes the handling with scientificmethods, and scientific methods like ”observing”, ”measuring”, ”testing hypotheses”,”modelling”, ”experimenting” and ”interpreting” help with learning concepts andprinciples of science ( learning OF science ). Getting to know scientific methods helpsunderstanding the nature of science ( learning ABOUT science ) and scientific methodsare a way towards ”goals” not specific for just one subject, e.g. learning how to solveproblems or acquiring critical faculties (
DOING science ). While solving the openproblems, students apply different scientific methods and therefore learn OF science,they also learn ABOUT science and - most important - they DO science. A widerange of literature about the subjects ”scientific methods and science education” and”nature of science” have been published.
Wynne Harlen , for example, discusses scientificmethods in the context of teaching, learning and assessing science ([14]).
McComas ([15]),
H¨ottecke ([16]) and
Neumann ([17]) introduce in the nature of science.Presenting, discussing and defending the problems in the contest itself finally fosters(self-)criticism as well as competencies concerning communication and decision-making.
4. Qualitative description of the phenomena
We observe three qualitative different behaviors of the ball that is hit by the jet of waterfrom above: First, damped oscillations can be seen in case of a weak water jet. Theresting position of the oscillations is the position where the ball is hit from the top ina symmetric way, i.e., where the jet impacts above the center point of the ball. If theball is displaced from the resting position there is a restoring force acting towards theresting position. Due to the damping the oscillations are damped and for weak waterjets there is no driving that is sufficient to uphold the oscillations. Second, in case of anincreased water flow, the oscillations persist due to a driving mechanism that we willdiscuss in this article. Third, if the water jet becomes too strong it is deflected fromthe ball and might splash in different directions. Thus the motion of the ball becomeschaotic and as a consequence hard to predict.Since the task of the competition concerns the oscillations, we especially studiedthe first and second case, where either damped or persisting oscillations are observed.The water from the jet flows along the surface of the ball until it hits the surface wherethe ball is placed on.In order to study how the water flows along the surface of the ball, we consider ahanging ball that is targeted by a vertical water jet in a slightly asymmetric way, i.e.,the water jet does not hit the ball directly on top but a short distance besides (see Fig.2(a) for a photograph and Fig. 2(b) for a sketch). The water then flows along all sidesof the ball and forms a new jet at the bottom of the ball. Interesting, the outgoing water scillations of a ball in a water jet
Water jet α in α out Figure 2. (a) The water of a jet flows along the surface of a hanging ball. If the waterhits the ball off center, a new water jet forms at the bottom of the ball and leavesthe ball in a direction opposite to the side where ball was hit by the initial jet. (b)Geometry of the incoming and outgoing water jet. The angle of the incoming jet istermed α in and of the outgoing one α out . Note that in the stationary case we observe α in = α out . However, in case of a moving ball, the α out is time-delayed with respectto α in , i.e., α out ( t ) = α in ( t − ∆ t ) with a delay time ∆ t . jet does not leave the ball in vertical direction but is clearly bent into the direction thatis opposite to the side where the incoming jet has hit the ball.In this article we will explain that the momentum carried by the outgoing water jetis essential for understanding the driving mechanism. Since the water jet is deflected atthe ball, there is a momentum transfer onto the ball which leads to the restoring forcetowards the resting position of the oscillations. We argue that the driving is due to atime delay of the restoring force as we will explain in Sec. 5. Note that we assumethat a similar momentum is transferred into the water layer if the ball is placed onto asurface.
5. Theory of the Oscillator
The motion of a ball that is displaced in an arbitrary direction in parallel to the surfaceis described by the equation m ¨ (cid:126)r ( t ) = (cid:126)F (cid:16) (cid:126)r ( t ) , ˙ (cid:126)r ( t ) (cid:17) − γ ˙ (cid:126)r ( t ) , (1)where (cid:126)r ( t ) is the displacement of the center of the ball from its resting position, m scillations of a ball in a water jet (cid:126)F (cid:16) (cid:126)r ( t ) , ˙ (cid:126)r ( t ) (cid:17) is the restoring force that in the harmonic approximation wouldbe (cid:126)F (cid:16) (cid:126)r ( t ) , ˙ (cid:126)r ( t ) (cid:17) = − k(cid:126)r ( t ), and γ ˙ (cid:126)r ( t ) with a friction coefficient γ is the damping force.Note that we neglect contributions due to the rotation of the ball because we do notobserve any rapid rotations. Slow rotations as for a ball that is not just displaced out ofthe resting position but that is rolled out by a small angle do not change the qualitativeform of the equation of motion in case of small displacements from the resting position.The restoring force mainly originates from the deflection of the water jet that incase of a ball that is fixed with a certain displacement (cid:126)r but with no velocity (i.e., ˙ (cid:126)r = (cid:126) (cid:126)F that depends on the absolute value of the displacement, i.e., (cid:126)F ( (cid:126)r ) = (cid:126)F (cid:16) (cid:126)r, ˙ (cid:126)r = (cid:126) (cid:17) = − f ( | (cid:126)r | ) (cid:126)r | (cid:126)r | . (2)Again, in a harmonic approximation f ( | (cid:126)r | ) = k | (cid:126)r | , which we will later assume forsimplicity in our more detailed analysis. Water jet
L r
Figure 3.
The outgoing jet that leaves the ball at the bottom is delayed with respectto the incoming jet. The delay can be estimated by using the delay length L ( (cid:126)r ( t )) andthe typical velocity v w of the water flowing around the ball. During the oscillation, the ball is obviously never at rest for any constantdisplacement (cid:126)r . Therefore, we have to think about how the force (cid:126)F (cid:16) (cid:126)r ( t ) , ˙ (cid:126)r ( t ) (cid:17) dependson the velocity ˙ (cid:126)r ( t ). In order to motivate such a velocity-dependent force, we takeanother look on Fig. 2(b). The restoring force is due to the deflection of the waterjet. To be specific, the momentum that is exerted on the ball in horizontal directionis opposite to the momentum that is carried away by the outgoing jet. Note however,that the outgoing water jet is delayed with respect to the incoming jet. The delay canbe roughly estimated by a delay length L ( (cid:126)r ( t )) as sketched in Fig. 3 and the typical scillations of a ball in a water jet v w of the water flowing around the ball, i.e., the time delay is∆ t = L ( (cid:126)r ( t )) v w . (3)As a consequence, the time-delayed force approximately is (cid:126)F (cid:16) (cid:126)r ( t ) , ˙ (cid:126)r ( t ) (cid:17) ≈ (cid:126)F (cid:16) (cid:126)r ( t ) − ∆ t ˙ (cid:126)r ( t ) (cid:17) = (cid:126)F (cid:18) (cid:126)r ( t ) − L ( (cid:126)r ( t )) v w ˙ (cid:126)r ( t ) (cid:19) . (4)For this approximation, we assumed that the time delay is small such that the velocity˙ (cid:126)r ( t ) can be considered approximately constant during the time delay. As we will showin the following, this time delay is the origin of the driving of the oscillations.In the harmonic approximation the equation of motion with a time-delayedharmonic restoring force is m ¨ (cid:126)r ( t ) = − k (cid:18) (cid:126)r ( t ) − Lv w ˙ (cid:126)r ( t ) (cid:19) − γ ˙ (cid:126)r ( t ) = − k(cid:126)r ( t ) + (cid:18) k L ( (cid:126)r ( t )) v w − γ (cid:19) ˙ (cid:126)r ( t )= − k(cid:126)r ( t ) − γ eff ( (cid:126)r ( t )) ˙ (cid:126)r ( t ) . (5)In the last step we introduced an effective friction constant γ eff ( (cid:126)r ( t )) = γ − k L ( (cid:126)r ( t )) v w .Therefore, the equation of motion corresponds to an oscillator with a harmonic restoringforce − k ( (cid:126)r ( t )) and a damping contribution − γ eff ( (cid:126)r ( t )) ˙ (cid:126)r ( t ). The effective frictionconstant γ eff ( (cid:126)r ( t )) can depend on the position, such that the equation becomesnonlinear. Furthermore, the γ eff ( (cid:126)r ( t )) might be positive indicating that there is anoverall driving instead of pure dissipation.Finally, the delay length L ( (cid:126)r ( t )) is given by (see Fig. 3): L ( (cid:126)r ( t )) = R [ π − arcsin ( | (cid:126)r ( t ) | /R )] ≈ πR − | (cid:126)r ( t ) | , (6)where R is the radius of the ball. Therefore, the equation of motion is m ¨ (cid:126)r ( t ) = − k(cid:126)r ( t ) − γ eff ( (cid:126)r ( t )) ˙ (cid:126)r ( t ) (7)with γ eff ( (cid:126)r ( t )) = γ − k πR − arcsin ( | (cid:126)r ( t ) | /R ) Rv w (8)or as an approximation γ eff ( (cid:126)r ( t )) ≈ γ − k πR − | (cid:126)r ( t ) | v w . (9) Equations (7) and (8) can be used to simulate the trajectory numerically. This has beendone using Runge Kutta Method of 4th order. In Fig. 4 the trajectory of a dampedand unbound oscillation is shown in black color. The derivatives are indicated by greenarrows. scillations of a ball in a water jet -4-3-2-1 0 1 2 3 4 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 x ' [ r / s ] x [r] -6-4-2 0 2 4 6-0.6 -0.4 -0.2 0 0.2 0.4 0.6 x ' [ r / s ] x [r] -4-3-2-1 0 1 2 3 4 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 x ' [ r / s ] x [r] (a) (b) (c) Figure 4. (a) Trajectory of a damped oscillation in phase space. The trajectory startsat (cid:126)r (0) = ( x ,
0) with x = 0 . R and zero velocity. (b) Trajectoriy converging againstan limiting cycle. The trajectories are started with x = 0 . R . (c) Trajectory ofan oscillator where the driving is stronger than the damping. The trajectory startswith x = 0 . R and zero velocity. The parameters of the simulation are kmγ = 19 . Rγ/ ( v w m ) = 10 . · − , (b) Rγ/ ( v w m ) = 8 . · − and (c) Rγ/ ( v w m ) = 7 . · − . In the following, we estimate the amplitude of oscillations in case of a steady state wherethe work added by the driving is equal to the dissipated work.In order to determine the work we integrate over half a period W = (cid:90) π/ω m ¨ (cid:126)r ( t ) ˙ (cid:126)r ( t ) dt (10)using an harmonic oscillation as a rough estimate: (cid:126)r ( t ) = (cid:126)A cos( ωt ) , (11)˙ (cid:126)r ( t ) = − (cid:126)Aω sin( ωt ) , (12)where (cid:126)A is the steady state amplitude and ω the circular frequency. Using Eqs. (7) and(9) one finds W = (cid:90) π/ω (cid:20) − k(cid:126)r ( t ) − (cid:18) γ − k πR − | (cid:126)r ( t ) | v w (cid:19) ˙ (cid:126)r ( t ) (cid:21) ˙ (cid:126)r ( t ) dt = A mω (cid:20) π (cid:18) kπRv w − γ (cid:19) + 23 A kv w (cid:21) . (13)A self-sustained oscillation is achieved if the energy does not change, i.e., if thework integrated over half a period vanishes. For W = 0 in Eq. (13) one obtains anestimate for the amplitude of a self-sustained oscillation A = 3 π (cid:16) πR − γv w k (cid:17) . (14)Note that the estimate given in this section is only valid for sufficiently smallamplitudes due to the approximation used in Eqs. (9), (11), and (12). scillations of a ball in a water jet
6. Experimental verification
In order to verify the calculations experiments were performed. First we investigate thefrequency dependence on the properties of the water jet.
The water jet is characterized by the height h w , the volume flow per time unit ∆ V ∆ t andthe nozzle diameter d N . Using the mass flow and the nozzle diameter the exit velocityof the water jet v exit = ∆ V ∆ t d N π can be computed. The height between exit nozzle andthe ball can then be used to derive the velocity from the water jet when hitting the ball v w : v w = (cid:113) v exit + 2 g h w = (cid:115)(cid:18) ∆ V ∆ t d N π (cid:19) + 2 g h w . (15) g denotes the gravitational constant. Multiplying the final water velocity v w with thevolume flow and the density of the liquid ρ results in the force of the water jet F w orequivalently the momentum per time: F w = v w ∆ V ∆ t ρ . (16)Using the assumption made in Fig. 2 we expect the jet to bend in a 90 ◦ angle if the ballis at the position | r | = R . This results in the full momentum transfer. The constant ofproportionality k in the harmonic force approximation of the restoring force therefore is k = F w R . (17)
The setup consists out of a flat slab with a ping pong ball on top being hit by a waterjet. The movement is tracked using a camera looking from the side onto the jet. Theposition is calibrated using the size of the ball which is 40 mm. The view of the camerais shown in Fig. 5.In order to excite oscillations of the ball a ruler is pushed onto the side and thenremoved quickly enough to allow the ball to oscillate freely. The motion of the ballthen either is damped until it comes to rest or in case of sufficient driving it continuesoscillating with a specific amplitude.
The video is analyzed using “OpenCV v2” [18]. The following function was fitted tothe measurement data: y ( x ) = A ( x ) sin(2 πf ( x ) · ( x + x off )) + y off , (18) scillations of a ball in a water jet Figure 5.
The view of the camera.
Where the amplitude and frequency approach a constant value according to anexponential ansatz A ( t ) = (A − A ∞ ) · e − γ f t + A ∞ (19)and f ( t ) = ( f − f ∞ ) · e − γ f t + f ∞ . (20)The result of fits is shown in Fig. 6. Note that in principle the plane of oscillation mightslightly change which would require an additional correction for long runs. P o s i t i o n [ r ] Time [s] MeasurementFit
Figure 6.
Tracked positions during an oscillation (violett) and a fit (green) accordingto Eqs. (18), (19), and (20).
The asymptotic amplitude A ∞ that is approached for t → ∞ corresponds to thesteady state amplitude calculated in Eq. (14) and depends on the product of γ and v W .Therefore, for a measured A ∞ the product of γ and v W can be determined: γv W = k (cid:18) πR − A ∞ π (cid:19) . (21) scillations of a ball in a water jet γ and v W individually, we compare our experimental resultsto the results of the numerical simulation that was used to generate the plots in Fig. 4.Starting with the phenomenological fit to the results of the experiment (see Fig.6) we extract the decay of the envelope function. Then, we performed numericalsimulations with different parameters, determined the maximum and minimum of thesimulation results and compare their position to the envelope function of the fit to theexperiment. To be specific, we vary v w and γ in the numerical simulations until thesum of the differences between the simulated extrema and the experimental envelopefunction are minimal. In Fig. 7(a) the comparison of the optimized simulations and theexperimental results are shown. -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 x [ / r ] t [s] fi tamplitude from eye-balled fi ttrack A m p li t u d e [ / r ] t [s]numerical amplitude fi tamplitude from eye-balled fi t (a) (b) Figure 7. (a) Comparison of the results from numerical simulations (stars) and thephenomenological fit curve to experimental data (magenta line). The parameters ofthe numerical simulation are varied until the deviations between the numerical extremaand the envelop function are minimal. (b) Example of the amplitudes obtained from afit to simulations (crosses) and the amplitudes according to the fit to the experimentalresults (line). The results are similar, but from (a) and (b) it is clearly visible thatthe simulation does not implement slipping and therefore does not fit perfectly. Mostimportant, the frequency change in the experiment in (a) is not reproduced by thenumerical simulations without slip.
In table 8 the parameters of simulations are shown that best fit the experiments. m e [g] F w [mN] γ [1/ s ] v w [ r/s ] γ · m e (cid:104) mN( r/t ) (cid:105) ± ± ±
10 10.8 ± ± ± ± ±
20 10.6 ± ± ± ± ±
13 6.6 ± ± ± ± ±
18 21.0 ± ± Figure 8.
The resulting values of simulations that best fit the results fromexperiments. m e denotes the mass of the ball, F w the water force, γ , v w the watervelocity on the surface of the ball, and γ · m e the water drag force. Using those values it is possible to do some plausibility checks of the theory. The scillations of a ball in a water jet γ · m e and the water velocity v w shouldonly depend on the geometry of the setup, the water velocity and the mass flow rate.This is confirmed when looking at the first two rows in table 8. The water force wasvaried using higher or lower flow with the same nozzle and therefore a faster or slowerwater jet at the exit of the nozzle. This can also be confirmed with looking at the watervelocity on the surface of the ball v w . Interestingly, the drag force decreases for largerwater force in the last row. This can be explained by the ball starting to lift up and slipwhich results in having less friction. The k value from equation 17 can be used to determine the frequency from our theory: km e = F w m e R = ω ⇒ f ∞ = 12 π (cid:114) F w m e R , (22)where m e effectively denotes all inertia effects, including inertia contributions due torotation. For our ping pong ball we assume that the mass is in a thin layer at Radius R .The inertia constant m e therefore is m e = m + I R = m . Five different water velocitieshave been used. The theory and the experiment are compared in Fig. 9. Note there areno free parameters used in this comparison. F r e q u e n c y [ H z ] Water Force [N]Frequencytheorytheory for m e = m Figure 9.
The asymptotic frequency f ∞ that is approached for long times dependingon water force F w . The measurement results are depicted in magenta. The theoreticalpredictions without slip are depicted in green. The blue dashed line represents thetheory for a slipping ball using the ball mass as the effective mass. For small water forces F w the experimentally determined frequency (depicted violettin Fig. 9) agrees with the theoretical prediction (green). However, for large forcesslipping occurs, such that m e = m no longer is a suitable assumption for the inertialcontribution. For a slipping ball the inertia is mainly due to the mass, because therotation does not occur, i.e., m e = m . Therefore, in Fig. 9 the prediction for m e = m is scillations of a ball in a water jet
7. Conclusions and Outlook
We studied the oscillations that occur if a ball on a flat surface is hit from aboveby a water jet. Obviously it is too complicated to try to describe all details of theoccurring forces: Such a detailed description would require a hydrodynamic explorationof the water film that surrounds the sphere as well as various considerations concerningfrictional and viscous forces and of course is far beyond what students (and even mostexperts in the field) can calculate. The major step towards solving the task is to find outwhat matters for the oscillations that should be studied. In this article we have shownthat the restoring force can be understood as momentum transfer on the ball due todeflection of the water jet on the ball. However, an instantaneous restoring force couldnot explain any steady state with oscillations. By taking the time delay of the restoringforce into account, we have shown that the time delay causes an effective driving forceleading to the observed self-sustained oscillations. Furthermore, we have compared ourtheoretical predictions with results from experiments.The way how this problem has been handled is typical for problems of the IPT. Animportant step is to find the correct level of abstraction such that important ingredients(e.g. the time delay) are preserved while complicated details (e.g. of the hydrodynamicforces) are not considered. Students that learn to attack the problems of the IPT in sucha way therefore are taught a lot about how they can handle complex research questionsthat they probably will deal with at later stages of their career.In future, the GPT will be supported by a network with locations at various Germanuniversities (see [4] for details). The network was founded in december 2019 and ismotivated by the successful network of the
German Young Physicists’ Tournament (GYPT) which is a similar competition for high school students [19] and that is used asnational qualification tournament for the
International Young Physicists’ Tournament (IYPT) [20]. The goal is to learn from the success story of the GYPT and the IYPT inorder to increase the visibility of the GPT and the IPT such that more students havethe possibility to participate in these competitions.
Acknowledgments