An experiment to address conceptual difficulties in slipping and rolling problems
AAn experiment to address conceptual difficulties inslipping and rolling problems
Alvaro Su´arez , Daniel Baccino , Arturo C. Mart´ı Departamento de F´ısica, Consejo de Formaci´on en Educaci´on, Uruguay Facultad de Ciencias, Universidad de la Rep´ublica, UruguayE-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
Abstract.
A bicycle wheel that was initially spinning freely was placed in contactwith a rough surface and a digital film was made of its motion. Using Tracker softwarefor video analysis, we obtained the velocity vectors for several points on the wheel, inthe frame of reference of the laboratory as well as in a relative frame of reference havingas its origin the wheel‘s center of mass. The velocity of the wheel‘s point of contactwith the floor was also determined obtaining then a complete picture of the kinematicstate of the wheel in both frames of reference. An empirical approach of this sort toproblems in mechanics can contribute to overcoming the considerable difficulties theyentail. a r X i v : . [ phy s i c s . e d - ph ] J u l n experiment to address conceptual difficulties in slipping and rolling problems Introduction.
Problems involving rolling without slipping are frequently includedin mechanics courses at upper secondary and introductory university levels. Althoughthe problems appear to be simple, they pose a number of difficulties for students becausethey require simultaneous application of static and dynamic concepts. Studies on rollingwithout slipping in the field of
Physics Education Research confirm these difficulties.Some studies demonstrate, for instance, that students have difficulty in recognizing thatthe direction of the static frictional force on a body that is rolling without slipping doesnot necessarily oppose the direction of rolling [1]. They also have difficulty determiningthe direction of velocity at different points on the wheel [2].Although such problems are frequently dealt with theoretically in textbooks [3],they are seldom studied experimentally. The absence of an empirical approach may havea negative effect on learning processes because students may be led to think that thetheoretical model has no connection with real life, without actually having tested theirpreconceptions against experimental evidence. In terms of experimental approaches, itis worth mentioning the study of the velocity of the center of mass of a cylinder duringrolling with and without slipping by means of video analysis [4] and proposal focusedon the transition between slipping and rolling [5].Here we describe an experiment to study the velocity distribution of the wheelviewed both from the laboratory frame of reference and from a relative frame of referencewith its origin at the wheel’s center of mass. The wheel initially is spinning freely, and isthen put in contact with a rough surface, so that eventually it is rolling without slipping.This analysis enabled students to visualize and gain intuition into the kinematics of arigid body that is rolling with and without slipping.
Experimental setup.
The system consists of a bicycle wheel of radius as depictedin Fig. 1. The rim was marked at equal intervals in order to facilitate automatic trackingof the wheel’s motion with Tracker software [6]. Motion was recorded at 30 fps usinga digital video camera (Kodak PlaySport) mounted on a tripod in such a way that itsoptical axis was at right angles to the plane of movement of the wheel. In order to obtainthe clearest possible image, we used spotlights to improve luminosity and increase thecamera’s shutter speed. We shall assume that the mass of the wheel is symmetricallydistributed around its physical axis.The wheel, initially spinning is then placed on a horizontal floor and the velocityof center of mass, initially equal to zero, increases with time, due to the force of kineticfriction acting in the direction of movement. This situation takes place over a veryshort time interval until the velocity of the point of contact with the floor is zero in thelaboratory frame of reference. Then, the wheel begins to roll without slipping so thatboth the velocity of the center of mass and the angular velocity remain constant. As canreadily be deduced from the equations of motion, when the wheel rolls without slipping,the resultant external force must be zero and therefore the static frictional force (theonly force acting in the direction of the horizontal axis) must also be zero. In practicethe wheel will not keep on rolling indefinitely because over longer timespans, the rollingresistance that we here chose to neglect, must be taken into account. n experiment to address conceptual difficulties in slipping and rolling problems Figure 1.
A wheel is made to spin at an angular velocity ω and is then placedin contact with the floor. Once the wheel is released, the center of mass begins toaccelerate due to the force of kinetic friction acting in the direction of movement. Velocity distributions.
The velocity at a given point on the rim can be thought,as shown in Fig. 2, as the addition of two vectors, one translational, in which all thepoints on the wheel have the same velocity as the center of mass, and one rotational,where the direction of the velocity of each point on the rim is tangential to the rim.
Figure 2.
Diagram of the velocity vector of a point Q on the rim of the wheelrepresented as the addition of the vectors of rotation and translation. The point P incontact with the floor is also indicated. In the laboratory frame of reference, while the wheel is slipping, the velocity of thecentre of mass is increasing and the angular velocity is decreasing, the velocity of thepoint P in contact with the floor is opposite in direction to the wheel’s motion and itdecreases until it becomes zero when the wheel is rolling without slipping. On the otherhand, when viewed in the frame of reference fixed to the wheel’s center of mass, point P describes a circular motion that is uniformly decelerated until the wheel is rollingwithout slipping, when P begins to describe uniform circular motion. Experimental results
We used Tracker to analyse the changing velocities of sixpoints on the rim from the time slipping started until rolling without slipping wasestablished. We did this in the laboratory frame of reference and in a frame of reference n experiment to address conceptual difficulties in slipping and rolling problems
Figure 3.
Velocity vectors corresponding to several points of the wheel (drawn withdifferent colours) in the laboratory frame of reference during slipping motion. Elapsedtime is indicated in each panel.
Making the wheel‘s center of mass the origin of a coordinate system, the sequenceof images in Fig. 4 shows how the profile of the velocities develops over time duringthe slipping phase in the frame of reference centred on the hub. In contrast to theprevious figure, in all these images the velocity vectors are tangent to the rim and ineach snapshot are approximately equal in magnitude. It can be also appreciated, inaccordance with the theory, that the modulus of the velocity diminishes as time elapses.To gain further insight, Fig. 5 shows the velocities of the same points, in thelaboratory frame of reference (left) and in the frame of reference centred on the hubof the wheel (right), but this time when the wheel is rolling without slipping. Directinspection of the left panel shows, as theory predicted, that the velocity of the point ofcontact with the floor is zero, while the velocity of the highest point is twice that of thecentre of mass [3]. On the other hand, the right panel shows that the modulus of thevelocity at the highest and lowest points are similar to each other and similar, also, tothe modulus of the velocity of the centre of mass in the laboratory frame of reference.
Conclusion.
We studied the changes in velocities over time of several points onthe rim of the wheel with respect to the two frames of reference described above. Byanalysing these changes, the students are able to visualize and better comprehend whatthe directions of the velocity vectors are at different points of the wheel. They are able n experiment to address conceptual difficulties in slipping and rolling problems Figure 4.
Similar to Fig. 3 but the velocity vectors are measured with respect to theframe of reference fixed to the hub of the wheel.
Figure 5.
Tracker screen snapshots showing in different colours the velocity vectorsat different points on the wheel with respect to the laboratory frame of reference (left)and with respect to the frame of reference with its origin at the center of mass (right).Below each panel, the values of the modulus of the velocity of the centre of mass ( A )and of the highest point ( F ) of the wheel are indicated. to observe, for example, how when the wheel is slipping the velocity of the point of thewheel in contact with the floor is in the opposite direction to motion, and they couldsee visual evidence for the direction of action of the kinetic frictional force. At the sametime, they are able to comprehend the nature of the motion of the wheel from the pointof view of a frame of reference centered on the wheel hub (the center of mass). Thisreinforced the idea of thinking about the set of velocities of points in the laboratory frameof reference as the addition of two velocity vectors, one translational and one rotational.This concept is fundamental to the understanding of rolling without slipping. Given allof the above, the analysis of the changes over time of the set of velocities can be a verypowerful tool to help students overcome certain conceptual difficulties associated withthe kinematics of a wheel that is rotating with and without slipping. n experiment to address conceptual difficulties in slipping and rolling problems References [1] Paulo Sime˜ao Carvalho and Adriano Sampaio e Sousa. Rotation in secondary school: teaching theeffects of frictional force.
Physics Education , 40(3):257–265, mar 2005.[2] Lorenzo G Rimoldini and Chandralekha Singh. Student understanding of rotational and rollingmotion concepts.
Physical Review Special Topics-Physics Education Research , 1(1):010102, 2005.[3] David Halliday, Robert Resnick, and Jearl Walker.
Fundamentals of physics extended . Wiley. com,2010.[4] VLB de Jesus and DGG Sasaki. Vıdeo-an´alise de um experimento de baixo custo sobre atritocin´etico e atrito de rolamento.
Revista Brasileira de Ensino de Fısica , 36(3):3503, 2014.[5] Alvaro Suarez, Daniel Baccino, and Arturo C Marti. Video-based analysis of the transition fromslipping to rolling. arXiv preprint arXiv:1901.10780arXiv preprint arXiv:1901.10780