Vocabulary and Experiences to Develop a Center of Mass Model

Vocabulary and Experiences to Develop a Center of Mass Model Taylor Kaar, Linda B. Pollack, Michael E. Lerner, and Robert J. Engels Citation: The Physics Teacher 55, 409 (2017); doi: 10.1119/1.5003741 View online: http://dx.doi.org/10.1119/1.5003741 View Table of Contents: http://aapt.scitation.org/toc/pte/55/7 Published by the American Association of Physics Teachers Vocabulary and Experiences to Develop a Center of Mass Model Taylor Kaar, Laurel School, Shaker Heights, OH Linda B. Pollack, Theodore Roosevelt High School, Kent, OH Michael E. Lerner, Beachwood High School, Beachwood, OH Robert J. Engels, St. Vincent-St. Mary High School, Akron, OH T he use of systems in many introductory courses is limited and often implicit. Modeling two or more objects as a system and tracking the center of mass of that system is usually not included. Thinking in terms of the center of mass facilitates problem solving while exposing the importance of using conservation laws. We present below three laboratory activities that build this systems thinking for introductory physics students. Our focus on center of mass was motivated by the new AP Physics 1 and 2 curriculum.1 The language of the exam, such as “the block-Earth-spring system,” confused our students, especially since we rarely used systems except in the energy unit. Systems thinking is not just limited to the Advanced Placement courses; Systems and System Models is a crosscutting concept in the Next Generation Science Standards (NGSS). In high school, NGSS states that students “can use models (e.g., physical, mathematical, computer models) to simulate the flow of energy, matter, and interactions within and between systems.”2 The need for students to have more fluency with defining systems and thinking about physical systems is obvious. The authors met at the Central Ohio Physics and Chemistry Modeling Workshops held at New Albany High School. As second-year participants at the 2016 summer workshop, we spent our three weeks at the workshop thinking through the concepts students would tackle, refining our vocabulary, and designing laboratory experiences. We developed a new modeling instruction unit that we’ve named the Center of Mass Model (CoMM). While CoMM is named after just one point in the system, the unit is designed to help students see how changing the system under study can make certain problems easier to solve. CoMM is not designed to be a standalone unit; systems thinking can integrate into all mechanics units. Our three major points of entry are after students have studied balanced forces, after students have studied projectile motion, and as students are studying momentum transfer. The first time the Center of Mass Model (CoMM) is introduced is after balanced forces. The topic is introduced with a video analysis of a single air puck sliding across the floor. (All videos, worksheets, and teacher support we created can be found here: http://bit.ly/1n4SCmd). Students can analyze the motion using either video analysis software or transparencies on a screen to determine position-vs.-time and velocity-vs.time graphs. This system is very easy to track and students can easily draw a free-body diagram showing the balanced forces and constant velocity. DOI: 10.1119/1.5003741 Fig. 1. Velocity-time graph of one puck in a rotating two-puck system. Fig. 2. Position-time graph of the center of mass of a rotating two-puck system. When the students are presented with a video of a rotating two-puck system connected by a single dowel rod, they fight the idea of tracking one puck. Even when told to find the position-vs.-time graph for one puck, students intuitively want to track the center of the puck-puck system. With encouragement, students obtain a velocity-time graph similar to the one shown in Fig. 1. Our students were uncomfortable with such a complicated velocity-time graph and were even more insistent that we should have been tracking the middle of the puck-puck system the whole time. Once students track the midpoint between the two pucks, they obtain positiontime graphs like the one shown in Fig. 2. This realization that selecting an appropriate point to represent the motion of the entire system simplifies the analysis of the motion and necessitates the definition of the center of mass. To further drive this point home, a four-puck square arrangement (see Fig. 3) can be used to demonstrate that the center of a symmetrical object is the center of mass. Furthermore, the square allows for the interesting discussion of “can THE PHYSICS TEACHER ◆ Vol. 55, O CTOBER 2017 409 Fig. 3. The four-puck arrangement. the center of mass exist at a place with no mass?” The idea of the center of mass being at the middle of a symmetrical object is enough for the AP 1 curriculum, but, depending on the instructor, class, and ability levels of the students, simple asymmetrical objects can be used as well to further understanding of the CoMM and how it can be used. Some vocabulary words are helpful here. It helps us to reinforce that an “object” has no internal structure for simplicity’s sake. Almost all of our students identified one puck as an object, ignoring the internal structure with batteries and fans. We define a “system” as one or more objects that we are interested in for a given problem. Here, the system the students agreed upon was a puck-puck or puck-rod-puck system, depending on whether they were willing to ignore the balsa wood. Finally the “situation” is all of the objects in the immediate environment. In this case, students mentioned that the floor, the air, the puck-rod-puck system, and the Earth are all necessary to include in the situation to explain the motion of the system of interest. Many teachers use “system schema”3 to visualize all of the interactions; we find it better to call these pictures “interaction diagrams,” leaving the word system to be just about the objects of interest. (Interaction diagrams will be explained in more detail using Figs. 10 and 11.) Furthermore, this expansion of vocabulary serves as a further demonstration about the different classes of forces. The ideas of negligible, internal, and external forces are present in this lab, but it is up to the students to really strike upon their key differences. Classroom discussions can lead to the realization of the negligibility of air resistance as a real force that does not have a meaningful impact on the system’s movement. Internal forces come from interactions entirely within our interaction diagram’s defined system border and can be ignored. It is only when an external force affects our objects and crosses into our system on the interaction diagram that we know the motion of the object will be affected. The center of mass can also be integrated into a unit on projectiles. Attaching two tennis balls together by a dowel rod can be used as a projectile. Figure 4 shows a graph that tracks 410 Fig. 4. Position-time graph of each end of a rotating twoball projectile. Fig. 5. Position-time graph for the center of mass for a twoball projectile. each tennis ball individually. Video analysis of the ball-ball system shows a complicated motion for each ball but the expected parabolic path for the center of mass. Figure 5 shows the tracking of only the center of mass. Again the students see how useful the center of mass is to analyze the motion of a system. After completing the momentum unit there is another opportunity for students to see how the use of systems and the center of mass can be used to derive physical quantities from analysis of the motion of a system of objects. In the lab the objects themselves are undergoing fairly complex motion that can’t be modeled by the students using previously developed models. Using a modified collision lab, students can observe the motion of two dynamics carts that are connected together by a rubber band, which collide with a hoop spring connected to a force sensor at one end of the track. The dynamics carts also have magnets in the opposing bumpers so that as the rubber band pulls them together they are repelled. See Fig. 6. This setup of the carts results in a translational oscillatory motion of each cart when they are set in motion along a level track. To analyze the motion of the carts, motion sensors are set up at each end of the dynamics track, and position readings of one of the sensors is reversed so that both are measur- THE PHYSICS TEACHER ◆ Vol. 55, OCTOBER 2017 Fig. 9. Total momentum of the system vs. time for the complex two-cart system Track normal normal magnetic Fig. 6. Full setup of momentum lab with two motion detectors and one force probe with hoop spring. Cart 1 gravity Cart 2 elastic gravity gravity Earth Fig. 7. Velocity-time graphs for the individual carts in the complex two-cart system. Fig. 8. Force-time graph for the complex two-cart system colliding with hoop spring. ing position in the same direction. The two-cart system is given a push and the system moves along the track so that it collides with the hoop spring/force sensor and reverses direction. We first show a velocity-vs.time graph for the carts and discuss the complexity of the motion (Fig. 7). At this point students are already familiar with one cart colliding with a hoop spring/force sensor and know that the change in momentum of the cart is equal to the area under the force-time graph (impulse-momentum theorem). Double hits of the hoop-force sensor by the cart increase the complexity of the situation but still can be analyzed using momentum and the center of mass. However, if the students consider the two carts, rubber band, and magnets as a system, the solution to the problem becomes apparent. Students first graph the force-time and find its area to calculate the total impulse on the system (Fig. 8). Then students graph the total momentum of the system by calculating the sum of the individual momenta of the carts Fig. 10. Interaction diagram after student suggestion for defining the system. (Fig. 9). They can readily see that the change in momentum (final minus initial value) of the system is equal to the area under the force-time graph. This demonstration provides an incredibly powerful moment to demonstrate the strength of the Center of Mass Model as a way to simplify otherwise complex systems. Note that impulse on the system can be found by multiplying the change in velocity of the center of mass by the system mass or by finding the change in the momentum of the system, which is the sum of the momenta of the two carts. When analyzing this situation, students often started with an interaction diagram. To analyze the motion of cart 1, the interaction diagram shows that there would be four forces acting on the cart: gravity from the Earth, normal force from the track, and both elastic and magnetic forces from cart 2. It quickly becomes apparent that this is too complicated to analyze. When one of us did this lab in class, a student said, “If it’s too complicated to analyze, make your system bigger.” As a class, we then drew our system as a dotted line around cart 1 and cart 2 in the interaction diagram, as shown in Fig. 10. The situation is now simpler to analyze, with only two balanced forces, the normal force and gravity, acting on the system. We have tried these labs in our classes. While students at first think that the idea of the center of mass is almost too obvious to mention, as the year progresses, they start to learn the power of this model. When one author presented the complex two-cart system lab, the students were fascinated by the complicated velocity-time graphs of each individual THE PHYSICS TEACHER ◆ Vol. 55, OCTOBER 2017 411 Physics teachers... get your students registered for the preliminary exam in the U.S. Physics Team selection process. All physics students are encouraged to participate in the American Association of Physics Teachers’ Fnet=ma Contest! The Fnet=ma Contest is the United States Physics Team selection process that leads to participation in the annual International Physics Olympiad. The U.S. Physics Team Program provides a once-ina-lifetime opportunity for students to enhance their physics knowledge as well as their creativity, leadership, and commitment to a goal. cart and by the simplicity after applying the Center of Mass Model. The jump to angular momentum was easier using this concept; the idea that only a net external torque could change the angular momentum of a system seemed much easier to grasp. The authors have implemented these ideas across a wide variety of schools (public and private, coed and single sex) and abilities (AP Physics C, AP Physics 1, Honors) and have seen consistent improvement in students’ ability to identify and work with systems. This understanding allows students to tackle ever more complex problems with confidence and continues to underscore the importance of being able to accurately define a useful system. Acknowledgments The authors would like to thank Dr. Kathleen Harper and Dr. Ted Clark from The Ohio State University, the Ohio Physics and Chemistry Modeling Workshops, New Albany High School, and the Improving Teacher Quality Program of the Ohio Board of Regents for their funding, use of their spaces, and their suggestions, which all contributed to this document. References 1. School Fee: $35 per school ($25 fee for teachers who are AAPT members) plus $4 per student for WebAssign or $8 per student for PDF download. Two or more teachers from the same school pay only one school fee. 2. 3. Registration begins October 1, visit: http://www.aapt.org/physicsteam One of the Five Big Ideas of the AP Physics 1 Curriculum is “interactions between systems can result in changes in those systems.” (AP Physics 1: Algebra-Based and AP Physics 2: Algebra-Based Course and Exam Description Including the Curriculum Framework Effective Fall 2014, accessed Aug. 3, 2016, from http://media.collegeboard.com/digitalServices/pdf/ ap/ap-physics-1-2-course-and-exam-description.pdf. http://www.nextgenscience.org/sites/ngss/files/Appendix%20 G%20-%20Crosscutting%20Concepts%20FINAL% 20edited%204.10.13.pdf , p. 8. Lou Turner, “System schemas,” Phys. Teach. 41, 404–408 (Oct. 2003). Taylor Kaar teaches at Laurel School, a private independent girls school in Shaker Heights, OH. [email protected] Linda B. Pollack teaches at Theodore Roosevelt High School in Kent, OH. Michael E. Lerner teaches at Beachwood High School in Beachwood, OH. Robert J. Engels teaches at St. Vincent-St. Mary High School in Akron, OH. 412 THE PHYSICS TEACHER ◆ Vol. 55, OCTOBER 2017