Understanding Kinetic Energy

Transcription

1 Got Science Supplemental Information Video Modules 1 & 2 Understanding Kinetic Energy There are many ways to explore the relationship of mass and speed to kinetic energy. Hot Wheels system was chosen rather than a professional system for studying force and motion, such as the systems sold by Pasco, for two reasons. First, a way to perform this investigation that was minimally expensive was offered so that it could be done even in classrooms with very limited resources. The components for this Hot Wheels system can be obtained for less than $100, which is substantially less cost than a Pasco system. Second, the students are thought to find the Hot Wheels system more engaging because many students are likely to be familiar with Hot Wheels products and the thrill of racing and playing with them. Nevertheless, a similar investigation can be performed with Pasco or other related types of professionally manufactured equipment. Indeed, such equipment provides a higher level of reproducibility and accuracy. If the teachers happen to have access to such equipment and have all of the components that they may need, they may decide to use that instead. Or the teachers may decide to use the Hot Wheels set up first so as to engage student interest and then move to a Pasco system to refine their observations. All of the information that they need to be able to acquire the components and construct a system based on a Hot Wheels track set is provided, as shown in these videos, and use that system to perform the investigation that was presented. Our initial tests with the Hot Wheels system showed that the pendulum s potential energy at its apex was markedly less than the kinetic energy of the car calculated from the car s mass and speed. This would make the system unsuitable for this demonstration. However, after some investigation, this inequality in the kinetic energy of the car and the potential energy of the pendulum was found to be most likely due to energy that was absorbed through deformation of the car and/or the pendulum in the collision. This energy absorption could be greatly reduced by attaching strong neodymium magnets to the face of the car and the pendulum bob, with the same poles of the magnets facing each other. Thus the repulsive magnetic fields mediate the collision and due to the elastic nature of this interaction, nearly all of the car s kinetic energy is transferred to the pendulum. Then, as the pendulum rises, its kinetic energy is gradually converted to potential energy until at its apex, all of its kinetic energy is converted to potential energy. The potential energy of the pendulum can be calculated from its height. Thus determining the pendulum s height at its apex can be used as a measure of the car s kinetic energy at the moment that it collided with the pendulum. This is the key feature of this demonstration providing a 1

2 way for students to visually observe the effects of variation in mass and speed on kinetic energy by watching and measuring the deflection of the pendulum. Constructing the Hot Wheels Track, Car, & Pendulum System The Hot Wheels Set Hot Wheels V-Drop Super Velocity Track Set (currently available from Amazon.com for $20) was used. Other Hot Wheels track sets also may work as long as the teacher has at least two meters of straight track. The teacher also will need at least one meter of track that is all at the same level and a section of track that can be elevated without raising the final one-meter section. The Hot Wheels V-Drop Super Velocity Track Set The V-Drop track set includes two cars, but additional cars that had flat tops and flat front ends were purchased. The flat roof facilitated attaching metallic lead sheets to increase the mass. The flat front end facilitated attaching the magnet used to reduce energy loss in collisions. Of course, it looks more appropriate if the car s front end is used, but the car could run in reverse if it is found easier to attach the magnet to the rear end of a car. These cars were tested and the one that was the fastest was selected. The fastest car was also likely to have the lowest friction. This reduces interference from friction in the relationship of mass, speed, and kinetic energy and thus makes this relationship more evident to students. The track was assembled as described in the instructions accompanying the Hot Wheels kit except that the start of the track was supported using a clamp on a ring stand rather than suspending it from a door. This allowed an easily and reliably change in the speed of the car by changing its launch height. 2

3 The mass of the car is varied by adding pieces of metallic lead sheets. Lead sheet 1/16 thick that can be easily cut to smaller sizes, can be obtained from chemical supply companies such as Fisher Science Education. The type of lead metal sheet used was Catalog No. S75144 and available at a price of $18.20 for one 500 g sheet from this vendor: Neodymium rare earth magnets The magnets were obtained from K & J Magetics, Inc. - Disc magnets that were axially magnetized were used; 1 inch diameter, 1/8 or 1/4 inch thick, N52 grade. The exact size of the magnets is not crucial and the teacher can experiment to determine what works best in their system. These magnets are very brittle and very powerful. It is important to use care in handling them to prevent two magnets from attaching to each other abruptly because the force involved may be sufficient to cause the magnets to break. Also, once two of these powerful magnets have attached to each other it can be extremely difficult to separate them. The easiest way to separate them is to slide them apart but even that can be extremely difficult. It is probably worthwhile to order some extra magnets due to the relatively high probability that some magnets will be broken during their use. Adhesive Putty Reusable adhesive putty such as BlueStik was used to attach magnets to cars and pendulum bobs and to secure metallic lead sheets to cars. The smallest amounts that were sufficient to securely make attachments were utilized. Use of excess putty should especially be avoided for attaching the magnets to the car and the pendulum since it might allow for energy absorption in the collision that would reduce energy transfer to the pendulum. If the pendulum bob is ferromagnetic no putty will be needed for that attachment. Some putty was placed at key points on both sides of the track throughout the level section to secure the track to the table and to minimize its movement during use. The Pendulum system For the pendulum support a piece of 3/4 inch plywood 12 inches high X 34 inches long was used to attach aluminum supports that were made using a T.I.G welder. These dimensions are not critical these just happened to be the dimensions of a piece that were available, so the teacher should feel free to adjust this for their circumstances. Indeed, this board is longer than what would be needed, but the extra length for the placement of the support brackets in locations were used and did not interfere with pendulum movement. Manufactured aluminum or wood shelf support brackets could be used instead of custom made supports and the shelf support brackets can be attached using screws. 3

4 Pendulum Support Board with Support Brackets The pendulum using a ball bearing such as those that can be obtained from McMaster-Carr was constructed. An example of a suitable ball bearing can be found at: The bearing shown on that web page is Part No K35, 9/32 thick, 7/8 OD with a 3/8 opening for the pendulum support rod and currently is listed at $5.32. The bearing was mounted by drilling or boring a hole of the appropriate size (7/8 for the McMaster-Carr bearing referenced above) in a block of aluminum. Three set screws on each of three sides secured the bearing in place. The block of aluminum was machined to create a 3/4 inch slot that allowed it to be mounted on the pendulum support board. 4

5 Bearing Holder Assembly Illustration By mounting the bearing in this aluminum block one is able to easily adjust the position of the bearing on the board. However, the bearing could also be mounted directly in the pendulum support board by drilling a hole of suitable size through the face of the board near the top of the board and drilling a hole through the edge of the support board to mount a set screw to hold the bearing in place. A pendulum support rod should be selected that has a diameter to fit the bearing opening (3/8 for the bearing referenced above). This support rod should be cut to an appropriate length (e.g., 3/4 to 1 inch) and press-fit into the bearing. A hole through the pendulum support rod was drilled to fit the pendulum rod. A brass pendulum rod 1/8 inch in diameter was used. Drill the hole in the pendulum support rod slightly larger than the diameter of the pendulum rod so that it is easy to mount and remove the pendulum rod and drill and tap a hole for a set of screws to secure the pendulum rod inside the pendulum support rod as shown in the photo above. The teacher could use a different type of metal for the pendulum rod but select a material that is light and stiff. It is better for the pendulum support rod to be as light as possible so that the center of mass of the pendulum is as close to the bottom as possible. This simplifies the calculations of the pendulum s potential energy. It is also beneficial for the pendulum rod to be as stiff as possible to prevent bending after collision that will absorb energy and thus reduce the conversion of car kinetic energy into pendulum potential energy. 5

6 Rectangular blocks of metal were used as pendulum bobs. A hole was drilled through the side of the bob to accommodate the pendulum rod and a set screw hole was drilled and tapped on one face of the bob to secure the bob on the pendulum rod. For complete transfer of momentum from the car to the pendulum bob, the bob must be the same mass as the car. The greater the difference in mass of the car and the bob, the more incomplete momentum transfer will be. When the mass of the car is varied, a bob could be used that is the average of these masses, but for more accurate results vary the bob mass with the car mass so that the bob in use always has the same mass as the car. This will also require using a different potential energy scale for each bob since the bob potential energy depends on its mass. Alternatively you could use a scale that is marked in height of the pendulum rather than in PE units and then the same scale could be used for all of the pendulum bobs. The size of the metal block needed for a pendulum bob can be determined once the car masses that will be tested are known, by using the density of the metal block material. The density can be determined by dividing the mass of the metal block the teacher starts with by its volume. Its volume can be determined by multiplying its length, by its width by its thickness. For instance, if one wants a bob that has a mass of 80 g, and they have metal stock that has a density of 8.40 g/cm 3 and is 2.4cm X 2.6cm X 20 cm long, they would cut a piece from this rod that is 80 g 8.40 g/cm 3 (2.4 cm X 2.6 cm) = 1.52 cm long. Be sure to orient the face of the pendulum bob so that it is exactly perpendicular to the travel of the pendulum. Thus the magnet on the front of the car and the magnet on the pendulum bob are exactly parallel to each other. This will help to maximize momentum transfer in the collision. The pendulum system was positioned so that the pendulum bob rested exactly at the end of the track as shown in the photo below of the pendulum potential energy scale. Potential Energy Scale A potential energy scale was made by attaching a white sheet of heavy paper to the pendulum support board using double-sided tape, in the area behind the swing of the pendulum. The center of gravity of the pendulum was determined by finding the point at which it balanced after removing it from the pendulum support rod. That point was marked on the pendulum and the pendulum was remounted in the support rod. The pendulum s arc of travel was marked by placing a pen or pencil next to the side of the pendulum at its center of gravity with the point of the pen or pencil touching the paper. The pendulum was then moved through its arc of travel with pen or pencil marking that arc. A horizontal line was drawn parallel to the bottom of the 6

7 pendulum support board at the point of rest of the pendulum center of gravity to represent zero potential energy. The height of the arc above that line and the mass of the pendulum is used to calculate the pendulum s potential energy at convenient intervals. Be sure to include the magnet when measuring the mass of the pendulum but be cautious because the powerful magnet may influence the balance so the teacher may need to place the pendulum in a tared container to isolate the balance from the magnet s field. The pendulum s potential energy PE is calculated using the equation: PE = pendulum mass X acceleration of gravity X height above rest level Example: PE = 80 g X 9.8 m/s 2 X m = 7.84 mj Note that millijoules (mj) were used as the unit for energy, which corresponds to g m 2 /s 2. Thus for the equations to be valid, the heights must be converted to meters and the velocities converted to meters/second. The Pendulum PE Scale The pendulum scale can be marked either with the height of the pendulum s center of gravity or the pendulum potential energy or both. One might mark pendulum height on the top side of the arc and pendulum potential energy on the bottom side of the arc. If the teacher labels the scale with only the height of the pendulum s center of gravity then the students can calculate the potential energy as part of the process of investigating the car s kinetic energy. The photo above shows the scale marked for PE in units of mj. To calculate at which heights to mark specific magnitudes of PE the following equation can be used: Height above rest point (mm) = PE pendulum mass (kg) 9.8 m/s 2 X 1000 mm/m 7

8 Example: Height = 5 mj 80 g 9.8 m/s 2 X 1000 mm/m = 6.4 mm An Excel spreadsheet will be provided titled Pendulum Height Calculations that the teacher can use to calculate the heights above the rest position at which to mark the various PE intervals. This spreadsheet can be found on our web site in the Supplemental Materials section for this set of videos. Be sure that the marks have sufficient visual contrast that they will be easily seen and read when the pendulum s apex is measured. Using the Hot Wheels System to Investigate Kinetic Energy An important and essential first step of any learning activity is to engage the interest of the learners. The student video that is provided is intended to help with this goal. It should give students a sense of how kinetic energy is related to the speed and mass of an object. But the student video is only intended to be an introduction sufficient to pique student interest and curiosity. Hopefully, they are then motivated to explore this relationship through hands-on inquiry. It is recommended to start with a showing of the assembled Hot Wheels system and a demonstration of how the pendulum s movement is an indication of the kinetic energy of the car. One can observe how the pendulum moves higher when the mass or speed of the car is increased. Students can then be asked to predict which will increase the motion of the pendulum more, doubling the cars mass or doubling its speed. If students comprehend the video, they should predict that speed will have the greater effect. They should then test this prediction with the system. Students can be challenge to determine the exact mathematical relationship of the car s mass and speed to its kinetic energy. They can do this by measuring the kinetic energy (KE) of the car while varying the mass and speed. Record the car s speed over the end section of track that is level. This section of the track should be at least one meter long. If the level section of track is not at least one meter long, adjust the position of the elevated portion of the track so that a longer portion of the track is level. Put a clear and easily seen mark on the track at the timing start point and measure the exact distance from this point to the pendulum. Set the car at the launch position and release it. Time the car s travel using a stop watch beginning as the front of the car reaches the timing start point and end the timing as the front of the car just reaches the pendulum. Record this time. Repeat this measurement at least three times to increase the accuracy of this measurement. It is expected that there is some variation in this measurement because of differences in the number of small collisions of the car with the sides of the track as it travels the track. These minor collisions represent variations in friction that slightly alter the car s speed. Students can be introduced to the concept of determining the arithmetic mean of multiple measurements as a method for reducing experimental uncertainty with this part of the procedure. While there is a delay in pressing the stopwatch button due to human reaction time, that delay should be about the same for both the start and finish so that should cancel in determining the overall time. If the teacher happens to have some other device for measuring the car s speed such as a motion sensor, they may use that instead to obtain measurements that may be more accurate. However, measuring the 8

9 travel time with a stopwatch has been found to provide adequate accuracy for observing the correct mathematical relationship of speed with KE. At the same time that some students are measuring the car s speed, others can also measure the apex of the pendulum s travel. One way this can be accomplished is by mounting a video camera on a tripod so that the video camera can record the pendulum s travel after the collision. Then by rewinding the recording and stepping through the recording of the pendulum s travel one frame at a time, the teacher can easily determine the magnitude of the pendulum s apex with a high degree of reliability. If a video camera is not available, then the apex must be judged by visual observation. If the teacher uses this method it is suggested that they assign two students to watch for each measurement and take the average of the two measurements. The teacher should neglect results if the car lost speed due to a bad launch or because it was excessively bouncing against the sides of the track. It is best to carefully adjust the track to minimize gaps between track sections and to align the track so that the car can travel as smoothly as possible to minimize the effects of friction. It may be helpful to have a student stationed near the pendulum to catch the pendulum after it reaches its apex and before it fully rebounds to where the car is resting so that it doesn t come in contact with the magnet on the car. It was often found that during the rebound, the pendulum and car magnets are able to move in such a way that they reorient themselves to maximize their attractions. Due to the strength of these magnets, the magnets are then strongly attracted to each other and once they attach to each other it can be very difficult to separate them. Thus for each measurement several students can be involved. One student can release the car from the launch point, one student can time the car s travel, one student can catch the pendulum after it reaches the apex and before it returns to the resting position, one student can operate the video camera or two students can visually observe the pendulum s apex, and one student can record the results. Another student can be responsible for varying the car s mass and another for varying the launch height. At least three measurements should be recorded for each set of values of the variables, mass and speed. An Excel spreadsheet titled KE Data Template is provided that can be used to record the data. This spreadsheet includes the formula for calculating the car s KE (equal to the pendulum s PE at its apex). It will work better to have measurements that include at least a doubling of the car s mass and a doubling of its speed so that the magnitude of the differences will enable observing noticeable differences in the mathematical relationships. When the teacher has recorded a complete set of measurements, they may ask their students which of the following equations best fit the data: KE = car mass X car speed KE = 1/2 car mass X car speed KE = 1/2 (car mass) 2 X car speed KE = car mass X (car speed) 2 9

10 They can check this by calculating the car s KE using each of these equations and determining which gives results that are closest to what they measure for the car s KE using the pendulum system. It is most likely that they will not observe perfect agreement due to experimental variation so they should decide which of these equations is the best fit which gives results that are closest to those they actually measured. This is the way that scientists select the best explanations for other phenomena that they observe. There is always some experimental variation that results in less than absolute perfect agreement so scientists search for the simplest explanation that best fits the evidence. However, in this case, the teacher should also call students attention to the fact that a very large number of other scientists have verified that KE does indeed equal 1/2 mass X speed 2 based on many thousands of highly accurate and carefully verified experimental measurements. Thus there is a very high level of confidence that this equation is the best fit for this data. If students fail to find this equation has the best fit for their data, because there is so much other evidence that indicates this equation is the best fit, they should search for an explanation in the nature of their data for the failed fit to this equation. A possible explanation is a pendulum that is not functioning well. For instance, it may not be swinging freely so it is not showing the car s full kinetic energy. Or the students may have made errors in measuring some of the variables such as the car s mass or speed or the pendulum apex. Teachers are encouraged to experiment with this system to find ways to make the measurements more reliable and reproducible and to further reduce friction and other complicating factors. Hopefully, they will find this system to be a useful way to guide Their students exploration of these concepts. Best wishes to the teachers and their students further inquiries into ways that nature behaves. 10