Inertial and non-inertial frames: with pieces of paper and in an active way Leoš Dvořák Department of Physics Education, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic Abstract A set of activities is described that can help the understanding of inertial and non-inertial reference frames. Using simple paper models, students can clarify and see on concrete examples how the same motion is described in different reference frames. Models enable the investigation of various types of reference frames: inertial, accelerated (moving in one direction) and rotated frames. Also, the motion of the object which is followed can be either uniform (with respect to inertial frames) or accelerated. This set of activities was piloted both in pre-service and inservice teacher training courses in Czech Republic. Keywords Classical mechanics, reference frames, simple models, active learning. Introduction Often, motion with respect to different reference frames is presented and described rather theoretically in textbooks and lectures for future physics teachers. Sometimes with nice drawings or photos, sometimes perhaps with some applets that can help understanding. In case of lectures, also experiments are of great help, of course. Nevertheless, often theoretical derivation is used as a simplest one. To overcome this formal approach, a set of activities was developed that could enable teachers and students to perceive directly how transformations between different reference frames work. The activities use very simple paper models on which students construct various types of motion and their descriptions, with respect to different reference frames step by step. The purpose of these activities is to give students some concrete experience how motion is described in different frames, to provide tasks that can stimulate discussions and, in general, to help students to have a better understanding of this area. To say this, we do not like to suggest that this approach should replace real experiments and video clips used in teaching this topic. Rather, it complements and supports them showing how the theory works or, to be specific, how transformations between different reference frames work. Models enable the investigation of various types of motion and reference frames. In the workshop, we went through most of these possibilities, starting from simple uniform rectilinear motion (adding of velocities in one dimension) and ending at rotational frames (where Coriolis, centrifugal and Euler acceleration can be demonstrated and investigated). Surely, in real school teaching such concentration of activities would not be useful. Rather, different activities can be used when it is appropriate in short blocks. Also, these activities can be used at different levels:
simple adding of velocities even at junior secondary level (pupils of age 12-15), activities including acceleration at senior secondary level (high school level) and activities concerning rotating reference frames at the end of high school level or at introductory university level. In fact, the basic idea of all these models is quite simple and teachers can easily modify the models and activities to suit the needs of their teaching. This set of activities was piloted in both pre- and in-service teacher training courses. At the workshop in Heureka Workshops 2013 conference in October 2013 in total more than fifty teachers participated at several runs of the workshop, with a positive feedback. Teachers appreciated that the models are very simple and low cost, can be easily modified and also that they can be used separately. Teachers reflecting on the workshop also stated that although the models and situations are basically simple, some parts of the activities were not so easy and really required them to think more deeply and to discuss some aspects of the tasks which they, in fact, also appreciated. A similar rather positive feedback concerning flexibility, range of applications, simplicity and clarity of models was kindly expressed by participants of the workshop at GIREP-MPTL 2014 conference. A detailed description of the models and their use was published in Czech in proceedings of Heureka Workshops 2013 conference (Dvořák 2013). Here, due to limited length of the paper, we describe the principle of this approach and illustrate it by several examples. Worksheets of all twelve tasks done at the workshop at GIREP 2014 (in English) are available from the author upon request. A common principle: simple paper models All we need are just few pieces of paper and some paper rulers such that are often available in some hobby markets. Alternatively, we can print the rulers at A4 sheets and cut them by scissors. Also, we need a sticky tape by which we will attach the rulers to a desk or to pieces of paper representing for example railway carriage. A paper tape which can be easily unstuck is ideal. All our models will have scale 1:100, i.e. each cm will represent one meter. One reference frame (an inertial one) will be represented by a desk. A ruler parallel to the edge of the desk forms a coordinate system. We can say that, for example, the desk represents straight rails and a landscape around rails, the ruler measures the coordinate along the rails. A model of a railway carriage is just a piece of paper (we used one third of A4 sheet) folded so that it can slide along the edge of the table, see Figure 1. Figure 1. Basic components of paper modelling of reference frames.
As it can be seen in Figure 1, we can have also a very simple paper model of a man walking or running along the rails second paper ruler attached to the desk enables to set position of the man. The prolonged part of the paper model of man enables to mark positions of the man with respect to the carriage, as Figure 2 shows. Figure 2. Position of the man in each second can be marked on the model of the railway carriage. Students work in pairs. One student sets position of the carriage (the position of its left edge that can be easily read on the ruler), the other sets the position of the man. Either of them marks the positions at the carriage or at other reference frames. Motion and our modelling starts at t = 0 s. Then positions are set and marked at each second. It is useful to write the values of time next to the marks as it is shown in Figure 2. Then, motion of the object with respect to different reference frames can be analysed as we shall see in the examples below. Basically, this is the principle of our modelling. It is very simple but it still requires students to realize which objects and reference frames move to which, on which rulers (i.e., in which coordinate systems) to read the values, to which piece of paper (i.e., to which reference frame) to draw the marks etc. And it visualizes the motion step by step, second by second. Also, it is not a passive visualisation done by some computer software; students have to do everything by their own hands. Perhaps we can even say that they are, in a sense, part of the modelling. Or, to express it in a way of software modelling, they themselves are the engine that does the modelling. Let us now look at several examples showing what situations can be modelled. Inertial frames: motion in 1D Perhaps the easiest possibility is shown in Figure 2. The railway carriage moves to the right with the speed 2 m/s, the man runs to the right with the speed 3 m/s (with respect to the rails). From the marks at the carriage one can clearly see (and measure) that the man moves with the speed 1 m/s with respect to the carriage. Various similar situations can be modelled, for example the case when the speed of the carriage is greater that the speed of the man. Also,
either the man or the carriage or both can move to the left. (It is useful to use markers or pencils of different colours to draw marks for different velocities.) This activity can be used as a starting one, in which just the principle of our paper modelling is introduced and the known formula for adding (or, rather, subtracting) of velocities is confirmed or students can discover the formula from the data they created. We can also let the man move with respect to the carriage and mark its position to the landscape around rails. The motion of the man can be uniform but also accelerated, as Figure 3a shows. Now we have one ruler (one coordinate system) attached to the carriage, the positions of the man with respect to the carriage are set using this ruler. In the case shown in Figure 3a, the acceleration of the man with respect to the carriage is 1 m/s 2. (It is convenient to have the table of distances from the starting position ready: 0, 0.5, 2, 4.5, 8,...) Position in each second is marked both to the carriage and to the landscape (an A4 sheet of paper attached to the desk). Figure 3a. Accelerated motion with respect to the railway carriage... What can be derived from marks at the landscape? As can be seen in Figure 3b, it is clear that the motion with respect to the landscape is non-uniform. Also, distance of positions in consecutive seconds can be measured. Each of these distances was travelled in one second therefore it represents an average velocity in the relevant second. For example, in case shown in Figure 3b, the average speeds are 2.5, 3.5, 4.5, 5.5,... (all in m/s). So, the average speed is increased by 1 m/s in each second. From this students can directly infer that the acceleration with respect to the landscape is 1 m/s 2, the same as the acceleration with respect to the carriage. (Of course, it must be so, both the landscape and the carriage are inertial frames, but here we see that the theoretical formula really works.) Figure 3b.... is accelerated also with respect to the landscape; the acceleration can be found from distances of the marks.
Inertial frames: motion in 2D Now, imagine that we look at the rails and the landscape from above. The railway carriage moves with constant speed (in the situation shown at Figure 4, it was 2 m/s). Somebody from the outside throws something (for example a parcel) perpendicularly to the rails. What is the velocity of that object with respect to the carriage? Figure 4 shows the motion of the object with respect to the carriage for two different speeds of the parcel (to the landscape, 1 m/s and 2 m/s in our case). Figure 4. An object thrown from outside in the direction perpendicular to the rails. Our model enables to see clearly that the direction of motion with respect to the carriage is not perpendicular to the rails; we can also directly measure the speed of the object with respect to the carriage. Again, many different situations can be modelled various combinations of speeds of the carriage and the object and directions of their velocities. Let s add one general technical note : It is clear that the precision of setting positions of the carriage and the man and of drawing marks is not very high, errors of about 1 mm or even more can be expected as common. That s why the values of velocities in the worksheet to the tasks are such that distances of most marks are at least 1 or 2 centimetres. When compared to these distances, errors of 1 mm or so do not disturb us too much and we can persuade students that such errors can be ignored. As an example of other situation in 2D, let us look at an accelerated motion of some object. Now we are looking at the carriage from side so in our model we will have two directions: direction of the rails and vertical direction. An object (a stone, a parcel or anything else) is dropped in a moving carriage. What is its motion with respect to the landscape?
Figure 5 shows both the model and the results for two different speeds of the carriage. Here, for the sake of clarity and simplicity, we take gravitational acceleration ten times smaller than the real acceleration on our Earth, i.e. 1 m/s 2. Figure 5. An object dropped in a moving carriage and its motion with respect to the landscape (side view). Non-inertial frames: rectilinear motion Paper models can be used also to model motion with respect to non-inertial frames. Now, the desk will represent inertial reference frame, our paper carriage an accelerated frame. Two simple tasks were part of the workshop: 1) accelerating carriage and the man standing still at the platform (which is at rest in inertial system) and 2) decelerating carriage and a man in it who is initially at rest with respect to the carriage and then continues to move with constant speed with respect to inertial reference frame (because no force acts to him, for example, he can stand on a skateboard). In both cases we see how the man moves with respect to the carriage, which can be a starting point of discussions on fictitious forces. Even more interesting example is a falling elevator. Our model can illustrate why in such elevator we are in a state of weightlessness. Our model is shown in Figure 6. At t = 0 the floor of the elevator is at height 20 m and its speed with respect to the Earth is zero it just starts to fall. The man is standing in the elevator on its floor and jumps up with the speed 1 m/s. (We can discuss with students the fact that the speed is the same with respect to both the Earth and the elevator at t = 0.) Using simple high school formula z = z 0 + v 0 t g/2 t 2 we can calculate the positions of both the man and the elevator in t = 0, 1, 2,..., 6 s. Again, for the sake of simplicity and clarity we take g = 1 m/s 2. It is useful for students to write down the coordinates of both the man and the elevator in a table setting of their positions is then easier.
Figure 6. A man jumping in a falling elevator. As can be clearly seen in Figure 6, the motion of the man with respect to the elevator is uniform (until the elevator hits the ground), as it is in absence of gravity. Rotating frames To model rotating frames, we need an axis around which a piece of paper our model of a merry go-around will rotate. Fortunately, there is no need to drill a hole into a desk to fix the axis. A drawing pin attached by a sticky tape to a desk will suit very well, see Figure 7. Figure 7. Drawing pin as an axis for our rotating frame ( merry-go-around ). A sheet of paper attached to the desk will represent an inertial reference frame we can say that it is an amusement park around our merry-go-around. It is useful to have a rose of directions on that sheet of paper an angular scale, in our case the differences of marked directions are 10º and 5º, see Figure 8. A model of merry-go-around can be a paper circle or some symmetric shape. (We chose a hexagon because it is easier to cut it.)
It is easy to demonstrate and measure an effect of Coriolis force in our model, see Figure 8. An object, for example a ball, is thrown from the centre of rotating frame. Its motion is uniform and rectilinear with respect to the inertial reference frame. So, if we attach a ruler to the desk, our object moves (in our case to the right) along the ruler with constant speed (in our case 2 m/s, which means in our scale 2 cm for each step). Simultaneously, our merry-go-around rotates with constant angular speed 10º/s. (So, in each step we rotate it by 10º.) In each step we draw a mark at the rotating frame. Figure 8. How to demonstrate an effect of Coriolis force on our paper model. After removing the ruler we can see the trajectory of the ball with respect to the merry-goaround. In Figure 9, we rather roughly connected the marks to make the trajectory clearly visible. Students see that the trajectory is not rectilinear it is as if something deflected it to the right from the direction of the velocity. This can be a starting point of discussion of Coriolis force. To show the variables on which Coriolis force depends we can change the angular velocity of rotation (also, to let the frame rotate in the opposite direction) and the speed of the ball. We can even measure the effect of Coriolis force quantitatively. However, it is necessary to have in mind that also centrifugal force influences the motion so satisfactory results concerning Coriolis force can be obtained only shortly after t = 0, when the distance from the axis is small and the ball moves more or less in radial direction. Figure 9. Result of modelling of the ball thrown radially from the axis: in rotating frame, the thrown object is deflected to the right.
Using similar modelling experiments we can demonstrate and measure also effects of centrifugal and Euler forces. One advantage of these models lies in the fact that students clearly see that with respect to inertial reference frame the motion is rectilinear and uniform. (They construct this motion by their own hands!) This can be a good starting point for discussing whether e.g. Coriolis or centrifugal force can act in inertial reference frames and what the nature of these fictitious forces is. Conclusions All activities described above are in principle quite easy. However, students have to realize with respect to which reference frame they should measure which position so, very concretely, which ruler to use to set some coordinate (of the man, of the carriage etc.). Also, when drawing marks showing positions in different times, they should realize to what frame to draw them. Even before starting to model any specific situation it is necessary to connect coordinate systems with relevant reference frames i.e. to physically attach the rulers to objects representing reference frames (the desk, the paper model of a carriage etc.) It occurred that students (future teachers, in our case) really had to concentrate on the tasks. On the other hand, just this enables students to better grasp the transformation between reference frames. During these modelling activities, they really see and feel the transformation formulas at work. As it was mentioned before, the models and activities can be adapted in various ways. They also enable students to express themselves even in some unexpected manners. For example, some teachers and future teachers drew some parts of landscape on a sheet of paper representing it making it less abstract and more enjoyable. Therefore, we can expect that teachers will create different ways how to use these types of models to suit the needs of their teaching and their students. In case you would like to try these activities, the author can provide upon request the English worksheets of all twelve tasks done at the workshop at GIREP 2014 (in pdf format) as well as simple templates of paper pieces of models described above (in cdr and pdf). References Dvořák, L. (2013). Inerciální a neinerciální systémy názorně. Dílny Heuréky 2013/Heureka Workshops 2013. Proceedings of the conference of the Heureka project, P3K, Prague, 14-34. (In Czech) Leoš Dvořák Department of Physics Education Faculty of Mathematics and Physics Charles University in Prague V Holešovičkách 2 180 00 Praha 8 Czech Republic e-mail: leos.dvorak@mff.cuni.cz