Visualizing the Movement of the Celtic Stone as High School Students Activity

The Celtic Stone (also known as a rattleback) is a semi-ellipsoidal shaped solid object which when spun rotates on its axis in a preferred direction. If spun in the opposite direction, it goes to the stop and reverses its spin to the preferred direction. As the movement of the stone is multidirectional it is a challenge to perform quantitative measurements of its motion characteristics. The work presents the experimental set-up and the procedure for collecting date of a rattleback motion as the high school students activity. Proposals for specific student actions and their benefits for learning were described in connection with the results obtained. Some preliminary results of video measurements performed with a specific metal rattleback and meant to visualize its motion are reported. Attempts to compare the results with predictions based on the numerical model of the situation are undertaken. The advantages and disadvantages of the measurement system are presented and discussed.


Introduction
The Celtic stone (also known as a rattleback) is a semi-ellipsoidal shaped solid object which moves in a way that offers a practical and engaging way to explore fundamental principles of physics, mechanics, and materials science.Due to its unique behaviour it serves both educational and scientific purposes, contributing to our broader understanding of the natural world.There is a number of movies generally available showing its motion [1] and a publication describing how to obtain such an object even from every day materials [2].Several investigation has been conducted to visualise such motion and to model behaviour of different Celtic stones [3,4] so the reproduction of the measurements could be treated as valuable high schools students activity.The activity is meant to shape key competencies [5] especially in area use of ICT technology and data elaboration.The Celtic stone as an experimental object for high school students is purposeful and justified because it brings specific educational benefits: carouses curiosity, has an challenging level of difficulty, stimulates student involvement.

Theoretical basis
The fascinating behavior of the Celtic stone is due to the asymmetric shape of the stone, which means that its center of mass is not directly above the point of contact with the ground [6].Main characteristics of the moving system are presented in the figure 1.The mathematical model for describing the equations of motion of a Celtic stone consists of formulas: • (1)-( 2) which define the motion of the center of mass relative to a stationary coordinate system.
• (4)-( 6) which represent kinematic differential equations expressing the relationships between the components of the angular velocity vector and the derivatives of the angles describing the orientation of the stone.• (7) that represents the assumption of continuous contact between the stone and the plane on which it lies.
̇=  1 cos +  3 sin, In the above figure and formulas the following notations is used: denotes the time derivative of a vector u in a stationary coordinate system ( 1 ,  2 ,  3 ).
•  �   denotes the time derivative of a vector u in a moving coordinate system.
• G is the origin of the stationary coordinate system.
• C is the origin of the movable coordinate system and is also the center of mass of the stone.
• O is the center of the top surface of the stone.
• A is the point of contact between the stone and the ground.
•  is a vector defining the relative position of the center of mass.
•  is a vector defining the relative position of point A.
•  is the unit vector of the surface normal ( 1 ,  2 ).
•  � =  �  is the ground reaction vector at point A with value  � .A hat above the quantity means a set of values in a form of a matrix.• m is the mass of the stone, g is the acceleration due to gravity, and mg is the gravitational force.
•  �  is the resultant (net) friction force at point A.
•  is the velocity vector of the center of mass.
•  is the angular velocity vector of the stone.
•  is the tensor of inertia of the stone with its center of mass C.
•   is a vector determining the position of the center of mass C relative to the stationary coordinate system G.
•  �  is the moment of kinetic friction.
•  �  is the moment of rolling friction of the stone.
•  is the angle describing the orientation of the stone relative to the axis  1 .
•  is the angle describing the orientation of the stone relative to the axis  2 .
•  is the angle describing the orientation of the stone relative to the axis  3 .
The work presents the results of measurements of changes in angle  i  values over time.The other degree of freedom of the vector k or the stone swaying both longitudinally and transversely are not analysed due to the fact that the camera is static and after a certain time the motion of the rattleback brings it out of the scene.Some further attempts are taken to overcame this obstacle in future experimental settings.
A set of equations even though they go beyond the reach of the high school students they could be used to illustrate the basic physics equations such as the second law of dynamics (equation 1 and 3), the definition of velocity (equation 2) and the balance of forces (equation 3).

Experimental set-up
The Celtic stone used in the reported measurements is made of steel and is wildly available as a toy-tool produced by Magic Makers ® .Information about the mass and dimensions (at the widest point) of the investigated stone is presented in Table 1.The uncertainty of weight is ± 0.1 g when the dimensions were measured with uncertainty ± 1.0 mm.Stones made of plastic are also widely available, but the use of a metal stone eliminates the unfavorable slipping when set in motion.
Data concerning the movement of the stone were collected using the video measurement technique with help of the video analyse software -Tracker [7].In order to use the auto-tracking function effectively, green dots were glued to the stone as markers.Short video clips were recorded simultaneously from two perspectives as shown on the figure 2. For the purpose of video recording two smartphone cameras were used.The rotational motion recordings had a resolution of 1920 x 1080 px and a frequency of 120 fps.The transverse vibration recordings had a resolution of 1280 x 720 px and a frequency of 120 fps.The stone rested on a sheet of graph paper, the smartphone recording the transverse vibrations was placed sideways 11 cm from the stone, and the smartphone recording the rotation was placed on a tripod 26 cm above the stone.It should be noted that the distances from individual elements of the system were chosen based on the size of the stone and the parameters of the video cameras.In the described arrangement they were positioned so that the entire stone was visible at all times during the movement.

Data collection
A series of simultaneous recordings were made from each of the two perspectives.Only videos with sharp images were selected for further analysis.Figure 3 and 4 presents some frames extracted from the analysed recordings.The recordings were imported into the Tracker program, which allows to automatically track the movement of a selected point in the recording frame by frame, saving its position relative to the selected coordinate system.The coordinate system for rotational recordings (the above perspective) was placed at the center of the stone.Due to slight changes in the position of the center of the stone resulting from its sliding on the surface, the values of the position of the center of the stone were collected using the program, and then the coordinate system was set according to the collected data.In the next step, the values of the position of the point on the end part of the stone were collected.On this basis, the program calculated the value of the rotation angle relative to the established coordinate system.The coordinate system for the transverse vibration recordings was set laterally, so that in the stone's equilibrium position (at rest), the horizontal axis intersected the center of the marker by which the value of the deflection from the equilibrium position was measured.
Two measurement series were made differing in the way the stone was set in motion at first.The first attend (series of measurements) was to set it in rotation (figure 5), while the second was to set it in transverse vibrations (figure 6).

Results
In order to calculate the angle of deflection from the equilibrium position in transverse vibrations from the obtained values of linear deflection, the law of cosines was used.The angle of rotation according to descriptions given in the theoretical bases is marked with the symbol , while the angle of deflection in transverse vibrations is marked with the symbol .
In the figures 5-6 the initial time was determined at the moment when transverse vibrations appear.Due to the fact that the measurements in the first series were performed from the moment the stone was set in motion, the stone had time to rotate even several times before transverse vibrations appeared.Hence such high values of the initial rotation angle.Some data collections from the recordings were completed earlier due to the marker disappearing from the frame.
For comparison, the modelling results obtained in [6] are presented in the upper right corners of selected graphs.These graphs show the modelling results for different initial conditions in different colours.The experiment was carried out for various values of initial conditions, but due to a space limitation only two examples are presented in the paper.The differences in angle values are a consequence of a different angular velocity and, consequently, of the angle covered by the stone in the experiment.The sharp break in data in figure 6 is caused by the fact that the stone leaves the space of video recording.

Educational benefits
Such "magical" objects as the Celtic stone arouse learners' interest.Its unusual behavior is complex, but that is why it adequately illustrates the need for an suitable selection of research tools and procedures.The understanding of the Celtic stone physical behavior is possible only when different methods of investigation are combined.
The mathematical model introduced to students illustrates that science knowledge can be used to predict the motion of even the most complex objects.It also brings the natural and obvious consequence that the model needs experimental confirmation.To do so students are forced to use their smartphones to record the movement of the object and to analyze its movement using easy-to-use, publicly available and free tool.
Comparison of students' results with those presented in scientific works serves to build research competences and feel satisfied with completing the task.Such research also enables students to develop methodological skills and to search for adequate solutions of experiential challenges.

Conclusions
The preliminary measurement results obtained in the student exercise formula describing time dependency of angle of rotation (4) and angle of deflection (6) are consistent -with limitations described above -with the results and theoretical predictions tasted with the numerical model of other authors [6].Analysis of the results can be used to determine the physical parameters of the stone such as its moment of kinetic friction or rolling friction of the stone but more recording devices are necessary to fill data gaps.
The Celtic stone's intriguing behaviour can be used to engage students and the general public in the basic ideas of physics.Demonstrations with the stone can make abstract concepts more tangible and enjoyable.Such a complex problem also forces to draw attention to a number of physical quantities and procedures allowing in-depth analysis of other science problems in the future.
Widely available modern multimedia technology such as video analyse procedures can stimulate high school students to examine certain interesting physics phenomena in detail and compere results with those obtained at sophisticated laboratories.This may encourage more people to take an interest in physics.The use of modern technologies in everyday education is one of the strategies to maintain the highest level of education.

Figure 1 .
Figure 1.Characteristics of the Celtic stone as in [6].

Figure 3 .
Figure 3. Successive frames of the recording from the above perspective.The red circle marks the location of the point at the end of the stone.The blue circle marks the location of the center of the stone.The pink lines represent a fixed coordinate system.

Figure 4 .
Figure 4. Successive frames of the recording from site perspective.The blue circle is the marked location of the item's center.The pink lines are the selected coordinate system.

Figure 5 .
Figure 5.The stone set in rotation.The graphs show the change in the rotation angle over time (above) and the change in the tilt angle over time (below).

Figure 6 .
Figure 6.The stone set into transverse vibrations.The graphs show the change in the rotation angle over time (above) and the change in the tilt angle over time (below).

Table 1 .
Mass and dimensions of the stone.