Study of circular and rotational motion with the help of experimentation and mathematical modeling enhanced by digital technologies

Circular and rotational motion belongs to basic topics studied at upper secondary school and courses on Mechanics at University. Many studies show problems in understanding basic concepts related to kinematics and dynamics of these motions. In this paper we present a series of inquiry activities aimed at developing the understanding of these concepts and the laws of circular and rotating motion with the effective use of digital technologies. The activity on the videomeasurement of a uniform and decelerating bicycle wheel motion analyses the basic kinematic concepts. In the experiments carried out with the help of a centripetal force apparatus with the rotating system of carriages, the dynamics of circular motion can be investigated by observing and analysing the effect of various parameters on the centripetal force acting on the carriage. In addition to that, a mathematical model for decelerated motion of a bicycle wheel and a carriage can be designed. By manipulating and simulating the model for different parameters the best correspondence with experimental data can be found. The activities can be implemented at confirmation or guided inquiry level. Applying experimental as well as modelling approach to answer research questions students learn how science and physicists work.


Introduction
Study of circular and rotational motion belongs to the basic topics of upper secondary physics curriculum as well as university course on mechanics.Even though this motion is among the phenomena that anyone experience every day, students experience many problems in learning the concepts involved in circular or rotational motion.Large number of studies were aimed at researching students´ difficulties and misconceptions in circular or rotational motion.The studies deal with understanding of various concepts, such as linear and angular speed and velocity, acceleration and forces acting on objects in circular or rotational motion [1] [2].Students usually assume that an object in uniform circular motion experiences constant velocity similar to an object in a uniform linear motion.This misconception leads to the absence of an acceleration that occurs in this case as a result of changing direction of velocity.In addition to this, there is a confusion between the linear speed and angular speed of a rotating object as well as difficulties in understanding conversion between linear and angular quantities.Significant difficulties are connected with the forces acting on the object in a circular motion [3] [4] [5].Students may think of a centripetal force as an extra force rather than a force resulting from the net force acting on the object.One of the central misconceptions involves confusion between centripetal and centrifugal forces.Students often believe that when an object is in circular motion, there is a real force acting on it in a radial direction pulling it out of the path, however the force responsible for keeping the object in circular motion is the centripetal force acting towards the centre of the circle [4].Many studies have revealed that students have difficulties in understanding circular and rotational motion and so that there are attempts to help students by designing activities that can overcome problems or help to avoid misconceptions in learning the corresponding concepts.The use of digital technologies and inquiry-based approach to teaching and learning these concepts were found to be effective in developing deeper understanding in this field.In the following sections a set of activities aimed at conceptual understanding of circular and rotational motion are described in detail.

Activities on investigation of rotational motion
In order to support the development of conceptual understanding of circular and rotational motion concepts we have designed three activities that investigate a rotational motion of a wheel and a system of two carriages positioned symmetrically on a rotating horizontal beam.They are designed as inquiry activities from simple activity aimed at understanding basic kinematics concepts up to more and more complex activities gradually involving dynamics of rotational motion introducing additional concepts and relationships.

How a bicycle wheel moves?
In order to understand the basic kinematic concepts of circular motion a simple activity on a videomeasurement of a rotating bicycle wheel motion is designed [6].In this activity students investigate two videorecordings of a wheel motion to collect data about the position of a selected point on the edge of the wheel and the angular displacement calculating linear and angular speed.This way they analyse how the wheel moves in detail in both cases.By fitting the angular displacement with an appropriate function students find out how the wheel moves (figure 1).In the observed short time period the motion is uniform for the first motion or uniformly decelerated for the second motion.

What force keeps the carriage in a circular motion and what does it depend on?
For detailed study of the dynamics of rotational motion the following experimental setup can be used.
It involves two carriages placed on a horizontal beam that can rotate about its centre on a spindle rod [7].A force sensor measures the centripetal force exerted on a carriage through a string that connects the carriage with the sensor.A rotational sensor is used to measure the angular displacement and the angular speed.Different masses can be added to the platform and positioned at variable radii (figure 2).The horizontal beam with carriages is set to a rotational motion while the force sensor collects data about the force that the carriage exerts on the string that is equal in magnitude as the force exerted by the string on the carriage.With the help of this experimental setup students answer the research question on what force keeps the carriage in a circular motion and what this force depends on.Before starting the experiment, the first part of this question needs to be answered.It is very important that students are really aware of what force keeps the carriage in circular motion and how this force is measured.Observing the experimental setup students realize that it is the string that does not allow the carriage to move away providing the centripetal force that keeps the carriage in the circular path.The force sensor measures the force that the carriage exerts on the string, however it is equal in magnitude with the force that the string exerts on the carriage (centripetal force).
The activity itself can be conducted at different levels of inquiry with more or less teachers´ guidance.
We have developed two variants, i.e. confirmation or guided inquiry (results are known or not known by students) that can be implemented depending on the intellectual level of students.In each case students are expected to follow the steps of the inquiry cycle framework for experimentation from conception, planning and design of the experiment, through implementation of the experimental design, data analysis and interpretation and communication of gained results [8].
In the experimental design it is crucial to decide about the studied relationship, i.e. which variables are manipulated (independent variables), what are the effected variables (dependent variables) and which variables are kept constant (control variables).In the following text we present three experiments that students are expected to design, as well as expected results, namely how different carriage masses, different radial distances and different angular speeds effect the centripetal force acting on the carriage.
In figure 3 typical experimental results can be seen.

Effect of angular speed on centripetal force (constant radius and carriage mass)
In this experiment the carriage with constant mass is placed to a certain distance from the centre.The force-angular speed diagram is displayed and indicates non-linear relationship.By fitting data with a quadratic function and displaying the force -squared angular speed the linear relationship can be seen.
In figure 4 seven experimental results for seven different carriage masses for the radius of 10 cm with corresponding fit functions can be seen.Figure 5 shows the corresponding forcemass diagram.

Effect of radius on centripetal force (constant angular speed and carriage mass)
In this experiment the carriage with constant mass is placed to a certain distance from the centre and set to rotational motion.This experiment with the same carriage is repeated for several radii of circular motion.Displaying the force -squared angular speed the linear relationship can be seen.In figure 6 six experimental results for six different radii for the carriage mass of 150.6 g with corresponding fit functions are displayed.In each set of data, the same angular speed and the corresponding centripetal force is selected.In figure 7 experimental results for six different radii with corresponding fit function under the constant angular speed and carriage mass can be seen.

Effect of carriage mass on centripetal force (constant radius and angular speed)
In this experiment the carriage with a certain mass is positioned to a selected distance from the centre and set to rotational motion.This experiment with the same carriage is repeated for several carriage masses keeping the carriage in the same radius of circular motion.In each set of data, the same angular speed and the corresponding centripetal force is selected.The force-mass diagram is displayed and indicates a linear relationship that is fitted by a linear function.In figure 5 experimental results for seven different carriage masses with corresponding fit function under the constant angular speed and radius can be seen.

Comparison of experimental and theoretical results
In case that the activity is conducted as a confirmation inquiry, students can compare the experimental data with the values resulting from the theory ( =  •  •  2 ).In table 1 details of the experiment on the effect of angular speed for seven different carriage masses can be seen.The coefficient k of the linear fit  =  in the Force-squared angular speed diagram (figure 4, kexper in table 1), represents the value of calculated  ℎ =  •  where m is mass and r is radius of circular motion.Students can follow with other experiments in the same sense.

Mathematical modelling of rotational motion
From the previous activities a new research question may arise.Having a closer look at the experimental results of angular speed of the decelerating wheel and decelerating carriages students may conclude that in case of carriages the speed does not decrease linearly compared to the wheel.In both cases there is a resistance to motion represented by a resistive torque acting on the rotating object.According to the rotational form of Newton´ s second law, the net torque  applied to an object (resistive torque in this case) causes angular acceleration  that is directly proportional to the net torque and inversely proportional to object´s moment of inertia I,  = /.This leads to solving a differential equation Applying this equation, students can develop a mathematical model of a concrete motion and test the model comparing it with the experimental data.The equation can be solved either analytically or numerically.

Mathematical modelling of a rotating wheel decelerated motion
In case the wheel motion, a constant resistive torque can be assumed producing a constant angular acceleration.Taking into account the direction of constant torque of magnitude c that opposes the angular velocity, the differential equation,

Mathematical modelling of the rotating system of carriages´ decelerated motion
In case the system of carriages, the net torque is apparently not constant since the angular speed does not decrease linearly.As suggested by Mungan [9] and others [10], the resistive torque varies linearly with the angular speed, with opposite direction to the angular velocity, where b is a constant coefficient.The angular acceleration is not constant; it changes with angular speed.
while  = . Solving the differential equation, the following analytical solution be obtained for angular speed  and angular displacement Equation ( 8) can be rewritten as In figures 10, 11 experimental results are fitted with functions based on the equations ( 9) and (10).Good correspondence is found for  0 = 15.41 rad • s −1 and  = 0.0534 s −1 .

Conclusion
We have developed a set of inquiry activities that are intended to be implemented at University Introductory Mechanics course (all activities) and upper secondary school (activities 1.1,1.2,1.3.1) at confirmation or guided inquiry level depending on the intellectual level of students.They are designed to gradually develop an understanding of concepts related to circular and rotational motion.With the help of digital technologies, such as experimental and modelling tools [7] [11], students can easily collect, process and analyse data as well as code mathematical models.Since they work in the same learning environment [11], the model can be easily compared with the experimental data in order to validate the designed theory.Applying experimental as well as modelling approach to answer research questions students learn how science and physicists work.

Figure 1 .
Figure 1.Angular displacement-time (blue) and angular speed-time (turquois) diagrams with fit functions (red) for uniform and uniformly decelerated motion of a bicycle wheel

Figure 6 .
Figure 6.Force-squared angular speed diagram for six different radii

Figure 7 .
Figure 7. Force-radius diagram for six different radii, m =150.6 g,  =10.12 rad/s results in the following analytical solution for angular speed and angular displacement () =  0 − , () =  0 +  0 of constant angular acceleration is  =   ,  0 and  0 are the initial values of angular speed and angular displacement.In figure8a mathematical model based on numerical solution of equation (1) can be seen.By setting the values of angular acceleration as well as the initial values the model results (red line in figure9) show good correspondence with the experimental data (turquoise or blue crosses).

Figure 8 .Figure 9 .
Figure 8. Model of a bicycle wheel uniformly decelerated motion

Figure 12 .
Figure 12.Model of carriages´ rotational motion

Table 1 .
Comparison of experimental and theoretical values