Verifying the equation for centripetal force: an augmented reality approach

According to Newton's second law, a net force acts on an object moving uniformly on a circular path, causing a centripetal acceleration (${a_r}$) of the object. This net force is traditionally called centripetal force. Since the centripetal acceleration is directed to the centre of the circular path, the centripetal force is also directed to the centre. Although during the object's circular motion the direction of the centripetal force is continually changing, the magnitude is constant:

$$\begin{equation}{F_{{\text{cp}}}} = m{a_r} = m{\omega ^2}r = m\frac{{{{\left( {2\pi } \right)}^2}}}{{{T^{\,2}}}}r\end{equation} \tag{ 1 }$$

where $m$ is the mass of the object, $\omega$ its angular speed, T the orbital period, and r the radius of the circular path [1].

Understanding the concept of centripetal force is of great importance for comprehending the dynamics of circular motion. However, past research shows that traditional instruction at all educational levels is relatively ineffective when it comes to developing an understanding of force and motion. This already holds for motion in one dimension and becomes even more evident when students start to learn about motion in two dimensions [2–4]. A common misconception about circular motion is that an object will continue to have a circular trajectory even after the centripetal force becomes zero, which is known as the notion of circular impetus [5]. Furthermore, students often misinterpret the general equation for centripetal force by claiming that a larger radius of the circular path is necessarily associated with an increased magnitude of the centripetal force [6]. Thereby, these students forget that the direct proportionality between centripetal force and radius only holds if the object's mass and angular speed are held constant.

This failing to take into account the 'third variable' is a well-known reasoning obstacle in many areas of physics [7–9]. A powerful approach to confronting this reasoning obstacle is to develop in students the habit of always thinking about all the variables that may be relevant to a physical situation. This could be most elegantly accomplished through learning activities in which students experimentally investigate the relationships between physical concepts. In such activities, students typically observe how varying one variable influences another variable, while holding all other variables constant. Besides helping the students to develop the habit of thinking about the 'third variable', conducting physics experiments helps them develop more vivid and intuitive mental models of the physical phenomena at hand. Both arguments are relevant for advocating an experimental approach to learning about centripetal force. In fact, the general equation for centripetal force includes more than two variables, and the concept of centripetal force is not intuitive to many students (e.g. it is not clear to them why this force does not always move the orbiting body to the centre of the circular path). Furthermore, it is highly desirable to start learning about centripetal force by letting the students vividly explore the factors that influence the magnitude of centripetal force. However, accurate experimental investigation of the centripetal force between two objects may require devices that are expensive or procedures that are demanding to implement in the typical school setting. The experimental verification of the general equation for the centripetal force is traditionally often performed through a conical pendulum experiment (figure 1). The magnitude of the centripetal force (here it is a component of the tension force) is varied by changing the mass of washers/screw nuts attached to the bottom end of the cord. To determine the dependence between F and the angular speed of the cork, period measurements are usually carried out, since the angular speed is directly related to the orbital period. In addition, by varying the radius of the orbit (based on the varying length of the cord above the top of the glass tube), the dependence between F and r can also be explored.

Zoom In Zoom Out Reset image size

However, from our experience with university students, we can say that they struggle to overcome several obstacles when conducting this experiment. On a conceptual level, students sometimes confuse the length of the cord above the glass tube with the radius of the cork's circular path. They also struggle very much with obtaining 'the right speed' of the cork necessary to keep the radius of the orbit constant during the period measurements, which can take quite a bit of lesson time. It also proves to be quite difficult to swing the cork and measure time at the same time, which means that students usually have to conduct this experiment by working in pairs. Another shortcoming is related to the occurrence of friction between the cord and the top of the glass tube. Due to all these practical and conceptual difficulties, verifying the relationship for centripetal force becomes a relatively stressful experience. This may be one of the reasons why some physics teachers avoid conducting the above-mentioned experiment in their physics classes. On the other hand, even in cases where physics experiments are conducted in the classes, students very rarely have the opportunity to investigate the relationships between physics concepts on their own. Therefore, besides the traditional physics experiment, it is advisable to use alternative methods for exploring relationships between physics concepts [10, 11]. This also holds for verifying the general equation for centripetal force. Apart from the conical pendulum context, a potentially productive context for discussing the centripetal force would be the motion of a satellite. For a satellite performing a uniform circular motion, the net force acting on it (i.e. the centripetal force) is the gravitational force between the satellite and the central body. A theoretical advantage of the satellite motion compared to the conical pendulum context is that for the students it is obvious what the radius of the satellite's circular path is, i.e. the chance of confusing the radius with another measure (e.g. the length of the string in the conical pendulum experiment) is minimal. To allow for expedited collecting of measurements about a satellite's motion, one can conduct a virtual experiment. Previous research has shown that virtual experiments are often at least as effective as traditional ones in developing conceptual understanding [12–15]. Students are particularly enthusiastic about conducting virtual experiments based on augmented reality applications (AR) that present digital content within the context of a real-world environment [16]. Concretely, past research showed that using AR in physics education is often associated with increased learning satisfaction, motivation, engagement, and development of understanding [12, 16–19]. Students often state that they like using AR environments more than traditional experimental setups, while teachers report that AR visualizations help students connect visual learning information to relevant physical objects [6, 20, 21]. Therefore, we decided to develop an Android-based AR simulation to help students 'experimentally discover' the mathematical equation for centripetal force. We named the application CEntripetalForceAR (CEFAR). Next, we will describe the most important technical and didactic features of CEFAR, as well as university students' experiences with using CEFAR for learning about centripetal force.

CEFAR can be easily downloaded to Android mobile devices via the quick response (QR) code (appendix). Mobile devices were chosen as a suitable tool for our AR simulation because they are accessible to students, easy to use, inexpensive, and previous research has discovered many learning advantages of mobile devices-based AR [22–24]. For the development of our Android-based AR simulation, we used Unity (real-time development game engine) and Vuforia (AR software development kit for mobile devices) [25, 26]. In developing the application, we have tried to take into account the most important principles of multimedia learning, such as the principles of interactivity, spatial contiguity, and segmenting principle [27]. In addition, to minimize irrelevant cognitive load, we attempted to avoid unnecessary graphical elements [28].

The final version of the mobile application is presented through figures 2(a)–(e). It provides a dynamic visualization of a satellite orbiting around a central body. Thereby, the only force acting on the satellite is the attractive gravitational force, which causes centripetal acceleration of the satellite:

$$\begin{equation}G\frac{{m{M_{\text{c}}}}}{{{r^2}}} = \frac{{m \cdot {v^2}}}{r} = \frac{{4{\pi ^2}}}{{{T^{\,2}}}} \cdot m \cdot r\end{equation} \tag{ 2 }$$

Zoom In Zoom Out Reset image size

where ${M_{\text{c}}}$ is the mass of the central body, $G$ the universal gravitational constant, T the orbital period, and $r$ the distance between the centres of the bodies, i.e. the radius of the circular path.

From equation (2) it can be easily derived that:

$$\begin{equation}v = \sqrt {\frac{{G \cdot {M_{\text{c}}}}}{r}} \end{equation} \tag{ 3 }$$

$$\begin{equation}T = \frac{{2\pi \cdot {r^{3/2}}}}{{\sqrt {G \cdot {M_{\text{c}}}} }}.\end{equation} \tag{ 4 }$$

It follows that for a fixed central body and radius of the circular path, the satellite's speed and orbiting period do not depend on the satellite's mass, as well as that period of motion increases with increasing orbital radius.

In CEFAR, the user can change the mass and orbital radius of the satellite using the Increase/Decrease buttons. Both quantities have minimal values present in the application. The minimal value for satellite's mass is 500 kg. For the orbital radius it is 6400 km which prevents the students from setting a value of r smaller than the radius of the planet. Additionally, the central body can be changed: Earth, Venus, and Mars are provided as options. Through these options, we have achieved a high degree of interactivity. Specifically, by changing the mass of the satellite m for a fixed r, the user can clearly observe that the speed of the satellite does not change, even though the magnitude of the centripetal force F (i.e. the gravitational force) changes. In addition, changing the mass m allows the user to investigate the relationship between centripetal force and the mass of the orbiting body. Furthermore, we can change the central body (i.e. the magnitude of the centripetal force F) and observe how this affects the orbital period of the satellite T, while r and m remain constant. By repeating this procedure for two different values of r, we can demonstrate the relationship between F and r (for constant values of m and T) as will be described later in detail. In addition, by changing the radius of the orbiting path while the masses of the two bodies remain constant, we can observe how the orbital radius is related to the speed of the satellite: smaller speeds are associated with a larger orbital radius and a larger orbital radius automatically means a smaller gravitational force. This application feature may help the students to overcome the earlier mentioned misconception that 'a larger radius of the orbit always implies a stronger force'. Finally, the central body can be completely removed at one moment, which means that the gravitational force on the satellite becomes zero, and the satellite after that moves along the direction of the velocity (i.e. tangential direction) it had right before the gravitational force decreased to zero. This may help students to overcome the misconception of circular impetus (figure 2(d)).

An advantage of virtual experiments is that the useful abstract representations of physical concepts can be overlaid onto the real-world representations of physical objects. In CEFAR, we have tried to follow the spatial contiguity principle by a good spatial organization of all relevant information, such as showing the centripetal force vector (red arrow), the velocity vector (blue arrow), and the numerical values of the relevant quantities, all close to each other (figure 2(b)). It can be easily observed that F is always perpendicular to v, which means that the work which Earth performs on the satellite is zero and consequently the satellite's speed does not change. Also, a Pause button is implemented to provide the user with greater control over her/his process of learning (figure 2(a)). Having control over learning is consistent with Mayer's cognitive theory of multimedia learning in which he suggests that students should control multimedia at their own pace [27]. For the same purpose, the Options button provides a Back option that allows the user to go back to the previous graphical scene. Additionally, the Colour button has been implemented to allow the user to change the colour of the force magnitude for the best viewing experience (figure 2(c)).

Next, we present some possible learning activities related to verifying the general equation for centripetal force ($F = \frac{{4{\pi ^2}}}{{{T^{\,2}}}}m \cdot r$), as well as related to helping students overcome some of the most common misconceptions about circular motion.

3.1. Identifying how centripetal force is related to orbital period

3.2. Dependence between centripetal force and orbital radius

The purpose here is to compare centripetal force measurements for two different values of the orbital radius while holding constant the mass of the satellite and orbital period, i.e. the orbital frequency. Concretely, the previous activity is repeated with the only difference that now the radius of the orbit is doubled:

• (a)  
  Using the T values obtained in activity 1, find corresponding f² values.
• (b)  
  Double the value of the orbital radius r for the satellite compared to activity 1 (e.g. set it to 13 200 km). Find the force F and orbital period T for the satellite orbiting the Earth. Repeat the procedure with the same radius of 2r for the planets Venus and Mars.
• (c)  
  From the values for T obtained in (b), find corresponding f² values.
• (d)  
  Based on the findings from activity 1 you know that $F \propto \frac{1}{{{T^{\,2}}}}$, which is equivalent to $F \propto {f^{\,2}}$. On the same plot, draw two $F\left(\, {{f^{\,2}}} \right)$ graphs, one for orbital radius r, and the other for orbital radius 2r.
• (e)  
  On the obtained graph, draw a vertical line at an arbitrarily chosen ${f^{\,2}}$ value. This line represents a constant value of the squared frequency (and a constant value of the orbital period).
• (f)  
  From the intersection of the above-mentioned vertical line with the two $F\left(\, {{f^{\,2}}} \right)$ graphs, find the two values of the centripetal force corresponding to the chosen value of ${f^{\,2}}$ . What can you say about F(r) dependence?

Table 2 presents possible measurements of the orbital period obtained for doubled orbital radius.

Table 2. Example of experimental measurements and calculations related to identifying how force depends on the period—radius 2r.

		F (N)	T (s)	T2 (s2)	FT2 (Ns2)
2r (13 200 km)	Earth	2287.49	15 093.42	2.28 × 108	5.21 × 1011
	Venus	1864.23	16 719.26	2.80 × 108	5.21 × 1011
	Mars	245.80	46 044.55	2.12 × 109	5.21 × 1011

Table 3 contains the values of the centripetal forces and the corresponding squared frequencies, obtained from the data in tables 1 and 2.

Table 3. Measurements and calculations related to identifying how force depends on the orbital radius.

		F (N)	${f^{\,2}}$ (s−2)
r (6600 km)	Earth	9149.94	3.51 × 10−8
	Venus	7456.92	2.86 × 10−8
	Mars	983.19	3.77 × 10−9
2r (13 200 km)	Earth	2287.49	4.39 × 10−9
	Venus	1864.23	3.58 × 10−9
	Mars	245.80	4.72 × 10−10

From the obtained plot (figure 3) it is evident that for a fixed value of frequency (i.e. period) and one and the same mass of the orbiting body, the centripetal force increases two times if the orbital radius increases two times.

Zoom In Zoom Out Reset image size

3.3. Identifying how the centripetal force depends on the mass of the orbiting body

Next, students may be asked to perform the following activity:

For the constant orbital radius and the Earth as the central body, please vary the satellite's mass, and for several different values write down the corresponding force. Enter the data for $F$ and $m$ into a spreadsheet. Create the $F\left( m \right)$ scatterplot. What can you conclude about the $F$–$m$ relationship?

From the obtained plot, the students can conclude that the centripetal force is proportional to the satellite's mass. An example of an obtained graph is given in figure 4.

Zoom In Zoom Out Reset image size

3.4. Determining the constant of proportionality

In the previous three activities, students could discover that the centripetal force is directly proportional to the mass of the orbiting body, to the orbital radius, and inversely proportional to the square of the orbital period. Next, students can determine the constant of proportionality within the equation for centripetal force by completing the following task:

• (a)  
  Please set the mass of the satellite to 1000 kg and vary the magnitude of the orbital radius around the Earth. For several different values, write down the corresponding force and period. Enter the data for F, T, and r into a spreadsheet. In two new columns, calculate the $mr$ and $F{T^{\,2}}$ values. Create the $F{T^{\,2}}\left( {mr} \right)$ scatterplot. What can you conclude about the $F{T^{\,2}}$–$mr$ relationship?
• (b)  
  Add a trend line to the graph and find its slope. Write down the slope value in terms of the number ${\pi ^2}$ . Write down the complete equation for F in terms of m, T, and r.

From the plotted graph and the trend line equation, students can relatively easily conclude that the slope is 4${\pi ^2}$ (figure 5), and conclude that $F{T^{\,2}} = 4{\pi ^2}mr$.

Zoom In Zoom Out Reset image size

3.5. Identify the relationship between the speed of the satellite and the radius of the orbit

3.6. Predicting effects of 'deactivating' the gravitational force

For the process of designing and improving applications, it is very useful to conduct user experience surveys. In the first step, we decided to investigate the CEFAR'S usability, which is defined in terms of International Organization for Standardization (ISO) ergonomics as the effectiveness, efficiency and satisfaction with which specified users can achieve specified goals in a particular environment [29]. To that end, 163 first-year students of chemical engineering and technology enrolled in physics course were asked to use CEFAR for conducting previously described learning activities. Thereafter, the students completed a survey consisting of three open-ended and ten Likert scale items from a standardized questionnaire, the system usability scale (SUS) [30].

SUS includes ten items providing a composite score that reflects the usability of the application as perceived by an individual user. If the total score is higher than 68, the system is considered suitable for use [31]. Questions from the SUS questionnaire with the percentage of answers are presented in figure 6.

Zoom In Zoom Out Reset image size

The CEFAR application achieved a total average score of 89.4, placing it in an excellent rating according to the SUS scoring methodology of Bangor et al [32].

The other part of the survey consisted of three open-ended questions:

Q1: What did you like best about the application?

As can be seen from figure 7, the majority of students liked the ability to visualize the physical problem and the ease of use. Here it is useful to note that visualization is well known for facilitating development of conceptual understanding and ease of use is important for preventing irrelevant cognitive load [28, 33]. Some other comments about what students like in application were: 'The way it evokes physics through the game', 'Experimentation', etc.

Zoom In Zoom Out Reset image size

Q2: What would you change about the application or in the ways of working with it?

We were particularly interested in receiving feedback which may help us to improve our application. To that end, we asked students what they would change about the application and ways of using it. Forty percent of the students stated that they would not change anything (figure 8). The biggest problem for the remaining students was that the application was only for the Android operating system, so they could not use their IOS mobile phones and had to borrow an Android phone from their friends or family for implementing this task. Some of the students suggested improvements in the design of the application. Particularly useful was the suggestion to add more planets. In fact, by adding more planets we would be in position to have more measurements in the first three learning activities, suggested above.

Zoom In Zoom Out Reset image size

Q3: Do you believe the application has helped you develop a deeper understanding of centripetal force? Describe your experience.

Eighty nine percent of the students stated that the application helped them to develop a deeper understanding of centripetal force (figure 9). Most of them answered in a similar fashion. They found the visualization of physical situations very helpful and also liked the opportunity to experiment. (e.g. 'Yes, because I saw in 3D what actually happens, which was easier to understand than just the theory.', 'Yes, experimentally I came up with an expression that defines the centripetal force.', 'Yes, because we see how centripetal force affects objects.', 'Yes, it is easier for me to visualize what is happening.') These students' answers are in line with the idea that exploring the cause and effect relationships within a visual context facilitates development of conceptual understanding of physics [34].

Zoom In Zoom Out Reset image size

Some students also noted that it was very helpful to see what would happen to the satellite's trajectory if at one moment the Earth simply disappeared.

Majority of the students who answered this question negatively stated that the application did not help them because they already understood the concept very well. (e.g. 'I think that it did not help me improve my knowledge, but to prove what I already know in a slightly different way (without deriving formulas and the like, you can come to every conclusion yourself).')

In this paper, we described a novel augmented-reality application that was developed for purpose of providing the students with the opportunity to experimentally verify the general equation for centripetal force within a vivid and practical environment. The CEFAR application can be easily downloaded to Android mobile devices and used in the physics classroom or distant learning settings. By allowing the students to experiment on their own, the use of CEFAR can help in strengthening the learners' scientific skills. The activities suggested in this paper may be particularly useful for developing conceptual understanding about centripetal force and strengthening skills related to certain aspects of mathematical modelling of experimental data. In our user experience survey which included 163 university students, CEFAR got an excellent usability rating. Students particularly liked the visualization of physical phenomena, the interactivity features, and the ease of use. Most of them believed that the application helped them to improve their understanding about centripetal force, which was mainly explained through the application's feature of visualizing cause and effect relationships. In our opinion, the CEFAR can also help the students to overcome some common misconceptions related to circular motion, such as the notion of circular impetus. In our future work, it would be certainly useful to implement some of the suggestions obtained through the user experience survey, such as the suggestion to increase the number of 'offered' planets and to design an IOS version of CEFAR. It would be also useful to develop an alternative version of CEFAR which would include random experimental error. Clearly, that alternative version of CEFAR would be more appropriate for organizing activities which require analysing experimental error, whereas the version presented in this paper is probably somewhat more appropriate when the primary goal is to develop conceptual understanding and experimental skills that do not include reasoning about experimental error [35]. When it comes to the suggested learning activities, an alternative to the confirmation inquiry approach presented in this paper would be to simply ask students to use CEFAR for purposes of verifying the general equation for centripetal force, i.e. to engage in open inquiry. Open inquiry would probably be a better option for strengthening students' scientific skills, particularly when it comes to the skill of planning and designing experiments. Another potential advantage of the open-inquiry approach is related to the development of creative thinking. However, more educational research is needed to explore the various effects of using different versions of CEFAR in learning physics.

All data that support the findings of this study are included within the article (and any supplementary files).