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Received: December 09, 2011 / Accepted: December 28, 2011 / Published: February 15, 2012.
Abstract: An activity of first year university students examining the velocity or acceleration of the rolling motion of cylindrical objects down a slope without slipping is reported. This activity contained the problem comparing the motion of the drums filled with three kinds of liquids, water, olive-oil and honey. At first students gave their prediction on the order of the velocity of three drums, next the demonstration and experiment were conducted, then the author explained about the reason of the order and the dynamics of these objects. It was clarified that observing the motion of bubble inside the drum is effective to understand the motion of liquid and the order of the velocity or acceleration of the drum. The problem comparing the motion of the drums half filled with three kinds of liquids, water, olive-oil and honey were also examined. It was clarified that the magnitude of the acceleration in this case strongly depends on the viscosity of the liquid and the configuration of the liquid surface inside the drum. These themes are effective to promote the students’concern to the rigid body dynamics.
Key words: Experiment, cylindrical object, viscous liquid, rolling motion.
1. Introduction??
In recent years Active-Learning method is widely accepted in the university physics course [1, 2], and the author has a sympathy with this method. However, the topics related to the moment of inertia have not yet taken up enough in the practice of this method though it is important for the understanding of rigid body dynamics. The author has conducted some trial in this field [3, 4]. This time the author conducted an educational activity on the theme “Research on rolling motion of cylindrical objects down a slope” in a preliminary seminar for three first-year students in the Department of the Mechanical Systems Engineering in a university. This theme consists of three parts:“Comparison between the velocity or acceleration of circular disk and the ring of rigid body” as PART I,“Comparing the motion of drum filled with three kinds of liquid” as PART II, and “Comparing the motion of drum half filled with three kinds of liquid” as PART III. The reason that the author set up this theme is as follows: at first the demonstration and experiment on these assignments (problems) can be easily conducted and second it seems that the PART II and III contain some unsolved fresh problems. The objective of this paper is to report the detail of this activity.
2. Method
The theme of this activity consists of above-mentioned three parts. The following learning method is adopted in each part: at first “Question and Prediction”, second “Demonstration or Experiment”and third “Explanations”. The “Question” and the method of “Demonstration or Experiment” are described in this chapter, while the students’predictions, the results of “Demonstration or Experiment” and “Explanations” are described in the following chapters.
2.1 Comparing the Velocity or Acceleration of the Circular Disk and the Ring of Rigid Body
At first the following question was given to the three students.
Question: The velocity or acceleration of which object is larger if a circular disk and a ring of rigid body are set up at the same level and are released at the same time? The students give their predictions to this question.
Next, a demonstration on this question was conducted as follows. A wooden circular disk of 95 mm diameter and 12 mm thickness and a wooden ring of 95 mm outer diameter, 85 mm inner diameter and 10 mm thickness on the market (Fig. 1) were used for the demonstration. They were set at rest at the same level on the two parallel guide lanes of aluminum (Fig. 2) and were released at the same time. The length of the lane was 2.0 m and the slope angle was set up about 5 degrees. The answer to the question was shown to the students by this demonstration. The author gave an explanation about the result of this demonstration.
2.2 Comparing the Velocity or Acceleration of the Drum Filled with Three Kinds of Liquid
At first the following question was given to the three students.
Question: When the drums filled with different kinds of liquids are rolling down a slope without slipping, the velocity or acceleration of which drum is the largest?
And which is the smallest?
The roughly estimated ratio of viscosity of each liquid was given to the students with the question as follows:
Viscosity of water:Viscosity of olive-oil:Viscosity of honey = 1:100:10000.
The mass of each liquid was also given to the students by measuring the total mass of each drum containing the liquid and the mass of the material of the drum. The concrete values were as follows:
Mass of water = 202 g, Mass of olive-oil = 178 g, Mass of honey = 290 g.
The students gave their predictions to the question.
Next, a demonstration was conducted as follows. The students had made each drums filled with liquid(Fig. 3) beforehand by the following procedure. At first an acrylic ring of 30 mm width and 3 mm thickness was glued to an acrylic plate of 100 mm diameter and 3 mm thickness with an adhesive agent. Some hours later each liquid was poured into the drum and another acrylic plate was glued to the acrylic ring. At this stage a weight was put on the second acrylic plate till the adhesion became perfect not to leak liquid. Hereafter the drum filled with water is called “Water” and so on. We compared motions of following each pair, Water vs. Olive-Oil, Water vs. Honey, and Olive-Oil vs. Honey in order to clarify the order of the velocity or the acceleration.
Furthermore, a supplementary experiment using video camera was conducted in order to clarify the detail of the motion of each drum. The author gave an explanation about the result of the experiment.
2.3 Comparing the Velocity or Acceleration of the Drum Half Filled with Three Kinds of Liquid
At first the following question was given to the three students.
Question: When the drums half filled with different kinds of liquids are rolling down a slope without slipping, the velocity or acceleration of which drum is the largest? And which is the smallest?
Next, a demonstration on this question was conducted as follows. We compare motions of following each pair, Water vs. Olive-Oil, Water vs. Honey, and Olive-Oil vs. Honey by using the drums half filled with each liquid made beforehand (shown in Fig. 4). The answer to the question was shown to the students by this demonstration. The author gave an explanation about the result of this demonstration.
3. Results
The results of above-mentioned activities were obtained as follows.
3.1 Comparing the Velocity or Acceleration of the Circular Disk and the Ring of Rigid Body
The predictions of students are as follow:
Student A: “Velocity or acceleration of both objects is same”;
Student B: “Velocity or acceleration of both objects is same”;
Student C: “Velocity or acceleration of the ring is larger than that of the disk since the air resistance to the ring is smaller than that to the disk”.
Results of demonstration:
It was observed that the disk moved faster than the ring as shown in Fig. 5a. No students predicted the correct result.
Explanation:
This theme is well known problem in the rigid body dynamics, however, almost all first year students in this period have not yet learned the concept of the moment of inertia. Then the author adopted the explanation based on the conservation of mechanical energy and the relationship between the translational speed of the center of the mass and that of rotational speed of each part of the object. Based on this explanation it is shown that translational speed of the center of the object V is
V?for the disk, wheregx,,?are the sliding distance, slope angle and gravitational acceleration (in Fig. 5b).
3.2 Comparing the Velocity or Acceleration of the Drum Filled with Three Kinds of Liquid
The predictions of students are as follow:
Student A: “Order of magnitude of velocity is Honey-Olive-Oil-Water due to the order of mass”.
Student B: “Order of magnitude of velocity is Honey-Olive-Oil-Water due to the order of viscosity”.
Student C: “Order of magnitude of velocity is Honey-Water-Olive-Oil due to the order of mass”.
Demonstration:
One static figure of the demonstration of Water vs. Olive-oil is shown in Fig. 6. The drum filled with water moved faster than that with olive-oil. It was clarified that the order or magnitude of acceleration is Water-Honey -Olive-Oil.
Supplementary Experiment:
The information about the position and the velocity of the center of the drum at every 0.1 s was obtained from the video picture for each object and the velocity-time graph was made (Fig. 7). It was clarified that in the case of Water and Honey the motion of the drum is approximately uniform accelerate motion while in the case of Olive-Oil complicated one. Approximate values of acceleration of each drum are shown in the figure.
Explanation:
It is available to observe the motion of bubble inside the drum in order to understand the order of acceleration. In the case of water the bubble remains at the top of the water, which means that the motion of water inside the drum is approximately translational motion. On the other hand in the case of honey the bubble rotates with almost same angular velocity as that of tape fixed to the drum, which means that the motion of honey inside the drum is approximately equal to that of rigid body rotation. It is not difficult to understand that the velocity of the drum filled with water is larger than that with honey due to the explanation in Part I. In the case of drum filled with olive-oil the motion of the bubble inside the drum is much complicated, namely it moves not only in the tangential direction but also radial direction, which makes the ratio of kinetic energy of translational motion in this case smaller than in other two cases.
3.3 Comparing the Velocity or Acceleration of the Drum Half Filled with Three Kinds of Liquid
The predictions of students are as follow:
Student A: “Order of magnitude of velocity is Honey-Olive-Oil-Water due to the order of viscosity or centrifugal force”.
Student B: “Order of magnitude of velocity is Water-Honey-Olive-Oil, same as the case of the drums filled with liquid”.
Student C: “Order of magnitude of velocity is Water-Olive-Oil-Honey due to the inverse order of viscosity”.
Demonstration:
It was clarified that the order or acceleration was Water-Olive-Oil-Honey.
Explanation:
In order to explain the result it is effective to observe the forms of the surface of the liquid. In the case of water the form of the surface is approximately straight and slightly inclined to the horizontal direction asshown in Fig. 8a. It suggests that the motion of the liquid is approximately translational. On the other hand in the case of honey the thick layer attaches to the side wall of the drum due to the effect of viscosity asshown in Fig. 8b. This mass distribution of the liquid gives the counter-clockwise torque to the drum and the motion of the drum becomes near the uniform velocity motion.
4. Discussion and Conclusions
In the case of Part II we explained the order of velocity or acceleration based on the behavior of the liquid inside the drum and the consideration of energy transformation. This consideration is endorsed by the following fact. The experimental values of acceleration of each drum Aex were compared with the calculated value of the acceleration of each drum assuming that the liquid motion is translational one Ac1 and rigid body rotation Ac2. In order to calculate Ac1 and Ac2 the mass of the liquid M, that of bottom part of the drum m1 and that of side part of the drum m2 were measured, and the expressions were adopted. These values are shown in Table 1 and it was clarified that Aex of Water is approximately equal to Ac2 and Aex of Honey is equal to Ac1.
After the class the author had students write reports on this activity. Students wrote the following impressions on this activity in their reports.
Student A: The experimental results were different from my predictions at any time. It was very good that we could clarify and understand the reasons through the seminar.
Student B: It would be available to know the detail of the motion of objects rolling down a slope.
Student C: I was surprised to know that potential energy is transformed into translational and rotational kinetic energy. I had never known that when the drum is filled with liquid whether the motion of the liquid is mainly translational or rotational depends on its viscosity. I want to know the behaviors of the drum filled with paste or sauce.
Summarizing, three subjects related to the motion of cylindrical objects rolling down a slope were assigned in the preliminary seminar for first year university student. First assignment was comparing the motion of a circular disk and a ring of rigid body. Second assignment was comparing the motion of drums filled with water, olive-oil and honey. It was clarified that the motion of water is approximately translational, that of honey rigid body rotation and that of olive-oil is complicated one by observing the motion of bubble inside the drum. The order of the velocities and the accelerations of the drums could be understood by these facts. Third assignment was comparing the motion of drums half filled with water, olive-oil and honey. It was clarified that the magnitude of the acceleration in this case strongly depends on the viscosity of the liquid and the configuration of the liquid surface inside the drum. These activities were effective in promoting students’interests and understanding on the motion of objects rolling down a slope.
References
[1] E.F. Redish, New directions of research on undergraduate physics education, in: Proceedings of ICPE2006, Tokyo, 2008, pp. 9-19.
[2] D.R. Sokoloff, R.K. Thornton, Active learning with interactive lecture demonstrations, in: Proceedings of ICPE2006, Tokyo, 2008, pp. 92-96.
[3] O. Hirayama, Development of demonstration apparatuses for rigid body dynamics for first-year student class in university, ABSTRACTS, ICPE2007, Marrakech, 2007, p. 40.
[4] O. Hirayama, Homework on the moment of inertia of a human arm, in: AIP Conference Proceedings, Volume 1263, ICPE2009 (Bangkok), 2010, pp. 110-113.
Abstract: An activity of first year university students examining the velocity or acceleration of the rolling motion of cylindrical objects down a slope without slipping is reported. This activity contained the problem comparing the motion of the drums filled with three kinds of liquids, water, olive-oil and honey. At first students gave their prediction on the order of the velocity of three drums, next the demonstration and experiment were conducted, then the author explained about the reason of the order and the dynamics of these objects. It was clarified that observing the motion of bubble inside the drum is effective to understand the motion of liquid and the order of the velocity or acceleration of the drum. The problem comparing the motion of the drums half filled with three kinds of liquids, water, olive-oil and honey were also examined. It was clarified that the magnitude of the acceleration in this case strongly depends on the viscosity of the liquid and the configuration of the liquid surface inside the drum. These themes are effective to promote the students’concern to the rigid body dynamics.
Key words: Experiment, cylindrical object, viscous liquid, rolling motion.
1. Introduction??
In recent years Active-Learning method is widely accepted in the university physics course [1, 2], and the author has a sympathy with this method. However, the topics related to the moment of inertia have not yet taken up enough in the practice of this method though it is important for the understanding of rigid body dynamics. The author has conducted some trial in this field [3, 4]. This time the author conducted an educational activity on the theme “Research on rolling motion of cylindrical objects down a slope” in a preliminary seminar for three first-year students in the Department of the Mechanical Systems Engineering in a university. This theme consists of three parts:“Comparison between the velocity or acceleration of circular disk and the ring of rigid body” as PART I,“Comparing the motion of drum filled with three kinds of liquid” as PART II, and “Comparing the motion of drum half filled with three kinds of liquid” as PART III. The reason that the author set up this theme is as follows: at first the demonstration and experiment on these assignments (problems) can be easily conducted and second it seems that the PART II and III contain some unsolved fresh problems. The objective of this paper is to report the detail of this activity.
2. Method
The theme of this activity consists of above-mentioned three parts. The following learning method is adopted in each part: at first “Question and Prediction”, second “Demonstration or Experiment”and third “Explanations”. The “Question” and the method of “Demonstration or Experiment” are described in this chapter, while the students’predictions, the results of “Demonstration or Experiment” and “Explanations” are described in the following chapters.
2.1 Comparing the Velocity or Acceleration of the Circular Disk and the Ring of Rigid Body
At first the following question was given to the three students.
Question: The velocity or acceleration of which object is larger if a circular disk and a ring of rigid body are set up at the same level and are released at the same time? The students give their predictions to this question.
Next, a demonstration on this question was conducted as follows. A wooden circular disk of 95 mm diameter and 12 mm thickness and a wooden ring of 95 mm outer diameter, 85 mm inner diameter and 10 mm thickness on the market (Fig. 1) were used for the demonstration. They were set at rest at the same level on the two parallel guide lanes of aluminum (Fig. 2) and were released at the same time. The length of the lane was 2.0 m and the slope angle was set up about 5 degrees. The answer to the question was shown to the students by this demonstration. The author gave an explanation about the result of this demonstration.
2.2 Comparing the Velocity or Acceleration of the Drum Filled with Three Kinds of Liquid
At first the following question was given to the three students.
Question: When the drums filled with different kinds of liquids are rolling down a slope without slipping, the velocity or acceleration of which drum is the largest?
And which is the smallest?
The roughly estimated ratio of viscosity of each liquid was given to the students with the question as follows:
Viscosity of water:Viscosity of olive-oil:Viscosity of honey = 1:100:10000.
The mass of each liquid was also given to the students by measuring the total mass of each drum containing the liquid and the mass of the material of the drum. The concrete values were as follows:
Mass of water = 202 g, Mass of olive-oil = 178 g, Mass of honey = 290 g.
The students gave their predictions to the question.
Next, a demonstration was conducted as follows. The students had made each drums filled with liquid(Fig. 3) beforehand by the following procedure. At first an acrylic ring of 30 mm width and 3 mm thickness was glued to an acrylic plate of 100 mm diameter and 3 mm thickness with an adhesive agent. Some hours later each liquid was poured into the drum and another acrylic plate was glued to the acrylic ring. At this stage a weight was put on the second acrylic plate till the adhesion became perfect not to leak liquid. Hereafter the drum filled with water is called “Water” and so on. We compared motions of following each pair, Water vs. Olive-Oil, Water vs. Honey, and Olive-Oil vs. Honey in order to clarify the order of the velocity or the acceleration.
Furthermore, a supplementary experiment using video camera was conducted in order to clarify the detail of the motion of each drum. The author gave an explanation about the result of the experiment.
2.3 Comparing the Velocity or Acceleration of the Drum Half Filled with Three Kinds of Liquid
At first the following question was given to the three students.
Question: When the drums half filled with different kinds of liquids are rolling down a slope without slipping, the velocity or acceleration of which drum is the largest? And which is the smallest?
Next, a demonstration on this question was conducted as follows. We compare motions of following each pair, Water vs. Olive-Oil, Water vs. Honey, and Olive-Oil vs. Honey by using the drums half filled with each liquid made beforehand (shown in Fig. 4). The answer to the question was shown to the students by this demonstration. The author gave an explanation about the result of this demonstration.
3. Results
The results of above-mentioned activities were obtained as follows.
3.1 Comparing the Velocity or Acceleration of the Circular Disk and the Ring of Rigid Body
The predictions of students are as follow:
Student A: “Velocity or acceleration of both objects is same”;
Student B: “Velocity or acceleration of both objects is same”;
Student C: “Velocity or acceleration of the ring is larger than that of the disk since the air resistance to the ring is smaller than that to the disk”.
Results of demonstration:
It was observed that the disk moved faster than the ring as shown in Fig. 5a. No students predicted the correct result.
Explanation:
This theme is well known problem in the rigid body dynamics, however, almost all first year students in this period have not yet learned the concept of the moment of inertia. Then the author adopted the explanation based on the conservation of mechanical energy and the relationship between the translational speed of the center of the mass and that of rotational speed of each part of the object. Based on this explanation it is shown that translational speed of the center of the object V is
V?for the disk, wheregx,,?are the sliding distance, slope angle and gravitational acceleration (in Fig. 5b).
3.2 Comparing the Velocity or Acceleration of the Drum Filled with Three Kinds of Liquid
The predictions of students are as follow:
Student A: “Order of magnitude of velocity is Honey-Olive-Oil-Water due to the order of mass”.
Student B: “Order of magnitude of velocity is Honey-Olive-Oil-Water due to the order of viscosity”.
Student C: “Order of magnitude of velocity is Honey-Water-Olive-Oil due to the order of mass”.
Demonstration:
One static figure of the demonstration of Water vs. Olive-oil is shown in Fig. 6. The drum filled with water moved faster than that with olive-oil. It was clarified that the order or magnitude of acceleration is Water-Honey -Olive-Oil.
Supplementary Experiment:
The information about the position and the velocity of the center of the drum at every 0.1 s was obtained from the video picture for each object and the velocity-time graph was made (Fig. 7). It was clarified that in the case of Water and Honey the motion of the drum is approximately uniform accelerate motion while in the case of Olive-Oil complicated one. Approximate values of acceleration of each drum are shown in the figure.
Explanation:
It is available to observe the motion of bubble inside the drum in order to understand the order of acceleration. In the case of water the bubble remains at the top of the water, which means that the motion of water inside the drum is approximately translational motion. On the other hand in the case of honey the bubble rotates with almost same angular velocity as that of tape fixed to the drum, which means that the motion of honey inside the drum is approximately equal to that of rigid body rotation. It is not difficult to understand that the velocity of the drum filled with water is larger than that with honey due to the explanation in Part I. In the case of drum filled with olive-oil the motion of the bubble inside the drum is much complicated, namely it moves not only in the tangential direction but also radial direction, which makes the ratio of kinetic energy of translational motion in this case smaller than in other two cases.
3.3 Comparing the Velocity or Acceleration of the Drum Half Filled with Three Kinds of Liquid
The predictions of students are as follow:
Student A: “Order of magnitude of velocity is Honey-Olive-Oil-Water due to the order of viscosity or centrifugal force”.
Student B: “Order of magnitude of velocity is Water-Honey-Olive-Oil, same as the case of the drums filled with liquid”.
Student C: “Order of magnitude of velocity is Water-Olive-Oil-Honey due to the inverse order of viscosity”.
Demonstration:
It was clarified that the order or acceleration was Water-Olive-Oil-Honey.
Explanation:
In order to explain the result it is effective to observe the forms of the surface of the liquid. In the case of water the form of the surface is approximately straight and slightly inclined to the horizontal direction asshown in Fig. 8a. It suggests that the motion of the liquid is approximately translational. On the other hand in the case of honey the thick layer attaches to the side wall of the drum due to the effect of viscosity asshown in Fig. 8b. This mass distribution of the liquid gives the counter-clockwise torque to the drum and the motion of the drum becomes near the uniform velocity motion.
4. Discussion and Conclusions
In the case of Part II we explained the order of velocity or acceleration based on the behavior of the liquid inside the drum and the consideration of energy transformation. This consideration is endorsed by the following fact. The experimental values of acceleration of each drum Aex were compared with the calculated value of the acceleration of each drum assuming that the liquid motion is translational one Ac1 and rigid body rotation Ac2. In order to calculate Ac1 and Ac2 the mass of the liquid M, that of bottom part of the drum m1 and that of side part of the drum m2 were measured, and the expressions were adopted. These values are shown in Table 1 and it was clarified that Aex of Water is approximately equal to Ac2 and Aex of Honey is equal to Ac1.
After the class the author had students write reports on this activity. Students wrote the following impressions on this activity in their reports.
Student A: The experimental results were different from my predictions at any time. It was very good that we could clarify and understand the reasons through the seminar.
Student B: It would be available to know the detail of the motion of objects rolling down a slope.
Student C: I was surprised to know that potential energy is transformed into translational and rotational kinetic energy. I had never known that when the drum is filled with liquid whether the motion of the liquid is mainly translational or rotational depends on its viscosity. I want to know the behaviors of the drum filled with paste or sauce.
Summarizing, three subjects related to the motion of cylindrical objects rolling down a slope were assigned in the preliminary seminar for first year university student. First assignment was comparing the motion of a circular disk and a ring of rigid body. Second assignment was comparing the motion of drums filled with water, olive-oil and honey. It was clarified that the motion of water is approximately translational, that of honey rigid body rotation and that of olive-oil is complicated one by observing the motion of bubble inside the drum. The order of the velocities and the accelerations of the drums could be understood by these facts. Third assignment was comparing the motion of drums half filled with water, olive-oil and honey. It was clarified that the magnitude of the acceleration in this case strongly depends on the viscosity of the liquid and the configuration of the liquid surface inside the drum. These activities were effective in promoting students’interests and understanding on the motion of objects rolling down a slope.
References
[1] E.F. Redish, New directions of research on undergraduate physics education, in: Proceedings of ICPE2006, Tokyo, 2008, pp. 9-19.
[2] D.R. Sokoloff, R.K. Thornton, Active learning with interactive lecture demonstrations, in: Proceedings of ICPE2006, Tokyo, 2008, pp. 92-96.
[3] O. Hirayama, Development of demonstration apparatuses for rigid body dynamics for first-year student class in university, ABSTRACTS, ICPE2007, Marrakech, 2007, p. 40.
[4] O. Hirayama, Homework on the moment of inertia of a human arm, in: AIP Conference Proceedings, Volume 1263, ICPE2009 (Bangkok), 2010, pp. 110-113.