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# Story of pendulum

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Story of pendulum using STELLA

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### Story of pendulum

1. 1. 1.0 Introduction. 1.0.1 Icon-based Model Building and Simulation Tool, STELLA STELLA is a model that is easy to use and provided endless opportunities to explore. It alsohave learning laboratory which give students, educators and researchers to study everything fromeconomics to physics, literature to calculus and also in biology. STELLA has an ability tostimulate learning and inspiring the exciting moments of learning. STELLA is used to simulate a system over time, jump the gap between theory and the realworld, enables students to creatively change systems, teach students to look for relationship,clearly communicate system input and outputs, and demonstrate outcomes. Download trial is fora month only. Students or educators that download for a trial should maximize the usage beforethe expired date. This is because there a lot of advantage using STELLA. 1.0.2 Simple Pendulum. A pendulum in simple harmonic motion oscillates about a central point. The pendulum is a body suspended from a fixed point and assembled from a string with a weight at one end. It swings freely to and fro under the action of gravity. From its rest position at a fixed period, the pendulum will swing through small displacement. The period is the time for pendulum to complete one or single oscillation. For example, pendulum move from one side and return to that side, that is dependent on the length of the string and the acceleration of gravity. This is πΏ given by π=2 π . T is a period for one oscillation, L is a length of a string of the π pendulum and g is the acceleration due to gravity. Notice that the weight of the bob does not show up in this equation. This means that no matter what the weight, a bob that is suspended on a certain length of string will take the same time to complete a cycles. The pendulum passes twice through the arc during each period. Galileo was the first to examine the pendulumsβ unique characteristics. Galileo interested in pendulums and he hit on the idea that pendulums have a constant period even when moving at different angles. This comes from his observation on motions of chandelier hanging in a cathedral and he noticed that it has a constant period even when moving at
2. 2. different angles. Hence, Galileo began conducting with pendulums in order to examine if their periods are indeed constant in 1602. He examined a variety of pendulums and claimed that the period of each is totally independent of the size of the arc through which it passes (displacement). Today, we know that the period of the pendulum will remain constant as long as the pendulumβs angle is not greater than about 20 degrees. If greater than 20 degrees, it is not completely precise. Galileo also noticed that the period of the pendulum is not dependent on weight of the material and pendulumβs period is influenced by its length alone. The longer length of the string of the pendulum, the longer its period. 1.0.3 Pendulum Story Background and Context By Using A Learning LaboratoryCreated With STELLA. Figure 1 (a) : A learning laboratory created with STELLA on pendulum story
3. 3. The simple pendulum is a small bob connected to the end of the string and is pulled ashort distance away from its rest position and released; it will begin to swing back and forth. Thetype of motion faces by simple pendulum is simple harmonic motion. It is assumed that the mass of the bob is concentrated at a point and that the mass of thestring is negligible. Time period, T (s) is the time taken for one oscillation. Frequency of theoscillation is the number of oscillation simple pendulum made in one second: = 1/ T. The law ofa simple pendulum is: οΌ The period of a simple pendulum of constant length is independent of bob or ball mass, οΌ The period of a simple pendulum is independent of the amplitude or displacement of oscillation, provided it is small, οΌ The period of a simple pendulum is directly proportional to the square root of length of the pendulum, οΌ The period of a simple pendulum is inversely proportional to the square root of the acceleration due to gravity. πΏ π=2 π π *where T (s) = time for 1 oscillation or cycle, L (m) = length of pendulum , g = acceleration due to gravity (m/s2) In this model, the relationship and the effect of different length of the string, bob massand initial starting position has on the motion of the pendulum are observed, ignoring the effectof friction and air resistance. Then, when other forces are taken into account, the motion changesare investigated.
4. 4. 1.1 To Study The Relationship of Initial Starting Position of The Pendulum With ThePeriod (T). In this experiment, there are four variables such as mass of ball (kg), initial displacement(m), string length (m) and time taken to complete one oscillation. In the first part of theexperiment, the constant variables are string length and mass of ball. The manipulated variable isthe initial displacement. While, the responding variable is the time taken to complete oneoscillation or period, T (s). The graphs below shows the different initial displacement(m) vs time(s). Figure 2 (a): Initial displacement of -0.20m
5. 5. Figure 2 (b): Initial displacement of 0.10m Before started, the string length and mass of ball were set at 1.0 m and 1.00 kg. Then,first initial displacement was set at -0.20 m and followed by -0.10m, 0.00m, 0.10m and 0.20m. Inthis part, only use gravity. Each one peak or loop show one complete oscillation. From the above graph in Figure 2 (a), at -0.20m initial displacement, the time taken forpendulum to complete two oscillations is four seconds. To complete only one oscillation, thependulum needs two seconds. Like in Figure 2 (a), graph in Figure 2 (b) also show four secondsto complete two oscillations and needs two seconds to complete one oscillation but its initialdisplacement is -0.10m.
6. 6. Figure 2 (c): Initial displacement of 0.00mFigure 2 (d): Initial displacement of 0.10m
7. 7. Figure 2 (e): Initial displacement of 0.20m However, when initial displacement is 0.00m as in Figure 2 (c), the graph is constant andhave no peak. From the graph, we know that as the time passes, the displacement is still zerobecause the pendulum is not swinging and have no displacement. No displacement means that nodistance and no direction by pendulum. In Figure 2 (d) and Figure 2 (e), the time taken for thependulum to complete two oscillations are four seconds and take only two seconds to completeone oscillation although the displacements are differ. The Figure 2 (a), Figure 2 (b), Figure 2 (d) and Figure 2 (e) have constant period of thependulum because they have only little bit different in displacement. The Figure 2 (c) only showconstant graph because of pendulum does not oscillate. From this result obtained, we know thatthe period of the pendulum or the time taken to complete oscillation for a pendulum will remainconstant as long as the pendulum angle displacement is not greater than 20 degrees.
8. 8. 1.2 To Study the Relationship of Mass of Ballβs Pendulum with Number of OscillationWithin 2 Seconds. By using Stella software, we also can study the relationship of ballβs mass with thenumber of oscillation within 2 seconds. We can set up experiment by set similar or fixedvariable that is string length and initial displacement. So, the manipulated variable is mass of balland responding variable is number of oscillation within 2 seconds. Every student has their own hypothesis. My hypothesis is mass of ballβs pendulum effectthe number of oscillation within 2 seconds. For me, as the mass of ballβs pendulum increases, thenumber of oscillation within 2 seconds also increases. The graphs below show the number ofoscillations by different mass of ball. To determine the number of oscillations, we just look atthe peak at graph. The one peak represents one oscillation; two peaks represent two oscillationsand so on. Figure 3 (a) : 0.01kg ball
9. 9. From the figure 3 (a), the number of oscillations in 2 seconds is three oscillations with theconstant or fixed displacement of 0.10 m. this is because there are three peak at the 2 seconds.The mass of ball used is 0.01kg and it is smallest mass used. In this experiment, we used gravityonly and not include friction and driving force. So, we need to use another mass of ball andrepeat the experiment with different mass of ball. The experiment is repeated by using 0.50kg,1.00kg, 1.50kg and 2.00kg. Figure 3 (b) : 0.50kg ball Figure 3 (c) :1.00kg ball Figure 3 (d) : 1.50kg ball Figure 3 (e) : 2.00kg ball
10. 10. By using 0.50kg of ball, the graph is still same with the previous graph. We can see thatin 2 seconds, the number of oscillations are also three although the mass of ball used is differentin 0.49kg. the graphs of 1.00kg, 1.50kg and 2.00kg ball also show same graph that have 3 peaksin 2 seconds. From all graphs, we know that the mass of ball whether high or less, it does noteffect the number of oscillations in two seconds. Hence, my hypothesis about the mass of ballβspendulum effect the number of oscillation in 2 seconds are not accepted. This STELLA software is interesting to use and make learning session fun. It attractstudents to explore the pendulum story by make experiment using STELLA software. Back toabove topic to explain the graph. The driving force for a pendulum used above is only gravity. Ifthe pendulum has twice the mass, gravity pulls twice as hard. Mass is also show how hard anobject resists the force it feels. For example, it more difficult to start motion for a bowling ballthan for a ping-pong ball. A pendulum that have a twice the mass feels twice the pull, but also has the twice theresistance to that pull. These two effects balance out and the twice mass still experiences thesame effect. The mass of ballβs pendulum does not affect how it moves. This is because the forceaccelerating the pendulum comes from its weight. The more massive the pendulum is, results in greater its weight, in strict proportion. Theinertia, that is, the resistance of the pendulum to acceleration by a force, is also in strictproportion to its mass. Since both the force and inertia vary in strict proportion to the mass, thetwo effects cancel out and the acceleration is independent of mass. Hence, the more massive ball,the more force is required to make it accelerate at a given rate. The more massive ball, the moreforce gravity exerts on it. The net result is that acceleration from the force of gravity does notchange as an objects mass changes.
11. 11. 1.3 To Study The Relationship Between String Length With Period , T (s). After studied the relationship of initial displacement with period and mass of ball withnumber of oscillation within time given, now relationship between string lengths with period alsocan be study by using STELLA. In this experiment, the motion of a swinging bob of a pendulumwill be measures using STELLA to prove or disprove the stated hypothesis. Equations relatingthe motion of each mass must be used to determine and compare their periods. πΏ π=2 π π Figure 4 (a) : 0.1 m string length
12. 12. Figure 4 (b) : 0.5 m string lengthFrom the above, these two graphs show different number of oscillations within 4 seconds. InFigure 4 (a), the number of oscillations within 4 seconds is more or less than 6 oscillations but infigure 4(b), the number of oscillations within 4 seconds is more or less than 3 oscillations. So, tocalculate the periods, we can use the equations, πΏ π=2 π πThe period, T for the 0.1m string length is 0.1π π=2 π π 9.8 2 π
13. 13. π = 0.63π This mean that the pendulum with string length of 0.1m have one complete oscillation at 0.63seconds. The period for 0.5m string length is also calculated by using that equation. Then, thenumber of oscillations for 4 seconds is, π‘πππ π‘πππππ= ππ’ππππ ππ ππ ππππππ‘πππ 4π 0.63π  = ππ’ππππ ππ ππ ππππππ‘πππ 4π ππ’ππππ ππ ππ ππππππ‘πππ = 0.63π ππ’ππππ ππ ππ ππππππ‘πππ = 6.35 ππ ππππππ‘πππ Figure 4 (c) : 1.0m string length
14. 14. Figure 4 (d) : 1.5m string lengthFigure 4 (e) : 2.0m string length
15. 15. In Figure 4 (c), the pendulum have one complete oscillation in 2 seconds and twocomplete oscillation in 4 seconds. Then, we can used below equation to calculate the period. 1.00π π=2 π π 9.8 2 π π = 2.00π This proved that STELLA shows an accuracy also in graph because T value get from the graphare same with the calculation. In Figure 4 (d) and 4 (e), the number of oscillation within 4 seconds are more than 1oscillation and less than 2 oscillation. So, to get the period of oscillation, we must use theequation that have been used before. The figure below shows the graph displacement versus time of combined string length.There are also table that include string length, period, mass of ball, initial displacement. Thetable below are used to compare the period of different string length used.Mass of ball (kg) Initial displacement String length (m) Period, T (s) (m)1.0 0.1 0.1 0.631.0 0.1 0.5 1.421.0 0.1 1.0 2.001.0 0.1 1.5 2.461.0 0.1 2.0 2.84 Table 1 : Period on different string length.
16. 16. Figure 4 (f) : Combined string length The pendulum will take more or less time to oscillate, depending on its length andacceleration due to gravity. From the experiment using STELLA, the longer the string length, thelonger period taken to complete one oscillation. The shorter the string length, the shorter periodtaken to complete one oscillation.
17. 17. 1.4 Conclusion Again, there is no relationship between mass and period, while for small amplitude ordisplacement, there is only the weakest relationship between the displacement and the period.Hence, the period, T must be proportional to square root of string length, L and inverselyproportional to square root of acceleration of gravity, g was found by Galileo. This STELLA software makes learning easier and interactive. By visual, we can look thedifference of the graphs when we used the different string length. The difference of the graphsare difference number of peaks resulted due to different string length. We are easily analyzed thegraph show and can test our hypothesis, and lastly make conclusion. Using of computer simulation programs in education helps students more familiar withuse of the computer in education. It offers the opportunity to students to experiment withphenomena which cannot normally be experimented with in the traditional way. Bork (1981)remarks: βSimulations provide students with experience that may be difficult or impossible toobtain in every day lifeβ. For example, it is not possible to experiment with an economic system.Only nature and content of the system that teacher can discuss in the class and of courseexperimenting be useful because can generate insight into the functioning of the economicsystem. Foster (1984) says that, simulation can be entertaining because of dramatic and game-likecomponents. Some students have no feeling to learn something that abstract. This make they donot have interest in learning. By use computer simulation in school in Malaysia, students aremore feeling for reality in some abstract fields of learning. The other advantage is teacher or trainee can just use computer simulation program tocarry out experiment and exercise as much as necessary. This is because the apparatus thatneeded in the experiment is too expensive. Simulation also often goes hand in hand withvisualization. The students can adjust the manipulated variable according to what they want toobserve. The results of changes are directly shown on the screen. This generally appeals to thestudents. Not only that, the flexibility of the computer simulation programs motivates studentwhether intrinsic or extrinsic. There are also disadvantages of using computer simulation programs in education.Computer simulation programs look well from a technical point of view, but they are difficult tofit into a curriculum. It also cannot be adapted into different student level within a class. Otherthan that, as it need computer to run the simulation programs, it also need electric source to turnon computer. This means that only main source is electric current and without electric source, thecomputer simulation programs cannot be run.
18. 18. Reference (s)Anonymous (2012). The Pendulum. Retrieved on November 17, 2012 from http://www.cs.wright.edu/~jslater/SDTCOutreachWebsite/pendulum_exp.pdfAnonymous (2012). Pendulum Motion. Retrieved on November 17, 2012 from http://www.physicsclassroom.com/class/waves/u10l0c.cfmClaver Pedzisai Bhunu (2010). A Mathematical Analysis of Alcoholism. Retrieved on November 17, 2012 from http://www.wjms.org.uk/wjmsvol08no02paper05.pdfGarrett, B. (2012). Simple Harmonic Motion β Part II. Retrieved on November 17, 2012 from http://www.educationalelectronicsusa.com/p/shm-II.htmRik Min (2012). Advantages and Disadvantages of Model-Driven Computer Simulation. Retrieved on November 17, 2012 from http://projects.edte.utwente.nl/pi/papers/simAdv.html