Exploring the Relationship Between the Kinetic Energy and the Number of Magnetic Groupings in a Gaussian Rifle. Toby McMurray

Transcription

1 Exploring the Relationship Between the Kinetic Energy and the Number of Magnetic Groupings in a Gaussian Rifle Toby McMurray Mr. Schmidt January 14th 2017

2 Table of Contents: Section Page Number Introduction.. 1 Experimental Design. 2 Research Question 2 Variables 2 Apparatus.. 3 Materials 3 Procedure.. 4 Analysis.. 4 Observations 4 Data Processing 5 Graph 1 7 Evaluation.. 8 Conclusion. 10

3 McMurray "1 Introduction: In this internal assessment, a Gaussian rifle (or Gauss gun) was created and used to explore the physical laws related to kinetic energy and the conservation of momentum. The mechanics of this contraption are very simple and straight forward. One metal ball is rolled towards a grouping of magnets, (these magnets have two or more metal balls on the other side), then the ball collides with the magnets, transferring its momentum to the outermost ball. This ball is then propelled to a second grouping of magnets and metal balls, identical to the first, for as many times as set up by the experimenter. After these balls have been propelled through the mechanism, the final velocity of the metal ball is found to be greater than the initial. From this, we can experiment with the physics by testing the affect of altering the number of magnetic groupings (being two magnets together with two or more balls on one side). Depending on the results, a number of conclusions can be drawn which will ultimately better our understandings of the physical laws being explored. This mechanism is fascinating, as magnets are being utilized to conserve momentum, and magnetic potential energy is being transferred into kinetic energy. The ball, initially with a very low velocity, is accelerated to a velocity great enough to do significant damage on an object. This form of weaponry is not of great enough strength to actually be used in combat, and is only used, typically, by those exploring the scientific laws this contraption illustrates. However, some other versions of similar mechanics are being explored for practical uses. A coilgun is a more efficient form of this Gaussian rifle, and it uses electro-magnetic coils to accelerate a projectile in a centralized barrel. This would be a form of weaponry with greater power and efficiency than conventional rifles, and it would not use any gunpowder. If there were a way to use the mechanics of the Gaussian rifle in weaponry, it could be much better for the environment. The current problem is that no form of weaponry, similar to a Gauss gun, such as a coilgun, or any variant of a railgun, is strong enough to do great damage. Some of these mechanisms are being researched as possible ways to improve space launches. This basic mechanism has lots of potential to be used in many areas of science experimentation, apart from the one explored in this internal assessment. There are many different types of Gaussian rifles, varying in size and power. For the purposes of this experiment, I used a size that would be large enough to get accurate readings with the LabQuest and photogate, but also a size that would not have as great safety dangers as larger Gauss guns do. As the size of the metal balls increases, the strength of the magnets must also increase for proper functionality, and many more safety precautions must be taken when using this equipment. I made sure to use magnets that are easily purchasable from any hardware store, and I made a mechanism with less strength than one that would be dangerous to me or anyone else nearby. Originally, I had used one magnet for each of my magnetic groupings, however, this proved to be impractical. Many of my magnets started to break, as they are very brittle and weak. Also, the magnetic force was not enough to get sufficient results for accurate analysis. To fix this, I restarted my experiments, using two magnets per grouping, and this reduced the amount shattered magnets during experimentation.

4 McMurray "2 Experimental Design: Research Question: How does the number of magnetic groupings affect the kinetic energy of the Gauss gun. (One magnetic unit, in this case, is defined as one pair of two magnets together, with two metal balls on one side.) Variables: Independent Variable: The number of magnetic groupings. Dependent Variable: The kinetic energy of the final metal ball. Controlled Variables: Table #1: Controlled Variables Controlled Variable Why it must be controlled How it was controlled The magnet strength. The lengths between magnetic groupings. Number of metal balls used in each magnetic grouping. The initial velocity. The stronger the magnet, the greater magnetic potential energy. If there were different magnets with varying strengths being used in the experiment, the recorded data would need to be analyzed in a much different way. If the distance between each grouping is not the same, the change in velocity between each grouping may be different. This may have a minute affect on the final kinetic energy, resulting in inconsistent data. The amount of balls in each magnetic grouping directly correlates to the kinetic energy of the projectile ball. The more balls per grouping, the greater the distance from the magnet, lowering the attraction force. The initial velocity of the ball would affect the final velocity and the kinetic energy. The consistency of the initial velocity simplified the calculations in the analysis. Two identical packages of Neodymium magnets were purchased, including six magnets of the same grade in each. Every unit was placed an equal distance away from each other. Each magnetic grouping has two metal balls on one side. The ball was dropped down a ramp from the same height during every trial.

5 McMurray "3 Apparatus: The apparatus used for my experiments is relatively simple. I used a wooden board that had been cut to size and carved out so that the sides are curved, giving every ball a guiding path. I needed a track for the metal balls to roll on, so that the acceleration of the ball could be linear. The magnetic groupings consist of two magnets with two metal balls on one side. In order to keep the initial velocity constant, a short ramp was used. This ramp gave the first metal ball in the system an initial velocity to eliminate the need to calculate the change in kinetic energy later in the lab. The photogate was placed right beside the last magnetic grouping, as seen in figure 1, in order to get the most accurate velocity measure Figure 1: One magnetic grouping next to a photogate Materials: - Metal balls - Neodymium magnets * - Scale - Track ** with ramp - LabQuest - Photogate - Tape Measure - Notepad for recording results * The track can be anything that will hold the magnets and metal balls. ** The magnets used in this experiment were of grade 35, with a diameter of 11.99mm, and a thickness of 2.30 mm.

6 Procedure: McMurray "4 I. Lay out track horizontally, level to the ground as not to cause any unwanted acceleration. II. Separate magnets into groups of two, attached by the faces of the magnets. III. Attach two metal balls to one end of the two magnets. See figure 1. IV. Place each magnetic grouping an equal distance from each other. V. Measure the initial velocity, using a LabQuest and a photogate, record. VI. Allow the metal ball the role down the ramp towards the first magnetic grouping. VII. Measure the velocity of the last ball using a LabQuest and a photogate, record. VIII. Repeat steps I. to VII. for each number of magnetic units. IX. Repeat all steps five times for a sufficient amount of data. Analysis: Observations: Table #2: Raw Data (Quantitative) Number of Magnetic Groupings Trial Number Final Velocity (m/s) ± m/s One Trial Trial Trial Trial Trial Two Trial Trial Trial Trial Trial Three Trial Trial Trial Trial Trial Four Trial Trial Trial Trial Trial Five Trial Trial Trial Trial Trial

7 McMurray "5 The initial velocity was calculated by taking the average velocity at the bottom of the ramp. The initial velocity was found to be: ± m/s The mass of one metal ball was calculated using a scale, it was found to be: 8.41 x10-3 ± 1.0 x10-5 kg Data Processing: The values recorded in table 2 were gathered from individual trials. To work with one number for each experiment, the average velocity can be found. This number will be a more accurate number, and will yield more accurate results during other calculations. Sample Calculation #1: Calculating the average final velocity using data from one magnetic grouping (total of the final velocities) average final velocity = (number of trials ) average final velocity = m/s Uncertainties: (final velocity from trial #1 + trial #2 + trial #3 + trial #4 +trail #5) = (number of trials) [(0.852) + (0.837) + (0.854) + (0.840) + (0.861)] = 5 [(± 0.001m/s) + (± 0.001m/s) + (± 0.001m/s) + (± 0.001m/s) + (± 0.001m/s)] 5 = ± m/s Therefore, the average final velocity of one magnetic grouping is ± 0.001m/s. These calculations were done for each number of magnetic groupings. The following chart organizes the average final velocity of each magnetic grouping.

8 McMurray "6 Table #3: The Average Final Velocity of each magnetic grouping Number of Magnetic Groupings Average final velocity of each magnetic grouping (m/s) ± m/s One Two Three Four Five From this, we can calculate the average kinetic energy, which is what we will be comparing to the number of magnetic units in this internal assessment. Sample Calculation #2: Calculating the average kinetic energy using the average final velocity of data from one magnetic grouping Kinetic Energy = (1/2) (mass of the metal ball) (velocity of the ball) 2 Ek = (1/2) (8.41 x10-3 kg) (0.849 m/s) 2 Ek = x 10-3 kg m 2 /s 2 Ek = x 10-3 J Uncertainties: [ (relative uncertainty of the mass) + ( 2 )(relative uncertainty of the velocity) ] = [ (± 0.119%) + (2)(± 0.118%) ] = [ (± 0.119%) + (± 0.236%) ] = ± 0.355% = ± 1.08 x 10-5 J Therefore, the average kinetic energy of the ball from one magnetic grouping is x10-3 ± 1.08 x10-5 J. In order to best compare the results, I should calculate the change in kinetic energy, subtracting the initial kinetic energy from the final. However, considering this internal assessment strives to only compare the relationship between the kinetic energy and the number of magnetic groupings, the numerical value is not as important as the trend it produces. Since there is a constant initial velocity, the initial kinetic energy of the ball would be the same for each experiment. This means the change in kinetic energy would have the same relationship as the final kinetic energy. Therefore the change in kinetic energy is not required to be calculated in order to find the correlation between these variables in this exploration.

9 McMurray "7 These calculations were done for each number of magnetic groupings. Each kinetic energy has a different amount of uncertainty. The following chart organizes the average kinetic energies of each magnetic grouping as well as the uncertainties. Table #4: The average kinetic energies of each magnetic grouping and their uncertainties Number of Magnetic Groupings Average kinetic energy of each magnetic grouping (J) One Two Three Four Five x x x x x 10-2 Relative uncertainty (%) iiiiiiiiiiii Actual Uncertainty ( ± J ) 1.08 x x x x x 10-5 Using these values, a relationship can be derived between the number of magnetic groupings and the kinetic energy of the final ball. To display this relationship, the data has been plotted using a graph. Graph #1: The Relationship Between the Number of Magnetic Groupings and the Kinetic Energy of the Final Ball Kinetic Energy of the Final Metal Ball ( J ) Number of Magnetic Groupings

10 McMurray "8 Evaluation: The relationship displayed in graph 1 is the correlation between the number of magnetic groupings and the kinetic energy of the last ball. This relationship is seen to be linear. This means the kinetic energy of the last ball does not increase exponentially with an addition of a magnetic grouping. This result matches exactly with the theoretical results that were expected to be found. To understand the theory behind this prediction, it is important to fully understand the mechanics of this magnetic system. The initial position for this mechanism is seen in figure 1. There are a certain amount of magnetic groupings, depending on the trial, and each has two metal balls on one side, and zero on the other. This is important because it is the side that has no metal balls which comes into contact with the oncoming ball. Since this mechanism is in a closed system, the energy within the system is being conserved. We know the initial metal ball has a very low velocity, and the final metal ball has a very high velocity, thereby the initial kinetic energy is much lower than the final. This means the change in kinetic energy must be positive. Therefore, by the laws of conservation of energy, the change in kinetic energy must be equal to the negative of the change in magnetic potential energy, seen through this equation: Ekinetic = - Emagnet potential The magnetic potential energy is decreased when a ball comes into contact with the magnets, on a side without any other metal balls. It takes more magnetic potential energy to hold two metal balls on one side than it does to hold one on each side. As this change in magnetic potential energy occurs as many times as there are magnetic groupings, the resulting change in magnetic potential energy is seen to be negative. Therefore, these results displayed in graph 1 are expected, as the relationship between the number of magnetic groupings and the kinetic energy of the final ball is linear. This is because, with every addition of a magnetic grouping, more magnetic potential energy is converted into kinetic energy, giving the ball a greater velocity. When the ball comes into contact with the magnetic grouping, the momentum of the first ball, is then transferred through to the magnets to the outer-most ball. Since the outermost ball is further from the magnets, it has less of an attraction force pulling it towards the magnet, and it is able to use the momentum transferred to it by the initial ball to leave its initial magnetic grouping, towards the next. This is known as the conservation of momentum, as the momentum of the first ball, being the mass multiplied by the velocity, is transferred to a new ball. It can be seen through this equation: m1v1 m2v2 = m1v1 1 m2v2 1 The masses of each ball are the same, therefore the only thing that changes is the velocity. The initial velocity of the outer-most ball is zero, and, in ideal conditions, the momentum of the

11 McMurray "9 initial ball will be transferred completely to outer-most ball, and the initial ball will have a new velocity, as it will be at rest. If a graph had been plotted displaying the relationship between the number of magnetic groupings and the final velocity of the metal ball, it would look like a exponential graph. This is because the velocity is increasing. Since the kinetic energy relationship is linear, and the velocity is squared in the kinetic energy formula, the relationship would be exponential. This being said, it is important to note that the momentum is not completely conserved as there is some lost due to friction. This is also true for the energy in this system. The friction between the metal balls and the track cause the ball to loose momentum, resulting in a lower final kinetic energy than in ideal conditions.

12 McMurray "10 Conclusion: In order to improve this lab, error must be reduced, experimentally, in order to obtain the most accurate results. Once these errors are eliminated, not only will the experiment become more accurate, it will also be more consistent, yielding the same results each time. The final velocities of each trial were not as consistent as possible. Each set of trials ranged more than 0.02m/s in both positive and negative directions from the average. The amount of random error was relatively low, meaning the main factors which affected the accuracy of the results were the systematical errors. The inconsistencies of the experiment prove there to be problems with the method of the lab. 1. The first possible source of error is the apparatus, more specifically, the way in which the magnets were positioned on the wood. There was no form of attachment of the magnets to the track. This means that the magnetic groupings would occasionally move towards the incoming metal ball. This is significant because it means that some of the magnetic potential energy that is supposed to be completely converted into the kinetic energy of the outer-most ball, is being converted into kinetic energy of the magnetic grouping. The grouping requires energy to move itself towards the incoming ball. This results in a lower velocity of the outgoing ball, therefore the ball will have a lower kinetic energy, affecting the final results. In order to eliminate this error, the magnets can be attached to the track. As long as the magnets do not move towards the incoming ball, the magnetic potential energy will not be lost in this unwanted way. 2. The track is also another possible source of error, in two possible ways. Firstly, if the track were to not be smoothed out properly, the friction against the rolling ball could be enough to seriously throw off the final results. Secondly, if the track has too minute of a curve, the metal ball that is in motion may have the opportunity to lose some of its velocity if it is not facing the correct direction. This loss of velocity results in a lower kinetic energy, directly affecting the final results. In order to eliminate these possible sources of error, a smooth track with minimal friction and a good curve, (to fit the metal balls properly), is to be purchased. All of these issues affect the results in the end of the lab. To minimize error, and to maximize accuracy, there are many solutions to those problems identified. As errors become eliminated, it becomes a more and more accurate lab, and there is an increase in consistency. The large amounts of systematical error in this experiment are what caused such a range in the trials. The results of this internal assessment are focussed solely on the relationship between the kinetic energy and the number of magnetic groupings, however, if there were more space, there are many other details that I would want to include. If space allowed for it, I would want to quantify the amount of magnetic force created by these Neodymium magnets. This would allow for more accurate discussions of results and predictions for other experiments. Also, I would want to include the graph displaying the relationship between the final velocity and the number of magnetic groupings, as previously stated. One last example of a possible addition to this internal assessment would calculating the ideal conditions of this magnetic system, then possibly calculating the coefficient of friction. There is much more that can be done with this topic, I have only scratched the surface.