The Pendulum Concept, Angular Displacement, Period of Motion of a Pendulum, Light vs Heavy Pendulums - Lab Report Example

Name Institution Lecturer Course Date Project Report of а Pendulum Introduction A pendulum comprises of a mass suspended from a fixed point in such a way that the mass is able to swing freely to and fro. The mass is given a specified displacement and allowed to swing towards the center of path of motion, which is perpendicularly below the fixed point from which the body is hung as shown in figure 1. Technically, the center of motion is known as the equilibrium position. Due to the momentum (made up of the mass and the speed of motion of the swinging body), the swinging body passes via the equilibrium position to the other side of the centre line until it reaches the maximum distance it can go on that side. The body then begins its way back via the center line to the other side of the equilibrium position. The distance the mass moves from one side of the equilibrium position to the other side of the equilibrium position is called the cycle of motion. The time the pendulum takes to complete one cycle is called the period. This process is repeated while the cycle length, the distance from one extreme of the equilibrium position to the other extreme end, reduces until the body comes to rest at the equilibrium position. Figure 1: A simple gravity pendulum illustrating the principle of pendulums (Parks 2) The pendulum concept has been used for a variety of practical applications. For example, the pendulum principle was applied, by Huygens, to clock mechanisms (Gindikin 83). The principle of pendulum operation has also been used for measuring seismic activities particularly through use of the horizontal pendulum seismograph (Wills 106). Most importantly, the equation of pendulum has been used in structural dynamics problems particularly in the determination and prediction of vibrations in mechanical structures (Landis et al. 168). A typical application of the pendulum equation for structural dynamics problems is the determination of breakdown of civil structures, such as roads, buildings and bridges, arising from external forces, such as earthquakes, automotives or sudden loading. Additionally, the pendulum motion can be used to determine how smooth or bouncy a ride can be when certain springs, of given mechanical properties, are used for a given car (Parks 2). Conclusively, the pendulum motion can be used as the basis for analyzing the vibration of complex structures. Some of the properties of a pendulum including period, amplitude, frequency and length relate in some way. This project is particularly aimed at establishing the relationship between the amplitude and period of a simple pendulum. Establishment of such a relationship is crucial in the various applications of the pendulum because it ensure quality analysis of mechanical vibrations, as well as quality determination of the various issues determined using instruments using the pendulum concept. Problem Statement It his experiments, Galileo concluded that there is no relationship between the amplitude and the period of the pendulum (Morgan). Galileo’s observations have sparked considerable debate with respect to whether Galileo meant there is no definite relationship between amplitude and period or that there is no significant difference between amplitude and period, which would then mean that period for all amplitudes is the same or differ slightly. An experiment by Morgan resulted into findings that were against Galileo’s findings and conclusion as Wright found out that different amplitudes result to different periods. In another experiment by Wright, the findings were different from Galileo’s and Morgan’s because Wright observed that were no significant changes in period arising from changes in amplitude for small angles of displacement, particularly for angles below 450. However, the period changes significantly for angles of displacement equal to or more than 450. However, there is no experiment available in literature, after Wright’s experiment, which considers the relationship between amplitude and period for angles > 450. Therefore, the purpose of this experiment is twofold: (1) To provide an answer to the debate, whether amplitude depends on period and (2) To expand Wright’s findings by determining if there is a definite relationship between period and amplitude for angles of displacement > 450. The Pendulum: Theory A simple pendulum is characterised by the following terms and their respective descriptions: Period (T): it is the time taken by a pendulum to undergo one complete cycle. In laboratory experiments, the period of a pendulum is obtained from the average of a number of complete and timed cycles, such as 20 cycles. As shown in later sections of this report, the period of a pendulum is determined using the equation; Angular displacement (θ): in order to initiate a periodic motion of the pendulum, the suspended body is usually given some displacement from the equilibrium position as shown in figure 2. The angle between the perpendicular line passing through the point of suspension and the line along the suspending string, upon displacement, is called the angle of displacement. The angle of displacement is as shown in figure 2. Length (l): this is the length if the string suspending the mass from the point of suspension, which is equal to the distance between the mass and the point of suspension. Amplitude (A): amplitude is the distance from the equilibrium position to the end of motion of the suspended mass, which is the point at which the motion reverses towards the equilibrium position. Frequency (f): frequency is the number of cycles within a second, which is determined as the inverse of the period of the pendulum. The frequency of the pendulum is determined thus; Since the period is determined using the equation; The frequency of vibration of the pendulum or any the mechanical system whose vibration is being analyzed using the pendulum concept is given thus; Galileo performed a number of experiments on the pendulum in which he came up with interesting findings. Galileo’s experiments have been repeated and reproduced with the aim of establishing the various characteristics of pendulums. Pendulums almost return to their release height Galileo, in his experiments, observed that pendulums usually return to almost the height at which they are released (Morgan). In Galileo’s experiments, different release heights were used and it was observed that return height was only slightly lower than the release height for all the pendulum lengths used (Morgan). At the time of Galileo, this observation did not have major implications. However, after some time, particularly with the onset of Newton, this observation was related with the principle of energy conservation, which states that there is always an equal and opposite reaction to every force (Shipman, Wilson and Higgins 59). The implication of this observation, on this law of Newton, is that the pendulum will always return to the release height if there is no resistance to the motion of the pendulum. Pendulums eventually come to rest at their equilibrium position but lighter pendulums come to rest sooner than heavier pendulums Another observation made by Galileo, regarding pendulums, is that all pendulums eventually come to rest at the equilibrium position. However, Galileo observed that pendulum of lighter weights come to rest faster than their heavier weigh counterparts. Similar results were observed with experiments on pendulums of equal length and displacement angle, but of different bob weights, which confirms Galileo’s observations (Morgan). This observation is based on the fact that heavier weights are capable of overcoming air resistance than lighter weights because heavier weights are associated with higher momentum that lighter weights. Since momentum is the product of speed and mass of the bob, heavier weights then have higher momentum than their lighter counterparts. The period of motion of a pendulum is independent of the weight of the bob used This discovery came after Galileo used bobs of different materials and weights, but giving them the same length and angle of displacement. The results were lack of significant difference between the periods of the different pendulums tested, having different weights (Morgan). Similar experiments by different experimenters confirmed Galileo’s findings, which has resulted to the rule, that the period of motion of a pendulum is independent of the weight of the pendulum (Morgan). The period of the pendulum does not depend on the amplitude of the motion of the pendulum It his experiments, Galileo concluded that there is no relationship between the amplitude and the period of the pendulum. The experiments involved swinging the pendulum from different angles of displacement, therefore achieving different amplitudes, and determining the period of the pendulum’s motion (Morgan). These observations of Galileo have, however, sparked considerable debate with some scholars debating whether Galileo meant there is no definite relationship between amplitude and period or that there is no significant difference between amplitude and period, which would then mean that period for all amplitudes is the same or differ slightly. In the effort to find an answer to the debate, Morgan conducted an experiment aimed at determining the relationship between amplitude and period. The experiment results were against Galileo’s findings and conclusion since Wright found out that different amplitudes result to different periods. In fact, Wright found out that pendulums of larger amplitudes resulted to longer periods that their shorter amplitude counterparts. Accordingly, it can be concluded that the interpretation of Galileo’s claim depends on how a scholar interprets Galileo’s claim. In a further effort to produce an answer to the debate on Galileo’s observation, Wright conducted an experiment aimed at determining the relationship between amplitude and period of a pendulum. The results were somehow interesting in that Wright observed that the relationship depends on the angle of displacement. The results of Wright’s experiment were that there are no significant changes in period arising from changes in amplitude for small angles of displacement, particularly for angles below 450. A graph of the relationship between the two pendulum properties would appear as shown in figure 2 according to Wright’s results. However, the period changes significantly for angles of displacement equal to or more than 45 degrees. However, Wright’s experiment did not consider determining the relationship between amplitude and period for angles equal to or greater than 45 degree. In other words, the experiment did not consider where there it is possible to predict the period of vibration for a given angle that is equal to or greater than 45 degrees. Figure 2: A graph showing the relationship between period and amplitude where amplitude is dependent (and represented) on the angle of displacement. The graph shows no significant change in period for changes in amplitude (displacement angles) for small angles (Wright.edu). There is a direct relationship between the length of the pendulum and the square of period of the pendulum In this experiment, Galileo used pendulums whose lengths were differing by factors of 2 and 4 in which the pendulums to be compared were released simultaneously (Morgan). The results of the experiment were that the length and period of the pendulum compared in accordance with the aforementioned relationship or equation. Afterwards, Galileo’s experiment was replicated and the results confirmed Galileo’s findings and conclusion on the relationship between the two properties of a pendulum, length and period (Morgan). Accordingly, this implies that the length of a pendulum, which is the length of the string used to attach the pendulum bob to the fixed point, is a crucial factor of a pendulum, which determines the operation of the pendulum. Galileo’s findings and conclusions have been confirmed by various experiments on the same, such as by Yin. Mathematically, this claim of Galileo can be represented thus; Therefore, Where k is a constant It can also be represented as, The relationship between period and length is as shown in figure 3 Figure 3: The relationship between the length of a pendulum and the period of oscillations (Wright.edu) Consequent experiments have resulted to an equation that relates period and length, which is represented as; Therefore, the constant, k, can be considered as This means, therefore, that the force of gravity, g, for a given place is a crucial factor that determines the operation of a pendulum and the dynamics of mechanical structures analyzed using the pendulum concept. In fact, Wright indicates that the displacement made by the pendulum’s weight is dependent on the length of the string and the acceleration due to gravity, which is usually quoted as 9.81N/m2. Other experiments have been conducted to determine the impact of friction on the pendulum. Friction is usually considered the only reason as to why a pendulum gradually comes to rest at the equilibrium position (Kirkpatrick and Francis 291). Friction of the medium in which the pendulum operates has the effect of reducing the amplitude of the pendulum’s vibrations as shown in figure 4. Figure 4: The amplitude of vibration (oscillations) reduces gradually with time due to the effect of friction of the media in which the pendulum vibrates (oscillates) (Wright.edu). Wright sought to explicate the impact of friction on the pendulum’s vibration by operating a simple pendulum in three different media, air, water and oil, which have different friction factors. The results were difference in the time taken before the pendulum came to stop in each of the three media. Therefore, if operated in a perfect vacuum condition, the pendulum will never come to rest unless stopped by an external force, such as being stopped by hand. Mathematical expressions The equation of motion for the pendulum that corresponds to the tangential direction of motion for a bob of a given mass is represented as; (1) If we take a small angle, such that, (1) can be simplified to (2) Further, if we define the simple harmonic motion as, where n is the pendulum’s natural frequency, and substitute for in (2), it can be simplified as; (3) The period (T) can be found using the equation, (4) Experiment Tools and Materials 1. Bob (200g, 500g and 750g) 2. Clamp 3. A strong nylon string (1500 mm long) 4. Meter rule 5. Compass (for measuring displacement angles) 6. A stop watch Experimental setup The experiment will be setup as shown in figure 1. A bob of 200 g mass will be attached at the end of the string and the length of the string between the point of attachment and the bob adjusted to 500 mm. Different angles of displacement ranging from 50 to 900 will be tested of which period for each displacement will be determined. For each angle, as shown in table 1, the test will be run 3 times. The pendulum will be displaced and allowed to swing freely. With the help of a stop watch, 20 cycles will be timed and the period determined from the results. This will be done for all the angles. The same procedure will be repeated for the bobs of 500g and 750g respectively. The length will then be adjusted to 1000 mm and the entire procedure repeated for all the three masses. Finally, the length will be adjusted to 1500 mm and the entire procedure repeated for all the angles. Figure 5: Experimental setup Results Experiment findings will be recorded as shown in table 1. Table 1: Results Length: 500 mm Mass: 200 g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Mass: 500 g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Mass: 750 g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Length: 1000 mm Mass: 200 g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Mass: 500 g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Mass: 750g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Length: 1500 mm Mass: 200g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Mass: 500 g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 Mass: 750 g Angle (degrees) 1st run 2nd run 3rd run Time for 20 cycles period Time for 20 cycles period Time for 20 cycles Period Average period 10 20 30 40 45 50 60 70 80 90 This will lead to an experimental matrix as shown in table 2, which applies for all the displacement angles tested Table 2: Experimental matrix Displacement angle = 100 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 200 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 300 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 400 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 450 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 500 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 600 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 700 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 800 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Displacement angle = 900 The variable Constant parameters Constant parameters M1 = 200 g L1 = 500 mm T = M2 = 500 g L1 = 500 mm T = M3 = 750 g L1 = 500 mm T = L1 = 500 mm M1 = 200 g T = L2 = 1000 mm M1 = 200 g T = L3 = 1500 mm M1 = 200 g T = Discussion and Conclusion Is the theoretical period equal to experimental period for all angles? Does change in displacement angle lead to change in period? Is it consistent for all angles of displacement? Is it possible to predict the period given the angle of displacement? Possible sources of errors a. Errors in measuring angles of displacement and lengths b. Errors in determining period by timing number of full cycles c. Interference from external factors, such as wind Works Cited Gindikin, Simon. Tales of Mathematicians and Physicists. Piscataway, NJ: Springer. 2007. Kirkpatrick, Larry D & Francis, Gregory E. Physics: A World View, 6th Edition. Thomson. 2007. Landis, Richard., Ertas, Atila., Gumus, Emrah & Gungor F. Free Pendulum Vibration Absorber Experiment using Digital Image Processing. In, D. Adams., G. Kerschen & A. Carrella. Topics in Non-Linear Dynamics, Volume 3: Proceedings of the 30th IMAC, A Conference on Structural Dynamics. Society of Experimental Mechanics. 2012. Morgan. Galileo’s Pendulum Experiments. 1995. Web. January 1, 2013 Parks, James E. The Simple Pendulum. Tennessee University Department of Physics and Astronomy. 2000. Web. January 1, 2013 Shipman, James T., Wilson, Jerry D & Higgins, Charles A. An Introduction to Physical Science. 13th edition. Cengage Learning. 2011. Wills, Graham. Statistics and Computing: Visualizing Time: Designing Graphical Representations for Statistical Data. Naperville, Illinois: Author. 2012. Wright. The Pendulum. N.d. Web. January 1, 2013 Yin, Cynthia L. How do Varying Amplitudes, Weights, and Lengths Affect the Period of Motion of a Pendulum? California State Science Fair. 2009. Web. January 1, 2013 Read More