Pendulum and Simple Harmonic Motion - Report Example

Pendulum and Simple Harmonic Motion Name Course Institution Instructor Experimental Date Submission Date Table of Contents Table of Contents 2 1.0 Abstract 3 2.0 Introduction 3 2.1 Aims 4 3.0 Theory 4 4.0 Materials and Method 7 4.1 Apparatus 7 4.2 Procedure 8 4.2.1 Results and Calculations 8 4.2.2 Calculations 9 4.2.3 Analysis 10 4.3 Task B: Period of pendulum Oscillation versus the length of the string 10 4.3.1 Calculations 11 4.3.2 Analysis 12 5.0 Conclusion 12 5.1 Possible Sources of Errors 12 5.2 Future Improvements to avoid these Errors 12 6.0 References 13 1.0 Abstract This report illustrates an experimental procedure that was used to estimate acceleration due to gravity. Various apparatus such as pendulum bob, string and stop watch were used. A pendulum bob was tied to the string to allow it oscillates freely. Adjustments were made on the length of the string used. After every adjustment, the pendulum bob was allowed to oscillate for a constant number of oscillations. Time taken to complete the required number of oscillations was then tabulated for each adjustment. Graphs of time in seconds against square root of the length in metres were then plotted for each series of tabulated values. Acceleration due to gravity could be estimated by using the calculated gradient and the equation; Where time period in seconds, Length of pendulum in metres, Acceleration due to gravity. 2.0 Introduction The two experiments were used to investigate motion of a simple pendulum. Motion of a simple pendulum can be used to investigate various physics’s formulations including acceleration due to gravity. The conventional accepted value of acceleration due to gravity is 9.806 m/s2 (NIST, 2012). Verification was possible through the calculated gradient values for the two graphs plotted. This report offers a detailed explanation of what a simple pendulum is and how it can be used to calculate the acceleration due to gravity. Various variables such as damping, mass of the pendulum bob, length of the string used, time and angle of displacement were looked into. For many centuries, diverse researches have been conducted by scientists such as Isaac Newton on the correlation between length and time (period) and values obtained were used to find precise value of acceleration due to gravity. The two experiments are replications of these past research experiments. The value obtained was compared to the accepted value; 9.806m/s2. 2.1 Aims The objectives of the experiment were to; Investigate the properties of a simple pendulum Familiarize with simple harmonic motion and its characteristics Determine the acceleration due to gravity by comparing length and period of oscillation 3.0 Theory A simple pendulum is a mechanical system exhibiting periodic motion. It is consisting of a bob of mass that is suspended on a fixed top point with the help of a light inextensible string of length as shown below in figure 1. Figure 1 The string is always assumed to be mass-less. The pendulum bob can be set into oscillation in two dimensions; Vertical plane Horizontal plane For this case, the motion was on a vertical plane and it is driven by gravitational force. Displacing the pendulum bob by an angle , it will undergo a simple harmonic motion. During that period of oscillation, two forces are acting on the pendulum bob. They are the tension force and the gravitational force. Gravitational force acting on the pendulum bob is always assumed to be acting uniformly and in the vertical direction. A pendulum bob attached to a string and displaced at an angle has an unstable equilibrium. The unstable equilibrium allows the pendulum bob to oscillate. When the bob is displaced by an angle and released to oscillate, the tangential component of yields a restoration force acting on the opposite direction of displacement. Restoration force opposes displacement from equilibrium position. The pendulum will oscillate displaying a simple harmonic motion. Restoring force is the vector sum of gravitational force on the pendulum bob and the tensional force on the string. Resolving the gravitational force into two components; Along the direction of radial and away from the suspension point Along the arc in the direction in which the pendulum bob moves Resolving the gravitational force along the direction of the arc gives the restoring force given by; Where is the restoring force, is the angle of displacement, is the mass of the pendulum bob and is the gravitational constant. in radians is given by; For small amplitudes, . Restoring force equation therefore becomes; . Substituting the value of in radians, this equation becomes; . For this case, restoring force is directly proportional to (displacement from equilibrium position) and it satisfies the relationship . Comparing and , then the value of . But the period of a simple harmonic motion is given by; where is the proportionality constant. Substituting the value of k, then we shall have; Therefore; The negative sign indicates that the direction of this force is opposite to displacement. The restoring force can be analyzed to determine the period of simple pendulum. The raw data obtained from the experiment should be able to give a square-root relationship between the length of string used and period of oscillation. The accurate value of can be estimated from the graph of time against square root of length. 4.0 Materials and Method 4.1 Apparatus Mass set Pendulum bob Stop watch Inextensible string of at least one metre A pendulum clamp A metre rule 4.2 Procedure 1. The pendulum bob was weighed on the mass scale and recorded. 2. The experiment was set up as shown in figure 2 below and used to measure the period of the pendulum by counting six complete swings. Initially, it began with a one metre of length string. Also, an approximately equal angle of displacement was used throughout the experiment. Figure 2 3. The results of elapsed time, number of swings and the length of the pendulum were tabulated. From these results, the period of oscillation was calculated. 4. The same procedure was repeated for other lengths below one metre and recorded. N/B; the length of the pendulum was measured from the pivot point to the centre of the bob. 4.2.1 Results and Calculations Task A: Period of pendulum Oscillation versus the length of the string Mass of the bob = 34.2 g. L(m) Square root of L (m1/2) Total time (s) Number of full swings Period of full oscillation T(s) 1.00 1.000 11.95 6 1.983 0.90 0.948 11.00 6 1.83 0.80 0.894 10.75 6 1.79 0.74 0.860 10.47 6 1.74 0.65 0.800 9.68 6 1.61 0.58 0.760 9.12 6 1.52 0.49 0.700 8.53 6 1.42 0.39 0.620 7.75 6 1.29 4.2.2 Calculations From the graph, the gradient can be calculated; Relating this to the equation, then we shall have; , = 9.87m/s2. 4.2.3 Analysis Percentage error = = The value obtained from this experiment is 9.872164 m/s2 which is comparable to the theoretical value of 9.81m/s2. An error of 0.61% is a reasonable value. 4.3 Task B: Period of pendulum Oscillation versus the length of the string Mass of the bob = 97.3 g. L(m) Square root of L (m1/2) Total time (s) Number of full swings Period of full oscillation T(s) 1.00 1.000 12.19 6 2.000 0.90 0.948 11.62 6 1.930 0.80 0.894 10.81 6 1.800 0.74 0.860 10.53 6 1.755 0.65 0.800 10.00 6 1.600 0.56 0.748 9.25 6 1.540 0.48 0.690 8.54 6 1.420 0.39 0.620 7.75 6 1.290 4.3.1 Calculations Calculating the gradient; Relating it with, then we shall have, . Therefore; 4.3.2 Analysis Percentage error = = The experimental value has an error of 9.28% when it is compared to the theoretical value. The error is reasonable value. 5.0 Conclusion From the two experiments, it can be deduced that the gravitational force is 9.81 m/s2 which is the theoretical value. Therefore, the objectives of the experiment were attained. The two experiments were related, only that the mass of the pendulum bob was changed. However, by changing the mass, there was no effect because there was minimal variance between the two. The period of the oscillation is independent of the amplitude and mass of the pendulum, but not the acceleration due to gravity. Deviation from the nominal theoretical value of 9.81 m/s2 is as a result of errors. 5.1 Possible Sources of Errors Parallax errors when measuring the length of the string Systematic errors on time due to uncertainty when recording 5.2 Future Improvements to avoid these Errors Taking readings at a perpendicular angle to the point Increasing the number of swings to reduce uncertainty errors 6.0 References Giordano, Nicholas. College Physics, Volume 1. Edition2. New York: Cengage Learning. 2012. National Institute of Standards and Technology. The NIST Reference on constants, Units and Uncertainty. 2012. Web. 6th Nov 2012. Pook, L.P. Understanding Pendulums: A Brief Introduction,Volume 12 of History of Mechanism and Machine Science. New York: Springer. 2011. Read More