The Relationship Between the Length and the Time of the String - Lab Report Example

Running head: Pendulum and Simple Harmonic Motion (SHM) Student’s name Institution Course Professor Date of Experiment Date of Submission Abstract This report describes experimental procedures that were used to determine the relationship between the length of the string and the time for one swing (the period) which was then used to calculate the gravitational acceleration. The length of the pendulum was varied and a change in the period was observed and tabulated. The equipments utilized in these experiments included; Pendulum weight, 1m string, a stand and clamp, Stopwatch and scales. Two graphs of time in seconds against the square root of the length of string in metres were plotted. The slope was then calculated in the graph and thereafter it was used to estimate the acceleration due to gravity. The calculated value of acceleration due to gravity in both experiment A and B were 9.216 m/s2 and 10.07m/s2 respectively. The two calculated values were comparable with the theoretical value of 9.81m/s2 of gravitational force. TABLE OF CONTENTS Abstract 2 TABLE OF CONTENTS 3 1.0Introduction 4 1.1Theory 5 2.0Methodology 7 3.0Procedures 7 3.1Experiment A 7 3.1.1Calculation and Results 7 3.1.2Discussion 9 3.2Experiment B 10 3.2.1Calculation and results 10 4.0Conclusion 12 5.0Experimental Errors 13 6.0References 14 1.0Introduction This report describes experimental procedures that were used to determine the relationship between the length of the string and the time for one swing (the period). The relationship then helped to determine the accurate value of acceleration due to gravity. The calculated gravitational acceleration was compared with the theoretical value of 9.806m/s2. In this experiment, the length of the pendulum was varied and a change in the period was observed and recorded. The length of the pendulum was set at 1m and for the successive measurements; the length of the string was reduced by 0.01m. A simple pendulum can be used to calculate the acceleration due to gravity. The aims of the experiment were to familiarize with the following; 1) The use of various length and tine measurements, 2) Taking accurate notes, 3) Calculating values and concepts of measurement significance and graphical methods i.e acceleration due to gravity To perform a simple harmonic motion, a simple pendulum is utilized. A simple pendulum consisted of a particle of mass, m that was suspended from the tripod stand by a string of length, l and negligible mass so that it could oscillate back and forth (Pook, 2011). The string is inextensible and has no resistance to bending and the suspension point is a clamp at the upper end. The gravity that acts on the mass is assumed to be uniform and acts vertically downwards. Simple string pendulum is isochronous for small amplitudes and the motion of the point mass will approximates to simple harmonic motion. fig 1 1.1Theory The motion of a simple pendulum, oscillation for small amplitudes approximates to simple harmonic motion. Simple harmonic motion is regarded as a linear system. A point mass, m can move along a straight line with a restoring force towards a fixed point that is proportional to the distance to the distance from the fixed point (Pook, 2011). The periodic motion of the simple pendulum can be described by an acceleration that is proportional to its displacement and is directed towards the centre of motion. In simple harmonic motion, the force is directly proportional to the displacement from the equilibrium position (Loyd, 2007). The period of the motion represents the time T for one complete vibration. Forces that exerts on the particles are its weight; W=-mg and the tensional force, T in the string. For example if the pendulum swings through a small angle from its equilibrium position, where , then it undergoes a simple harmonic motion (Jerry &Cecilia ,2009). The restoring force which is tangent component to the path and acts opposite to the displacement of the particle brings the particle towards the equilibrium position (Jha, 2005). This force is expressed as Where F is restoring force, θ is the angle of displacement, g is the gravitational constant m is the mass of the pendulum. The equation of restoring force for small amplitudes i.e can be expressed as. The radians, θ can be expressed as where x represent displacement from the equilibrium position. Since the restoring force is directly proportional to the displacement from the equilibrium position. Thus the equations becomes and . When the two equations are compared, the value of k will be expressed as. The period of a simple harmonic motion is given by; where is the proportionality constant. Substituting the value of k, the equation is Where T is the periods in seconds, l is the length in metres and g is the gravitational acceleration in m/s2. The period of a pendulum depends on its length and is independent of its mass and the amplitude of oscillation. For instance, the period T of the swindling pendulum is proportional to the square root of the length l of the pendulum (Jerry & Cecilia, 2009). 2.0Methodology Materials and equipment employed during the experiment were Pendulum weight 1m string A stand and clamp Stopwatch Scales The clamp was used to secure the position of the tripod stand as the pendulum was swinging. 3.0Procedures 3.1Experiment A The pendulum bob was weighed on the scale and recorded. Then the time period of the pendulum was measured by counting seven completed swings. The results indicating the elapsed time, number of swings and the length of the pendulum were tabulated. Care was taken to ensure that the length of the pendulum was measured from the pivot point to the centre of the metal weight. The period of oscillation was then calculated from the results. The same procedure was repeated for the other lengths of the string. 3.1.1Calculation and Results Experiment A: Period of pendulum oscillation versus the length of the string Length (m) Square root of L (m 1/2) Total time (s) Number of full swings Period of full oscillation T(s) 1m 1 14.31 7 2.044 0.9m 0.948 13.47 7 1.924 0.8m 0.894 12.80 7 1.828 0.7m 0.836 11.97 7 1.71 0.6m 0.774 10.95 7 1.564 0.5m 0.707 10.14 7 1.564 0.4m 0.632 9.18 7 1.311 The mass of the metal =42.9 g 3.1.2Discussion From the graph, the slope (gradient) can be calculated by =2.07 To find the acceleration due to gravity, the equation, was used = 9.216 m/s2 The percentage error 6.055% 3.2Experiment B The same procedure was used as above in experiment B 3.2.1Calculation and results Experiment B: Period of pendulum oscillation versus the length of the string 1ength (m) Square root of Length (m1/2) Total time (s) Number of full swings Period of full oscillation T(s) 1m 1 14.41 7 2.058 0.9m 0.948 13.72 7 1.96 0.8m 0.894 12.66 7 1.808 0.7m 0.836 12.20 7 1.742 0.6m 0.744 11.49 7 1.641 0.5m 0.707 10.21 7 1.458 0.4m 0.632 09.06 7 1.294 Mass of the metal= 145.5 g 3.2.2Discussion From the graph, the slope (gradient) can be calculated by =1.98 To find the acceleration due to gravity, the equation, was used = 10.07 m/s2 The percentage error =2.68% 4.0Conclusion The calculated value of acceleration due to gravity in both experiment A and B were 9.216 m/s2 and 10.07m/s2 respectively. The two calculated values were comparable with the theoretical value of 9.81m/s2 of gravitational force. The motion of the simple pendulum is approximated by the simple harmonic motion. Also the period, T of the mass attached to the pendulum of the length l with gravitational acceleration, g is given by From the calculated results, it was realized that the period of oscillation is independent of the amplitude and mass of the pendulum but not the acceleration due to gravity. However the variation witnessed between the theoretical value and calculated values were due to errors. Indeed the objectives of the experiments were achieved. 5.0Experimental Errors Possible sources of errors during the experiment included; parallax errors during the measurement of the length of the string and systemic errors on time. However, increasing the number of swings will reduce the errors. 6.0References Jerry D. W &Cecilia A. H, (2009).Physics Laboratory Experiments, Cengage Learning. Loyd, D. H, (2007).Physics Laboratory Manual, Volume 10, Cengage Learning. National Institute of Standards and Technology. The NIST Reference on constants, Units and Uncertainty. 2012. 07th November 2012 Pook, L.P, (2011).Understanding Pendulums: A Brief Introduction, Springer. Jha, D.K (2005).Text Book of Simple Harmonic Motion and Wave Theory, Discovery Publishing House, 2005 Read More