The Simple Pendulum - Lab Report

Lab Number: 4 The Simple Pendulum Objective: To observe the variables that determines the period of the oscillation of a pendulum. In addition this experiment was conducted, to show that the period of the oscillation of a pendulum is independent of the mass of the bob.When conducting this experiment, experiments are observing three variables to demonstrate if they affect the period of pendulum. The variables are the following: the amplitude (height) of the pendulum, the length of the string, and the mass of the bob. Background: Within the system of a simple pendulum, it consist of a bob that has small mass that is strung by a non-stretching “massless” string that holds a length of L. The period T of the oscillation is the amount of time it takes for the bob to travel from one position to another and back again. The swinging eventually stops because of air friction that causes the mass to slow down in velocity and eventually stop. The variables that determine the period of oscillation of a pendulum are not only the amplitude (height) of the pendulum, the length of the string, and the mass of the bob, but also the acceleration due to gravity, g. Principles: A pendulum is similar to a swing that is seen in a playground. Similar to the swing, the pendulum starts at rest in an centered position, to be moved from a certain angle in one direction then to be released and swing to one extreme position and then back again. A mass is suspended from a string or rod of negligible mass at a given length of L so that it may swing freely. When the mass is displaced from its resting position and released it will be accelerated back towards its resting position by gravity. Due to this acceleration of gravity, the mass will surpass its resting position and continue to swing back and forth from one extreme position to the other and back again, also known as the oscillation. The amplitude of this oscillation is the angle between pendulum at any of the extreme positions and its resting position or the angle at which the pendulum is pulled away from its resting position. The time taken for one complete oscillation cycle is the time taken for the mass to swing left and right is known as the period, T, of oscillation and be calculated using the formula: =2Lg Where T= period of oscillation m= mass of the bob L= length of the non stretching “massless” string g= acceleration of gravity By units: mm/s2=s2=s Apparatus:   Table clamp  Steel rod  Pendulum clamp (silver)  Pendulum bob (various sizes)  String  Electronic balance  Master photogate timer (set to pendulum mode)  Meter stick Pend. Protractor Procedure: Students are to first retrieve the items from the front of the class and when this is received, students are to begin prepping the lab area. To begin the experiment first turn on the photogate and make sure that the mode is in pendulum mode. When this has been verified, place the first bob on the scale for its mass to be calculated. The mass is 30.1g. Having found the mass of the bob, set the pendulum in resting position so that the bob is directly in the center of the photogate. Measure the length of the string, L, 34.1 cm and make sure to have accurate measurements as the string length must be fixed. Procedure A: The first part of the experiment states that experimenters are to find the period of the pendulum when the amplitude is at the following 's: 5, 10, 15, 20, 25, and 30. Using the protractor, measure the with the string,when the angle is authenticated, release the bob at , but before this is done make sure that the photogate has been calibrated for pendulum mode. When the photogate has been double checked, release the bob. When the bob comes to a complete stop check the photogate for the reading of the period (T) in seconds. For Procedure A the period should be constant despite that the amplite keeps changing, and when graphed the graph will depict a zero slope. Procedure B: For the second part of the experiment the L, is no longer the fixed length of 34.1 cm, but instead differentiating with each change in height length of the string, however to have passable measurements the changes in length should not be overly drastic. In addition instead of having multiple ’s the is now at a fixed 10. Before commencing each experiment make sure that the photogate has been calibrated correctly before beginning each new length. The length of the string should be measured or converted into meters. The resulting graph for this procedure is an curve due to formula: =2Lg as the x-axis is taken into respect to the Length of the string. Thus, this gives you the period in relation to the length of the string. When the variables are then placed into the PHY 116 Pendulum Experiment simulator the gravitational acceleration or g is found to be 9.691 0.03352. Procedure C: For the final procedure the is once again at a fixed 10with L at a fixed .371m. Yet in this part of the experiment, experimenters are to find the period of various bob masses at a fixed amplitude. Record each mass and period at the fixed length and amplitude. Similar to the first graph, this one should be one with zero slope and a horizontal line. Experimental Data: Procedure A (T vs ) Amplitude Period (T) in seconds 5 1.15 10 1.18 15 1.18 20 1.18 25 1.18 30 1.18 m=30.1g Length of string= 34.1cm L=34.1cm Mass of the bob: 30.1g Procedure B (T vs L) String Length (L), in Meters Period (T) in Seconds 0.421m 1.31s 0.405m 1.29s 0.350m 1.19s 0.293cm 1.09s 0.230cm 0.970s 0.157cm 0.830s Amplitude of the angle: 10 Mass of the bob: 30.1g g is found to be 9.691 0.03352. Procedure C (T vs M) Mass (M) of the bob in g Period (T) in seconds 30.1g 1.23s 71.4g 1.23s 3.4g 1.21s 28.4g 1.22s Amplitude of the angle: 10 L is fixed at 0.371m Discussion: The period of oscillation (T) is the time taken for the bob to complete one oscillation cycle by swinging both to the left and right of the resting position. From the plots of period vs amplitude, period vs length and period vs mass experiments observed that the period was affected only by the length of the string. As the length of the string was increased the period also increased. This was as expected since increasing the length of the string will lower the rate at which the pendulum swings back and forth, this rate is known as the frequency of the pendulum. Increasing the length causes the pendulum to take more time to complete one oscillation. The mass of the bob did not affect the period because the acceleration towards the resting position was due to gravity which is constant which will cause objects of different masses with similar shape to fall at the same rate. The amplitude of the pendulum also did not affect the period because increasing the amplitude allows the bob to continue accelerating for a longer period. This would allow a larger amplitude to move faster and cover a greater amount of distance in the same time it would take a smaller amplitude to cover a smaller distance. This would result in both amplitudes having the same period of oscillation. Conclusion: We found that the pendulum goes slower than simple pendulum theory at larger angles. In conclusion, the period of oscillation is dependent on the length of the pendulum and is not greatly affected by the mass or amplitude.The gradient of the slope in the in the plot of period vs length will be equivalent to the acceleration. Since the acceleration was due to gravity we expect the gradient to be 9.8. However, the gradient was observed to be 9.681 which had a percentage difference of 1.11% or a decrease of 0.109 units. This difference could have been caused by systematic and gross errors which may have occurred throughout the experiment. Some of these errors may have included, errors associated with the photogate, ruler and protractor. The angle of the pendulum may have been shifted slightly before it was released. Also, slight air resistance may have also influenced the results.