PENDULUM PERIODS First Last Partners: student1, student2, and student3 Governor s School for Science and Technology 520 Butler Farm Road, Hampton, VA 23666 April 13, 2011 ABSTRACT The effect of amplitude, length, and mass on the period of a simple pendulum was studied. Pendulums with varying amplitudes of release, lengths of string, and masses on the string were tested to see its effect on the period. It was found that a systematic variation on the amplitude or mass of the pendulum has no significant affect on the the period, but a decrease or increase in the length of the string caused a decrease or increase in the period, respectively. 1. Introduction Pendulums have long been a common apparatus in scientific classrooms and laboratories. They have even been used for clocks and astronomical measurements. One of the ways to describe a pendulum s motion is the period, or how long it takes for the pendulum to complete one complete swing. There are several criteria that could affect the period of a pendulum, which include the amplitude of the pendulum swing, the length of the pendulum, and the mass (or bob) of the pendulum. However, current theory suggests that only the length of the pendulum will affect the period of the pendulum. In this experiment we attempt to demonstrate this effect. The remainder of the report is organized as follows. In Section 2 we discuss the pendulum model. The experimental apparatus and procedures are discussed in Section 3 and the analyses are discussed in Section 4. We discuss and conclude our studied in Section 5. 2. Pendulum Governing Equations If we consider a mass m hangs at the end of a long thread (see Fig. 1) and set it swinging with a small amplitude, the mass (bob of the pendulum) will move back and forth in a simple harmonic oscillation (motion). The inertia is the mass of the pendulum and the spring is in the gravitational attraction between the bob and the earth. The tangential component changes direction every time the mass passes through the midpoint of its swing and thus always acts to restore the mass to its central position. Thus, we have the restoring force F = mg sin(θ), (1) where m, g, and θ are the mass of the bob, the acceleration due to the Earth gravity, and the amplitude or the angle that is displaced away from its equilibrium position. The minus sign in equation (1) indicates that the force is a restoring force. If the angle θ is very small, which we assume here, then sin(θ) θ, hence, F = mg sin(θ) mgθ. (2) From Figure 1, the displacement angle (θ) from equilibrium can be determined by using arc length (x) and the length of the string (L) to obtain θ = x/l. Upon substituting θ into equation (2) yields F = mg x ( mg ) L = x. (3) L
2 Fig. 1. The dynamics of a simple pendulum. Equation (3) has a form of Hooke s law, F = - k x, with the force constant k = mg/l. For a simple harmonic motion the period is given as m T = 2π k. (4) Substituting k = mg/l into equation (4) the period for a simple pendulum is then given as L T = 2π g. (5) Note that the period of a simple pendulum in equation (5) is independent of its mass and angular displacement. 3. Experimental Apparatus and Procedures The bob was attached to two strings to assure motion along in one direction (x) (see Fig. 1). The Vernier Photogate was attached to a stand to ensure no movement throughout the experiment (see Fig. 2). The Vernier Photogate is a device to measure the period of one complete cycle as the bob passes through it with two consecutive swings. The Photogate was connected to a Window PC and Logger Pro was used to record and analyze the data. The data was collected for five complete cycles and then averaged to find the average period. There are three parts to this experiment. Part I - A bob of constant mass was released from different amplitude to see the effect on the period. The length of the string was also constant. The angle of release was measured with a protractor from the support point at the angles of 10, 15, 20, 25, and 30 degrees. Part II - The length of the string and the release angle were kept constant while varying the mass of the bob. Three trials were conducted with masses of 100, 200, and 300 grams. Part III - A bob of constant mass was released from the same angle, but with each trial the string was lengthened by 10 centimeters. Six trials were conducted with lengths of 97, 87, 77, 67, 57, and 47 centimeters.
3 Fig. 2. The simple pendulum apparatus. 4. Data Analysis, Results, and Uncertainties The experimental data for part I, II, and III are shown in Table 1, 2, and 3, respectively. The raw data of the experiment are shown in Appendix A. The data for the period was collected every other time the bob passed through the Photogate since two executive passing indicates one complete cycle or one period. Part I - Amplitude. The results of the first experiment, where we studied the effect of amplitude of swing on a pendulum s period, are shown in Figure 3. The result indicates that as the amplitude increases the period remains fairly constant. However, the result also shows a systematic increase in the period as the amplitude increases (see Table 1). This variation could be explained due to air resistance as the bob flew through space. Air restricting the movement of the bob would make the period longer. The percent difference between any two experimental value is given by %Difference = E 1 E 2 1 2 (E 100. (6) 1 + E 2 ) Using equation (6) the % difference between two consecutive experimental values is less than 0.4% and the % difference between the smallest and largest experimental values is less than 2% as shown in Table 1. These results indicate that our data is at least 98% accurate. Hence, our result indicates that the period is independent of the pendulum mass is therefore consistent with equation (5). Table 1. Period vs Amplitude Amplitude( o ) Period(s) %Difference 10 1.958 0.357 15 1.965... 20 1.970... 25 1.982... 30 1.993...
4 Table 2. Period vs Mass Mass(g) Period(s) %Difference 100 1.972... 200 1.971... 300 1.974... Table 3. Period vs Length Length(cm) Period(s) %Difference 97 1.989... 87 1.882... 77 1.784... 67 1.663... 57 1.520... 47 1.352... Fig. 3. The filled circles are data and the solid line is a line joining two points. This figure shows that the period remains constant as the amplitude increases.
5 Part III - Mass. The results of the third experiment, where we studied the effect of the mass on a pendulum s period, are shown in Figure 4. The result indicates that as the mass increases the period Fig. 4. The filled circles are data and the solid line is a line joining two points. The figure shows that the period remains constant as the mass increases. remains the same. Using equation (6) the % difference between two consecutive experimental values is less than 0.05%, and the % difference between the smallest and largest experimental values is less than 0.1% as shown in Table 2. These results indicate that our data is at least 99% accurate. Hence, this suggests that the mass did not affect the period of the pendulum s swing, and is therefore consistent with equation (5). Part II - Length. The results of the second experiment, where we studied the effect of the length of the swing on a pendulum s period, are shown in Figure 5. The result indicates that as the length of the string Fig. 5. The filled circles are data and the solid line is a line joining two points. This figure shows that as the length of the string increases the period increases. increases, the period also increases. With each shortening of 10 cm, the period length decreases by about 0.1 second. Using equation (6) the % difference between two consecutive experimental values is more than 5%, and the % difference between the smallest and largest experimental values is more than 38% as shown in Table 3. These results indicate a significant different to show that the period does depend on the length. Hence, this suggests that the length of the pendulum do affect the period of the pendulum s swing, and this is also
6 consistent with equation (5). Using the same data in Table 3, these results are plotted with T vs. L 2 and T 2 vs. L as shown in Figures 6 and 7, respectively. Of the plots comparing the period to the length (Figs. 5, 6, and 7), Figure 7 Fig. 6. The filled circles are data and the solid line is a line joining two points. This figure shows that the period also varies as the length square. Fig. 7. This figure shows that the period square varies proportional to the string length. The filled circles are data and the solid line is a linear fit. provides a plot which is closet to a direct proportion with a constant slope. Using equation (5) we can square both side of the equation to give a linear relationship between T 2 and L T 2 = 4π2 g L, (7) with a slope of m = 4π2 g. From Figure 7 a linear fit to the data gives a slope value of 0.04215 s2 /cm. Setting the slope 0.04215 (s 2 /cm) = 4π2 g, we obtain the Earth gravitation constant to be 9.355 m/s2, where the accepted value of g is 9.807 m/s 2. The % error between an experimental quantity, E, with a theoretical or accepted value, A, is given by %Error = A E A 100. (8) Using equation (8) the % error for the experimental value of g is less than 5%. This indicates that our experimental value of g is at least 95% accurate.
7 5. Discussion and Conclusion From this experiment we have shown that as we change the amplitude or the mass of the pendulum the period remain constant. However, there is small systematic deviation in the period when we increase the amplitude if the swing. Our data show that as the amplitude increases there is a slight increases to the period. However, this slight deviation can be explained due to presence of air resistance. The effects of air resistance decreases as the amplitude decreases. Moreover, the maximum deviation is less than 2%. Based on our analysis the experimental data of the period that we have recorded for the variation of the amplitude or the mass is at least 98% to 99% accurate. This provides confidence that our data are good. Our experiment also shows that period does depend on the length of the pendulum, and from our analysis we also obtain the Earth gravitation constant g with a value of 9.355 m/s 2 with 95% confidence. Overall, the results of our experience support the established knowledge that both the mass and amplitude have no affect on the period, and that the period square is directly proportion to the length of the pendulum. The author would like to thank... 6. Acknowledgements Becker, P. A., Das, S., & Le, T. 2008, ApJ, 677, L93 Becker, P. A., Das, S., & Le, T. 2009, in preparation REFERENCES This preprint was prepared with the AAS LATEX macros v5.2.