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Experiment #1: The Simple Pendulum Student Name PHY 119 L__ Due Date: TA Name:

1. Abstract In class we observed that the period of the pendulum appeared not to be impacted by the angle of deflection or by the mass of the bob, but rather by the length of the pendulum. We set out to test this the following week by recording the period of a pendulum at 10 different lengths from 0-100 cm. We used a meter stick to measure the length and a cell phone stopwatch to measure time. We recorded the time elapsed over 10 periods to minimize the impact of human error on our calculations. Graphing our results revealed a near linear relationship between L and T 2 which agreed with the known formula for the period of a pendulum, T = 2 π √ L g . 2. Introduction The purpose of this experiment was to determine the relationship between the length of a simple pendulum and the period. We recorded the period of a simple pendulum at 10 different lengths from 0-100 cm. We recorded the length (L) with a meter stick and the time elapsed in 10 periods (10T) with a cell phone stopwatch. We used excel to calculate T, L 2 , and T 2 and to create the graphs to visualize our results. 3. Methods and Materials Equipment :  Ring Stand  Steel Ball  String  Meter Stick  Stopwatch 4. Experimental Procedures Make 10 pairs of measurements of L and T in which L is varied over the range from about 10 cm to about 100 cm in intervals of 10 cm. For each chosen value of L, you should measure the corresponding value of T. The length L is thus the independent variable, and T is the dependent variable. 1. Measure L from the fixed point of support to the center of the mass. For each measurement of L, estimate your uncertainty by repeating the measurement several times. Both lab partners should make independent measurements. Your best value for L is your average value (using data from both partners). Estimate the uncertainty in the average L from the scatter of the measurements. 2. Use a cell phone stopwatch to measure T, or use an internet stopwatch. Make sure the program measures time intervals to the nearest .01 (hundredth) seconds. 3. Test to see whether varying the initial swing angle θ has an effect on T. First, perform a measurement of T when θ is about 10° and then repeat with a value of θ of about 20°. Can you observe a difference? Repeat with θ = 60˚ 4. Make graphs of your data relating T to L . Try potting T vs. L , T vs. L 2 and T 2 vs. L. Label the graphs. Determine the errors in T 2 and L 2 , following the write up on error analysis. Put error bars on all three graphs (You might neglect error bars which are smaller than your Experiment #1 Page 2 of 4 Student Name

plotted points, although in general you should plot graphs on a large enough scale so that error bars are visible, if practicable.) 5. From your results, determine which equation (if any) is valid: T = kL, T = kL 2 , or T 2 = kL . ( k is called a proportionality constant). 6. Determine k as the slope of the relevant graph. The final error in k is given by the maximum and minimum slopes consistent with your data. 7. Theory (to be discussed later in the course) tells us that k = 4π2/g. What value of g do you get from your data? What is your uncertainty in g? 5. Measurements, Results, and Calculations L L 2 10T T T 2 7.8 60.84 8.6 0.86 0.7396 23.2 538.24 9.88 0.988 0.976144 30.3 918.09 10.99 1.099 1.207801 44.2 1953.64 13.29 1.329 1.766241 58 3364 15.38 1.538 2.365444 68.7 4719.69 16.69 1.669 2.785561 72.9 5314.41 17.23 1.723 2.968729 80.5 6480.25 18.16 1.816 3.297856 86.4 7464.96 18.81 1.881 3.538161 96.9 9389.61 20.01 2.001 4.004001 fig. 1 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 L vs. T L cm T sec fig. 2 Experiment #1 Page 3 of 4 Student Name

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0 1000 2000 3000 4000 5000 6000 7000 8000 900010000 0 0.5 1 1.5 2 2.5 L2 vs. T L cm2 T sec fig. 3 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 L vs. T2 L cm T sec2 fig. 4 T 2 2 − T 1 2 L 2 − L 1 = 4.004 − .7396 96.1 − 7.8 = 0.03664 sec 2 cm = 3.664 sec 2 m 6. Analysis and Discussion Because the relationship between L and T 2 , or rather √ L and T, was nearly linear (apart from one major outlier at L = 7.8 ) this shows that the relationship between the length of the pendulum and the period follows an altered square root curve. This is what we expected to see after viewing the relationship between L 2 and T. The collected data, however, has a high level of error. The trendline seen in fig. 4 has a slope of 3.664. The expected slope was approximately 4.1, showing a percent error of 11.9%. This could’ve been improved by increasing the number of periods in each measurement, or by more precisely measuring the length of the pendulum. For example, a mark at the center of the steel ball would’ve allowed up to more accurately and consistently measure L. Experiment #1 Page 4 of 4 Student Name

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