Pendulum Lab Summer 1

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Feb 20, 2024

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Report for Experiment #12 The Simple Pendulum Lab Partner: TA: May 17, 2023

Introduction This experiment was conducted on the simple pendulum – also known as the ideal pendulum – to better understand it and test how different variables, mass and L , play a role in the movement of the pendulum. The simple pendulum is composed of a mass that is suspended by a string and concentrated at the point of suspension, or fulcrum. At rest, the mass is vertically aligned below the fulcrum. When pushed or pulled at an angle θ, the mass swings. This figure shows a simple pendulum where the mass has been moved to an angle θ away from the fulcrum and equilibrium point. It should be noted that the mass, in this case bob, is a distance of x = θL from the equilibrium point. In theory, there are only two forces that act on the bob at this moment. The first is gravity, F=mg , towards the ground. The second is the string tension force, depicted as F T , which is in the direction of the fulcrum. In this figure, gravity has been broken down to two parts: mg cosθ and mg sinθ . Fortunately, F T and mg cosθ cancel out, proven by Newton’s third law. This leaves the restoring force acting on the bob to be F = – mg sinθ . When the angle is small, sinθ approximates to be θ (radians) and the force can be shown as: F =− mgθ =−( mg / L ) x Additionally, when g and L are constants, the following equation applies and demonstrates simple harmonic motion: d 2 θ d t 2 = − g L θ There is one other known equation that acts as the foundation for this experiment as it mathematically articulates the period of the oscillations done by the pendulum: T = 2 π ω = 2 π √ L g It should be emphasized that this equation does not factor mass into period at all, and the only factors that affect the period of oscillation are gravity, and length of the pendulum. Investigation 1

For this experiment, the equipment included a digital scale, two spherical bobs (one wood, one brass), a digital caliper, a rod, a clamp, a meter stick, a short rod with string clamp, a protractor, and some string. The rod was attached to the desk in a vertical position with the rods stabilizing the set up and supplying a place to attach the string. A piece of string was cut to be a little longer than 90 cm. A knot was tied at one end of the string and was then used to attach the string to the rod and secure the string at a specific height and place. Another knot was tied on the opposite end to attach the string to one of the bobs. The digital scale was used to mass the identical bobs and then the digital caliper was used to record the diameter of the bobs. The lighter, wooden bob was then attached by the hook to the string by the small knot at the bottom. The length of the string from the top, the fulcrum, to the center of the bob was measured and found to be 92.5 cm. The length of the string in this investigation was represented by L in the data table and its error was represented as δL . A 10º angle was measured at the fulcrum and the bob was moved such that the string and bob followed this 10º angle to the right of the fulcrum. The bob was released and the time it took to complete 10 full oscillations was recorded using a digital stopwatch. This measurement was repeated 4 times. The times recorded were represented by t rec in the data table and the average of these measurements was taken and represented by t avg and its error was represented by δt avg . From these measurements, the measured period, T , and its error δT , were calculated. The values for T 2 were found by squaring T and δT 2 was calculated. The entire procedure was repeated for the heavier, brass bob and L was kept constant. Table 1 - Recorded and Derived Data From 4 Trials with the Wooden Bob Trials Mass (g) t rec (s) t avg (s) 𝛿 t avg T avg (s) 𝛿 T T 2 𝛿 T 2 1 6.8 18.76 18.8125 0.082298 1.8813 0.0082298 3.53910 0.008749 2 18.61 3 18.89 4 18.99

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Table 2 - Recorded and Derived Data From 4 Trials with the Brass Bob Trials Mass (g) t rec (s) t avg (s) 𝛿 t avg T avg (s) 𝛿 T T 2 𝛿 T 2 1 245.8 19.43 19.1975 0.083404 1.91975 0.0083404 3.6854 0.008689 2 19.04 3 19.19 4 19.13 The derived quantities such as the error values were calculated in multiple ways. Period ( T ) was found by dividing t rec by 10. Averages were found by adding up relevant quantities and dividing by the number of data points used. Error of length was found by dividing the smallest unit of measurement on the meter stick by 2. The quantities for 𝛿 t avg were found using the following equation: The values for 𝛿 T avg were found by dividing these number by 10. Following this equation: δT = δ t avg 10 The values derived for 𝛿 T 2 avg were calculated using the following equation: δ T 2 = 2 δT T

0 0.05 0.1 0.15 0.2 0.25 0.3 1.86 1.87 1.88 1.89 1.9 1.91 1.92 1.93 f(x) = 0.16 x + 1.88 Average Period of 10 Oscillations by Mass Mass (kg) Period (s) Figure 1 - Comparing How Mass Affects the Period of Oscillations Between Wooden and Brass Bobs y = (0.16109 ± 0.04903)x + (1.88015 ± 0.00847) From the graph above, the equation for the linear line of best fit was extracted and is presented as in the caption. The slope represents the correlation between the mass of the bob and the period for 1 oscillation when the bob is released from a 10º angle. The difference in seconds between the oscillations of the two bobs is 0.0385 seconds and the slope is 0.16109 seconds/kg. These represent fairly small differences and largely concur with the equation used to represent period and the idea that mass has no effect on period. However, the differences and positive slope are not negligible, and can likely be attributed to error that took place throughout the experiment. The measured error is included in the equation under the graph and was calculated using the IPL slope calculator. For any directly measured quantities there were several possible sources of error. Human error is likely a large source of deviation in this experiment as the bobs may not have been released at a perfect 10º angle or maybe have been released with a some horizontal push, causing the pendulum to oscillate in circles, as opposed to back and forth directly. There may also have been some systematic error in the times recorded as two different people started the stopwatch and released the bobs, causing them to be uncoordinated.

Investigation 2 The setup for the second investigation was identical apart from the length of the string and only the brass bob was used. The string was shortened to 61.3 cm and the brass bob was attached. The procedures from investigation one were repeated with the shortened string. Four trials were completed and data from each trial was recorded. This process was repeated after the string had once again been shortened, to 30 cm this time. Once again, data from four trials was recorded. Any derived data was calculated in an identical fashion to Investigation 1. Table 3 - Recorded and Derived Data From 4 Trials with the Brass Bob on 60 cm Pendulum L (m) 𝛿 L (m) t rec (s) t avg (s) 𝛿 t avg (s) T rec (s) T avg (s) 𝛿 T avg (s) T 2 (s) 𝛿 T avg 2 (s) 61.3 0.05 15.85 15.8025 0.0765 1.585 1.58025 0.00765 2.4972 0.009685 15.58 1.558 15.85 1.585 15.93 1.593 Table 4 - Recorded and Derived Data From 4 Trials with the Brass Bob on 30 cm Pendulum L (m) 𝛿 L (m) t rec (s) t avg (s) 𝛿 t avg (s) T rec (s) T avg (s) 𝛿 T avg (s) T 2 (s) 𝛿 T avg 2 (s) 30 0.05 10.99 11.305 0.11934 1.099 1.1305 0.011934 1.27803 0.021112 11.25 1.125 11.5 1.150 11.48 1.148

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 f(x) = 3.97 x + 0.04 Comparing Period Squared (seconds squared) to Length (meters) of the Setup L (m) T^2 (s^2) Figure 2 - Correlation Between T 2 and L The data and corresponding line of best fit show that period squared, and length of the pendulum are correlated and linearly proportional. As such, period and length must be correlated, although not linearly proportional. This fits with the basic knowledge stated in the introduction and the equation for period listed in the introduction. The y-intercept is listed as (0, 0.039) which is obviously not possibly give the circumstances meaning that error was introduced at some point to the trials. The same sources of error would likely be involved in both investigations. However, an additional error source likely affecting Investigation 2 could be a lack of accuracy in the measurements of L for the two changes in length of the pendulum. Conclusion This experiment examined and investigated the relationship between length of the pendulum, mass, gravity, and the period of oscillation of a simple pendulum which exhibits simple harmonic motion. The results of Investigation 1 confirmed the equation stated at the beginning of the experiment and the theory that mass had no effect on the period of oscillation in an ideal pendulum. Investigation 2 confirmed that if gravity and angle are constant and length is the only manipulated variable, a change in length will have a direct effect on the period of oscillation.

Students are not infallible mechanisms in experiments and humans do not possess the accuracy of machines. As such multiple mistakes and external factors could have skewed the results. The oscillations of the pendulum were visually cyclical and therefore not perfect representations of a simple pendulum with idyllic behavior. In future, to minimize error, the bob might be released more horizontally in line with the equilibrium point in an effort to minimize cyclical oscillations. The release of the bob and the start of the stopwatch could be more coordinated, perhaps released by the same student in order to get a more accurate result. Questions 1. Using the equation below, it is possible to calculate the length necessary for T to equal 1s. T = 2 π ω = 2 π √ L g 1 second / 2π = (L / g) 1/2 (1/2π) 2 = L / (9.8 N) ω g L = .25 meters 2. The ratio of the lengths for the 1s and 2s pendulums is 1:4 T = 2 π ω = 2 π √ L g 2 second / 2π = (L / g) 1/2 (1/π) 2 = L / (9.8 N) L = 1 meter 3. 2.75% (1 + θ 2 /16 + 11θ 4 /3072) = 1.019 (1 + θ 2 /16 + 11θ 4 /3072) = 1.047

4. As we have seen based on the data collected in this experiment, if length increases, the period will also increase. This means the clock would count more than one second for every oscillation, and would undercount time. 5. The astronaut will find that in space, where gravity is much lower, the period will increase drastically. This is because if gravity decreases, the value of L/g will increase and period will thus increase. In space the count will undercount time. Acknowledgments I’d like to thank my lab partner for staying behind to do the lab with me and for his patience throughout the process. I’d also like to thank the TA for putting the time in to allow us to do the lab and for answering any question we had along the way.

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