The Simple Pendulum_Hypothesis Testing_PHYS 130

The Simple Pendulum Purpose: To investigate how the period of a pendulum is affected by its mass, angle of swing, and length. Discussion The simple pendulum is of historic and basic importance. Its regular swing, discovered by Galileo Galilei, makes it an accurate and simple timekeeper and, in the hands of Isaac Newton, resulted in the first evidence that inertial and gravitational masses are proportional. Until relatively recently, the motion of a pendulum provided the most accurate and convenient method for measuring the local gravitational pull of the Earth. Although “pendular” motion had been likely observed for centuries, Galileo (1564-1642) was intrigued by the back and forth motion of a suspended weight. Legend has it that he began his studies of pendulums after watching a ceiling lamp swing back and forth in the cathedral of Pisa (no doubt during a particularly long winded sermon!) Using his pulse to time the motion of the swinging lamp, the crude measurements indicated the period (the amount of time required for one swing) remained constant although the swinging motion varied as time passed. The motion of the pendulum bob posed interesting problems. What causes it to swing back and forth? What keeps it moving after it is released? Which factors influence the time it takes to complete one swing (called the period)? Throughout his experimental work, the pendulum was never very far from Galileo's thought. Although he did not address the first two questions, he did figure out the third which led to the development of the pendulum clock, although it would take another 60 – 70 years before it became a practical time keeping device. Model Testing When setting up a simple pendulum, you have three variables you can adjust: (1) the bob mass, (2) the pendulum length, and (3) the pendulum swing amplitude ( θ ). The “physics analysis” of the motion of the simple pendulum as it swings forth and back makes the following prediction for the period measurement: T = 2 π √ L g In this model, T ≡ period ,L≡length, ∧ g≡gravity constant . You can see that the model predicts that the period is independent of the amplitude of the swing ( θ ) and the mass of the pendulum bob ( m ). But under what conditions, if any, is this model valid? It is your goal to test the limits of this model to discover when it is appropriate to use in an experimental setting. (That is, when can we use the formula to get reliable and valid results from measurements made in an experimental system?)

Link for online simulation: https://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html PROCEDURE: When the pendulum simulation has opened “Click” on the lab window . Click on Period Timer. Make sure that speed is set to Normal and Friction (slider) is set to None. To get a period measurement, simply pull the pendulum to the side and release it. Click on the period timer to get the measurement.

Part 1 – Period vs. Mass (Length, Gravity & Amplitude constant) 1. Set the pendulum length to 1.0 m and mass to 0.3 kg. Pull the pendulum to the side until the amplitude of the swing is 10 o . Release the pendulum and “Click” the pendulum timer once. When the pendulum has completed one oscillation the timer will display the period. Record this in the data table. 2. Change the mass to 0.6 kg and again measure period. Remember to keep the amplitude and length unchanged from one trial to the next. Repeat for each bob mass listed in the table below. Data Table 1: Period as a Function of Mass Pendulum Bob Mass (kg) Period (s) % Difference [ high value − lowvalue average of values ] x 100% 0.3 2.0099 2.0099-2.0099 2.0099 0*100 = 0% DIFFERENCE 0.6 2.0099 0.9 2.0099 1.2 2.0099 1.5 2.0099 Analysis Part 1: The model of a simple pendulum predicts that the period will not depend on the mass of the pendulum bob. Does you data support this prediction? Defend your answer by citing the data that supports it. Percent differences are used to evaluate whether the data shows a significant effect when the variable of interest is changed. For most physical science experiments, less than 2% difference between the values is not considered significant. The period time (s) is not dependent on the mass of the pendulum bob as the mass of the pendulum bob increased the period time (s) remained constant.

Data Table 2: Period as a Function of Amplitude 1. Set the pendulum length to 1.0 m and mass to 1.0 kg. Pull the pendulum to the side until the amplitude of the swing is 10 o . Release the pendulum and “Click” the pendulum timer once. When the pendulum has completed one oscillation the timer will display the period. Record this in the data table. 2. Change the amplitude to 20 o and again measure period. Remember to keep the mass and length unchanged from one trial to the next. Repeat this for amplitude listed in the table below. Pendulum Amplitude Period (s) % Difference [ high value − lowvalue average of values ] x100% 10 o 2.0099 2.1059-2.0099 2.0672 =.04644*100 4.644% 4.7%Difference 20 o 2.0215 30 o 2.0410 40 o 2.0689 50 o 2.1059 Analysis Part 2: The model of a simple pendulum predicts that the period will not depend on the amplitude of the pendulum swing. Does you data support this prediction? Defend your answer by citing the data that supports it. My data does not support the hypothesis “The model of a simple pendulum predicts that the period will not depend on the amplitude of the pendulum swing” when the pendulum release angle increases the period time length increases by a “significant” percentage of time

Application The simple pendulum provides a remarkably simple way to determine the gravity constant of any planetary body (or moon): T = 2 π √ L g →T 2 = 4 π 2 ( L g ) →g = 4 π 2 ( L T 2 ) So a precise measurement of pendulum length and the resulting period can be used to calculate the gravity constant g . (Remember that this model only gives a reliable value under certain experimental conditions.) In the simulation, switch your location to Planet X: Set your pendulum length to 1.0 m and the bob mass to 1.0 kg. Select a pendulum amplitude to use for the period measurement and record in the box below. Remember, you want to get the best measurement for the gravity constant on Planet X, and you have discovered that the Period formula is not valid for all amplitudes. Choose an angle for which the Period formula provides reliable results. Measure the period as a function of length and complete the data table below. Remember to keep the mass and amplitude of the pendulum constant from one trial to the next. Pendulum Length (m) Period (s) Gravity Constant (m/s 2 ) g = 4 π 2 ( L T 2 ) % Difference [ high value − lowvalue average of values ] x 100% 0.20 .7464 14.172 14.185-14.172 14.181 =.00092*100 .09% Difference 0.40 1.0553 14.179 0.60 1.2923 14.183 0.80 1.4921 14.185 1.00 1.6682 14.185 Average Gravity Constant = 14.181 Do you believe your measurement yielded a reliable value for the gravity constant of Planet X? Defend your answer by citing the data that supports it. Amplitude = 5 Yes, as we noted previously amplitude does not have a significant affect on period length below a 7 degree angle. However, we know pendulum length will affect the period time. However, we calculated that as length changed the Gravity constant remained significantly the same across 5 examples. When calculating the percent difference of the results we found that the difference was less than 2% resulting in a .09% difference.