Galileo s Pendulum: An exercise in gravitation and simple harmonic motion

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1 Galileo s Pendulum: An exercise in gravitation and simple harmonic motion Zosia A. C. Krusberg Yerkes Winter Institute December 2007 Abstract In this lab, you will investigate the mathematical relationships governing the motion of a simple pendulum a bob swinging back and forth on a string due to the presence of gravity and use your deepened understanding of this motion to first determine the acceleration due to gravity at Earth s surface, and second, to create a precise time-keeping device. Until as recently as 1930, pendulum clocks were the most accurate time-keeping devices around! Objectives Upon completion of this lab, you will understand: the components of a pendulum. that the motion of a simple gravity pendulum can be approximated as simple harmonic motion for small angles. that gravity acts as a restoring force that causes the mass to oscillate about its equilibrium position. the relationship between period and frequency. how the pendulum s period depends on mass, oscillation amplitude, length, and gravitational acceleration. Vocabulary Upon completion of this lab, you will be familiar with the following terms: pendulum; pivot; bob; simple harmonic motion; period; frequency; restoring force; equilibrium position 1

2 Materials rope/string; weights of different masses; meter sticks; stands; stop watches; calculators; post-it boards; markers A bit of history Figure 1: It is this chandelier in the Pisa Cathedral that is rumored to have inspired Galileo to investigate the properties of pendulums. In the 17th century, Galileo the same guy who was forced into house arrest by the Roman Inquisition for defending heliocentrism (the theory that the Sun is at the center of the solar system) became one of the first people to study the physical properties of pendulums. According to legend, Galileo s interest in pendulums was sparked when he was watching the swinging motion of a chandelier in a cathedral in Pisa, Italy, and timed its period with his pulse. His subsequent studies of pendulums led him to discover the relationship between a pendulum s period and its mass, amplitude, and length. These investigations encouraged Galileo to suggest that pendulums could be used as timekeeping devices, and the first patent for a pendulum clock was filed in 1656 by another famous physicist, Christian Huygens. 2

3 Figure 2: Here, in the Panthéon in Paris, hangs Foucaults original pendulum that proved Earth s rotation about its own axis. In 1851, the French physicist Léon Foucault presented a public display of a large pendulum consisting of a 28-kg bob and a 67-m wire in the Panthéon in Paris as an experiment to demonstrate the rotation of the Earth. This type of pendulum now named a Foucault pendulum works because the pendulum is free to oscillate in any vertical plane, and because of the rotation of the Earth underneath the pendulum, the direction of oscillation rotates with time. At the North or South poles, the plane of oscillation of the Foucault pendulum undergoes one full rotation in one day (you can picture the pendulum s plane of oscillation staying fixed with respect to the stars while Earth rotates underneath). At the equator, on the other hand, the plane of oscillation stays constant. At intermediate latitudes, the time for the plane of oscillation of the pendulum is longer than at the poles, but depends on the exact latitudes. Procedure Part I: The relationship between period and mass Goal: Students will determine the relationship between a pendulum s period and its mass. 1. Draw a diagram of a generalized pendulum in your lab notebook. Make sure that you identify the pivot, the bob, and the equilibrium point. 2. Draw a diagram of the experimental setup in your lab notebook, and label the components appropriately. 3. Create a data table in your lab notebook. The first column should read Mass, the second should read Time for 10 oscillations. 4. Attach each weight in your weight collection to the string in turn, and for each weight, time how long the pendulum takes to make 10 oscillations. Write this 3

4 number down in the table in your lab notebook. Remember to include units in your data table: to make it easy for yourself, include it in brackets along with the title of each column, e.g., Mass (g). Make sure that you let go of the pendulum at the same height each time, and that the length of string is the same each time: we want to keep all variables constant except mass. 5. Add a third column to your data table, and title it Period. The period is the time it takes for the pendulum to perform one oscillation, i.e., the time it takes for the pendulum to return to its starting position. To obtain this number, divide the number in the previous column by 10. This number is actually the average period over 10 oscillations. 6. Add a fourth column to your data table, and title it Frequency. The frequency is the number of oscillations the pendulum performs per second; it is the inverse of the period: f = 1 P where f is the frequency and P is the period. The unit of frequency is the hertz (Hz). Calculate the frequencies for the periods you obtained in the previous column. 7. Answer the following questions in your lab notebook: Does the period change significantly as mass changes? What are the advantages and disadvantages of timing 10 oscillations rather than 1? Part II: The relationship between period and amplitude Goal: Students will determine the relationship between a pendulum s period and the amplitude of its oscillations. 1. Create a second data table in your lab notebook. The variable we re testing now is amplitude, so title the columns appropriately. 2. Using a meter stick, let go of the pendulum at different initial heights (amplitudes), and time how long the pendulum takes to make 10 oscillations. Write this number down in the table in your lab notebook. You should collect at least 7 data points, i.e., release the pendulum at 7 different initial heights. Make sure to use the same mass and length of string each time, since we only want to vary one variable, amplitude. 3. Determine the period of the pendulum for each height using the procedure from 4. Determine the frequency of the pendulum for each height using the procedure from 4

5 5. Answer the following question in your lab notebook: Does the period change significantly as amplitude changes? Part III: The relationship between period and length Goal: Students will determine the relationship between a pendulum s period and its length. 1. Create a third data table in your lab notebook. The variable we re testing now is length, so title the columns appropriately. 2. Using different lengths of string, time how long the pendulum takes to make 10 oscillations. As before, you should collect at least 7 data points, i.e., time the pendulum s period for 7 different string lengths. Make sure to get a wide range of lengths, from really short to as long as you can manage with the setup. Make sure to use the same mass and initial height each time, since we only want to vary one variable, length. 3. Determine the period of the pendulum for each length using the procedure from 4. Determine the frequency of the pendulum for each length using the procedure from 5. Answer the following question in your lab notebook: Does the period change significantly as amplitude changes? Part IV: Determine the acceleration due to gravity Goal: Students will use their data from Part III to determine the acceleration due to gravity, g. 1. Using the data table you created in Part III of this activity, make a plot of period versus length in your lab notebook. Make sure you title your plot, label your axes, and include units. 2. Describe the general appearance of this plot. Does it resemble any function you ve seen before? 3. The equation for the period of a simple gravity pendulum is the following: l P = 2π g where l is the length of the pendulum and g is the acceleration due to gravity. If this looks confusing, don t worry too much about it. If you ve done some algebra, you can see that the period only depends on the square root of the length (both 5

6 π and g are constants). We re now going to use this equation to determine the acceleration due to gravity, g. Remember that we ve done this once already this year, so this will be a way to test the result we obtained in a different experiment. Using the data that you collected in Part III of this activity, make a second plot, but this time, plot period versus the square root of length ( l). To do this, add another column to your data table and title it Length, and fill it in by calculating these values. By plotting the data this way, the plot will show a linear relationship (straight line). Again, make sure you title your plot, label your aces, and include units. Find the slope of this line. The slope, let s call it m, is equal to m = 2π g So, to determine the acceleration due to gravity, we solve this equation for g: g = 4π2 m 2 Use this equation to calculate the value of the acceleration due to gravity. 4. Answer the following question in your lab notebook: If the experiment were carried out on the Moon, how would the period of the pendulum change? What if it were carried out on Jupiter (though we all know Jupiter has no solid surface to stand on, so this would be a difficult experiment to carry out in practice!)? Part V: Challenge Using your data, design and construct a pendulum that you can use to accurately measure a time interval of 30 seconds. Test your pendulum clock against a watch or clock. 6

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