Washington University in St. Louis Introductory Physics Lab Pendulum Lab Fall 2015 A Pendulum and a Pulse Pre-­‐Lab: Galileo, Springs, and Pulses A Bit of History Though you may not appreciate it, a ball on a string a wonderful thing. Until the 1930s, the pendulum clock was the most accurate timepiece ever created. Léon Foucault used a very long pendulum to demonstrate the rotation of the earth. And Newton used a pendulum to test the Equivalence Principle, which would become one of the postulates of Einstein’s theory of General Relativity a couple centuries later. These are just three historical examples of the greatness of this seemingly simple apparatus. A current example of the magic of the pendulum can be found in the Wash U Physics Department. Professor Bender has used coupled pendulums to demonstrate PT-­‐
symmetry breaking, a concept he usually applies to abstract quantum mechanical systems. Check out the lab website for links to details. Current
Research
But before all of these came Galileo, the first person to do rigorous scientific studies on the pendulum. Most of the properties of pendulums that introductory students discuss in the 21st century were fleshed out by Galileo at the beginning of the 17th century. His student and biographer Vincenzio Viviani wrote of how Galileo first became interested in the pendulum. Roger G. Newton recounts the tale in his book Galileo’s Pendulum: He was seventeen and bored listening to the Mass being celebrated in the cathedral of Pisa. Looking for some object to arrest his attention, the young medical student began to focus on a chandelier high above his head, hanging from a long, thin chain, swinging gently to and fro in the spring breeze. How long does it take for the oscillations to repeat themselves, he wondered, timing them with his pulse. To his astonishment, he found that the lamp took as many pulse beats to complete a swing when hardly moving at all as when the wind made it sway more widely. While this basic story is easy to find in the literature, it is almost always accompanied by the disclaimer that it is fictional. Springs This lab is about the pendulum, but your Pre-­‐Lab exercises will concern a mass-­‐spring system. It turns out that for small oscillations, both pendulums and mass-­‐spring systems undergo simple harmonic motion. What that means for the pendulum is that if you plot the angle as a function of time, you get a sinusoidal wave. For the mass-­‐spring system, if you plot the position of the mass as a function of time you will also see a sinusoid. 1 Washington University in St. Louis Introductory Physics Lab Pendulum Lab Fall 2015 The oscillations of a mass-­‐spring system can be described using just a few variables: •
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𝑚, the mass of the mass 𝑘, the force constant of the spring 𝐴, the amplitude of the oscillation 𝑇, the period of oscillation 𝑓, the frequency of oscillation (or 𝜔 , the angular frequency of oscillation) What follows is a little virtual experiment where you will briefly explore the relationships among some of these quantities. Do This: Go to the Pre-­‐Lab Links page on the lab website to find the PhET Masses & Springs simulation. Get the applet up and running on your computer. Do This: In the top right corner of the applet window is a slider that controls friction. To make things easier, turn the friction all the way down to “none”. Do This: In the bottom right corner of the window there is a checkbox next to the word “Stopwatch”. Check that checkbox. Do This: Connect the 250-­‐g mass to Spring 2 using your mouse. Get the mass oscillating. Read This: The period is the time it takes for the system to complete one full cycle. In PL1 you are asked to find the period of the mass-­‐spring system. If your response is not within 5% of the correct value, your response will be marked as incorrect. Further, any value that you calculate using this incorrect period will likely be incorrect. With that in mind, be very careful when you measure the period. You may want to look back to Part II of the Measurement Lab for inspiration. PL1. What is the period of the oscillation? PL2. Using your response to PL1, what is the frequency of the oscillation? PL3. Using your answer to PL1 or PL2, what is the angular frequency of the oscillation? PL4. Using your answer to PL1, PL2, or PL3, what is the force constant of Spring 2? PL5. Why don’t the directions above instruct you to use the ruler to make sure the oscillations have a specific amplitude? Read This: This particular applet can help you learn a lot more about springs. For example, you could explore Hooke’s law or the beautiful ballet of energy transformations that take place as the mass oscillates. We won’t force you to do any of that, but don’t forget that this resource is out there. 2 Washington University in St. Louis Introductory Physics Lab Pendulum Lab Fall 2015 Pulses We now move from springs to hearts. It just so happens that these two topics are not as unrelated as they may seem at first glance. A beating heart can be modeled as a mass on a spring with some damping. In fact, there is a group at Wash U that uses such a model in a non-­‐invasive diagnostic technique. See the lab website for links to details including a video about this work. Current
Research
In this lab, it will be highly important for you to know how to check your pulse. In case you don’t know how to perform this important life task, never fear! We are here to help you out! Do This: Read the article about pulses found on the Pre-­‐Lab Links page. Then answer the following questions. PL6. When is your heart rate slowest? PL7. When is your heart rate fastest? PL8. Which finger should you NOT use to check your pulse? PL9. According to the National Institutes of Health, USA, what is the typical pulse rate for people over 10 years of age? Read This: In addition to traditional heart rate monitors, new technology is also emerging which makes monitoring your heart rate extremely easy. A particularly fascinating development is an app that can use a webcam to check your heart rate! If you’re interested, there are optional links on the Pre-­‐Lab Links page that will direct you to information about this app. End of Pre-­‐Lab 3 Washington University in St. Louis Introductory Physics Lab Pendulum Lab Fall 2015 Part I: Your Pendulum and Your Pulse The Story Apocryphal stories like Galileo’s chandeliers are not uncommon in the history of science (think Newton’s apple). Even though we know this particular story isn’t true, it’s interesting to ask ourselves if it could be true. Can a pulse actually be used as a timer to discover the properties of the pendulum? Equipment •
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6 pendulum bobs (wood, white plastic, black plastic, aluminum, steel, brass) Thread Scissors Protractor •
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Heart Meter stick Digital scale Support stand Tape
1. Your First Experiment S1
Synthesis Question 1 (40 Points): Devise, execute, and analyze an experiment to answer the question “How does varying _____ affect the period or frequency of a pendulum?” But there’s a twist: you may only use your pulse to keep time. That is, your heart is your only clock. You may in no way refer to any other clocks as you do your experiment. (That means you may not act like you know your pulse rate from the Pre-­‐Lab because you used a clock to find that.) A good answer should contain the following: •
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A clear statement of what relationship is being studied. A clear description of your procedure. Diagrams that show your setup and give an indication of what all variables mean. Data table that organizes your measurements in a clear way. The independent variable should span as wide a range of values as is reasonable given the available equipment. A plot that makes your data easy to understand. Curve fits could be nice. Check out Appendix A and Appendix B for advice on fitting curves to non-­‐linear data. A conclusion with several sentences of English. 2. Your Second Experiment S2
Synthesis Question 2 (40 Points): Repeat Synthesis Question 1, but choose a different property to modify. You are still answering the question “How does varying _____ affect the period or frequency of a pendulum?” but you are filling in the blank with a different word. 4 Washington University in St. Louis Introductory Physics Lab Pendulum Lab Fall 2015 3. Addressing Uncertainty One reason that you were asked to use your heart as a timepiece in this lab was to do a little Mythbusting with the Galileo story. In addition, using your heart leads to some interesting experimental constraints that you will address in this final question. S3
Synthesis Question 3 (20 Points): In this lab you used a timepiece with a large uncertainty. Explain how you designed your procedure to reduce the uncertainty in your results. Further, answer the question at the end of the story: Can a pulse actually be used as a timer to discover the properties of the pendulum? Or is the uncertainty too much to deal with? Time to Clean Up! Please clean up your station according to the Cleanup! Slideshow found on the lab website. 5 Washington University in St. Louis Introductory Physics Lab Pendulum Lab Fall 2015 Appendix A: Nonlinear Curve Fits in Logger Pro Introduction As described in documents on the Reference page of the lab website, you can fit a curve to data in Logger Pro by clicking . Often in this course you’ll use a linear fit. However, there are many other mathematical models you can use. If you search this list, you will see many functions that you are familiar with. Square Roots There are some other common functions, though, that appear to be missing. One such function is the square root. There are two easy ways to fit a square root curve to data in Logger Pro. One method is to use the variable power curve. Select this model and then set the value of the power to one-­‐half (0.5). A second method is to use the power curve. When Logger Pro performs a fit with this model, the power is one of the fitting parameters. If a square root curve is a good model, then the power of the fit will naturally be found to be close to 0.5. Additionally, data that can be modeled as a square root can be modeled as a parabola when you switch the x-­‐ and y-­‐ axes. Fitting a parabola to your data is easy, but such a plot may be less clear to the reader. 6 Washington University in St. Louis Introductory Physics Lab Pendulum Lab Fall 2015 Appendix B: Plotting Non-­‐Linear Data as a Line Linear fits are really easy to analyze. In fact, there is a common trick physicists use to turn non-­‐linear data into a straight line. Consider an experiment where we drop a ball from a known height and use a stopwatch to measure how long it takes to fall to the ground. We repeat this from several different heights with a goal of determining the acceleration due to gravity. It can be shown that the time (𝑡) is related to the height (ℎ) by 2ℎ
𝑔
𝑡=
where 𝑔 is the acceleration due to gravity. We gather data and plot the time as a function of height. The resulting plot is nonlinear. We could fit a square-­‐root curve to the data and find the acceleration due to gravity. Unfortunately, we would have to define the function ourselves. This is not an insurmountable problem by any means, but there is a trick that most physicists would turn to first. If we square both sides of the equation above, we come up with the following: !
𝑡 ! = ℎ !
Now we don’t have that clumsy radical! Do you see that plotting 𝑡 ! as a function of ℎ should result in a !
straight line with a slope of ? !
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It looks just like 𝑦 = 𝑚𝑥 + 𝑏 where 𝑡 ! is like 𝑦, is !
like 𝑚 , ℎ is like 𝑥 , and 𝑏 is equal to zero. The resulting plot is linear. It’s often easier to analyze the linear plot both qualitatively and quantitatively. 7