Determination of the Properties of the Pendulum

Determination of the Properties of the Pendulum Purpose In this activity you will investigate how hanging mass, length and starting angle affect the period of a pendulum. Based on your data, a mathematical model will be developed for the period of a pendulum. Note this activity can and should be done prior to any class discussion of details. Overview Students will play the role of ‘scientist’ by doing this set of experiments before the details of a pendulum are discussed or studied in class. Students will be expected to do a short series of controlled experiments to determine what, if any, effect mass, length and starting angle have on the period of a swinging pendulum.

For example, by keeping the mass and initial angle fixed, students will do a series of trials of measuring the period of oscillation as they vary the length of the pendulum. By plotting the period as a function of length, students should find a non-linear graph, and that a power law fit works best and has a power near 0.5 – the period is proportional to the square root of length. This will not be exact, especially if stop watches are being used, but anything close to a power of 0.5 will be acceptable and will increase the student ‘buy in’ to the eventual equation for the period of a pendulum, T = 2π[L/g]½. Similar experiments will be done with varying mass and varying initial angle, while holding the other parameters constant.

Based on the collected data, students will find best-fit functions and create an initial empirical equation for the period of a pendulum. They should find that mass does not have a significant effect on the period, but that the period depends on something close to the square-root of the length of a pendulum (using a power law fit to good data). There is also a small, but measureable, effect of starting angle on the period of the pendulum that can be included.

The one aspect of this system we cannot easily test is the strength of gravity. In the end, students will be asked if there should be a direct or inverse relationship between gravity and period, and it should be inverse. The teacher will provide the correct g-½ relationship once the experiment and analysis is completed. One way students can investigate this is through computer simulations that allow the user to vary the acceleration of gravity. One example is the pendulum computer experiments found at http://phet.colorado.edu/en/simulations/category/physics,

Student Outcomes Learner objectives Students will:  Discover and appreciate the complexity that is often not initially present in everyday, seemingly simple events or phenomena;  Identify and investigate individual physical quantities that may have an impact on the nature of a basic pendulum;  If they go on and actually investigate this experimentally, students will begin to learn what a controlled experiment is, how to go about designing and running controlled experiments, learn and create measuring techniques appropriate to , take and organize and record numerous data points, and analyze data;  Learn something about periodic motion phenomena;  If experiments are done, analyze data to find best-fit mathematical functions between the radius and/or depth of a hydraulic jump and any one of numerous possible parameters, and then compare their findings and experimental model to more accepted theoretical models.

Next Generation Science Standards  HS-PS2-1 Analyze data to support claim Newton’s 2nd law of motion describes macrocosmic objects behavior and motion  HS-PS3-3 Design, build, refine a device that works within given constraints to convert one form of energy into another form of energy  HS-PS2-4 Use mathematical representations of Newton’s law of gravity to describe and predict gravitational force between objects (weight of pendulum bob, and how it drives the motion of a pendulum)

 PS2.A Forces and Motion; PS2.B Types of interactions

 PS3.A Definitions of energy; PS3.B Conservation of energy and energy transfer

 ETS1.C Optimizing the design solution (break a problem into simpler ones and approach systematically to see relevance/importance)

 Science and Engineering Practices: Planning and Carrying Out Investigations  Science and Engineering Practices: Developing (mathematical) Models  Science and Engineering Practices: Using Math and Comp. Thinking  Science and Engineering Practices: Constructing Explanations and Designing Solutions (this will be a phenomenon students know nothing about, and will try to construct some understanding from experimentation and observation)

This activity addresses the following skills from the Computational Thinking (CT) STEM Taxonomy chart: - 1a Collecting Data: See Appendix A - 1d Manipulating Data: See Appendix B - 1e Analyzing Data: See Appendix C - 1f Visualizing Data: See Appendix D - 3a Using Computational Models to Understand a Concept: See Appendix E - 3c Assessing Computational Models: See Appendix F

Time This experiment will take 1-2 class periods (on order of 45 minutes), depending on the length of a period.

Level This experiment is appropriate for high school introductory and advanced physics classes.

Materials and Tools In a standard pendulum experiment, the basic materials will be a stand or support on which to hang a pendulum, string and a set of masses to use as bobs, stop watches or other timing devices or an electronic force sensor that can be used to extract the period of oscillation, a balance to measure mass, a protractor or other device for measuring the initial release angle, and a meter stick.

If students have access to video cameras, this provides an alternative way of measuring period and angle, where one can film the starting point with a protractor in the picture, and then do a frame-by-frame playback to measure the period of oscillation. The other method uses an electronic force sensor (such as purchased from Vernier or PASCO) – hang the pendulum directly from the force sensor, and when it records it produces a force vs time graph. This will be a periodic, sinusoidal plot, and the peak to peak time is the period of oscillation. These two methods are more accurate than using a stop watch. Most students have video capabilities with their cell phones. Note that these techniques work well for oscillating spring experiments (simple harmonic motion, waves), too.

All introductory physics textbooks have more detailed information about a pendulum.

Preparation The main preparation for labs of this type should begin with the first day of class. Students need to be prepared to actually do the science, and not sit back and digest what is provided from a science teacher. This lab should be the first step of a unit on periodic motion or perhaps circular motion, depending on when the teacher introduces the pendulum.

The lab itself is fairly routine for a physics classroom. Materials for the pendulum should be set out at lab stations. The teacher will need to decide if students are to use stop watches, which will require multiple trials, or set up motion sensors or force sensors or photogates, which will require computers and the appropriate software to run and collect data. That is the only physical preparation that needs to be done.

But if this type of lab experience is new to students, the teacher should prepare the class by explaining the purpose of doing the lab first, prior to having any class discussions or textbook readings or demonstrations. Students will be in the position of scientist, and must develop and collect data from the experiments to come up with their own mathematical model. The student experimental model will then be compared to the accepted model in the textbook, and conclusions and deeper scientific discussions will result from that comparison, as details and the physics behind the pendulum are studied.

A new mindset will be developed for and executed by the students.

Prerequisites In a normal high school laboratory experience, students will have been exposed to and studied a topic prior to the experiment, and the experiment is more of a confirmation of the concepts and ideas discussed in class. This is not meant to be a confirmation experiment, but rather the first step in learning about a pendulum and periodic motion.

Students do not need any prior knowledge about details of periodic motion, the pendulum or any analysis details about the circular motion of a pendulum. Students should be starting with little, if any, prior knowledge – they are in the role of scientists trying to make the discoveries of the details on their own, from real data.

The only prerequisite students should have is knowledge about what a controlled experiment is, and how to go about controlling physical quantities while only varying a single quantity to test how affects, in this case, the period of a pendulum. Students should also be aware of how to write up a lab report, using whatever criteria the teacher requests.

Background A pendulum is a classic example of periodic motion, which is a motion that is repetitive, redundant, keeps repeating itself, keeps going back and forth... A pendulum only works when there is gravity, and of course can be used as a clock. It is also an example of vertical circular motion, which means there is a net centripetal force, mv2/R, created between the tension in the string and gravity.

For this activity, do not get into any other details about the pendulum. If this is truly new for students, then they will be in a position of investigating the unknown, that is they are in the role of the scientist who needs to discover the basic mathematical relationships and rules for a swinging pendulum. Once students have their mathematical model for this system, it can be compared to the known model in a textbook.

Teaching Notes To do this activity, or any labs similar to it, the teacher should place a focus on developing an attitude and mindset amongst students that this is going to be the norm in the class. If possible, as many labs should be done with the approach that the lab and the collected data make up the very first part of new units of study. With Next Generation Science Standards, a large part of the new approach to science education is getting students to be the scientist. This will begin to take teachers away from beginning a new unit with a lecture or demonstrations that spell out what the details are, or simply writing down notes that repeat what is in the textbook and have students read about a phenomenon , and finally allowing the lab to simply confirm what was already stated in class – instead, put the lab first, let students discover the relationships, patterns, and details of a phenomenon, and then go to the textbook and compare what students conclude with the accepted. If there are differences, this will provide a rich set of questions and discussion points to figure out why the experimental work might differ from what professional scientists have concluded. Students will learn about and actually live the scientific process with this approach, and along the way it builds up the mindset of how to tackle the unknown and how to solve problems of any kind by looking for data and evidence first, from which conclusion should be based.

For this lab, the only demonstration and information the teacher should provide upfront is what a pendulum is, and define that the length of a pendulum is from the point it hangs to the center of mass of the bob. Also, the teacher needs to be sure students understand what the definition of the period of oscillation is, which is the time for one full swing (i.e. one round trip). But that is about it.

Teachers should set the expectation that procedures will no longer be provided for labs like this, at least not to the point of ste-by-step procedures. Students should try to figure out the best techniques, and determine which parameter(s) are being controlled and which is being varied. Students should be allowed to fumble a bit, use trial and error – after all, this is what professional scientists do when they are investigating the unknown. Experimentation is largely troubleshooting and trial and error.

While students are doing the experiment, it is valuable to have conversations with lab groups. Ask why they are making the measurements they are making. How and why they chose certain values for length or mass or angle. Ask students what some uncertainties are with their measurements or procedures. Let them talk through their thinking, fix misconceptions, ensure they are doing multiple time trials if using stop watches. Make sure they are using Excel properly and making good, labeled graphs, and are finding the best-fit functions correctly. But this is a process that is more inquiry based and student centered. After doing a few labs this way, students will catch on to the expectations and it will be more comfortable and accepted.

If interested, an extension can be made for students and this lab. Once they have a mathematical model for their pendulum, challenge them to use it as a metronome. That is, anyone should be able to ask them to set up the pendulum to have a specific, arbitrary period so it could be used as a metronome. One can assess this by measuring with a stop watch (recommend timing 10-20 full swings to get the period if using a stopwatch), photogate or force sensor, how accurate the metronome, and therefore the mathematical model, is. Assessment Teachers should use whatever their normal grading criteria/rubric is for laboratory reports. While it is nice when students get results that agree well with accepted mathematical results, one should anticipate some discrepancies in this lab, particularly if students are using stop watches to measure the period of a pendulum.

Teachers should assess how students are going about the process of finding a mathematical model in this activity. In some cases, students will have little if any experience doing this type of lab. There is much to learn about creating the initial research questions and purpose, developing experimental designs and measuring techniques, and data collection and organization for analysis – these are all important features the teacher will need to ensure students are learning, in addition to the physics principles of a pendulum.

If the teacher decides to have students test their model as a metronome, assess how accurate it keeps time relative to a standard time keeping device or technique.

Additional Information There is much information about the simple pendulum and the physical pendulum in any standard introductory physics textbook. Depending on the level of the class and students, teachers (such as AP level) will want to get into the derivation of the classic result, T = 2π[L/g]½. This will involve the small angle approximation and some knowledge of simple harmonic motion. A sample how-to video is at: http://docvphysics.blogspot.com/2010/04/how-to-get-simple-harmonic- motion.html

If a class wants to do something similar with a physical pendulum, such as a oscillating stick, a sample how-to video of the theory is at: http://docvphysics.blogspot.com/2012/04/shm-of-oscillating-stick-due-to- spring.html Below is a sample student lab sheet, outlining the goals of the experiments as well as some possible analysis questions.

Determination of the Properties of the Pendulum

Purpose: In this activity you will investigate how hanging mass, length and starting angle affect the period of a pendulum.

Materials: Pendulum apparatus Stop watch or other timing device/technique Meter stick Various masses for pendulum bob Balance Protractor or other means of measuring angle

Background: A pendulum is a classic example of periodic motion, which is a motion that is repetitive, redundant, keeps repeating itself, keeps going back and forth…oh, I’ll stop now. A pendulum only works when there is gravity, and of course can be used as a clock. It is also an example of circular motion, which means there is a net centripetal force, mv2/R. Because we began studying circular motion and will be getting deeper into gravity, we will try to figure out the basic properties of a pendulum and see how we can combine gravity and circular motion together in a few different ways, whether it is a pendulum, an amusement park ride, or satellite motion. Keep the force diagram for a pendulum in mind (what does it look like?).

Research Questions: What effect(s), if any, do mass of the bob, length of the pendulum, and angular amplitude of a pendulum, have on the period of the pendulum?

Develop Hypothesis: State what you think the effect of mass, length and angular amplitude have on the period of the pendulum.

Procedure: You will be trying to determine the best way of obtaining answers to the questions below. Keep in mind, you should always be thinking of estimating errors on all measurements (in any experiment you ever do!) as well as how to minimize those errors. Think about what might be the best way of measuring a given quantity. Also, do not forget to include units on all measurements, along with a reasonable estimate of the uncertainty of all measurements! For anything measured in trials, think standard deviation. For your write-up: Purpose, Materials, Procedure, Data Tables, Questions and Analysis (with appropriate graphs; these should be titled and labeled with quantities being graphed and units! Graphs need to be done on the computer.). Do a group report, and consider Google Docs if that works out for the group. Definitions: Period = time it takes the pendulum, when released from rest, to swing over and back to where it started; or the time for one round-trip. Length = length from the point where the string is free to swing to the center of mass of the bob at the end of the string. Mass: we assume massless string.

Questions/Analysis: 1. For a constant length and constant mass of the pendulum, gather data for a graph of period as a function of angle (as measured from the vertical). Your measurements should range from a few degrees (small angles) up to 90o. Estimate errors on measurements and include them as error bars on your graph. The error bar for period measurements (some number of trials) should be the standard deviation of the mean. Find the best-fit function for the graph.

2. Interpret your graph of period vs angle in a couple of sentences; what relationship(s), if any, do your data suggest? Direct or inverse? Linear, flat or something different? Include in the discussion how your interpretation depends largely on the error bars.

3. For a constant length and constant small angle (less than 10 degrees) of the pendulum, investigate whether the period of the pendulum is dependent on the mass of the bob. Use five different masses to do this. Make sure to adjust the length of string so the overall lengths of the pendulum are always the same. For instance, when you hang a larger mass on the string, you will need to shorten the string a bit since the object will be longer (think about where the center of mass is). Try to estimate errors to see if any discrepancies are significant or not. Graph period as a function of mass, and explain your data and draw conclusions about such a dependency based on the data. Include estimated error bars on the periods, and find a best-fit function for the data.

4. For a constant mass and constant small angle (less than ten degrees), collect data for a graph of period as a function of length of the pendulum. Plot your data using Excel and obtain a best-fit function to the data to get an idea of the relationship. Include the point (0.01,0.01) on your graph and in your fit. This is close to (0,0), but should allow you to have all fit options available in Excel.

5. From your graph of period vs. length, what type of relationship is suggested for period as a function of length? Is it linear or non-linear? Explain your answer in a few sentences based on your fit to the data.

6. If you were to do this experiment on the moon, what changes in your data would you expect to find, particularly in your period measurements? Make sure to explain your answers in some detail; you don’t have to do any calculations necessarily, but conceptually what might you expect would change? 7. There are only three significant forces acting on the weight at any time, those being tension, gravity and air friction. Which of these are constant, which are non-constant? If any are non-constant, when are they the strongest, and when are they the weakest? Explain, and include a force diagram/free-body diagram.

8. Using your graph in #4, how long should a pendulum be in order to have a period of 1.0 sec? How long should a pendulum be to have a period of 2 seconds? Use interpolation and/or extrapolation to estimate your answers.

9. Solve the following problem: For a pendulum of length 75 cm, what is the maximum speed (neglecting air resistance, etc) of the bob if the pendulum was started at rest at a 12o angle relative to the vertical? Where in the swinging motion is the maximum speed reached? Why? Hint: Think back to energy conservation. PE = mgh and KE = ½ mv2

2 2 10. What exactly does Fcentripetal = mv /R mean? What does the value of mv /R tell us, and what does it depend on specifically for a pendulum? Think about what force or forces keep the pendulum moving in that portion of a circle. Consult our notes and the book for assistance.

Write a brief, to-the-point (i.e. one paragraph) summary of your findings/conclusions about what the period of the pendulum depends on and how it depends on specific quantities.

*Use and combine the results of your best-fit functions for your three graphs to obtain a mathematical model for the period of your pendulum as functions of length, mass, and angular amplitude.*

Post-Lab: Do computer experiments with PhET. Go to http://phet.colorado.edu/en/simulations/category/physics, and then select Pendulum Lab and run it. You can vary the same parameters that you just did in this experiment. Try it, and compare the simulated results with your observations and measurements with a physical experiment.

Are there any significant differences between the simulated and physical experiments?

One advantage the simulation has is you can vary gravity. What happens to the period of a pendulum on the moon? Jupiter? No gravity? Use the simulation to check your answer on lab question #6. You can also vary air friction, so make a prediction of what happens to the period of a pendulum with air friction, and then test your prediction with the simulation and report what happens. Was your prediction correct?

How does your final mathematical model for the period of a pendulum compare to the accepted model, T = 2π [L / g]1/2 ?

Above and Beyond: Explore the derivation of the period of a pendulum by using the small-angle approximation to Newton’s 2nd law for the tangential component of the motion.

Remember that the general equation for SHM is d2x/dt2 = -ω2x.

Then investigate other, similar small-angle oscillations, such as a physical pendulum of a meter stick hanging by one point. Explore the derivation of the period of this nd pendulum using the small-angle approximation to Newton’s 2 law for rotations, τnet = Iα = I d2θ/dt2.

Use the double pendulum simulation at http://www.myphysicslab.com/. This is difficult to setup and study in a physical lab.

For the Teacher:

This student lab sheet has students use stop watches to measure the period of the pendulum. It is meant to be used for more advanced students (AP level) because it requires them to do multiple trials and find uncertainties. This is done by calculating standard deviations for each set of trials. Students then use these uncertainties to make error bars on their graphs, and to think about if the range of values of error bars could make them think twice about the best-fit functions Excel provides. You will need to decide if you want your students to do similar work.

Teachers have the option of hanging the pendulum on an electronic force sensor, which will allow students to get more accurate period measurements and avoid doing error analysis and error bars on graphs.

Lab groups are asked to do a group report using Google Documents for this lab. Of course, teachers can choose for students to write their own individual reports.

The Post-Lab is optional, depending on the teacher’s needs for their students. Keep in mind that some teacher may simply want students to use the PhET Pendulum Lab. Non-calculus classes will avoid Above and Beyond.