THE SIMPLE PENDULUM: A CLASSIC CASE STUDY Originally developed by Dr. Larry Silverstein

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1 THE SIMPLE PENDULUM: A CLASSIC CASE STUDY Originally developed by Dr. Larry Silverstein This Experiment/Activity draws together, and exemplifies, many threads and activities of science, among them: 1. Observation. 2. Sophisticated Observation (choosing what to measure carefully). 3. Reality versus Intuition. 4. Unexpected results. 5. The development of new knowledge. 6. The need for careful design of the observation process, including: (a) Collection and recording of Data. (b) Analysis of Data. (c) Extraction of Patterns, if any. (d) Presentations of patterns in graphical form and symbolic form. (e) Predicting future results. 7. Why are Pendulums important? Here are some of the reasons: (a) Accurate Timekeeping (b) Serve as a model for other kinds of oscillatory/vibratory motion (c) Have implications for the behavior and stability of structures (d) Have implications for the description/understanding of atomic theory Pendulums draw together many basic principles and concepts which are essential to our understanding of the universe and how we acquire knowledge about it. These include: (1) Dynamics: Force, motion, velocity, acceleration, Newton's Laws, gravitation, free fall, etc. (2) Work, Energy, Power, Conservation of Energy, Transformation of Energy, etc. 8. A little history: Galileo (1580's): Observes chandelier motion in churches. Begins to make careful observations. Makes more formal studies/measurements. Identifies important patterns in the data through simple mathematical analysis. Huygens (1656): Develops the first truly accurate large clocks, based on the regularity of Pendulum motion. Pendulum Physics (rules and regularities of pendulum behavior) develop from This experiment will replicate some of that development. 9. The most important characteristic of Pendulum behavior is the Period, T, which is the time for a full oscillation cycle (back-and-forth swing), in seconds. Alternatively, we sometimes refer to the Frequency f, which is the reciprocal of Period. (f= 1/T). f has units of cycles/second, or Hertz (Hz). 10. Our intuition might reasonably lead us to suspect, assume, anticipate that T might depend upon 3 possible "variables" (under certain circumstances, they are called "parameters"). These parameters are: (1) Mass m of the pendulum bob; (2) Release angle A; (3) Length of Pendulum L.

2 11. A useful, simple investigation is to conduct a set of simple physical measurements to determine how T varies (if at all), with variations of m, A, L. 12. It is important that we design the experiment to be sufficiently accurate and precise. This means that we need to think about, and estimate (if possible) the influence of sources of errors, accuracies and imprecisions. We might, for instance suspect (correctly) that under certain circumstances friction and air resistance may play an undesirable role in confusing the measurements. 13. Overall, it has been found that to obtain good results, it is essential that we provide: (a) A stable pivot point (without wobble) (b) A pivot with as little friction as possible (c) Pendulum bob masses which are accurately and precisely specified (d) An accurate way of measuring: (1) Angle of release, A (which should ideally be small, <10 degrees) (2) Mass of bob, m (3) Pendulum Length, L (4) Period, T 14. Once we provide these necessities, we can then allow certain parameter values to vary in a controlled fashion, and examine the resulting effect, if any, on Period T. 15. Your Instructor will provide your Team with equipment and guidance as your investigation proceeds, and will advise and mentor you through any experimental difficulties which you may encounter (and there will always be some difficulties even in the simplest of experiments. That is one of the lessons to be learned here.) 16. Team members will collaborate during the performance of the experiment, and jointly record data in tabular form. Each member will create, and fill in, his/her data sheet. 17. Each Team member will then individually be responsible for a personal report describing/explaining: Experimental Design; Data Recording; Data Analysis/Calculations; Interpretation of Results; Conclusions. Below is the attached Student Handout with suggested Answers (in blue). Note: Student Handout is also listed as a separate document without the answer.

3 THE SIMPLE PENDULUM: A CLASSIC CASE STUDY This activity represents a hands-on team-effort to assemble a simple pendulum apparatus and then create your experimental design and strategically collect data that will explore the nature of the system. You will be provided with equipment that your team will use to assemble the pendulum apparatus for subsequent investigations. Your instructor (that is me), will mentor you through any experimental difficulties which you may encounter. Remember, there will always be some difficulties even in the simplest of experiments. That is one of the lessons to be learned here! You are expected to collaborate with your team members and jointly record data in tabular form. Each member will create, and fill in, his/her data sheet. As your Data Analysis Assignment each team member will be individually responsible for a report describing/explaining the following: Experimental Design & Data Recording Data Analysis & Calculations Interpretation of Results Conclusions The report (up to 4 typed pages) is due next week. Questions listed below should be addressed within your report.

4 Questions 1: Recall the key steps associated with the Scientific Methods and outline them. Pose a question of interest about a system under study (hypothesis); Design an experiment and collect the data (testing the hypothesis); Present the data in a table or graph to observe the trends/patterns; Using pervious knowledge and imaginative thinking to interpret the data; Modify the hypothesis, if needed, and verify the new hypothesis by further testing (collecting data again). Question 2: Once you build the pendulum apparatus you will release pendulum bob (weight) and measure the Period (the time for one full oscillation cycle). List three factors/parameters that would influence the Period. 1) Mass of the weight attached; 2) length of the string; 3) release angle Questions 3: Provide a prediction of how each of the three parameters that you have listed would influence the Period. You will subsequently test your prediction (tentative hypothesis). Answers depend on student s intuition about the system and previous knowledge. Many students might assume that mass will influence the T, with larger mass leading to shorter T (faster oscillatory motion). Remember when collecting data and experimenting, you need patience & perseverance. Question 4: Think about how you want to measure the Period (T). Would it be beneficial to measure the total elapsed time, t, for 30 full swing cycles? If so, why? What type of error could you minimize in this fashion? Measure Period T by measuring total elapsed time, t, for 30 full swing cycles. This will provide an averaging effect which will tend to smooth out fluctuations due to vibrations, air currents, rough release of bob, etc. Then calculate: Period T = t/30. This data collection practice would minimize the random error.

5 CASE I: Only Mass m varies CASE II: Only Angle A varies CASE III: Only Length L varies L A m t (30 cycles) T= t/30 Display the tabulated results in a graphical form, and comment. For CASE III, which you may find particularly interesting, show a carefully plotted graphical display of values of T versus values of L. Some typical experimental plots are sketched below. Your task will be to make more careful (accurate) graphs. T ~ L In CASE III, you should try to show, by appropriately interpreting the data, that (as Galileo discovered around 1580): The Period T depends upon (varies as) the SQUARE ROOT of Pendulum Length L. This means, for example, that each time L doubles, T increases by a factor of the square root of 2, which is Similarly, as L triples, T increases by a factor of the square root of 3, which is