- Regression Modelling - A Historic Note Next to the famous leaning tower of Pisa is the beautiful Santa Maria Assunta Duomo or cathedral. The famous tower is the bell tower of the Duomo. It is said that the teenage Galileo discovered the isochronism of the pendulum after he had observed a lamp swinging in the Duomo. It is also claimed that during the Mass he timed the oscillations of the lamp using his pulse. Many years later he used the oscillation of a pendulum to time the movement of balls rolling down ramps. Definitions A simple pendulum consists of a small mass called the bob oscillating to and fro at the end of a light string. The period (T) of a simple pendulum is the time taken for one swing or oscillation of the bob. That is, the time taken to move from position A to B, back through A to C and then back to A. The amplitude of oscillation is shown as the angle θ. If the amplitude of oscillation is small (less than 10 ), the bob moves with simple harmonic motion. The length (l) is measured from the support to the centre of the bob. Isochronal: recurring at regular time intervals. Purpose It has been shown that the period of a simple pendulum is independent of the mass of the bob and of the amplitude (so long as that is small). There is a simple rule relating the Period (T) to the Length (l). The purpose of this lesson is to find that rule. Data Sources You may be expected to collect your own data by measuring real pendulums or your teacher may have data already collected by another class. It is important to include measurements taken from very long pendulums. Because we don t want you climbing in high places that may prove dangerous, I have asked friends to do some of these measurements for you. Copyright 2007, Hartley Hyde Page 1 of 8
For example, people who live in Auckland New Zealand visited the Sky Tower in the centre of Auckland. From the 192 metre bungy jumping platform one of them suspended a pendulum 191 metres long and those below timed two oscillations. Then they timed another two before he dropped the string and jumped. Measuring the Period of a Pendulum With some care it is possible to get very accurate results when measuring the period of a pendulum. You will need string, a small weight (such as a fishing line sinker), a stop watch, something to support the pendulum and a ruler. If outdoors, you should avoid windy days. A firm support is essential. If the support moves as the pendulum swings then you are measuring a compound pendulum instead of a simple pendulum. The point of suspension must be clearly defined. If you use a knot to tie the string to a nail, you need to take care that the loop of string isn t swivelling and that the string doesn t ride up on the knot more one side than the other. If you have access to a laboratory, the best method is to cut a cork or stopper vertically and clamp the string between the two halves using a retort clamp. The clamp can then be fitted to a retort stand. By loosening the clamp, you can easily change the length of the pendulum. Make sure the stand doesn t move. Let the bob hang still and put a marker just below it. Now move the bob to one side (less than 10 ) and let it swing freely. Start the stop-watch as the bob moves from left to right past the marker. Count ten whole oscillations. Count only as the bob moves from left to right past the marker. Stop the watch as the bob moves from left to right past the marker at the end of the tenth oscillation. For short pendulums that move quickly you may choose to use twenty oscillations. For very long pendulums you may use less than ten. Don t forget to measure the length of your pendulum. Measure from the support, (the bottom of the cork or knot) to the middle of the bob. Table of Results Fill in your results in the table on the next page. Your teacher will show you how to fill in the top section of the table. Do not use a calculator to find the numbers in the last column. You can do calculations like these quicker in your head. Copyright 2007, Hartley Hyde Page 2 of 8
Table of Results Length of Pendulum Number of Time in seconds Time for one in metres Oscillations (n) for n oscillations oscillation 15.42 10 78.9 15.42 10 78.6 24.26 5 49.2 24.26 5 49.7 52.73 5 73.1 52.73 5 72.7 113.8 5 107.0 113.8 5 107.2 191.2 2 55.3 191.2 2 55.8 Copyright 2007, Hartley Hyde Page 3 of 8
Questions about the method Why do we time ten or twenty oscillations instead of just one? Why do we time each length twice? Why do we time from a marker in the middle of the swing instead of at the end? a. b. Entering your data Switch on your ClassPad o and tap on the Spreadsheet icon If your spreadsheet has data from a previous investigation, check if you need to save this before you clear it from the screen. To clear the screen, tap on the File menu and select New. You will be asked if you are sure and you then tap OK. Tap on the cell A1. We are going to type the word Length into this cell. To do this: press the k button; select the 0 keyboard; to get the capital letter L, tap the V key and then the L; then tap the keys for the rest of the word Length ; check your typing in the formula bar and then tap the s key or press E. In the same way, type the word Period into cell B1. In the first column, under the heading Length, type the lengths from the first column of your data table, starting at cell A2 and working down. Just type the values; we know that all the measurements are in metres. In the second column, under the heading Period, type the times from the fourth column of your data table, starting at B2 and working down. Just type the values; we know that all the measurements are in seconds. When you have finished, check your screen horizontally against your data table. Make sure that each pair of values in your spreadsheet matches the measurements from your table. Then save your spreadsheet as Pendulum. Copyright 2007, Hartley Hyde Page 4 of 8
Drawing a Graph Select all of the cells into which you typed values. To do this tap on cell A2 and drag down and across until you reach the last value that you typed in column B. Tap the $ to the right of the graphing icon in the tool bar. The graphing icon is always the one after the A icon. It may use any of the thirteen different icons shown in the drop down menu: it depends on which type of graph you last drew. Tap on the scatter graph icon X. Your screen will split and a scatter graph will appear. Length is on the X-axis and Period is on the Y-axis. Checkpoint The data points should follow a smooth curve. If one is seriously out of alignment with the others, you should check that you have typed your data accurately. If you make changes to your data, the data points should move accordingly. Finding a rule Tap on the graphing screen and you will see the spreadsheet menu bar and tool bar change to the graphing screen menu bar and toolbar. Tap on Series in the menu bar and from the drop down menu tap on Trend and this will offer a further menu from which you tap on Power. Copyright 2007, Hartley Hyde Page 5 of 8
Choosing Power asks the ClassPad to draw the best curve of the form y = a x b through your data points. Just like this: If you tap on the curve, the equation of the curve appears in the formula bar at the bottom of the screen. This means that using my measurements the rule relating the variables is given by y = 1.974 x 0.5042. The independent variable is the one we control: in this case we set the Length of the pendulum to a particular value. This variable is plotted on the X-axis. The dependent variable is the result of an experiment: in this case the Period values depend on the Length of the pendulum. This variable is plotted on the Y-axis. ClassPad assumes that you will enter your dependent variable to the right of your independent variable. Because we put Length in the left column and Period in the right column, ClassPad has assumed that the Length is our independent variable on the X-axis and the Period is our dependent variable on the Y-axis. Thus my measurements generated the rule: Period = 1.974 Length 0.5042. Copyright 2007, Hartley Hyde Page 6 of 8
Now it is your turn. Find the Power curve that fits your data points and tap on the curve. Express your rule for the Period of your pendulum in terms of its Length and write your answer in this box. Period = Length Given the accuracy of the tools we were using, you should round your results to two decimal places. Does this simplify your rule? A subsequent lesson will consider a way of showing that this is the best rule. Period = Length Checkpoint If you study physics, you will eventually be able to derive a formula for the Period of a pendulum from some basic principles of physics. Scientists have been able to check this theory with very accurate measurements and the formula for the Period of a pendulum is accepted to be Period = 2π Length g Where g is the local acceleration due to gravity. Use your results to find the value of g at the place where your measurements were taken. g = ms -2 The first way in which a pendulum was used was to check people s pulse rate. Suggest some other ways people might use a pendulum. Checkpoint Copyright 2007, Hartley Hyde Page 7 of 8
Checkpoints If you can, negotiate a lesson in a science laboratory so that your students are able to measure their own pendulums comfortably with as little hassle as possible. If you are also a science teacher this may be easy, however, we are all aware of the accommodation pressure on science facilities in some schools. Even if there are no free laboratories on the same line, it is usually possible to borrow sets of pendulum equipment for use in your classroom. The quickest way to run the experiment is to form eleven groups, each with a different length pendulum, for which they measure the period twice. As a last resort, you are welcome to use data that my class gathered in 1995. It is available in The Australian Mathematics Teacher, CACTUS pages for August 1995. An update, relating the lesson to the ClassPad is in print for the August 2007 edition. Questions about the method (Page 4) Why do we time ten or twenty oscillations instead of just one? It spreads reaction time errors across ten swings and gives us an extra decimal place for our time measurements. Why do we time each length twice? It helps average out any small errors and mis-copies are more obvious. Why do we time from a marker in the middle of the swing instead of at the end? a. The bob moves more quickly past the marker and the moment is better defined. b. As the pendulum looses energy, an end marker would have to shift inward. The Scatter Plot (Page 5) Hold the display sideways and check that the scatter looks like half of y=x 2. The Rule (Page 7) Since g π 2 it follows that the Period 2 Length and that g is approximately 10 ms -2 Uses for a pendulum include timing devices and anything that depends on the accurate measurement of variations in the gravitational field strength, such as finding mineral deposits or underground aquifers. Be prepared for suggestions that crystal pendants can determine the gender of anything from eggs to unborn children and a myriad of dousing concepts. Copyright 2007, Hartley Hyde Page 8 of 8