`Periodic Motion - The PendulumDeborah Lojkutz                Joliet West High School                               401 N. Larkin Ave.                               Joliet IL 60435                               (815) 727-6950Objectives:To introduce the concept of periodic motion and relate it to the movement of a pendulum.  To discover that the period of a pendulum is dependent on the length of the pendulum and independent of the bob and the amplitude.  Materials Needed:Each group needs a stop watch and pendulum with a different bob.Materials for pendulum - string bob - infant stacking rings provide colorful bobs of different sizes and mass right angle clamp ringstand rodFor class graphs - two pieces of end roll paper approximately 21/2 meters long,a meter stick, markers and masking tapeStrategy:Begin the class period with a discussion of what the students think periodic motion is.  After a few minutes, bring out a pendulum from behind the lab table and use it as an example of periodic motion.  Point out its various parts - bob, length, pivot point.  Demonstrate what is meant by period and amplitude.  Spend a few minutes discussing the accuracy of measuring a single period.  The students should realize that timing how long it takes for ten cycles and dividing by 10 will lessen the effects of reaction time and result in a more accurate measurement of the period. Break up the class into groups.  Each group is given a pendulum with a different bob but all pendulums are 1 meter in length.  (A different option would be to have the students construct their own pendulums 1 meter in length.  If doing so,make sure to discuss that the length of the pendulum is measured from the pivot point to the center of gravity of the pendulum bob.)  Each group is to find the period of their pendulum by timing it for 10 cycles and using an amplitude of 10 cm.  After doing so, they are to experiment with other amplitudes (5 cm, 15 cm, 20 cm, etc.) to determine if the amplitude effects the period.  All groups record their data in the class data table on the board under the following headings: Color of bob, Time for 10 cycles, Period, Effect of changing the amplitude.   When all groups have recorded their data, call the class together for a discussion of the results.  It should be apparent that the shape and mass of the bob and the amplitude have no effect on the period.  Small differences can be explained by experimental error.  If the students are not sure that the rings are actually different masses, bring out a scale and prove it.  If you are using infant stacking rings, there will be about a 50% difference between the largest and smallest ring's mass. Each group is given a different length of string to create a new pendulum.  Lengths should vary from 25 cm to 2 m.  As before the groups will find the period of their pendulum.  This time they will graph their results on a length versus period graph. While the students are finding the period of their pendulums, hang a piece of end roll paper about 21/2 meters long on a wall and label the axes.  The vertical axis is marked off to the actual length of the pendulum.  The horizontal axis is the period marked in a convenient scale. When the students have found the period of their pendulum they should remove it from its support bar and hang it on the graph at its corresponding period.  Remind the students that the actual length of the pendulum is measured from the pivot to the center of gravity of the bob.  When using rings for bobs, the center of gravity is at the center of the ring, therefore it is important that the centers of the rings be lined up on the horizontal axis.  By using this self graphing technique, it is not necessary for the students to measure the pendulum's length and the effect of the length of the pendulum on the period is shown quite dramatically. Once all groups have added their pendulums to the graph discuss the results.  The graph should look like a y-parabola.  If it is not obvious that it is a parabola remember that the origin is a point on the graph - zero length will have zero period.  With a marker sketch the curve on the graph.  Discuss with the students the shape of the graph and what it represents mathematically.  Hopefully they will come up with the idea that there is a direct relationship between the length and the square of the period.  (This depends on their level of math ability.)  If this relationship is not obvious, lead the students by a discussion of what needs to be done to straighten out the graph.  This approach usually gets to the idea of squaring the period.  The students should now verify these predictions by squaring their period and regraphing on the second end roll graph.  The students should transfer their pendulums from the first to the second graph.  The resulting graph should be a straight line through the origin.  At this point the class can discuss the results that the square of the period is directly proportional to the length of the pendulum.  This would be a good point to start a discussion of the equation and theory of a pendulum. Conclusion:This activity will take more than the usual lab period.  A good breaking point would be after finding the effect of the bob and the amplitude.  This activity can be used with elementary students up to the first graph.  Evaluation:The student's understanding of this material can be evaluated by having them use the graph to predict what the period of a pendulum will be for a specific length.  They can then experimentally verify their prediction. `