Periodic Motion - The Pendulum

Deborah Lojkutz                Joliet West High School
                               401 N. Larkin Ave.
                               Joliet IL 60435
                               (815) 727-6950

Objectives:

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
 rod

For class graphs - two pieces of end roll paper approximately 21/2 meters long,
a meter stick, markers and masking tape

Strategy:

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.