An Application of Exponential Decay
On this web page you will investigate how the
exponential function can be used to model the motion of a pendulum.
This exploration will reinforce the concepts in Lesson 5.4 of Discovering
Advanced Algebra: An Investigative Approach.
Sketch
This sketch shows a pendulum near a motion sensor. Press
Swing Pendulum and
observe the motion of the pendulum. You will see that it gradually
slows down due to
friction. The trace of the pendulum's path is shown; you can erase the
trace by clicking the red X in the lower right-hand corner.
Investigate
- Press Swing
Pendulum. Sketch a graph on which
the x-axis represents time and the y-axis
represents the distance of the bob to the motion sensor. There is a
point on each swing where the pendulum bob is farthest from the motion
sensor. Estimate and plot the data points corresponding to those
extremes.
- What kind of function do you think best models the
extreme points you have plotted? Explain.
Sketch
When you swing the pendulum in this sketch, you will
also the data points corresponding to the times when the bob is
farthest from the motion sensor. There is also a graph of the
exponential function y = 3sx and a slider that
controls the value of s.
Investigate
- What value of s
makes the graph of y = 3sx pass through the
extreme points?
- What's a good mathematical model for the extreme
points of the pendulum? Explain.
- Why do you think that the swing lengths of a
pendulum decrease?
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