# Harmonic Motion Lessons

## VelocityLab Investigation of Damped Harmonic Motion

This investigation shows how VelocityLab allows for a quick and easy demonstration of damped harmonic motion.  The photo below shows the experiment setup as performed by the author.  A jellied cranberry sauce can was selected as there is virtually no sloshing of the cranberry sauce as the can oscillates back-and-forth on a curved piece of laminate flooring.  The center of the flooring is clamped down to the table with an adjustable wrench.  The ends of the laminate flooring are raised a little with some small wood blocks.  The cranberry sauce can is shown at rest at the VelocityLab zero position.  PocketLab has been mounted to  one end of the can with some Velcro.

The video below shows a VelocityLab video combined with data for a typical run.  Note at the start of the video how the author carefully rolls the can to the top left side of the curved surface in a way that preserves the zero position and makes positions to the left of center negative while those to the right are positive.

The figure below shows a combined graph of position, speed, and acceleration vs. time. The graph was obtained by using Excel to massage the data in the pos_vel_acc.csv file produced by VelocityLab.  A number of questions about the video and graph are worth discussing with the students including:

1. What can be said about speed and acceleration when the cylinder is at maximum amplitude on either side of the curved surface?

2. What can be said about the speed and acceleration when the cylinder is rolling while at the center of the curved surface?

3. What happens to the magnitude of the maximum amplitude of position, velocity, and acceleration with each successive oscillation of the cylinder?

## Quantitative Experiment to Determine the Relationship Between a Pendulum's Length and Period

PocketLab is a perfect device for determining the quantitative relationship between the length of a pendulum and its period of oscillation.  Pendulums of known lengths were made from balsa wood strips such as those available from Michaels and other hobby stores.  The photo below shows six such pendulums of lengths 15, 30, 45, 60, 75, and 90 cm alongside a meter stick.  The picture shows that PocketLab was taped with double-stick mounting tape to the pendulum whose length is 45 cm.

The photo below shows the apparatus setup. The balsa wood pendulum with PocketLab attached is hung with masking tape from a ring stand supported by the weight of several books to keep it stable.  The orientation of PocketLab shows that it will be swinging in the XZ plane.  Therefore, the Y angular velocity data will provide information necessary to compute the period of the pendulum.

The video below shows the PocketLab graphs superimposed on the actual moving pendulums.  The Y angular velocity, shown in blue, contains the data of interest in the analysis.

The Excel graph below shows the Y angular velocity in deg/s obtained from the gyroscope data file for the pendulum of length 15 cm.  (Note that half the length (3.2 cm) of PocketLab is subtracted from 15 cm, giving a length of 11.8 cm to the approximate center of mass of PocketLab.)   As shown in red in the graph, the period is calculated by averaging the time for ten complete oscillations.  This process is repeated for all six pendulum lengths.  All gyroscope data for these six pendulums can be found in the attached gyroscope Excel file.

Below is shown an Excel chart the summarizes the results of the experiment.  When a power regression type is applied to the data, it is seen that the power turned out to be 0.4773, very close to the theoretical 0.5 expected for such a pendulum.  It can be concluded that the period of a simple pendulum is proportional to the square root of the length of the pendulum.

## PocketLab on an Oscillating Cart

An oscillating cart with a PocketLab provides an interesting way to study Newton's Second Law of Motion as well as some principles of damped harmonic motion.  The apparatus setup is shown in the figure below.  The small dynamics cart that can quickly be made from parts included in the PocketLab Maker Kit is shown in its equilibrium position.  Rubber bands are attached to each side of the cart and to two ring stands weighted down with some heavy books.  It is best to use rubber bands that provide as small Newton/meter as possible.  PocketLab is attached to the cart with its x-axis parallel to the rubber bands.

The close-up in the figure below shows that two small pieces of wire are threaded into holes in the cart with the rubber bands attached.  The ends of each wire are twisted together to tighten the rubber band on the cart.

The movie below shows a typical run, with 20 data points per second and acceleration selected in single-graph mode.  The red trace on the graph is the acceleration of interest, namely acceleration in the X-direction.  The blue and green traces, representing acceleration in the Y and Z directions, are quite erratic due to slight jiggling of the cart, and are not of interest here.  The red curve shows a very regular pattern, in which it can be observed that the magnitude of the acceleration is greatest when the cart is at each end of its swing and zero in the center of the swing.  It is also noted that the magnitude of the acceleration decreases with time in a pattern that suggests exponential decay.

The image shown below contains a graph of x acceleration from data produced by the PocketLab app.  The graph was obtained by importing data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com).  A model involving the sine function and exponential function was created.  It is seen that the model (the black curve) follows the red acceleration X curve very well.

Students can therefore conclude that this oscillating cart has a negative exponential decay with individual cycles characteristic of the sine function.  There are two constants of particular interest in the model equation shown in the gif image: Accel X = A*exp(-B*x)*sin(Cx+D)+E.  The constant C in the fit is 2*Pi/Period, from which we see that the period is 2*Pi/D = 2*Pi/11.61 = 0.54 s.  This agrees very well with the period obtained by direct observation of the graph.  The constant B in the fit is the reciprocal of the so-called lifetime.  Any exponential decay is characterized by its lifetime, which is the amount of time required for the amplitude to decay to 37% of its initial value.

By loading the cart with different masses and collecting PocketLab data on the resulting accelerations, students should be able to verify Newton's Second Law of Motion (Fnet = ma), showing that acceleration is inversely proportion to mass if the net force is held constant.

This experiment also provides a nice way to determine the period when the period of the oscillation is quite small, and difficult to measure with a stop watch.

### Negative Exponentially Damped Harmonic Motion from a PocketLab Pendulum

This experiment allows one to do a quantitative investigation of the damped harmonic motion of a swinging pendulum.  The pendulum is a piece of wood about a yard long from a Michael's hobby shop one end of which has been attached to a PocketLab by a rubber band.  The other end is taped to the top of a doorway, allowing the resultant pendulum to swing back-and-forth as shown in the image below.

The y-axis is perpendicular to the XZ plane of the swinging pendulum.  Therefore, the main item of interest is the magnitude of the angular velocity vector in the Y direction (shown as a blue curve in the movie).

The image shown below contains a graph of the Y angular velocity (shown in blue).  The X and Z angular velocities (shown in red and green in the video, respectively) are small and erratic due to slight wobble in the swinging pendulum and are not included in the graph.  The graph was obtained by importing the data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com).  A model involving the sine function and the exponential function was created. It is seen that the model (the black curve) follows the blue angular velocity curve very well.

Students can therefore conclude that this damped harmonic motion has a negative exponential decay with individual cycles characteristic of the sine function.  There are two constants of particular interest in the model equation shown in the gif image: Angular Velocity = A*exp(-C*x)*sin(Dx+E)+B.  The constant D in the fit is 2*Pi/Period, from which we see that the period is 2*Pi/D = 2*Pi/3.657 = 1.72 s.  This agrees very well with the period obtained by direct observation of the graph.  The constant C in the fit is the reciprocal of the so-called lifetime.  Any exponential decay is characterized by its lifetime, which is the amount of time required for the amplitude to decay to 37% of its initial value.

You will note from observing the movie that the angular Y velocity is zero when the pendulum is at the two ends of its swing, and is a maximum at the middle of the swing, both of which are expected.  This is easier to see if you view the movie frame by frame.

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