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Resonance and Damped Harmonic Motion

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Submitted by Rich on Thu, 06/20/2019 - 02:47

Introduction

Resonance can be defined in a number of ways.  The most common definition is that resonance occurs at the frequency at which forced oscillations produce maximum amplitude.  When the driving forces of oscillation are removed, friction gradually decreases the amplitude.  This is known as damped harmonic motion.  Most young children experience resonance as well as damped harmonic motion in schoolyard playgrounds.  They experience resonance while pumping the swing at the right frequency--the natural frequency of the swing.  They experience damped harmonic motion when they stop pumping and the swing gradually comes to a stop.

In the case of children swinging and other similar phenomena, resonance occurs because the system stores and readily transfers energy between kinetic and potential energy.  A child  at the top of a swing cycle has stored energy as gravitational potential energy.  At the bottom of the swing cycle, the energy has been converted to kinetic energy.  Children quickly learn that relatively small pumps at the correct time intervals result in large increases in the swing's oscillation amplitude.

Resonance and Damped Harmonic Motion with Springs and the PocketLab TurboTrack

The PocketLab TurboTrack kit combined with 5 N/m springs from Vernier Software & Technology make for a great way for students to study resonance and damped harmonic motion.  Figure 1 contains an overhead view of the experiment setup.  Two Mini HotRods on three 1-foot pieces of Hot Wheels track are connected by three 5 N/m springs.   Voyagers are inserted into the slot in each of the HotRods.  Only one Voyager need be connected to the PocketLab app, but a Voyager in each HotRod keeps the  mass of each the same.  The Voyagers are oriented so that the IR rangefinder windows face the rangefinder reflectors.  Each reflector is an 8½" x 11" piece of card stock that has been folded in half.

Resonance experiment setup
Figure 1 - Overhead view of the experiment setup

Figure 2 is a close up of the IR reflector holder.  It is 3D printed and contains a slot for holding the folded piece of card stock.  In addition, there is included a peg on which the end of a spring can be easily inserted.  The STL file is included with this  lesson.  A small wood block mounted to the table top  with a 3M damage-free hanger keeps the track from slipping when performing the experiment. 

IR reflector track connector with peg
Figure 2 - IR reflector track connector with peg

Figure 3 shows an overhead view of the Mini HotRod.  A 3D printed bumper (red) with a loop provides a loop on the right side of the HotRod to accompany the loop that is built into the left side of the HotRod.  The spring easily connects to the loops.  The STL file for the bumper with a loop is included with this  lesson. 

Mini HotRod overhead view
Figure 3 - Mini HotRod overhead view

Activity 1 - Damped Harmonic Motion

Set up the apparatus as shown in Figure 1.  In this activity you will be investigating damped harmonic motion.  The video below shows how easy it is to produce this type of harmonic motion.  While holding HotRod B in place, HotRod A is pushed aside and then released.  Use the rangefinder on HotRod A to obtain a graph of position vs. time.  Import this data into a spreadsheet and use it to answer the following questions:

  •  What happens to the amplitude (height of the wave) as time goes on?  Why is this happening?
  •  Is the period (time for one complete back-and-forth swing) of oscillation independent of the amplitude?  If so, what is the value of the period in seconds?

Activity 2 - Resonance

The video below shows how to produce resonance.  The spring is removed from HotRod A and then stretched to where HotRod A is located when at equilibrium.  Grab the spring and move your hand slightly back and forth at a rate that gives maximum amplitude to the motion of Cart B.  Use the rangefinder on HotRod B to obtain a graph of position vs. time.  Import this data into a spreadsheet and use it to answer the following questions:

  • What happens to the amplitude of the swing as time goes on?  Why is this happening?
  • Is the period fairly constant as time goes on?   If so, what is the value of the period in seconds?
  • How does the period for resonance compare to the period for damped harmonic motion that you obtained in Activity 1?

Activity 3 - Oscillation of a Pair of Coupled Mini HotRods - Mode 1

We'll now turn our attention on the behavior of the two mini HotRods when they are connected by three springs as shown in Figure 1.  In this case the period of oscillation  depends upon the initial conditions, such as the location of each HotRod when they are released.  The first mode of oscillation to be investigated is shown in the video below.

In the video, with HotRods A and B at their equilibrium separation,  they are moved to one side and then released simultaneously.  Be sure to keep them at their equilibirum separation while moving them to one side. Use the rangefinder on one of the HotRods to obtain a graph of position vs. time.  Import this data into a spreadsheet and use it to answer the following questions:

  • Is the motion damped?  How can you tell from your graph?
  • Is the period fairly constant as time goes by?  If so, what is the value of the period in seconds?
  • How does the period of this mode of oscillation compare to the periods from Activity 1 and Activity 2?

Now unhook the end of the spring from the peg where it connects HotRod B to the end of the track.  Keeping the spring stretched close to the peg's location, move your hand back-and-forth slightly so as to produce resonance for this mode of oscillation.  How does the period for this resonance compare to the natural damped period of oscillation for this mode of vibration?   

Activity 4 - Oscillation of a Pair of Coupled Mini HotRods - Mode 2

Repeat the procedures of Activity 3, but use the vibration mode shown in the video below.  As seen in the video, the HotRods are moved closer to each other, allowing the middle spring to "unstretch" some.  The HotRods are then released simultaneously.  Use the rangefinder on one of the HotRods to obtain a graph of position vs. time.  Import this data into a spreadsheet and use it to answer the following questions:

  • How does the period of Mode 2 compare to that of Mode 1?
  • How does the period of damped Mode 2 compare to resonant Mode 2?

 

For high school and AP students, in order to explain the differences in Mode 1 and Mode 2 from basic physics principles, this lesson can be used:

Periodic Motion of a Pair of Physics Carts: Experiment and Theory.

Activity 5 - Mystery Mode of Vibration

Our final activity is a quick challenge for you.  We are not showing you a movie, as your job is to predict what will happen if you begin with these initial conditions:  Hold HotRod A steady at its equilibrium position while moving B away from A.  Then release the two carts simultaneously.

Do your predictions agree with what happens when you actually perform this experiment with the HotRods?

If you want to perform this mystery mode by collecting position data for both HotRods simultaneously on the same device, take a look at this Phyphox experiment.  It allows connecting two Voyagers to the same device.

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