In this laboratory you will study wave motion in one dimension only, to learn about several of their characteristic behaviors.  You will explore the motion of waves on a string.

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  This log will form the basis of your lab report.  Address the points highlighted in blue.  Answer all questions.

Exploration

Use an on-line simulation from the University of Colorado PhET group to explore the behavior of waves on a string.

(a) Explore the interface.  Try the different controls and click "Help" to discover features you otherwise may miss.

(b) Investigate the behavior of a wave pulse.

 Pulse No End Amplitude: 50 (arb. units) Pulse Width: 50 (arb. units Damping: 0 Tension: high Ruler and timer: checked
Observe this pulse and measure its speed in cm/s.
Vary (one at a time) amplitude, pulse width, damping, and tension, and describe what happens.
Return to the original pulse but change the end to a loose end and the to a fixed end and describe what happens.

(c) Investigate the behavior of a traveling wave.

 Oscillate No End Amplitude: 50 (arb. units) Frequency: 50 (arb. units Damping: 0 Tension: high Ruler and timer: checked
Observe this wave (wave 1). Describe your observations.
 Measure the amplitude of the wave in cm. Measure the wavelength in cm. Measure the period in s. Find the speed of the wave in cm/s. Enter all numbers into a table in Excel.
 wave 1 wave 2 wave 3 wave 4 wave 5 amplitude A wavelength λ period T frequency f speed v

 Move the amplitude slider to 25 (wave 2) and then to 75 (wave 3), and describe what changes. With the amplitude slider at 50, move the frequency slider to 25 (wave 4) and then to 75 (wave 5), and describe what changes. Paste your table into your log. Discuss the relationships between wavelength and frequency, period and frequency, amplitude and frequency and speed and frequency. Can you change the speed of the wave?  What can you do to produce a wave that moves with approximately 1/4 the speed of wave 1. Describe what happens when you include damping.

(d) Investigate the behavior of a standing wave.

 Oscillate No End Amplitude: 20 (arb. units) Frequency: 50 (arb. units Damping: 5 Tension: high Ruler and timer: checked
Describe the wave.  What is its maximum amplitude in cm?  Does this maximum amplitude change when you change the frequency slider?
Change the end to a fixed end, the frequency to zero, and click the reset button.
 Slowly increase the frequency.  You can type an integer into the slider box if it is hard to control the slider with the mouse.  Wait several seconds to see what happens to the maximum displacement from equilibrium of the wave for each frequency setting. For which frequencies is the maximum displacement from equilibrium largest (much larger than that of the traveling wave)?  In other words, for which frequencies do you hit resonance? What can you say about the wavelength associated with those frequencies? At one of those frequencies, decrease the damping to zero and describe what happens.

Experiment:

Standing waves of many different wavelengths can be produced on a string with two fixed ends, as long as an integral number of half wavelengths fit into the length of the string.  For a standing wave on a string of length L with two fixed ends

L = n(λ/2), n = 1,2,3,... .

 Fundamental: L = λ/2, n = 1, 1/2 wavelength fits into the length of the string. Second harmonic: L = λ n = 2, one wavelength fits into the length of the string. Third harmonic: L = 3λ/2, n = 3, 3/2 wavelengths fit into the length of the string.

For a string the speed of the waves is a function of the mass per unit length μ = m/L of the string and the tension F in the string.

In this experiment waves on a string with two fixed ends will be generated by a string vibrator.  The waves will  have a frequency of 120 Hz.  Their wavelength is given by λ = v/f.   You will analyze video clips in which the string tension F is fixed and the length of the string is being varied.  You will measure the length of the string when the string supports a standing wave such that 1, 2, or 3 half wavelength of a wave fit into the length of the string.  Then 120 Hz is a natural frequency of the string and the vibrator drives the string into resonance.  The amplitude increases and the standing waves can easily be observed.

#### Procedure:  (hints)

You will analyze three video clips, string_x.mp4, x = 1 - 3.  To play a video clip or to step through it frame-by-frame click the "Begin" button.  The "Video Analysis" web page will open.

## Begin

In the video clips the vibrator is mounted onto a rod which is fixed to the table with a clamp.  A pulley is mounted onto another rod on a movable stand.  One end of a string is attached to the vibrator.  The string is passed from the vibrator over the pulley and a mass is attached to its other end.  The string is level.

The string is a string with two fixed ends.  The amplitude of the vibrator arm is so small compared to the amplitude of the string at resonance, that the vibrator is very close to a node.  The other node is the top of the pulley.

Step through the video clip string_1.mp4 frame by frame.  You will see the vibrator drive the string into resonance when the length of the string becomes equal to exactly 1/2 wavelength of a standing wave.  Locate the frame in which this happens and hold the clip at that frame.
Choose to track the x-coordinate and calibrate x.  Use the 0.2794 m long marker for the calibration.
Click "Start Taking Data".
Click on the node at the vibrator  for your first data point.  Then click "Step Down" once, to return to the same frame and click on the node on top of the pulley for your second data point.   Find the distance between the two nodes L (the difference between the two x values).
Open Excel and construct a spreadsheet as shown below.   Enter L, λ and the string tension F given in the clip.  For the fundamental λ = 2L.
 clip f (Hz) L (m) λ (m) v=λf (m/s) v2 (m/s)2 F (N) string_1 120 string_2 120 string_3 120
Repeat the experiment using the video clips string_2.mp4 and string_3.mp4.  These clips show the second and third harmonic, respectively.  For the second harmonic λ = L and for the third harmonic λ = 2L/3.
For waves on a string we have F = μv2.  This tension is provided by a hanging mass, F = mg.  Calculate v and v2 and plot F versus v2.  Use Excel's regression function to find the slope of the best fitting straight line to this plot.  This slope is equal to the mass per unit length of the string μ.  The uncertainty in the slope equals the uncertainty in μ.

Answer the following questions:

 What is the mass per unit length μ of the string in units of kg/m and what is the uncertainty in this value as determined from your measurements. Refer to the video clip string_1.mp4.   How much hanging mass would be needed to produce a resonance of the fundamental standing wave, if the string had same length L but only half the linear density μ? Refer to the video clip string_1.mp4.   How much hanging mass would be needed to produce a resonance of the fundamental standing wave, if the string had same linear density μ but twice the length L?

#### Laboratory 10 Report

Convert your log into a lab report. Make sure you have addressed all items in blue, and you are describing your  procedures, results and conclusions in complete sentences

Save your Word document (your name_lab10.docx), go to Blackboard, Assignments, Lab 10, and attach your document.