About your report: All data for this lab will be recorded in WebAssign for reasons that should be apparent after you read the instructions.

 Goal

To use the phenomenon of standing waves to determine the speed of waves on a long, helical spring to within 1%

 Equipment

• helical spring
• C clamp (optional)
• stopwatch (manual or software)
• meter stick or tape measure
• glove or sock

 Introduction

You'll create standing waves with various numbers of antinodes on a long, helical spring  For each wave pattern, you'll measure the frequency. Then you'll plot frequency vs. the number of antinodes and determine the relationship between them. By comparing the fit results to the theoretical expectations, you'll determine the speed of the waves on the spring.

 Prelab

Write your answers to the following in the WebAssign assessment L147PL.

1. View this animation of the expected standing wave patterns on the spring. Then do the following.

1. Let L represent the distance between the ends of the spring. In the middle column of the table below, give the number (or fraction) of wavelengths between the fixed ends of the spring for each harmonic. In the 3rd column, give the number that must be multiplied by L in order to give the wavelength. View this animation if you need help.

Harmonic	L = ___ ∙ λn	λn = ___ ∙ L
1
2
3
4

2. Generalize your results from the table by writing the general formula for λ_n in terms of n and L.

3. You now have a general equation that relates λ_n to L and n. Next use v = f_nλ_n and your result from the previous step to write the general equation for the frequency f_n of the nth harmonic in terms of n, L, and the speed v of the waves on the spring.

2. For the conditions of the experiment, explain why the speed of the waves on the spring is independent of f_n (or nearly so). Read the instructions below so that you know what the conditions of the experiment are.

 Preliminary Data

Submit your prelimimary data in WA L147D.

You'll need to have a long, unobstructed area about 5 m long in order to oscillate the spring. A hallway is a good location. One end of the spring must be kept fixed. This can be done by having an assistant hold the end of the spring for you. but a better method is to use a C clamp to clamp one end to a table. You'll oscillate the other end. But first stretch the spring horizontally to such a distance that there's some sag but not so much that the spring hits the floor when oscillated. Once you've decided on how far to stretch the spring, keep the distance between the two ends of the spring constant while taking data. Also, keep the string stretched by the same amount. That is, don't take in or let out coils. (Do you know why?) Place something on the floor where you're standing to mark your position. Use a meter stick or tape measure to measure the distance from the fixed end of the spring to the point where you'll stand.

You'll be creating standing wave patterns on the spring. Do this by oscillating your end of the spring vertically. Begin with a pattern having 2 antinodes (loops), since that one is fairly easy to produce. You'll know when you have a good pattern, because you won't have to move your hand very far up and down in order to maintain the standing wave. If you're moving your hand more than a few inches, then you don't have a standing wave. All that should be required is a bit of wrist action to keep the pattern going. Remember, the two ends of the spring are nodes, and that means the vertical spring motion is minimal at those points. Now try producing patterns with 3 and 4 antinodes.

See the following video to get an idea of how to set up the spring and also how to oscillate it correctly in order to produce standing waves: Streamed / RealPlayer / Flash / MP4. If you have to move your hand very far to produce the wave, then it's not a standing wave.

In order to determine how many successive oscillations you need to time in order to achieve a 1% mean deviation criterion, take 5 trials of the time for a single oscillation. Make these preliminary measurements on the 4-antinode standing wave pattern, since that's the one for which consistency in timing is typically the most difficult to achieve. Enter your data in the WebAssign L147 form and determine the number N of successive oscillations needed to achieve the 1% mean deviation criterion. When you've done that, you'll be ready to collect the data that you'll use to meet the goal of the experiment. (Note that the symbol N is for number of oscillations and n is for number of antinodes.)

 Final Data

Review the introduction for the plan of the experiment. In order to achieve the 1% mean deviation criterion, use the value of N that you determined in the prelab. Collect 5 time trials for each value of n from n = 1 to 4. Record your data and calculations in the WA L147 form.

 Analysis

Enter results and discussion in WebAssign L147.

1. Use Logger Pro to draw a graph of frequency f_n vs. number of antinodes n. Do a fit that's appropriate to the kind of relationship that you expect. Don't force the fit to go through the origin, since that's not a data point. You know from previous labs that it's a good practice to allow for the possibility of a non-zero intercept. That may help to identify systematic errors that tend to increase or decrease all values of the dependent variable by the same amount or percentage.

2. Add error bars to your graph as follows:  Create a new manual column in your LP data table for the percentage mean deviation. Then double click on the frequency column in your data table. Click the Options tab. Select Error Bar Calculations and Percentage. Then click Use Column and select the % mean deviation column. Click OK to add the error bars to your graph. This is a common way to visually display expected uncertainties.

3. Do the following in a text box.

   1. Construct the matching table

   2. Write the equation of fit.

   3. Tell what the significance of the slope is. Review this if you're not sure what is being asked for.

4. Upload your file to WA L147.

5. Use a coefficient from your fit together with a result from the prelab theory to calculate the speed of waves in the spring.

 Discussion

1. Let's revisit a question from the prelab: Why is the speed of the waves on the spring independent of frequency? The reason has to do with keeping the tension and linear density constant, since those are the factors on which the wave speed depends. Tell what you did to keep the tension and linear density about the same for different standing wave patterns.

2. Suppose that the distance from the oscillator's hand to the other end of the spring were kept fixed, but more tension was applied to the spring by stretching it more (pulling in coils).

   1. How would the wavelength of the standing wave patterns change?

   2. How would the speed of the waves change?

   3. How would the frequency of any given standing wave pattern change?

3. You used an indirect method to determine the speed of waves on the spring. This method involved using your fit to the graph of frequency vs. antinode number. There are at least two other experimental methods for determining the wave speed. Suppose you have at your disposal a meter stick, a stopwatch, and a spring scale. Describe a method using any or all of these items of equipment to determine the speed of waves on the spring. (In case you're tempted to say that you would measure the frequency and wavelength and multiply them, that method is in principle no different than the one used in L147.)

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