About your report: All data for this lab will be recorded in WebAssign for reasons that should be apparent after you read the instructions.
To use the phenomenon of standing waves to determine the speed of waves on a
long, helical spring to within 1%
 helical spring
 C clamp (optional)
 stopwatch (manual or software)
 meter stick or tape measure
 glove or sock
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.
Write your answers to
the following in the WebAssign assessment L147PL.

View this animation of the expected standing wave patterns on the spring.
Then do the following.
 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 = ___ ∙ λ_{n} 
λ_{n} = ___ ∙ L 
1 


2 


3 


4 



Generalize your results from the
table by writing the general formula for λ_{n} in terms of n and L.

You now have a
general equation that relates λ_{n} to L and n. Next use v = f_{n}λ_{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.

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.
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.
Safety considerations:
Wear long pants to protect your legs from rebounding springs. Wear a glove or a sock on the hand that holds the spring. Keep a firm grip on the spring when oscillating it and
make sure that the
other end of the spring is fixed firmly in place. There's no need to produce
large amplitude oscillations. There's more energy in the spring with greater
amplitude. That means if the spring hits you or someone else, there's greater
potential for injury. While oscillating the spring, watch out for people who may
be walking by to make sure they stay out of harm's way. 
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 4antinode 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.)
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.
Enter
results and discussion in WebAssign L147.

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 nonzero 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.

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.

Do the following in a text box.

Construct the matching table

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

Upload your file to WA L147.

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

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.

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).

How would the
wavelength of the standing wave patterns change?

How would the speed
of the waves change?

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