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PHY 133 Lab 9 ­ Standing Waves [Stony Brook Physics Laboratory Manuals]
Stony Brook Physics Laboratory Manuals
PHY 133 Lab 9 - Standing Waves
The purpose of this lab is to study transverse standing waves on a vibrating string.
Equipment
electric motor with flag and metal reed
variable power supply for motor
photogate
pulley
meter stick
various masses (at least 50g and 100g)
golden “stretchy” string
white “non-stretchy” string
clip
digital scale
Introduction
In this lab, you will study the transverse standing waves formed along a vibrating taut string attached to a rotating electric motor. For
your measurements, you will use two different kinds of string: a golden “stretchy” (or elastic) string, or a white “non-stretchy” (or
inelastic) string. At certain speeds, the motor will create transverse standing wave patterns along the string length between the motor
and the pulley. The string is held taut by the motor on one end and the mass dangling below the pulley at the other end.
As you may know, “transverse” indicates that the displacement of the wave medium (the string) from equilibrium is perpendicular to
the direction of propagation of the wave (which is parallel to the string). “Standing” indicates that the waves do not appear to be
moving in either direction; rather, a seemingly stationary pattern appears due to constructive and destructive interference of two
counter-propagating waves. Hence, the anti-nodes (the “maxima” in the amplitude of transverse oscillations) and the nodes (the
“minima” in the amplitude of transverse oscillations) do not move. These transverse standing waves will only appear under certain
conditions, which is what you will investigate in this lab.
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In this lab, a standing wave pattern is produced by an electric motor that vibrates one end of the string up and down. As this
happens, the string displacement is sent from one end of the string to the other. At this other end, which is fixed, the incoming wave
reflects and bounces back in the other direction. When it reaches the end with the motor, it is reflected back, and this repeats again
and again. However, if the reflected wave traveling to the left is in phase with the original wave traveling to the left, then their
amplitudes constructively interfere. Similarly, the leftward moving wave and a rightward moving wave may interfere destructively.
The combination of all of these interactions yields a standing wave pattern, as shown in the diagram below, which looks like a
stationary wave pattern, rather than many separate waves traveling to the left and to the right. Depending on the speed of the electric
motor, different frequencies can be generated along the length of the string, so long as there are nodes at the two fixed endpoints.
This constraint gives us a relationship between the wavelength λn of the nth mode and the length L of the string:
2L
λn =
n
As shown in the diagram below, the higher the mode number n, the shorter the wavelength λn of the standing wave pattern. In this
lab, you will generate many standing wave modes like these, while exploring the relationship between the string's properties and those
of the standing wave pattern produced.
(As you might have guessed, the physics to be discussed in this lab applies very well to stringed musical instruments like the guitar,
violin, and cello. When playing one of these instruments, a musician is controlling the frequency of sound produced by the standing
waves formed along the strings, generated by their fingers, a pick, or a bow. By striking a string freely, it is able to vibrate along the
entire length, in the “lowest order mode,” which has the lowest frequency (or “note”) given the length, tension, and mass of that
string. However, when the string is lightly pressed down along the neck of the instrument, the effective string length shortens, and
the string no longer vibrates in its lowest order mode. Instead, it vibrates primarily at a higher frequency (called a “harmonic”) and
produces a higher pitch of sound. If you're familiar with stringed musical instruments, you should be able to anticipate the outcome
of the investigations you'll conduct in the lab today!)
Procedure
In this lab, you will first predict what the traveling wave velocity v should be for the golden stretchy string, using the stretched linear
mass density μ of the string and the tension T along the string due to a hanging mass M . Next, you will excite multiple orders
(labeled as “order n”) of standing waves along the golden stretchy string, and calculate the traveling wave velocity using experimental
data. (You can then compare your experimental value to your predicted value of wave velocity.) Lastly, you will investigate the
dependence of the wave velocity on the tension in the white non-stretchy string, and use your data to estimate the gravitational
acceleration g.
Part I: Calculation of wave velocity v from linear mass density μ and tension T
In this part of the lab, you will calculate the traveling wave velocity v from the tension T in the stretched golden string when a mass
M = 150 g is attached to its end. You will also need to compute its stretched linear mass density, μs for this calculation. When a
string is fixed at both ends (with node points), at certain vibration frequencies fn , traveling waves going left and right along the
string add constructively and destructively to form standing wave patterns. The mode number n associated with each frequency
corresponds to the number of maximum amplitudes (or anti-node points) between the two fixed ends. An example with 11 nodes
(not counting the end points) and 12 anti-nodes is shown in the picture below.
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The equation for the velocity of these counter-propagating waves along the string is:
−−
T
v = √
μ
(1)
where the linear mass density μ for a string is the ratio of its mass m and its length L′ :
m
μ =
(2)
′
L
The tension T in the string is created by the weight of the hanging mass M , so that:
T = Mg
(3)
If the string is stretchy (or elastic), then its stretched length L′ depends on T . Since the stretching of the string does not change its
total mass m, but it does change its length L′ , its stretched linear mass density μs is not equal to its un-stretched linear mass density
μ. As the string stretches, then, you should expect that μs < μ.
To predict a value for the velocity v along the stretched golden string, do the following:
Measure the mass m of the golden string using the digital scale in the lab room. Assume an uncertainty of σ m = 0.1 g.
Loop one end of the golden string, and hang a mass M = 150 g from it. (Assume an uncertainty for this mass of
σ M = 0.1 g. Pass the free end up and over the pulley, towards the electric motor. It is essential that you then pass the
string through the hole in the motor reed, then through the hole in the motor post, and clamp it to the side of the
post using the clip. Do NOT tie the string end directly to the motor reed! The mass should now hang slightly below the
pulley.
Measure the length L from the end of the string at the hole in the motor reed to the end of the string where it first touches
the pulley. This portion of the string is where your standing waves will appear. Also, obtain a value for the full length of the
stretched string L′ by measuring the distance from the top of the pulley down to the loop where the mass is hanging. Adding
this distance to L should give L′ . Assume uncertainties of σ L = σ L = 1 cm.
Using equations (2) and (3) above, calculate the values of tension T and stretched linear mass density μs , as well as their
uncertainties, σ T and σ μ . You will need to use the uncertainty propagation formulas from the uncertainty guide. How does
the golden string's stretched linear mass density μs compare to the “accepted” value of its unstretched linear mass density of
−3
μ = 2.7 × 10
kg/m? As mentioned before, why should it be that μs < μ?
Lastly, using equation (1) above, calculate the predicted value of wave velocity v (and its uncertainty σ v ) along the stretched
golden string.
′
s
Part II: Measurement of wave velocity v using the frequencies fn of standing wave patterns
In this part of the experiment, you will now use the electric motor to excite standing wave patterns with frequencies fn along the
horizontal length L of the stretched golden string. The integer n is the number of anti-nodes (or half-wavelengths, as shown in the
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figure above) between the two fixed endpoints. Then, by plotting the frequency fn vs. mode number n, you will determine the
traveling wave velocity and compare it to the value predicted in Part I.
The excitation frequency fn is determined by the DC voltage applied to the motor. Higher voltages produce higher frequencies,
since more energy is sent into the string. First, position the photogate so that its beam is blocked by the small metal flag on the
motor when it is vertical, which will happen once per cycle. Note: Because this occurs once per cycle, you will NOT need to
find and enter into LoggerPro a characteristic distance d for this experiment! By combining the equation for the wavelength
λn
of the nth mode (given in the Introduction) with the wave equation v
, we can write an expression for the frequency fn :
= fλ
v
fn = (
)n
(4)
2L
Now, you're ready to collect some data:
With the photogate aligned properly, connect the photogate output wire to the input of the interface box labeled
“DIG/SONIC 1.” Turn on the computer and double-click the Desktop icon labeled “Exp8_Period.” A “Sensor
Confirmation” window should appear, so make sure that “Photogate” is selected, and click “Connect.” A LoggerPro window
with a spreadsheet (having columns labeled “State” and “Pulse Time) on the left and an empty graph of Pulse Time (or
Period) vs. Time on the right should appear. The “Pulse Time” is the period of a single rotation of the electric motor.
Test the photogate by passing your finger through its beam. The red LED on the photogate should flash on/off when the
beam is blocked/unblocked.
Turn on the power supply, and turn both the coarse and fine tuning voltage knobs fully counter-clockwise (to zero). You
must make sure that the current knob is turned up (clockwise) enough to run the motor. If the current knob is set too low,
then increasing the voltage will not increase the speed of the motor!
Now, with the motor barely turned on, you should have no wave pattern along the string. Turn up the coarse voltage knob
until a single anti-node appears to vibrate on the string. Then, adjust the fine voltage knob until this amplitude is maximized.
This is the n = 1 mode, vibrating at a frequency f1 .
Once you see a stable standing wave pattern, click the green “Collect” button in the LoggerPro window. The time values
appearing in the spreadsheet are the period values of the motor, which we assume is also the period of the waves along the
string. The frequency fn at which the stretched golden string is vibrated to produce the nth standing wave mode is the
inverse of the period Tn : fn
=
1
Tn
. Do not confuse the period values Tn with the string tension T from before!
The graph of Pulse Time (or Period) vs. Time on the right should fill with data points along a horizontal line. If no data
points are appearing, then right-click near the y-axis of the graph, and click “Autoscale” (not from 0). This should change the
scale of your y-axis to better show the period values measured by the photogate. Note: The plot may now be “zoomed
in,” so even a large variation in these data points does not imply the values are spread far apart. Look at the y-axis
to verify this.
Under the “Analyze” tab in LoggerPro, click “Linear Fit.” A small box should appear on the plot, containing the linear fit
information. You can click and drag the end points of your selection for the fit if necessary. Record the mean of these “Pulse
Time” (or period) values, which will be the period value for T1 .
Now, increase the coarse and fine voltage knobs until a stable standing wave pattern appears with two anti-nodes, indicating
mode n = 2 , and repeat the steps above to find the period T2 .
Repeat these steps until you have five period values for modes n = 1 through n = 5 .
To find an estimate of the wave velocity using these data, you will need to plot fn vs. n and determine its slope, then apply equation
(4) above. Use the uncertainty propagation methods to find the uncertainty σ v in your estimate. For plotting, you should use the
web-based Plotting Tool. (Note: Assume no uncertainty in your values of fn or n.)
Is your experimentally-determined estimate for the wave velocity v consistent with the predicted value from Part I? Use the “overlap
method” with each of the uncertainty ranges to do this. Regardless of your findings, what potential sources of error may have
influenced your results?
Part III: Verifying the dependence of wave velocity v on tension T , and obtaining an estimate of g
In this last part of the lab, you will measure the dependence of one of the frequencies fn on the string tension T in a non-stretchy
string. You will vary the tension by suspending various amounts of mass M from the string. During this part of the experiment, you
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must keep the standing wave pattern with exactly 1 anti-node, so that you examine the n
= 1
mode with frequency f1 only.
Because the elastic golden string would change its length for different amounts of mass attached, and thus its linear mass density
would vary, you must first replace the golden string with the white non-stretchy string. For various attached masses M , and thus
various tensions T , the linear mass density of this white string will always remain constant.
The equation for the standing wave frequency f1 (for mode n
combining equations (1), (3), and (4) above:
fn
2
= 1
) may be expressed in terms of the hanging mass M by
g
=
2
M
(5)
4μL
For data collection for this part, do the following:
Measure the mass m of the white string using the digital scale in the lab room. Assume an uncertainty of σ m = 0.1 g.
Loop one end of the white string, and hang a mass M = 50 g from it. (Assume an uncertainty for this mass of σ M = 0.1
g. Pass the free end up and over the pulley, towards the electric motor. Again, it is essential that you pass the string
through the holes in the motor reed and the motor post, then clip it to the motor post. Do NOT tie it to the hole in
the motor reed! The mass should now hang slightly below the pulley.
Measure the length L from the end of the string at the hole in the motor reed to the end of the string where it first touches
the pulley. (It should be the same value you measured in Part I.) Also, obtain a value for the full length L′ of the string by
measuring the distance from the top of the pulley down to the loop where the mass is hanging. Adding this distance to L
should give L′ . Assume uncertainties of σ L = σ L = 1 cm.
Using equation (2) above, calculate the linear mass density of this white non-stretchy string, using its mass m and its full
−3
length L′ . Compare this value to its “accepted” value of μ = 0.47 × 10 kg/m. They should be close; if they are not,
try measuring the length L′ again, and recalculate your value of μ.
Align the photogate with the flag on the motor as in Part II, and repeat the same process as in Part II to create a stable
n = 1 standing wave pattern. Using the same procedure as before, obtain the period T1 , and take the inverse of this value
to get the frequency f1 .
Increase the hanging mass to M = 100 grams, make slight adjustments to the voltage knobs on the power supply to create
a stable n = 1 standing wave pattern with maximum amplitude, and repeat the measurement process to find the period T1
and frequency f1 .
Repeat this process for hanging masses M of 150, 200, and 250 grams. Record all mass M values and their corresponding
standing wave frequencies f1 .
′
2
To find an estimate of the gravitational acceleration g, you need to plot f1 vs. M and determine its slope, then apply equation (5)
above. Use the uncertainty propagation methods to find the uncertainty σ g . For plotting, you should use the web-based Plotting
Tool. (Note: Assume no uncertainty in your values of f1 or M here.)
Is your experimentally-determined estimate for the gravitational acceleration g consistent with its accepted value of 9.81 m/s2? Does
your estimate equal this value within experimental uncertainty? Regardless of your findings, what potential sources of error may have
influenced your results?
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