PHYS 1420: College Physics II
Fall 2009
Lab 02: Waves on a String
Introduction
A standing wave bridges the gap between
our concepts of simple harmonic motion and
waves. A standing wave occurs when a
wave interferes with itself. The wave is
generated by a vibrating source: in our
case, the 60Hz electrical input from the
wall socket creates the SHM of the metal
blade. The stationary blade vibrates with a
constant frequency, and by attaching a
string to the blade, we can create a wave
that travels the length of the string. When
the wave reaches the end of the string, it
reflects and heads back towards the source.
As the he incident and reflected wave occupy the same string at the same time,
the combination of the two waves creates the pattern you see above. A string
fixed at both ends will have nodes at either end. By adjusting the length of the
string, the tension in the string, or the frequency of the vibration, it is possible to
establish multiple resonances. Notice that this occurs when there is a whole
number of half wavelengths between the endpoints of the string. We will be
adjusting the tension in the string.
Objectives
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Observe a standing wave pattern on a vibrating string
Distinguish between a wave’s nodes and anti-nodes
Identify and measure the wavelength of a standing wave
Measure the speed of a wave using several separate methods
Compare the results and discuss their accuracy
Equipment
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C–clamp and right-angle clamp
48” ring stand
SuperPulley
Short rod (from photogate)
Meter stick
Triple-beam balance
50g hanger with slotted masses
Experimental Procedure
1.
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6.
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8.
9.
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String Vibrator with string
Ring Stand
Measure and cut 2m of string from the spool.
Record the mass (m).
Attach the string vibrator to the ring stand and clamp the stand to the tabletop. It will be easiest to make
observations if you can see the standing wave while looking down on the tabletop from above. The blade vibrates in
the direction perpendicular to the flat side, which means that you want the vibrator mounted so that the flat blade
is vertical.
Clamp the pulley to the ring stand at the opposite end of the table.
Attach the string to the vibrator blade and tie a loop at the other end. Suspend the 50g hanger over the pulley.
Before you plug in the vibrator, adjust the position of the pulley so that the string is level. Measure and record the
distance d from the blade to the pulley (about 1.75m, but measure as accurately as possible).
Plug in the power supply and observe the vibrating string. The frequency (f) of the vibration is constant at 60Hz.
If there is no node at the blade end of the string, you should increase the tension until you have a good node at both
fixed ends. When you have a good resonance, record both the amount of mass at the hanger (M) and the number of
half–wavelengths on the string (n).
Use the meter stick to carefully measure the wavelength (λ). Because you have multiple wavelengths on the same
string, measure at least three times and calculate the average.
Increase the tension by adding mass to the hanger until you have decreased the number of half–wavelengths on the
string by one. Record the mass (M) and n. Make the appropriate measurements of the new wavelength.
Continue to increase the hanging mass M and decrease the number of ½–wavelengths until you have a single node at
either end of the string. Do not forget to measure and record the wavelength at each step.
PHYS 1420: College Physics II
Fall 2009
Data
Arrange your data into a neat table if you have not already. Make sure that you
have recorded mass in kilograms and length in meters. You may want to use an
Excel spreadsheet to perform your data reduction. If you create a spreadsheet,
make sure to label your quantities adequately (and save the sheet frequently).
Data Reduction
1.
Calculate the linear mass density of the string:
µ=
2.
where m is the measured mass and l is the entire 2.0m length of string.
Calculate the wave speed based on the tension in the cord:
€
3.
m
l
v=
T
=
µ
Mg
µ
where T is the tension (equal to the weight of the hanger, Mg) and µ is the
linear mass density. Calculate the speed for each resonance you established.
Calculate the wave speed based on the wavelength and frequency:
v = λf
€
where λ is the average wavelength you measured, and f is a constant 60Hz for each resonance. Again, these calculations
can be completed in a spreadsheet. Make sure that you label and save. The example sheet shown below is not complete.
Analysis
4.
Compare the two wave speeds for each resonance. The
speeds should be the same regardless of the method of
determination. Calculate the percent difference
between the values for each resonance:
€
 v −v 
%diff =  1 1 2  •100
 2 (v1 + v 2 ) 
5.
Which speed is more accurate? Yes, trick question. We
don’t have a good basis (yet) for drawing any
conclusions. Check the values for the wavelength–based
Recalculate the wavelength based on the
€speed.
resonance:
λ=
d
n
( )
1
2
where d is the length of string you measured from blade to pulley and ½n is the total number of wavelengths on the
string. Recalculate the wave speed using this new wavelength.
6.
Check the value of the linear mass density µ. Skipping several steps of algebra, if you set
€
you will derive the equation of a line:
T
µ
1
T = ( 4 µf 2 d 2 ) 2 
n 
v = λf =
where y = T = Mg, the tension in the€string, x = (1/n2) and the slope = 4µf2d2. Prepare a graph of T vs (1/n2), and
find its slope.
7. Extract the value of the linear mass density µ from the slope. Compare this calculated value to the value you
measured. Why would these values be different?
€
8. Recalculate the wave speed for each resonance, using the measured value for the tension, but the newly extracted
value of the linear mass density µ.
9. You should have four values of wave speed for each resonance you observed. By inspection, can you now draw any
conclusion about which method is most accurate? Or maybe you can see most clearly which method is least reliable.
10. Comment on random vs systematic error. For example, what does your graph indicate to you? Does your data
appear compromised? When you examine the wave speeds, can you see evidence of random variations? Can you see
any systematic trends? If you observe systematic error, explain its source. Is there any method for minimizing or
eliminating this error?