Physics 2310 Lab #1 Standing Waves on a String
[Pages:4]Physics 2310 Lab #1 Standing Waves on a String
Objective: The objective of this experiment is to investigate the phenomena of standing waves on a string produced by fixing it at one end and driving the opposite end with a harmonic force. The conditions that result in resonance will be investigated.
Apparatus:
Pasco Variable Frequency Wave Driver with string Pasco Student Function Generator 50 g weight hanger and slotted weight set 2 table clamps short rod pulley Digital multimeter 1 or 2 meter stick
Pre-Lab Exercise: A string is oscillating in a standing wave pattern. The third harmonic is exhibited. a. How many nodes are present? How many antinodes are present?
b. If the string is 2.2 m long and has a mass of 9.5 g, what is its mass per unit length, ?? c. Suppose only 1.2 m of the string is oscillating in the third harmonic, and the string
has a tension of 2.0 N. What is the frequency of oscillation of the string? d. Write the equation for the standing wave pattern on the string in the form of equation
1 (below) with appropriate numbers substituted and simplified as much as possible.
Theory: The Standing Wave Pattern: A transverse wave traveling along a string is reflected from the fixed (pulley) end and
returns interfering with the wave continuously generated by the wave driver. At resonance, a standing wave pattern will be established. There are antinodes at intervals along the pattern, nodes at the ends of the string, and there may be nodes periodically occurring within the pattern as well.
The equation for a transverse wave traveling in the positive x direction along a string is
y1 = Asin (kx -t ),
where k = 2 , = 2 f , is the wavelength, and f the frequency of the
oscillation.
The wave reflected from the fixed end is described by
y2 = Asin (kx +t );
the reflected wave travels in the negative x direction.
The combined wave will be the sum of the original wave and the reflected wave
y = y1 + y2 = Asin (kx -t ) + sin (kx +t ) .
Expanding the sine functions as the sum or difference of two angles yields:
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y = A[sin kx cost - cos kxsint + sin kx cost + cos kxsint],
which simplifies to
y = (2Asin kx)cost .
(1)
The factors in parentheses are time independent and represent the distribution of displacement
along the string. That is, the variation of the amplitude of the cost term.
We now analyze the "amplitude," 2Asin kx , for nodes. The nodes will occur when
2Asin kx = 0 , which will occur when the phase kx is equal to 0, or a multiple of .
kx = n , n = 0,1, 2,K
2 x = n The end points of the string are nodes, thus x = 0, n = 0 and at x = L where n will equal the number of segments or loops in the standing wave.
Substituting x = L, and solving for provides the condition for the wavelength for each pattern
or harmonic of the vibrating string.
2L=n
2L
n = n
(2)
The nodes divide the string into half wavelength segments (/2), so if the frequency (f) is known,
the speed of a wave can be determined by the fundamental wave velocity equation
v= f
(3)
The frequency of the wave can be found by substituting equation (2) into equation (3) and
solving for f:
n fn = v 2L
If we also note that
f1
=
v
1 2L
,
then
fn = nf1
(4)
Velocity of a transverse wave in a string:
The velocity of a wave traveling on a string is given by
v= T
(5)
?
where T is the tension in the string and ? is the mass per unit length (kg/m) of the string.
Overview: In this lab a string is attached to a mechanism that drives waves at a particular frequency. The wave speed is determined by the mass per unit length of the string, and the
SA496 9/5/2016 5:58 PM Comment [1]: Changed to kg/m
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tension of the string. The wavelength of the wave can be determined by the length of the string that participates in the oscillation, and the number of nodes present. The frequency will be adjusted to set up the fundamental mode of oscillation (also known as the first harmonic), the second harmonic, third, etc. The properties of each wave, such as the number of nodes present, the wavelength of the wave, the frequency of oscillation, and the wave speed, will be studied.
SA496 9/5/2016 6:14 PM
Comment [2]: Changed the picture to show the new lab setup. Can change this for a picture that has less table and a better look at the equipment.
Figure 1. A picture of the apparatus for generating standing waves on a string.
Procedure: Examine the wave driver, and make sure that the oscillating pin is in the locked position
during set up.
Set up:
1. Measure the mass and length of the string, and record these values into an excel spreadsheet. 2. Tie two loops in the string: one loop at an end, and the other about 1.5 m from the first. 3. Assemble the two table clamps to the table at each end of the table with the pulley inserted in
one clamp and the wave driver secured to the other clamp. 4. Assemble the string between the two supports with 200 g total mass applied to the end of the
string placed over the pulley. This mass supplies the tension (T) in the string. (For a 200g mass, what is the tension in the string?) 5. Attach two wires to the function generator (check to be sure the switch on the function generator is set for a sine wave) and attach them to the two inputs of the wave driver. DO NOT PLUG IN OR TURN ON THE FUNCTION GENERATOR UNTIL TOLD TO DO SO BY YOUR INSTRUCTOR!
SA496 9/5/2016 6:00 PM
Comment [3]: Changed the comment about needing a digital voltmeter.
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6. Turn on the function generator, select the X1 multiplier range on the function generator, set the dial for 1 Hertz. Make sure the amplitude setting on the function generator is near the maximum value.
Data collection:
7. Adjust the frequency of the function generator until the string oscillates in its fundamental mode. Record the oscillation frequency, the number of nodes, and the total length of the string that participates in the oscillation. Determine the distance between the nodes, and the wavelength of the oscillation. Record all information into an excel spreadsheet.
8. Repeat step 9 for the second, third, etc. harmonic. Measure at least the first 10 harmonics, recording all of the information listed in 9. (This set of runs is called case 1.)
9. Reduce the tension to one half of its value for the new length and repeat steps 9 and 10 for the first 6 harmonics. (This set of runs is called case 2.)
Analysis of the Data
1. For cases 1, and 2, and for each frequency find the wavelength of the wave, and the value of n that describes the wave (see equation 2).
2. For case 1, plot the 1/ versus frequency of the wave; the slope of the line should be equal to the wave speed. Compare the wave speed determined from the slope of the line with that obtained using equation 5.
3. Repeat step 2 for case 2, and find the slope of the line. 4. What is the ratio of wave speeds for case 1 compared to 2 (from the graphs). Is this ratios
equal to the ratio of the predicted wave speeds (Apply equation 5 here)? 5. Are the measured frequencies for case 1 equal to nf1, where n is the number of the harmonic? 6. What is the ratio of the frequency of the second harmonic for case 1 compared to case 2? Is
this ratio the same for the third harmonic? the fourth? the fifth? Is there a pattern here? Make sure that you discuss experimental sources of error in your laboratory report.
Post-lab Questions
1. Assuming that the temperature is 40?C, calculate the speed of sound in a closed tube. 2. With one end of the tube being closed in this experiment, would you expect to measure
higher or lower fundamental frequency (n=1) compared to both ends being open? Explain your answer.
SA496 9/5/2016 6:06 PM Comment [4]: I deleted and changed this up a bit.
SA496 9/5/2016 6:09 PM Comment [5]: Changed to ? because at 1/4, or 50g, the resistance of the pulley was no longer negligible. I was also very difficult to see the lower harmonics. SA496 9/5/2016 6:09 PM Comment [6]: Deleted step 10 because it seemed redundant.
SA496 9/5/2016 6:13 PM Comment [7]: Changed the order here because I think that is what needs to be done to get the correct units out of the slope ((1/s)/(1/m)) -> m/s.
SA496 9/5/2016 6:37 PM Comment [8]: Added some post-lab questions
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