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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 pendulum clamp 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, µ? 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

( )1 siny A kx tω= − ,

where 2k πλ

= , 2 fω π= , λ is the wavelength, and f the frequency of the

oscillation. The wave reflected from the fixed end is described by

( )2 siny A kx tω= + ;

the reflected wave travels in the negative x direction. The combined wave will be the sum of the original wave and the reflected wave

( ) ( )1 2 sin siny y y A kx t kx tω ω= + = − + +⎡ ⎤⎣ ⎦ .

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Expanding the sine functions as the sum or difference of two angles yields: [ ]sin cos cos sin sin cos cos siny A kx t kx t kx t kx tω ω ω ω= − + + ,

which simplifies to ( )2 sin cosy A kx tω= . (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 cosωt term. We now analyze the “amplitude,” 2 sinA kx , for nodes. The nodes will occur when

2 sin 0A kx = , which will occur when the phase kx is equal to 0, or a multiple of π.

, 0,1,2,kx n nπ= = K

2 x nππ

λ=

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 l provides the condition for the wavelength for each pattern or harmonic of the vibrating string.

2 L nλ

=

2

nLn

λ = (2)

The nodes divide the string into half wavelength segments (l/2), so if the frequency (f) is known, the speed of a wave can be determined by the fundamental wave velocity equation v f λ= (3) The frequency of the wave can be found by substituting equation (2) into equation (3) and solving for f:

v2nnfL

=

If we also note that 11v2

fL

= , then

1nf nf= (4)

Velocity of a transverse wave in a string: The velocity of a wave traveling on a string is given by

v Tµ

= (5)

where T is the tension in the string and m is the mass per unit length (m/l) of the string.

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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 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.

Figure 1. A picture of the apparatus for generating standing waves on a string. Note that the function generator

requires a digital volt meter to determine the frequency output of the function generator. 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 approximately 1.1 m apart with the pulley

inserted in one clamp and the pendulum clamp extending at a right angle over the table at the other end.

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. Approximately 10 cm from the pendulum clamp, attach the string to the wave driver. 6. Attach two wires to the sine wave output of 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!

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7. Attach two more wires from the DC volts output of the function generator to the V-W and COM inputs on the digital voltmeter and select the 20 V DC (20 DCV) setting on the meter. Turn the meter on if there is a separate on-off switch. NOTE: When reading the frequency (voltage) displayed on the digital meter you must use the same multiplier as is used on the function generator to obtain the correct order of magnitude of frequency.

8. Unlock the wave driver, select the X1 multiplier range on the function generator, set the dial for 1 Hertz, select an output in the middle of the output range. AFTER THE SET UP IS INSPECTED BY YOUR INSTRUCTOR, turn on the function generator.

Data collection: 9. 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.

10. 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.)

11. Reduce the tension to one fourth 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.)

12. Repeat steps 9 and 10 for this new tension. Analysis of the Data 1. For cases 1, and 2, and for each frequency find the wavelength λ of the wave, and the value

of n that describes the wave (see equation 2). 2. For case 1, plot the frequency of the wave versus 1/λ; 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.